(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1544777, 51417]*) (*NotebookOutlinePosition[ 1549022, 51532]*) (* CellTagsIndexPosition[ 1548089, 51507]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "On the sample distribution of the adjusted coefficient of determination ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " in OLS" }], "Title"], Cell["\<\ Thomas Colignatus, July 2 2006 http://www.dataweb.nl/~cool\ \>", "Text"], Cell[CellGroupData[{ Cell["Summary", "Subsection"], Cell[TextData[{ "In specification search, one invariably meets with the coefficient of \ determination ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ". 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Minimizing the sum of squares of the errors ", StyleBox["e'.e ", FontSlant->"Italic"], "(see below for all notation) is equivalent to maximizing ", Cell[BoxData[ \(TraditionalForm\`\(\(R\^2\)\(.\)\(\ \)\)\)]], "Including more explanatory variables increases the value of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ". Thus there is a temptation to include variables, even at random. The \ \"adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "\", denoted as ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ", is a first step for correction for inclusion of variables. Maximizing ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], "is equivalent to minimizing the estimated variance of the error ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " = ", StyleBox["e'.e / ", FontSlant->"Italic"], "(", StyleBox["n - k", FontSlant->"Italic"], ") (Green (1993:245)). ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " rises if variables are dropped that have a ", StyleBox["t", FontSlant->"Italic"], "-value less than 1 (Green (1993:193)). More sophisticated would be the use \ of the Amemiya and Akaike criteria (Green (1993:245) or Campos, Ericsson and \ Hendry (2005)). Currently we only consider ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ". " }], "Text"], Cell[TextData[{ "True significance always relates to the content of the analysis, but it \ can be useful to have information on statistical significance as well. If \ some search criterion has not been increased by a statistically significant \ amount, then the search doesn't seem successful. An expression of \ significance can thus be based upon the sample distribution of ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " or ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ". Below, the focus is on the sample distribution of ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ". " }], "Text"], Cell["\<\ R.A. Fisher 1915 derived the distribution for the case when the explanatory \ variables are normally distributed (random causes). Currently, we are \ interested in the case of Ordinary Least Squares where the explanatory \ variables are given (fixed causes). Note that the latter case still has one \ random element, the error, but we are interested in the relation between the \ explanators and the dependent variable.\ \>", "Text"], Cell[TextData[{ "While ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " is a sample statistic, the question arises whether there is a parallel \ for the population. Earlier, Barten (1962) suggested an expression (discussed \ by Theil (1971:179)), that later was revised by Press and Zellner (1978) to \ be ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "= ", StyleBox["B", FontSlant->"Italic"], " / (", StyleBox["B", FontSlant->"Italic"], " + ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "), with ", StyleBox["B", FontSlant->"Italic"], " = \[Beta]", StyleBox["'X'X.", FontSlant->"Italic"], "\[Beta]", StyleBox[" ", FontSlant->"Italic"], ". Defining \[Theta] = ", StyleBox["B", FontSlant->"Italic"], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ", a useful expression is ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "= \[Theta] / (\[Theta] + ", StyleBox["n", FontSlant->"Italic"], "), or \[Theta] = ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "). Another useful definition is \[CurlyPhi] = \[Theta] / ", StyleBox["n = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ") or ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " = \[CurlyPhi] / (\[CurlyPhi] + 1)", StyleBox[".", FontSlant->"Italic"], " We take ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " as an estimator of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " and can show that it is almost unbiased. Basically, we have an unbiased \ estimator ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " for ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ", an unbiased estimator ", StyleBox["b ", FontSlant->"Italic"], "for \[Beta], giving ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " = ", StyleBox["b'X'X.b - k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], StyleBox[" ", FontSlant->"Italic"], "as unbiased E[", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], "] = ", StyleBox["B", FontSlant->"Italic"], " (see below)", StyleBox[", ", FontSlant->"Italic"], "and thus it is easy to define ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], " = ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " ", StyleBox["/ ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " = (", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ") ", StyleBox["/ ", FontWeight->"Bold"], "(", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "), which gives us an estimator ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], " which indeed can be shown to be just ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ". The question that remains to discuss is what the distribution of this ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], " is and how it relates to ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "." }], "Text"], Cell[TextData[{ "Since ", StyleBox["B ", FontSlant->"Italic"], "is conditional to the sample ", StyleBox["X, ", FontSlant->"Italic"], "the (sample, \"parent\") value ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "(", StyleBox["X", FontSlant->"Italic"], ") also depends upon the sample, so that ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " can best be called a \"sample parameter\". There is only a true \ population value for some limit value of ", StyleBox["X ", FontSlant->"Italic"], "that may be difficult to define. Though we consider ", StyleBox["X", FontSlant->"Italic"], " fixed, it might be a more relevant question how we selected the ", StyleBox["X, ", FontSlant->"Italic"], "since this affects the value of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "(", StyleBox["X", FontSlant->"Italic"], ") and the distribution of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], StyleBox[".", FontSlant->"Italic"], " That said, having some value ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], "(", StyleBox["n, k", FontSlant->"Italic"], ") and its distribution, we still could determine the probability how much \ a new ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], "(", StyleBox["m, \[Kappa]", FontSlant->"Italic"], " ) differs from it (or vice versa)." }], "Text"], Cell[TextData[{ "Carapeto and Holt (2003) clarify that P[", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " < ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], "] = P[1 - ", StyleBox["n/", FontSlant->"Italic"], "(", StyleBox["n-k", FontSlant->"Italic"], ")", " (1 - ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ") < 1 - ", StyleBox["n/", FontSlant->"Italic"], "(", StyleBox["n-k", FontSlant->"Italic"], ")", " (1 - ", StyleBox["x", FontSlant->"Italic"], ")] = P[", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " < ", StyleBox["x", FontSlant->"Italic"], "], for any ", StyleBox["x ", FontSlant->"Italic"], "in [0, 1] and its transformed ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], ", so that ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " have the same distribution. Yet it appears that ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " is a more useful estimator of the relevant sample parameter ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", and, to determine its properties, it is sensible to work directly with \ its distribution. " }], "Text"], Cell[TextData[{ "Finally, there is a curious notion in the literature that the definition \ of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " would be arbitrary and that, hence, all ensueing deductions would have \ arbitrary value. Supposedly, one could design various other formats and \ transformations, such as e.g. ", Cell[BoxData[ \(TraditionalForm\`\[Eta]\^2\)]], " = \[Theta] / (\[Theta] + ", StyleBox["n - k", FontSlant->"Italic"], "), and there would no sensible or rational criterion about what would be \ the best representation of the \"true parameter\" behind the multiple \ correlation coefficient. This notion is curious, since the given definition \ of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " depends upon the population parameters \[Beta] and \[Sigma], and hence is \ a derived sample parameter that clearly is as objective as those population \ parameters that one already has accepted. Thus it is strange to accept \ \[Beta] and \[Sigma] on the one hand, and to reject them on the other hand. \ Next, it can be shown that E[", StyleBox["y'.y", FontSlant->"Italic"], "] = ", StyleBox["B", FontSlant->"Italic"], " + ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = (\[Theta]", " + ", StyleBox["n", FontSlant->"Italic"], ") ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ", so there is a rationale to take this as the base of a ratio; and then ", "the given definition of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " nicely captures the same intuition, already expressed in ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ", that we want to summarize the relative weight of \[Beta] versus \ \[Sigma]. Finally, the value of the definition of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " is enhanced by the observation, shown below, that E[", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " ] \[TildeTilde] ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ". Thus the definition of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " seems as arbitrary as the definition for the true mean of a sample." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Derivations", "Section"], Cell[TextData[{ "The ordinary least squares (OLS) model is ", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["X", FontSlant->"Italic"], " \[Beta] + \[Epsilon] = ", StyleBox["X", FontSlant->"Italic"], " ", StyleBox["b", FontSlant->"Italic"], " + ", StyleBox["e", FontSlant->"Italic"], ", where \[Beta] are fixed population parameters, \[Epsilon] ~ N[0, ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox["I ", FontSlant->"Italic"], "] and ", StyleBox["b", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], ".", StyleBox["X'.y. ", FontSlant->"Italic"], "The idempotent and symmetric projection is ", StyleBox["P", FontSlant->"Italic"], " ", StyleBox["=", FontSlant->"Italic"], " ", StyleBox["X.", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], ".", StyleBox["X' ", FontSlant->"Italic"], "so that ", Cell[BoxData[ \(TraditionalForm\`y\&^\ = \ P\ \ y\)]], ", while ", StyleBox["M ", FontSlant->"Italic"], "= ", StyleBox["I", FontSlant->"Italic"], " -", StyleBox[" P ", FontSlant->"Italic"], "so that ", StyleBox["e = M.y = M.\[Epsilon] ", FontSlant->"Italic"], "since ", StyleBox["M.X = 0. ", FontSlant->"Italic"], "There are ", StyleBox["N", FontSlant->"Italic"], " observations, ", StyleBox["K", FontSlant->"Italic"], " explanatory variables. The variables can best be measured in deviation \ from their means, so that, with the model redefined, the total degrees of \ freedom are ", StyleBox["n", FontSlant->"Italic"], " = ", StyleBox["N", FontSlant->"Italic"], " -1, ", StyleBox["X ", FontSlant->"Italic"], "has ", StyleBox["k", FontSlant->"Italic"], " =", StyleBox[" K", FontSlant->"Italic"], " - 1 columns, while the error has ", StyleBox["n - k", FontSlant->"Italic"], " degrees of freedom. The unbiased estimated variance then is ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "= ", StyleBox["e 'e", FontSlant->"Italic"], " / (", StyleBox["n - k", FontSlant->"Italic"], "). " }], "Text"], Cell[TextData[{ "As we have ", StyleBox["y ", FontSlant->"Italic"], "~ N[", StyleBox["X", FontSlant->"Italic"], "\[Beta], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ", StyleBox["I ", FontSlant->"Italic"], "] and ", StyleBox["e ", FontSlant->"Italic"], "~ N[0, ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ", StyleBox["M", FontSlant->"Italic"], "] we also have ", StyleBox["Xb ~ ", FontSlant->"Italic"], "N[", StyleBox["X", FontSlant->"Italic"], "\[Beta], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ", StyleBox["P", FontSlant->"Italic"], "], noting that ", StyleBox["X.(b", FontSlant->"Italic"], " - \[Beta]) =", StyleBox[" P", FontSlant->"Italic"], " \[Epsilon]." }], "Text"], Cell[TextData[{ "We find ", StyleBox["b", FontSlant->"Italic"], " - \[Beta] = ", Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], ".", StyleBox["X' \[Epsilon] ", FontSlant->"Italic"], " so that Var[", StyleBox["b", FontSlant->"Italic"], "] = E[ (", StyleBox["b", FontSlant->"Italic"], " - \[Beta])'.(", StyleBox["b", FontSlant->"Italic"], " - \[Beta]) ] = ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], " and b ~ N[\[Beta], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], "]. If we take ", Cell[BoxData[ \(TraditionalForm\`V\)]], " = Diag[ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], "] which takes only the diagonal elements, then ", Cell[BoxData[ \(TraditionalForm\`V\^\(\(-1\)/2\)\)]], ". (", StyleBox["b - ", FontSlant->"Italic"], "\[Beta]", StyleBox[" ", FontSlant->"Italic"], ") ~ N[0, ", StyleBox["I ", FontSlant->"Italic"], "], then, following \"Student\", we can perform t-tests on the significant \ difference of the parameters from zero, with the F-test from R.A. Fisher \ providing a joint test. When \[Beta] \[NotEqual] 0, the off-diagonal elements \ become important. The following statistic ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " provides joint information on those \[Beta] \[NotEqual] 0." }], "Text"], Cell[TextData[{ "Since ", StyleBox["X'e", FontSlant->"Italic"], " = 0 the total sum of squares SST = ", StyleBox["y'.y = b'X'X.b + e'e = y'.P.y + e'e =", FontSlant->"Italic"], "SSR + SSE, with SSR = ", StyleBox["b'X'X.b = y'.P.y ", FontSlant->"Italic"], "as the sum of squares of the regression and SSE the sum of squares of the \ error (or the sum of squares of the residuals). All these sums of squares are \ nonnegative. The coefficient of determination is ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = SSR / SST (i.e. the square of the multiple correlation coefficient)." }], "Text"], Cell[TextData[{ "The target value of SSR is ", StyleBox["B", FontSlant->"Italic"], " = \[Beta]", StyleBox["'X'X.", FontSlant->"Italic"], "\[Beta]", StyleBox[" ", FontSlant->"Italic"], "and the \"population\" coefficient of determination is ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "= ", StyleBox["B", FontSlant->"Italic"], " / (", StyleBox["B", FontSlant->"Italic"], " + ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "). Note (again) that ", StyleBox["B ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " are", " sample functions but restricted by the population parameters. It may be \ useful to distinguish any sample from a fixed observed value ", StyleBox["P", FontSlant->"Italic"], ", so that we can consider E[SSR | ", StyleBox["P", FontSlant->"Italic"], "] etcetera. Since the latter condition is obvious, it is not necessary to \ always mention it." }], "Text"], Cell[TextData[{ "To determine the sample distribution of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " we must investigate the distribution of its components, check their \ possible dependence and then consider their ratio." }], "Text"], Cell[CellGroupData[{ Cell["On SST", "Subsubsection"], Cell[TextData[{ "We first observe that E[SST] = E[ ", StyleBox["y'y ", FontSlant->"Italic"], "] = E[(", StyleBox["X", FontSlant->"Italic"], " \[Beta] + \[Epsilon])'.(", StyleBox["X", FontSlant->"Italic"], " \[Beta] + \[Epsilon])] = (", StyleBox["B", FontSlant->"Italic"], " + ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ") so that E[", StyleBox["y'y ", FontSlant->"Italic"], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "] = \[Theta] + ", StyleBox["n = n ", FontSlant->"Italic"], "(\[CurlyPhi] + 1) = ", StyleBox["n ", FontSlant->"Italic"], "/ (1 ", StyleBox["- ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "). Note the infinities when ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = 0 or ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " = 1." }], "Text"], Cell[TextData[{ "We have", StyleBox[" y", FontSlant->"Italic"], " ~ N[", StyleBox["X", FontSlant->"Italic"], "\[Beta], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox["I ", FontSlant->"Italic"], "] and, using \[Xi] = ", StyleBox["X", FontSlant->"Italic"], "\[Beta] / \[Sigma], then ", StyleBox["y / ", FontSlant->"Italic"], "\[Sigma] ~ N[\[Xi], ", StyleBox["I", FontSlant->"Italic"], "] and we find ", StyleBox["y'y / ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox[" ", FontSlant->"Italic"], "~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["n, ", FontSlant->"Italic"], "\[Xi]'\[Xi]] which is the non-central chi-square with non-centrality \ parameter \[Lambda] = \[Xi]'\[Xi] . The expectation of this distribution is \ ", StyleBox["n + \[Lambda] ", FontSlant->"Italic"], "and thus we verify that \[Xi]'\[Xi] = \[Beta]", StyleBox["'X'X.", FontSlant->"Italic"], "\[Beta]", StyleBox[" ", FontSlant->"Italic"], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = \[Theta]. Hence ", StyleBox["y'y / ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox[" ", FontSlant->"Italic"], "~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["n, ", FontSlant->"Italic"], "\[Theta]], where \[Theta] = ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ")." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["On SSR", "Subsubsection"], Cell[TextData[{ "Subsequently, SSR = ", StyleBox["b'X'X.b", FontSlant->"Italic"], " = (", StyleBox["P \[Epsilon] ", FontSlant->"Italic"], " + ", StyleBox["X", FontSlant->"Italic"], "\[Beta])'. (", StyleBox["P \[Epsilon] ", FontSlant->"Italic"], " + ", StyleBox["X", FontSlant->"Italic"], "\[Beta]) = \[Epsilon]' ", StyleBox["P", FontSlant->"Italic"], " \[Epsilon] + 2 \[Beta]'", StyleBox["X' \[Epsilon] + B", FontSlant->"Italic"], ". The expected value of the SSR is E[SSR] = E[", StyleBox["b'X'X.b", FontSlant->"Italic"], "] = E[\[Epsilon]' ", StyleBox["P", FontSlant->"Italic"], " \[Epsilon] ] + ", StyleBox["B, ", FontSlant->"Italic"], "since E[\[Epsilon]] = 0. An idempotent matrix has only characteristics \ roots 1 or 0, hence the trace of ", StyleBox["P ", FontSlant->"Italic"], "is equal to its rank, and thus E[", StyleBox["b'X'X.b", FontSlant->"Italic"], "] = ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " + ", StyleBox["B.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "Using the unbiased ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ", and using the independence of ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " and ", StyleBox["b", FontSlant->"Italic"], " (Green (1993:185)), we find unbiased estimator ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " = ", StyleBox["b'X'X.b", FontSlant->"Italic"], " - ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ", since E[", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], "] = E[", StyleBox["b'X'X.b", FontSlant->"Italic"], " - ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "] = E[", StyleBox["b'X'X.b", FontSlant->"Italic"], "] - E[", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "] = (", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " + ", StyleBox["B", FontSlant->"Italic"], ")", StyleBox[" - ", FontSlant->"Italic"], " ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = ", StyleBox["B.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "Thus, this other allocation of squares ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " = ", StyleBox["b'X'X.b", FontSlant->"Italic"], " - ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " = ", StyleBox["y'.y", FontSlant->"Italic"], " - ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " gives us an unbiased estimator of ", StyleBox["B", FontSlant->"Italic"], ". This ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " can be called the \"adjusted SSR\" (SSRadj). Note that ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " / ", StyleBox["y'.y", FontSlant->"Italic"], " = (", StyleBox["b'X'X.b", FontSlant->"Italic"], " - ", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ") / ", StyleBox["y'.y", FontSlant->"Italic"], " = (", StyleBox["b'X'X.b ", FontSlant->"Italic"], "/ ", StyleBox["y'.y)", FontSlant->"Italic"], " - (", StyleBox["k", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " ", StyleBox[" ", FontSlant->"Italic"], "/ ", StyleBox["y'.y", FontSlant->"Italic"], ")", " = ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " - ", StyleBox["k / ", FontSlant->"Italic"], "(", StyleBox["n - k", FontSlant->"Italic"], ")", StyleBox[" e'e", FontSlant->"Italic"], " / ", StyleBox["y'.y", FontSlant->"Italic"], " ", " = ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " - ", StyleBox["k / ", FontSlant->"Italic"], "(", StyleBox["n - k", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "(1 - ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ", which expression is", " commonly called the adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "." }], "Text"], Cell[TextData[{ "This estimator ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " for ", StyleBox["B", FontSlant->"Italic"], " thus inspires the consideration of the estimator ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " ", StyleBox["/ (", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " + ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ") = (", StyleBox["y'.y", FontSlant->"Italic"], " - ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ") / ", StyleBox["y'.y", FontSlant->"Italic"], " = 1 - ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " / ", StyleBox["y'.y", FontSlant->"Italic"], " = 1 - ", StyleBox["n/(n-k)", FontSlant->"Italic"], " SSE / SST = 1 - ", StyleBox["n/(n-k)", FontSlant->"Italic"], " (1 - ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ", the adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " in other formula formats. Note that ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], "= (1 - 1/Ftest) ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " where the Ftest-value = (", StyleBox["n - k", FontSlant->"Italic"], ") / ", StyleBox["k", FontSlant->"Italic"], " . ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "/ (1 - ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "), which finds its use in testing the hypothesis that \[Beta] = 0 or ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = 0. Note also that ", StyleBox["-k / (n - k) ", FontSlant->"Italic"], "\[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " \[LessEqual] 1 and that ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " > 0 iff ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " > ", StyleBox["k", FontSlant->"Italic"], " / ", StyleBox["n. ", FontSlant->"Italic"], "Restated, ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = 1 - ", StyleBox["(n - k) / n", FontSlant->"Italic"], " (1 - ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ") and hence E[", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "] = 1 - ", StyleBox["(n - k) / n", FontSlant->"Italic"], " (1 -E[ ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], "]). Below we will show that E[", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " ] \[TildeTilde] ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " and hence E[", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "] \[TildeTilde] 1 - ", StyleBox["(n - k) / n", FontSlant->"Italic"], " (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "). As we have 0 \[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " \[LessEqual] 1 then it is logical that E[ ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "] \[GreaterEqual] ", StyleBox["k", FontSlant->"Italic"], " / ", StyleBox["n", FontSlant->"Italic"], ". We can already show that ", Cell[BoxData[ \(TraditionalForm\`lim\+\(\[Theta]\ \[Rule] \ 0\)\)]], " E[ ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "] = ", StyleBox["k / n. ", FontSlant->"Italic"], "Note that SSR = ", StyleBox["b'X'X.b", FontSlant->"Italic"], " \[TildeTilde]", StyleBox[" B + k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], StyleBox[" ", FontSlant->"Italic"], "and certainly in expectation. Hence ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " \[TildeTilde]", StyleBox[" ", FontSlant->"Italic"], "(", StyleBox["B + k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], StyleBox[") / ", FontSlant->"Italic"], "(", StyleBox["B + k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], StyleBox["+ (n - k) ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "). ", StyleBox[" ", FontSlant->"Italic"], "For ", StyleBox["B = ", FontSlant->"Italic"], "0 the latter collapses to ", StyleBox["k / n", FontSlant->"Italic"], ". ", StyleBox[" ", FontSlant->"Italic"], "If \[Beta] = 0 or ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " = 0 then the SSR ", StyleBox["b'.X'.X.b ", FontSlant->"Italic"], "of OLS can still generate a nonnegative value such that 0 < ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " < ", StyleBox["k", FontSlant->"Italic"], " / ", StyleBox["n, ", FontSlant->"Italic"], "but we would take this as an indication that the true correlation is close \ to zero. Alternatively said, a condition for a successful regression (i.e. \ showing a contribution of the regressors) is that ", StyleBox["b'.X'.X.b ", FontSlant->"Italic"], "> ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ", or ", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " > 0." }], "Text"], Cell[TextData[{ "It follows that we actually are less interested in ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " and more interested in ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ". Thus, our definition of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " suggests an expression for a sample parameter ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", subsequently ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " appears to be an estimator of that sample parameter, and hence ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " drops from our considerations. " }], "Text"], Cell[TextData[{ "We would like E[ ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ". If it would be possible to do so, we would take the expectation of the \ numerator independently from the denominator, so that E[ ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "] = ? = E[", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " ] ", StyleBox["/ E", FontSlant->"Italic"], "[", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " + ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "] = ", StyleBox["B", FontSlant->"Italic"], " / (", StyleBox["B", FontSlant->"Italic"], " + ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ") =", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\[Rho]\^2\), "TraditionalForm"]}], TraditionalForm]]], ". The terms are dependent however, and we cannot simply do that. But we \ can show below in a graphical manner that E[ ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "] \[TildeTilde] ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", in fact very close." }], "Text"], Cell[TextData[{ "As we have ", StyleBox["b'X'X.b", FontSlant->"Italic"], " - ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], StyleBox[" = y'.y", FontSlant->"Italic"], " - ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " we can consider the distribution of either side of the equation." }], "Text"], Cell[TextData[{ "Considering ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ", we know that", StyleBox[" e'e", FontSlant->"Italic"], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ~ ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]], ", so that ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " is a ", StyleBox["n / (n - k) ", FontSlant->"Italic"], "sum of those terms, so that ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ~ ", StyleBox["n / (n - k)", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The ratio", "Subsubsection"], Cell[TextData[{ "To determine the sample distribution of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " we consider the distribution of its components again. " }], "Text"], Cell[TextData[{ "Using ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= 1 - ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], " / ", StyleBox["y'.y ", FontSlant->"Italic"], "we find (with both ", StyleBox["n ", FontSlant->"Italic"], "degrees of freedom):" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ~ ", FontSlant->"Italic"], " 1 - ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], " / ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["n, ", FontSlant->"Italic"], "\[Theta]], where \[Theta] = ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ")." }], "Text"], Cell["\<\ However, these chisquares are dependent, and thus we cannot use known \ relations (such as on the F-distribution).\ \>", "Text"], Cell[TextData[{ "A successful strategy is to derive the distribution of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " from that of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ", since ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "= ", StyleBox["b'X'X.b ", FontSlant->"Italic"], "/ (", StyleBox["b'X'X.b + e'e", FontSlant->"Italic"], ") and we know that ", StyleBox["b", FontSlant->"Italic"], " and ", StyleBox["e ", FontSlant->"Italic"], "are independent. " }], "Text"], Cell[TextData[{ "We have ", StyleBox["Xb ~ ", FontSlant->"Italic"], "N[", StyleBox["X", FontSlant->"Italic"], "\[Beta], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ", StyleBox["P", FontSlant->"Italic"], "] and let ", StyleBox["P C", FontSlant->"Italic"], " = ", StyleBox["C \[CapitalLambda],", FontSlant->"Italic"], " where ", Cell[BoxData[ \(TraditionalForm\`C\^\(-1\)\)]], " = ", StyleBox["C ' ", FontSlant->"Italic"], "are orthogonal eigenvectors, and where the diagonal matrix ", StyleBox["\[CapitalLambda] ", FontSlant->"Italic"], "has", " ", StyleBox["k", FontSlant->"Italic"], " non-zero eigenvalues. In fact, since", StyleBox[" P ", FontSlant->"Italic"], "is idempotent", StyleBox[" ", FontSlant->"Italic"], "there are ", StyleBox["k", FontSlant->"Italic"], " values 1. Use ", StyleBox["z", FontSlant->"Italic"], " = ", StyleBox["C.X.b ", FontSlant->"Italic"], "/ \[Sigma] and \[Zeta] = ", StyleBox["C.X.", FontSlant->"Italic"], "\[Beta]", StyleBox[" ", FontSlant->"Italic"], "/ \[Sigma] so that ", StyleBox["z", FontSlant->"Italic"], " ~ N[\[Zeta], ", StyleBox["\[CapitalLambda] ", FontSlant->"Italic"], "]. Thus ", StyleBox["z'z ~ ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["k, ", FontSlant->"Italic"], "\[Zeta] '. \[Zeta]]. Obviously ", StyleBox["z'z ", FontSlant->"Italic"], "= (", StyleBox["b'.X' C'.", FontSlant->"Italic"], ").", StyleBox["C.X.b ", FontSlant->"Italic"], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = ", StyleBox["b'X'X.b ", FontSlant->"Italic"], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox[" ", FontSlant->"Italic"], "and \[Zeta] '. \[Zeta] = ", StyleBox["B", FontSlant->"Italic"], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = \[Theta], so that ", StyleBox["b'X'X.b ", FontSlant->"Italic"], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ", StyleBox["~ ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["k, ", FontSlant->"Italic"], "\[Theta]]. We used the property of the eigenvalue decomposition, but we \ don't have to actually calculate them, since they disappear again." }], "Text"], Cell[TextData[{ "Thus ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = ", StyleBox["b'X'X.b ", FontSlant->"Italic"], "/ (", StyleBox["b'X'X.b + e'e", FontSlant->"Italic"], ") ~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["k, ", FontSlant->"Italic"], "\[Theta]] / (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["k, ", FontSlant->"Italic"], "\[Theta]] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]], ") where \[Theta] = ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ")." }], "Text"], Cell[TextData[{ "Thus ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = ", StyleBox["f ", FontSlant->"Italic"], "/ (", StyleBox["f", FontSlant->"Italic"], " + 1) = 1 / (1 + 1 / ", StyleBox["f ", FontSlant->"Italic"], ") where ", StyleBox["f ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ") so that ", StyleBox["r = f (n - k) / k ", FontSlant->"Italic"], " ~ F[", StyleBox["k,", FontSlant->"Italic"], " ", StyleBox["n - k, ", FontSlant->"Italic"], "\[Theta]] which is the noncentral F-distribution, with in the numerator a \ noncentral chi\[Hyphen]square random variable with noncentrality parameter ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)], "InlineFormula"], " = \[Theta] and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\<\"n\"\>", "TI"], "1"], TraditionalForm]], "InlineFormula"], "= ", StyleBox["k", FontSlant->"Italic"], " degrees of freedom and in the denominator a central chi\[Hyphen]square \ random variable with ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\<\"n\"\>", "TI"], "2"], TraditionalForm]], "InlineFormula"], " = ", StyleBox["n - k", FontSlant->"Italic"], " degrees of freedom. We actually have already seen that transformation, \ see \[CurlyPhi], ", "though \[CurlyPhi]", " relates to the sample parameter of adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " while ", StyleBox["f ", FontSlant->"Italic"], "is a sample statistic of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " itself", ". The Jacobian then is \[PartialD]", StyleBox["r / ", FontSlant->"Italic"], "\[PartialD] ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = 1 / ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", FormBox[\(R\^2\), "TraditionalForm"]}], ")"}], "2"], TraditionalForm]]], " ", StyleBox["(n - k) / k", FontSlant->"Italic"], " . " }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Derivation using ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Section"], Cell[TextData[{ "In the following, we will use the labels \"density\" and \"expectation\" \ for expressions in terms of \[CurlyPhi], and the labels \"dens\" and \"exp\" \ for expressions in terms of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["Using the noncentral F distribution", "Subsection"], Cell["The following functions are useful:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Rho2\)\)], "Input"], Cell[BoxData[ \("Rho2[\[Theta], n] gives \[Rho]2 = \[Theta] / (\[Theta] + n), with \ \[Rho]2 the sample parameter for R2\nRho2[\[CurlyPhi]] gives \[Rho]2 = \ \[CurlyPhi] / (\[CurlyPhi] + 1) "\)], "Print", CellTags->"Info3360749948-3223356"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Phi\)\)], "Input"], Cell[BoxData[ \("Phi[\[Rho]2] gives \[CurlyPhi] = \[Rho]2 / (1 - \[Rho]2) with \[Rho]2 \ the sample parameter for R2"\)], "Print", CellTags->"Info3360749955-3021122"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?Theta\)\)], "Input", Evaluatable->False], Cell[BoxData[ \("\"\)], "Print", GeneratedCell->False, CellAutoOverwrite->False, CellTags->"Info3360749975-5495875"] }, Open ]], Cell[TextData[{ "Note that we might replace ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], " by Floor[", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], "] so that some functions simplify. 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\(TraditionalForm\`\[Lambda]\)], "InlineFormula"], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\<\"n\"\>", "TI"], "1"], TraditionalForm]], "InlineFormula"], " degrees of freedom to a central chi\[Hyphen]square random variable with \ ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\<\"n\"\>", "TI"], "2"], TraditionalForm]], "InlineFormula"], " degrees of freedom.\" " }], "Text", CellTags->{"S5.97.1", "1.15"}], Cell[TextData[{ "Note that the f-value that is submitted to the function in ", StyleBox["Mathematica", FontSlant->"Italic"], " must itself be corrected for the degrees of freedom (which causes double \ entry)." }], "Text", CellTags->{"S5.97.1", "1.15"}], Cell["For us, the substitutions result into:", "Text"], Cell[CellGroupData[{ Cell["\<\ PDF[NoncentralFRatioDistribution[k, n - k, n \[CurlyPhi]], r2 / (1 - r2) * (n - k) / k] / (1 - r2)^2 * (n - k) / k\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(k\^\(k\/2 - 1\)\), " ", \(\((n - k)\)\^\(\(n - k\)\/2 + 1\)\), " ", \(\((\(\((n - k)\)\ r2\)\/\(k\ \((1 - r2)\)\))\)\^\(\(k - 2\)\/2\)\), " ", \(\((\(-k\) + n + \(\((n - k)\)\ r2\)\/\(1 - r2\))\)\^\(\(-n\)/ 2\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ \((n - k)\)\ r2\ \[CurlyPhi]\)\/\(2\ \((1 - r2)\)\ \((\(-k\) + n + \(\((n - k)\)\ r2\)\/\(1 - r2\))\)\)\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{\(\((1 - r2)\)\^2\), " ", RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}]}]], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "When we use simplification, then ", StyleBox["Mathematica", FontSlant->"Italic"], " increases complexity by creating factors like (", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "- 1)^n, which are, indeed, in the complex plane - which we are not \ interested in. For this reason we use a2 = 1 - r2. (PM. Note that this is \ only a simplification procedure in ", StyleBox["Mathematica", FontSlant->"Italic"], " and not a change of distribution that would require a Jacobian.)" }], "Text"], Cell[CellGroupData[{ Cell["\<\ res = PDF[NoncentralFRatioDistribution[k, n - k, n \[CurlyPhi]], r2 / a2 * (n - k) / k] / a2^2 * (n - k) / k\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(k\^\(k\/2 - 1\)\), " ", \(\((n - k)\)\^\(\(n - k\)\/2 + 1\)\), " ", \(\((\(\((n - k)\)\ r2\)\/\(a2\ k\))\)\^\(\(k - 2\)\/2\)\), " ", \(\((\(-k\) + n + \(\((n - k)\)\ r2\)\/a2)\)\^\(\(-n\)/2\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ \((n - 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2)\)\)\), " ", \(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(r2\^\(k\/2 - 1\)\), " ", \(\((a2 + r2)\)\^\(\(-n\)/2\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ r2\ \[CurlyPhi]\)\/\(2\ \((a2 + r2)\)\)\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}]], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ And now we can safely substitute 1 - r2 and find the much desired density, in \ an acceptable form.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(res3\ /. \ a2\ \[Rule] \ 1\ - \ r2\)], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(\((1 - r2)\)\^\(1\/2\ \((\(-k\) + n - 2)\)\)\), " ", \(r2\^\(k\/2 - 1\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ r2\ \[CurlyPhi]\)\/2\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}]], TraditionalForm]], "Output"] }, Open ]], Cell["The Estimate` package has already defined this funtion.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredDensity[Phi, \ r2, n, k, \[CurlyPhi]]\)], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(\((1 - r2)\)\^\(1\/2\ \((\(-k\) + n - 2)\)\)\), " ", \(r2\^\(k\/2 - 1\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ r2\ \[CurlyPhi]\)\/2\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}]], TraditionalForm]], "Output"] }, Open ]], Cell["It can be verified that it is a density.", "Text"], Cell[CellGroupData[{ Cell["\<\ Integrate[RSquaredDensity[Phi, r2, n, k, \[CurlyPhi]], {r2, 0, 1}, \ Assumptions \[Rule] (n > 2 && k > 0 && k < n && ((n | k) \[Element] Integers) && (\[CurlyPhi] \ \[Element] Reals) && \[CurlyPhi] >= 0)]\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`1\)], "Output"] }, Open ]], Cell[TextData[{ "The density in terms of parameter ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredDensity[r2, n, k, \[Rho]2]\)], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\)\), " ", \(\((1 - r2)\)\^\(1\/2\ \((\(-k\) + n - 2)\)\)\), " ", \(r2\^\(k\/2 - 1\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ r2\ \[Rho]2\)\/\(2\ \((1 - \[Rho]2)\)\)\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}]], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?RSquaredDensity\)\)], "Input"], Cell[BoxData[ \("RSquaredDensity[R2, n, k, \[Rho]2] gives the density (PDF) of R2 with \ sample parameter \[Rho]2, and n = N - 1 for N observations, and k = K - 1 \ other regressors than the constant\nRSquaredDensity[Phi, R2, n, k, \ \[CurlyPhi]] gives the density of R2 with parameter \[CurlyPhi] = \[Rho]2 / \ (1 - \[Rho]2) \nSee RSquared for the definition of terms\n\ RSquaredDensity[Plot (, \[Rho]2)] plots for {0, 1}, with \[Rho]2. If the \ latter is not specified the option AdjustedRSquared in the options is taken.\n\ RSquaredDensity[Set, r2_, \[Rho]2_] can be set by the routine RSquared[y, \ yest, K] or SetR2Plots"\)], "Print", CellTags->"Info3360824400-6333766"] }, Open ]], Cell[TextData[{ "For example, when ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\ = \ \(50 %\ \ then\ \ \[CurlyPhi]\ = \ 1\)\)]], ". 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r2, n, k, \[CurlyPhi]], {r2, 0, 1}, \ Assumptions \[Rule] (\[CurlyPhi] \[Element] Reals && r2 \[Element] Reals && n > 2 && k > 0 && \ k < n && ((n | k) \[Element] Integers) && (0 <= r2 <= 1) && \[CurlyPhi] >= 0 \ && Re[\[CurlyPhi]] >= 0 && Im[\[CurlyPhi]] == 0)]\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[\(2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\ \/2\)\)\ \((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\ \(\[CapitalGamma]( k\/2)\)\ \(\[CapitalGamma](\(n - k\)\/2)\)\ \((2\ \((k - n)\)\ \(\[CapitalGamma](n\/2 + 1)\) + n\ \((2\ \(\[CapitalGamma]( n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\))\)\), RowBox[{"n", " ", RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}], " ", \(\[CapitalGamma](n\/2)\)}]], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ ev2 = FullSimplify[ev, Assumptions -> (n > 2 && k > 0 && k < n && (n | k) \[Element] Integers && \ \[CurlyPhi] \[Element] Reals && Re[\[CurlyPhi]] >= 0)]\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \ \[CurlyPhi]\)\/2\)\)\ \((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma](n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\)\)], "Output"] }, Open ]], Cell["\<\ We are however not interested in the complex plane. Hence we define a general \ function that allows us to select the real parts, the imaginary part and the \ absolute value. The best shortcut is to take the real part.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredExpectation[Phi, \ func, \ n, \ k, \ \[CurlyPhi]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`func( 2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\ \ \((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma]( n\/2)\) + 2\ \(\[CapitalGamma]( n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\))\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredExpectation[n, \ k, \ \[Rho]2]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`Re( 2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\)\ \((\(-\(\(n\ \[Rho]2\)\/\(1 - \ \[Rho]2\)\)\))\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma]( n\/2 + 1, \(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\))\))\))\)], "Output"] }, Open ]], Cell[TextData[{ "The expected value of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " has a lower limit ", StyleBox["k / n", FontSlant->"Italic"], " but we cannot find this formally." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Limit[RSquaredExpectation[Phi, Identity, n, k, \[CurlyPhi]], \[CurlyPhi] -> \ 0, Assumptions \[Rule] (n > 2 && k > 0 && k < n && ((n | k) \[Element] Integers) && (\[CurlyPhi] \ \[Element] Reals) && Re[\[CurlyPhi]] >= 0)]\ \>", "Input", Evaluatable->False], Cell[BoxData[ FormBox[ RowBox[{"Limit", "[", RowBox[{\(2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \ \[CurlyPhi]\)\/2\)\)\ \((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma]( n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\)\), ",", \(\[CurlyPhi] \[Rule] 0\), ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{\(n > 2\), "\[And]", \(k > 0\), "\[And]", \(k < n\), "\[And]", RowBox[{\((n | k)\), "\[Element]", TagBox["\[DoubleStruckCapitalZ]", Function[ {}, Integers]]}], "\[And]", RowBox[{"\[CurlyPhi]", "\[Element]", TagBox["\[DoubleStruckCapitalR]", Function[ {}, Reals]]}], "\[And]", \(Re(\[CurlyPhi]) \[GreaterEqual] 0\)}]}]}], "]"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "The formal Limit does not work but in numerical cases it does. Note that \ ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " \[Rule] 0 iff \[CurlyPhi] \[Rule] 0. " }], "Text"], Cell[CellGroupData[{ Cell["\<\ RSquaredExpectation[Phi, Identity, 15, 3, \[CurlyPhi]]\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`\(64\ \@\(2\/15\)\ \[ExponentialE]\^\(\(\(-15\)\ \ \[CurlyPhi]\)/2\)\ \((\(-3\)\ \(\[CapitalGamma](15\/2, \(-\(\(15\ \[CurlyPhi]\ \)\/2\)\))\) + 2\ \(\[CapitalGamma](17\/2, \(-\(\(15\ \[CurlyPhi]\)\/2\)\))\) \ - \(405405\ \@\[Pi]\)\/32)\)\)\/\(170859375\ \((\(-\[CurlyPhi]\))\)\^\(15/2\)\ \)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Limit[%, \[CurlyPhi] -> 0]", "Input"], Cell[BoxData[ \(TraditionalForm\`1\/5\)], "Output"] }, Open ]], Cell[TextData[{ "We recognise this as ", StyleBox["k / n", FontSlant->"Italic"], ", and remember that ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " > 0 iff ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " > ", StyleBox["k", FontSlant->"Italic"], " / ", StyleBox["n", FontSlant->"Italic"], ". When E[", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " ] \[TildeTilde] ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " and 0 \[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " \[LessEqual] 1 then it is logical that E[ ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "] \[GreaterEqual] ", StyleBox["k", FontSlant->"Italic"], " / ", StyleBox["n", FontSlant->"Italic"], "." }], "Text"], Cell["Let us switch to \[Rho]2.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredExpectation[n, \ k, \ \[Rho]2]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`Re( 2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\)\ \((\(-\(\(n\ \[Rho]2\)\/\(1 - \ \[Rho]2\)\)\))\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma]( n\/2 + 1, \(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[Rho]2\)\/\(2\ \((1 - \ \[Rho]2)\)\)\)\))\))\))\)], "Output"] }, Open ]], Cell[TextData[{ "This is a plot. The diagonal gives the true value, and E[", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "] deviates." }], "Text"], Cell[CellGroupData[{ Cell["\<\ pexpr2 = Plot[{rho2, RSquaredExpectation[15, 3, rho2]}, {rho2, 0, 1}, \ AxesLabel -> {\"\\!\\(\[Rho]\\^2\\)\", \"Exp[\\!\\(R\\^2\\)], n = 15, k = \ 3\"}];\ \>", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.588604 [ [.21429 .00222 -12.5 -16 ] [.21429 .00222 12.5 0 ] [.40476 .00222 -12.5 -16 ] [.40476 .00222 12.5 0 ] [.59524 .00222 -12.5 -16 ] [.59524 .00222 12.5 0 ] [.78571 .00222 -12.5 -16 ] [.78571 .00222 12.5 0 ] [.97619 .00222 -5.5 -16 ] [.97619 .00222 5.5 0 ] [1.025 .01472 0 -8 ] [1.025 .01472 16.5 8 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One wonders whether one couldn't find a simpler expression for the \ density such that \n", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%1 R\^2\ \(f( R\^2, \ \[Rho]\^2)\) \[DifferentialD]R\^2\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " + ", StyleBox["k/n", FontSlant->"Italic"], " (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "), of course with the proper confidence intervals." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Confidence interval", "Subsection"], Cell[TextData[{ "If we can find the CDF then we can find the confidence interval for \ \[Rho]2 given an observed value of R2 as CDF[0, R2 ; \[Rho]2Lower] = 1 - \ 0.025 to CDF[0, R2 ; \[Rho]2Higher] = 0.025. However, ", StyleBox["Mathematica", FontSlant->"Italic"], " fails the integral for the CDF." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[\ RSquaredDensity[Phi, \ r2, n, k, \[CurlyPhi]], \ {r2, \ 0, \ x}, Assumptions \[Rule] \((n > 2 && k > 0 && k < n && \((\((n | k)\) \[Element] Integers)\) && \((\[CurlyPhi] \[Element] Reals)\) && \[CurlyPhi] \[GreaterEqual] 0)\)]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(\((1 - r2)\)\^\(1\/2\ \((\(-k\) + n - 2)\)\)\), " ", \(r2\^\(k\/2 - 1\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n\ r2\ \[CurlyPhi]\)\/2\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(k\/2, \(n - k\)\/2\), ")"}]], ",", \({r2, 0, x}\), ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{\(n > 2\), "\[And]", \(k > 0\), "\[And]", \(k < n\), "\[And]", RowBox[{\((n | k)\), "\[Element]", TagBox["\[DoubleStruckCapitalZ]", Function[ {}, Integers]]}], "\[And]", RowBox[{"\[CurlyPhi]", "\[Element]", TagBox["\[DoubleStruckCapitalR]", Function[ {}, Reals]]}], "\[And]", \(\[CurlyPhi] \[GreaterEqual] 0\)}]}]}], "]"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ For a numerical case, suppose that for n = 15 and k = 3 we observe a value R2 \ = .75. Then we can find a numerical solution:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredCI[ .75, \ 15, \ 3, \ .05]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{0.30453998946255506`, 0.8409777116841824`}\)], "Output"] }, Open ]], Cell["\<\ Note that we better use the adjusted R2 to find the confidence interval. But \ we cannot simply substitute it in the above function since that function uses \ the distribution of R2. 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To settle the issue, we would like more formal \ statements about the closeness.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(AdjustedR2Expectation[n, k, \[Rho]\^2] \[TildeTilde] \ \[Rho]\^2\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1 - \(n\ \((1 - Re(2\^\(n\/2 - 1\)\ \ \[ExponentialE]\^\(-\(\(n\ \[Rho]\^2\)\/\(2\ \((1 - \[Rho]\^2)\)\)\)\)\ \ \((\(-\(\(n\ \[Rho]\^2\)\/\(1 - \[Rho]\^2\)\)\))\)\^\(\(-n\)/2\)\ \((\((k - \ n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma](n\/2 + 1, \(-\(\(n\ \ \[Rho]\^2\)\/\(2\ \((1 - \[Rho]\^2)\)\)\)\))\) - k\ \(\[CapitalGamma](n\/2, \ \(-\(\(n\ \[Rho]\^2\)\/\(2\ \((1 - \[Rho]\^2)\)\)\)\))\))\)))\)\)\/\(n - k\) \ \[TildeTilde] \[Rho]\^2\)], "Output"] }, Open ]], Cell["\<\ The appendices contain various efforts to get rid of the complex terms, to \ start with; all failed.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Confidence interval", "Subsubsection"], Cell[TextData[{ "Earlier we considered the confidence interval for ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " assuming that ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = .75 had been observed for ", StyleBox["n ", FontSlant->"Italic"], "= 15 and ", StyleBox["k ", FontSlant->"Italic"], "= 3. Using the distribution for adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " ", "we can now compare the original confidence interval with the one that we \ get when using the adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " and its density." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquaredCI[ .75, \ 15, \ 3, \ .05]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{0.30453998946255506`, 0.8409777116841824`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AdjustedRSquared[ .75, \ 15, \ 3]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`0.6875`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AdjustedRSquaredCI[%, \ 15, \ 3, \ .05]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{0.30453998946255384`, 0.8409777116841824`}\)], "Output"] }, Open ]], Cell["which is rather consistent.", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Substituting the expected value into the density", "Subsection"], Cell[TextData[{ "When we have an observed ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], " then a next question is what the most likely distribution is from which \ it comes. A shortcut is to accept ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " as the almost unbiased estimator and plot densities assuming this value \ for ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ". This is formally doubtful and numerically risky but it allows quick \ visual comparison and gives an impression of the stituation." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?SetR2Plots\)\)], "Input"], Cell[BoxData[ \("SetR2Plots[r2adjobs, n, k] sets RSquaredDensity[Set,...] and \ AdjustedR2Density[Set, ...] so that they can be used for plotting. 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{176.938, 0}} -> {-0.39579, -1.13261, \ 0.00555487, 0.037421}}] }, Open ]], Cell[TextData[{ "PM. Now we assume normal errors and then derive the sample distribution of \ ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], ". We might also work around, assume a Beta distribution for some ", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], StyleBox[", ", FontSlant->"Italic"], "and then derive the distribution of the errors. Perhaps this gives \ formulas that are more manageable and still acceptable. Our objective is to \ make probability statements on the goodness of fit, and not to impose, at all \ cost, normality on the error." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " routine to combine it all" }], "Section"], Cell["\<\ We can combine various results of the above discussion into a single routine \ RSquared.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Options[RSquared]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{ZeroTest \[Rule] True, RSquaredDensity \[Rule] True, Fit \[Rule] True, Chop \[Rule] 1.`*^-6, SignificanceLevel \[Rule] 0.05`, ConfidenceLevel \[Rule] 0.95`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?RSquared\)\)], "Input"], Cell[BoxData[ \("RSquared is used in the output of regression functions to identify the \ square of the multiple correlation coefficient.\n *** AddedUsage by \ Cool`Estimate`: *** \nRSquared[y, yest, K] for vector y and its estimated \ value yest, with K explanators (including a constant), develops various \ statistics around the (centered !) coefficient of determination R^2 (the \ square of the multiple correlation coefficient). Both y and yest are rescaled \ to yd and yestd, i.e. in deviations around their (own) mean. If N = \ Length[yd] then the total degrees of freedom are n = N-1. Since a constant is \ assumed, k = K - 1 are the remaining degrees of freedom for the explanators. \ The degrees of freedom for the error are n - k = N - K. Note that if the \ original estimate (with K) did not include a constant, the refit will include \ a constant, and proper K at input is K+1. Thus, when the routine refits, it \ is under the assumption that the original estimate was merely badly \ calculated, and the routine does not adjust K itself. \nIt should hold that \ Add[e = yd - yestd] = 0 and that yestd . e = 0. If Fit -> True, the OLS fit \ yd = b yestd + e is applied to arrive at a rescaled yestd2 = b yestd. Then \ the total sum of squares SST = yd.yd, SSE = e.e and SSR = SST - SSE = \ yestd'.yestd (or yestd2). If not Fit -> True, the routine gives a warning \ when Add[e] <> 0 or e'yestd <> 0 so that SSR <> yestd'.yestd.\nR2 = 1 - SSE / \ SST\nAdjustedRSquared = 1 - (SSE / (n-k)) / (SST / (N-1)) \n = 1 - n/(n-k) \ (1 - R2) = R2 - k/(n-k) (1 - R2)\nR2[h] = h R2 + 1 - h, where h = 1 gives R2, \ while h = n/(n-k) gives R2Adj\nDegreesOfFreedom -> n - k\nEstimatedVariance = \ s^2 = SSE / (n-k) the variance of the error (unbiased estimate in the \ standard linear model)\nF-test = ( (SST - SSE) / k)) / (SSE / (n-k)), for the \ hypothesis that k non-intercept explanators are zero\nSSR = SST - SSE is the \ (regression) explained sum of squares\nSSRAdj = SSR - k SSE / (n-k), so that \ R2Adj = SSRAdj / SST\nIf RSquaredDensity -> True then RSquaredDensity[Set, \ r2, rho2] and AdjustedR2Density[Ser, r2adj, rho2] have been set, and \ RSquaredDensity[Plot] and AdjustedR2Density[Plot] plot the {0, 1} domain\n\ RSquared[h] has been defined as R2[h] above, when the routine has run\nThe \ option ConfidenceLevel -> cf: if cf is a number then RSquaredCI is called \ with 1-cf; and if cf = {left, right} then RSquaredCI[..., cf]. Note that the \ confidence interval for Rho2 is the same whether calculated with the density \ for R2 or the density for R2adj"\)], "Print", CellTags->"Info3360824314-7600670"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Application of the routine", "Section"], Cell[CellGroupData[{ Cell["Arbitrary example", "Subsection"], Cell[TextData[{ "This gives some arbitrary ", StyleBox["y", FontSlant->"Italic"], " and equally arbitrary ", Cell[BoxData[ \(TraditionalForm\`y\&^\)]], ". The conditions for an OLS estimator are not fulfilled so that ", Cell[BoxData[ \(TraditionalForm\`y\&^\)]], " is rescaled." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquared[\ {5. , 3, 2, 4, 2}, {4.5, 3, 3. , 3.9, 1.5}, 2] // MatrixForm\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(RSquared::"fit"\), \(\(:\)\(\ \)\), "\<\"yest adjusted by \\!\ \\(TraditionalForm\\`\\(1.013986013986014`\\\\ \\\"yestd\\\"\\)\\)\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\("RSquared[h]" \[Rule] 1 - 0.22161661867544225`\ "h"\)}, {\(AdjustedRSquared \[Rule] 0.7045111750994104`\)}, {\(Coefficient \[Rule] 1.013986013986014`\)}, {\(EstimatedVariance \[Rule] 0.5023310023310023`\)}, {\(FRatioCriticalValue \[Rule] 7.708647422176785`\)}, {\(FRatioTest \[Rule] Reject(Hyp(0))\)}, {\(N \[Rule] 5\)}, {\(RSquared \[Rule] 0.7783833813245578`\)}, {\(RSquaredCI \[Rule] {0.19621322653470086`, 0.9089984855720361`}\)}, {\(FRatioTest(Value) \[Rule] 10.536890951276105`\)}, { RowBox[{"ANOVATable", "\[Rule]", TagBox[GridBox[{ { FormBox["\<\"\"\>", "TraditionalForm"], FormBox["\<\"Sum of Squares\"\>", "TraditionalForm"], FormBox["\<\"DoF\"\>", "TraditionalForm"], FormBox["\<\"Mean Square\"\>", "TraditionalForm"]}, { FormBox["\<\"SSR\"\>", "TraditionalForm"], "5.293006993006993`", "1", "5.293006993006993`"}, { FormBox["\<\"SSE\"\>", "TraditionalForm"], "1.5069930069930069`", "3", "0.5023310023310023`"}, { FormBox["\<\"SST\"\>", "TraditionalForm"], "6.8`", "4", "1.7`"}, { FormBox["\<\"SSRAdj\"\>", "TraditionalForm"], "4.790675990675991`", "1", "4.790675990675991`"}, { FormBox["\<\"SSEAdj\"\>", "TraditionalForm"], "2.0093240093240086`", "4", "0.5023310023310023`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{"SSR", "SSE", "SST", "SSRAdj", "SSEAdj"}, {"Sum of Squares", "DoF", "Mean Square"}}]]]}]} }], "\[NoBreak]", ")"}], MatrixForm[ {"RSquared[h]" -> Plus[ 1, Times[ -.22161661867544225, "h"]], AdjustedRSquared -> .70451117509941041, Coefficient -> 1.013986013986014, EstimatedVariance -> .50233100233100225, FRatioCriticalValue -> 7.7086474221767851, FRatioTest -> Reject[ Hyp[ 0]], N -> 5, RSquared -> .77838338132455775, RSquaredCI -> {.19621322653470086, .90899848557203611}, FRatioTest[ Value] -> 10.536890951276105, ANOVATable -> TableForm[ {{5.2930069930069932, 1, 5.2930069930069932}, { 1.5069930069930069, 3, .50233100233100225}, {6.7999999999999998, 4, 1.7}, {4.7906759906759913, 1, 4.7906759906759913}, { 2.0093240093240086, 4, .50233100233100225}}, TableHeadings -> {{ "SSR", "SSE", "SST", "SSRAdj", "SSEAdj"}, {"Sum of Squares", "DoF", "Mean Square"}}]}]], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Above routine sets the density function. Any value of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " can be entered but the default is ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], "." }], "Text"], Cell[TextData[{ "PM. 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If Fit \[Rule] True (default) then the estimate is automatically \ rescaled, under the assumption that the problem is merely numerical precision \ and rounding errors (otherwise raise K by 1). If the option Fit \[Rule] False \ is used then the test on those conditions will indicate whether these hold. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RSquared[{5. , 3, 2, 4, 2}, {4.5, 4, 3.5, 2, 1}, 2, \ Fit\ \[Rule] \ False] // MatrixForm\)], "Input"], Cell[BoxData[ \(TraditionalForm\`RSquared::"yse" \(\(:\)\(\ \)\) "(yest - Mean[y])'.e should be zero, isn't; use the option Fit -> True"\ \)], "Message"], Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-5.`\)}, {"1.`*^-6"} }], "\[NoBreak]", ")"}], MatrixForm[ {-5.0, .99999999999999995*^-6}]], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ If one wishes, once can force the use of the raw input, neglecting the zero \ conditions. 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For specification search it seems advisable that they are \ more standardly included. Of course, the discussion above makes some \ assumptions here, and these may not be suitable, but presently they seem \ consistent.\ \>", "Text"], Cell["\<\ The advantage of the discussion above is not merely that these relations have \ been stated but also that they are available as routines, with numbers with \ known accuracy and with formulas and graphs (that are the (temporary) human \ advantage over digital machines). \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Appendices", "Section"], Cell[CellGroupData[{ Cell["Graphical decomposition", "Subsection"], Cell["\<\ The density of R2 consists of a beta distribution (depending upon n and k) \ times another term (depending upon rho2). 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same result", "Subsection"], Cell["\<\ The awkward result on the expectation causes us to look for alternative ways \ to derive our results. It appears that we find the same results, which adds \ to our confidence, but, the results remain awkward.\ \>", "Text"], Cell[TextData[{ "We had ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " = ", StyleBox["b'X'X.b ", FontSlant->"Italic"], "/ (", StyleBox["b'X'X.b + e'e", FontSlant->"Italic"], ") ~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["k, ", FontSlant->"Italic"], "\[Theta]] / (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["k, ", FontSlant->"Italic"], "\[Theta]] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]], ") where \[Theta] = ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ")." }], "Text"], Cell[TextData[{ "Subsequently we follow the same calculation scheme in MATH2011 (see the \ references). There, the property is derived, using ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[p]\)\(\ \)\)\)]], " = \[Gamma][2, ", StyleBox["p", FontSlant->"Italic"], " / 2], that for two variates y1 ~ \[Gamma][2, n1 / 2] and y2 ~ \[Gamma][2, \ n2 / 2] then ", "y1", " / (y1", StyleBox[" + ", FontSlant->"Italic"], "y2) ~ Beta[n1, n2]. For us, one of the distributions is a non-central \ chisquare, but the derivation scheme is the same. Thus we select u1 = y1 / \ (y1 + y2) and u2 = y1 + y2, so that y1 = u1 u2 and y2 = u2 (1 - u1) and the \ Jacobian is u2 (which is, given the squares, always nonnegative). Later we \ substitute n1 = ", StyleBox["k", FontSlant->"Italic"], " and n2 = ", StyleBox["n - k", FontSlant->"Italic"], ". A special condition is n2 \[LessEqual] n1 or ", StyleBox["n - k", FontSlant->"Italic"], " \[LessEqual] ", StyleBox["k", FontSlant->"Italic"], " or ", StyleBox["k", FontSlant->"Italic"], " \[GreaterEqual] ", StyleBox["n", FontSlant->"Italic"], " / 2." }], "Text"], Cell[CellGroupData[{ Cell["?ChiSquareDistribution", "Input"], Cell[BoxData[ RowBox[{"\<\"ChiSquareDistribution[n] represents the chi-square \ distribution with n degrees of freedom.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Statistics`NormalDistribution`", Active->True, ButtonStyle->"AddOnsLink"]}]], "Print", CellTags->"Info3358864142-7079604"] }, Open ]], Cell[CellGroupData[{ Cell["?NoncentralChiSquareDistribution", "Input"], Cell[BoxData[ RowBox[{"\<\"NoncentralChiSquareDistribution[n, lambda] represents the \ non-central chi-square distribution with n degrees of freedom and \ non-centrality parameter lambda.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Statistics`ContinuousDistributions`", Active->True, ButtonStyle->"AddOnsLink"]}]], "Print", CellTags->"Info3358864142-1416755"] }, Open ]], Cell[TextData[{ "PM ", StyleBox["Mathematica", FontSlant->"Italic"], " notes: \"The sum of the squares of ", Cell[BoxData[ FormBox[ StyleBox["n", "TI"], TraditionalForm]], "InlineFormula", GridBoxOptions->{ColumnWidths->Automatic}], " normally distributed random variables with variance ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2 = 1\)], "InlineFormula", GridBoxOptions->{ColumnWidths->Automatic}], " and nonzero means follows a ", StyleBox["noncentral chi\[Hyphen]square distribution", "TB"], " ", StyleBox["NoncentralChiSquareDistribution[", "MR"], StyleBox["n", "TI"], StyleBox[", ", "MR"], StyleBox["lambda", "TI"], StyleBox["]", "MR"], ". The noncentrality parameter ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\)], "InlineFormula", GridBoxOptions->{ColumnWidths->Automatic}], " is the sum of the squares of the means of the random variables in the \ sum. Note that in various places in the literature, ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]/2\)], "InlineFormula", GridBoxOptions->{ColumnWidths->Automatic}], " or ", Cell[BoxData[ \(TraditionalForm\`\@\[Lambda]\)], "InlineFormula", GridBoxOptions->{ColumnWidths->Automatic}], " is used as the noncentrality parameter.\"" }], "Text"], Cell[CellGroupData[{ Cell["\<\ g1[u1_, u2_] = PDF[NoncentralChiSquareDistribution[n1, \[Theta]], u1 u2] * PDF[ChiSquareDistribution[n2], u2 (1 - u1)] * u2\ \>", "Input"], Cell[BoxData[ FormBox[ RowBox[{\(1\/\(\[CapitalGamma](n2\/2)\)\), RowBox[{"(", RowBox[{\(2\^\(\(-\(n1\/2\)\) - n2\/2\)\), " ", \(\[ExponentialE]\^\(1\/2\ \((\(-u1\)\ u2 - \[Theta])\) - 1\/2\ \((1 - u1)\)\ u2\)\), " ", "u2", " ", \(\((\((1 - u1)\)\ u2)\)\^\(n2\/2 - 1\)\), " ", \(\((u1\ u2)\)\^\(n1\/2 - 1\)\), " ", TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_0\)\(F\&~\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox["\[Null]", InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric0F1Regularized, Editable->False], ";", TagBox[ TagBox[ TagBox[\(n1\/2\), Hypergeometric0F1Regularized, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric0F1Regularized, Editable->False], ";", TagBox[\(\(u1\ u2\ \[Theta]\)\/4\), Hypergeometric0F1Regularized, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQRegularized[ #, #2, #3]&], Editable->False], Hypergeometric0F1Regularized]}], ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Where the symbol ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\_0\)\(F\&~\_1\)\)]], "is the Hypergeometric0F1Regularized function as implemented in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "Integrating out u2, we have the density of ", Cell[BoxData[ \(TraditionalForm\`\(\(R\^2\)\(.\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell["\<\ dens[u1_] = Integrate[g1[u1, u2], {u2, 0, \[Infinity]}, Assumptions \[Rule] (n1 > 2 && n2 > 2 && ((n1 | n2) \[Element] Integers) && 0 <= u1 <= 1 && \ (\[Theta] \[Element] Reals) && Re[\[Theta]] >= 0 && Im[\[Theta]] == 0)]\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(\(-\[Theta]\)/2\)\), " ", \(\((1 - u1)\)\^\(n2\/2 - 1\)\), " ", \(u1\^\(n1\/2 - 1\)\), " ", \(\[CapitalGamma](\(n1 + n2\)\/2)\), " ", TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_1\)\(F\&~\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(\(n1 + n2\)\/2\), Hypergeometric1F1Regularized, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1Regularized, Editable->False], ";", TagBox[ TagBox[ TagBox[\(n1\/2\), Hypergeometric1F1Regularized, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1Regularized, Editable->False], ";", TagBox[\(\(u1\ \[Theta]\)\/2\), Hypergeometric1F1Regularized, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQRegularized[ #, #2, #3]&], Editable->False], Hypergeometric1F1Regularized]}], \(\[CapitalGamma](n2\/2)\)], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Where the symbol ", Cell[BoxData[ \(TraditionalForm\`\(\[InvisiblePrefixScriptBase]\_1\)\(F\&~\_1\)\)]], "is the Hypergeometric1F1Regularized function as implemented in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["We can check formally that the integral is 1.", "Text"], Cell[CellGroupData[{ Cell["\<\ Integrate[dens[u1], {u1, 0, 1}, Assumptions \[Rule] (n1 > 2 && n2 > 2 && ((n1 | n2) \[Element] Integers) && 0 <= u1 <= 1) && \ (Re[\[Theta]] >= 0)]\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`1\)], "Output"] }, Open ]], Cell["We even find the expected value.", "Text"], Cell[CellGroupData[{ Cell["\<\ expval[u1_] = Integrate[u1 dens[u1], {u1, 0, 1}, Assumptions \[Rule] (n1 > 2 && n2 > 2 && ((n1 | n2) \[Element] Integers) && (0 <= u1 <= 1) && \ (\[Theta] \[Element] Reals) && (Re[\[Theta]] >= 0) && Im[\[Theta]] == 0)]\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`\(-2\^\(1\/2\ \((n1 + n2 - 2)\)\)\)\ \[ExponentialE]\^\ \(\(-\[Theta]\)/2\)\ \((\(-\[Theta]\))\)\^\(1\/2\ \((\(-n1\) - n2)\)\)\ \((n2\ \ \(\[CapitalGamma](\(n1 + n2\)\/2)\) + n1\ \(\[CapitalGamma](\(n1 + n2\)\/2, \(-\(\[Theta]\/2\)\))\) - 2\ \(\[CapitalGamma]( 1\/2\ \((n1 + n2 + 2)\), \(-\(\[Theta]\/2\)\))\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Change to ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]] }], "Subsubsection"], Cell[TextData[{ "Taking n1 = ", StyleBox["k", FontSlant->"Italic"], " and n2 = ", StyleBox["n - k, ", FontSlant->"Italic"], "and using proper variable r2 to indicate the ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ". Also use \[CurlyPhi] = \[Theta] / ", StyleBox["n", FontSlant->"Italic"], ". " }], "Text"], Cell[CellGroupData[{ Cell["\<\ g1[r2_, n_, k_, \[CurlyPhi]_] = g1[r2, u2] /. {n1 \[Rule] k, n2 \[Rule] n-k, \ \[Theta] -> n \[CurlyPhi]} // FullSimplify\ \>", "Input"], Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox[ RowBox[{\(2\^\(\(-n\)/2\)\), " ", \(\[ExponentialE]\^\(1\/2\ \((\(-u2\) - n\ \[CurlyPhi])\)\)\), " ", \(\((r2\ u2)\)\^\(\(k - 2\)\/2\)\), " ", \(\((u2 - r2\ u2)\)\^\(\(n - k\)\/2\)\), " ", TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_0\)\(F\&~\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox["\[Null]", InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric0F1Regularized, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric0F1Regularized, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric0F1Regularized, Editable->False], ";", TagBox[\(1\/4\ n\ r2\ u2\ \[CurlyPhi]\), Hypergeometric0F1Regularized, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQRegularized[ #, #2, #3]&], Editable->False], Hypergeometric0F1Regularized]}], \(\((r2 - 1)\)\ \(\[CapitalGamma](\(n - k\)\/2)\)\)]}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Integrating with these variables gives the same result as when we would \ directly substitute them in the result above, which indicates that the \ domains of n1 and n2 might be less relevant.\ \>", "Text"], Cell["\<\ The main effect of this alternative route is that the Beta expression now is \ expressed in Gamma's.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ densityR2[r2_, n_, k_, \[CurlyPhi]_] = Integrate[g1[r2, n, k, \[CurlyPhi]] , \ {u2, 0, \[Infinity]}, Assumptions \[Rule] (n > 2 && k > 0 && k < n && ((n | k) \[Element] Integers) && 0 <= r2 <= 1 \ && (\[CurlyPhi] \[Element] Reals) && (Re[\[CurlyPhi]] >= 0) && \ (Im[\[CurlyPhi]] == 0))]\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(-\(\(n\ \[CurlyPhi]\)\/2\)\)\), " ", \(\((1 - r2)\)\^\(1\/2\ \((\(-k\) + n - 2)\)\)\), " ", \(r2\^\(k\/2 - 1\)\), " ", \(\[CapitalGamma](n\/2)\), " ", TagBox[ TagBox[ RowBox[{\(\(\[InvisiblePrefixScriptBase]\_1\)\(F\&~\_1\)\), "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(n\/2\), Hypergeometric1F1Regularized, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1Regularized, Editable->False], ";", TagBox[ TagBox[ TagBox[\(k\/2\), Hypergeometric1F1Regularized, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1Regularized, Editable->False], ";", TagBox[\(\(n\ r2\ \[CurlyPhi]\)\/2\), Hypergeometric1F1Regularized, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQRegularized[ #, #2, #3]&], Editable->False], Hypergeometric1F1Regularized]}], \(\[CapitalGamma](\(n - \ k\)\/2)\)], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "If we take ", StyleBox["k ", FontSlant->"Italic"], "> 0", " the integral is 1." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Integrate[densityR2[r2, n, k, \[CurlyPhi]], {r2, 0, 1}, Assumptions \[Rule] (n > 2 && k > 0 && k < n && ((n | k) \[Element] Integers) && \[CurlyPhi] \ >= 0)]\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`1\)], "Output"] }, Open ]], Cell["And the expected value is not different.", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Note on the use of ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has been extremely helpful in deriving these relations (1) by being a \ database on known mathematical relations, (2) for symbolic manipulation, (3) \ by supporting quick numerical examples, with worriless accuracy, (4) by its \ graphical interface, (5) by providing speed and convenience, (6) by being \ flexible so that routines could be taylored. Needless to say, pen and paper \ proved indispensable as well, but those didn't provide the other points \ above." }], "Text"], Cell[TextData[{ "The focus was on deriving these relations. Much work is not shown here. \ For example, at one stage, I used ", StyleBox["Mathematica", FontSlant->"Italic"], " to derive the distribution of N[\[Mu], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "] / (N[\[Mu], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "] + ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[", StyleBox["n", FontSlant->"Italic"], "]) but later I realized that I hadn't properly thought about \ interdependence, that is, I assumed independence for the derivation, but the \ assumptions didn't fit the problem. One can wonder whether the good features \ of ", StyleBox["Mathematica", FontSlant->"Italic"], " provide a sink, i.e. that one stops thinking properly and proceeds with a \ line of thought since it is so easy (that is, relatively) to continue. \ However, once one reaches a result, it is standard to review all steps, and \ errors are identified. Alternatively, if dead alleys are not investigated, \ one might as well give up the whole endeavour. Thus in this case I think that \ it is fortunate that ", StyleBox["Mathematica", FontSlant->"Italic"], " provided the environment to identify an intermediate result, so that it \ was possible to confidently reject it, and to refocus on the main line of \ reasoning. " }], "Text"], Cell[TextData[{ "I did notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " is not strong on integrals of standard statistical functions. For example \ the expectation of ", StyleBox["k x ", FontSlant->"Italic"], "when ", StyleBox["x ~ ", FontSlant->"Italic"], "N[\[Mu], ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "] makes one eventually abort the operation." }], "Text"], Cell[TextData[{ "Also, \"explanations\" provided by ", StyleBox["Mathematica", FontSlant->"Italic"], " can be non-informative. For example the \"explanation\" of ", Cell[BoxData[ \(TraditionalForm\`tanh\^\(-1\)\)]], " leaves a person nonplussed, while it would be possible to give a small \ explanation." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?ArcTanh\)\)], "Input"], Cell[BoxData[ RowBox[{"\<\"ArcTanh[z] gives the hyperbolic arc tangent of the complex \ number z.\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"ArcTanh", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3358101636-6182888"] }, Open ]], Cell["The following can be noted as well. ", "Text"], Cell[CellGroupData[{ Cell["\<\ pdf = PDF[NoncentralFRatioDistribution[n1, n2, \[Lambda]], x n2 / n1]\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(\[ExponentialE]\^\(\(-\[Lambda]\)/2\)\), " ", \(n1\^\(n1/2\)\), " ", \(n2\^\(n2/2\)\), " ", \(\((\(n2\ x\)\/n1)\)\^\(\(n1 - 2\)\/2\)\), " ", \(\((x\ n2 + n2)\)\^\(\(-\(1\/2\)\)\ \((n1 + n2)\)\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(\(n1 + n2\)\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(n1\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n2\ x\ \[Lambda]\)\/\(2\ \((x\ n2 + n2)\)\)\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(n1\/2, n2\/2\), ")"}]], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Domain[NoncentralFRatioDistribution[n1, n2, \[Lambda]]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`Interval[{0, \[Infinity]}]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is curiously unable to show what this default function integrates to. As \ it is a PDF, one would expect the value 1." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ pdf, \ {x, \ 0, \ \[Infinity]}, \ \[IndentingNewLine]Assumptions \[Rule] \((n1 > 0 && n2 > 2\ && \((\((n1 | n2)\) \[Element] Integers)\) && \((\[Lambda]\ \[Element] \ Reals)\)\ && \ \((\[Lambda] \[GreaterEqual] 0)\))\)]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(Integrate::"gener"\), \(\(:\)\(\ \)\), "\<\"Unable to check \ convergence. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Integrate::gener\\\"]\\)\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{\(\[ExponentialE]\^\(\(-\[Lambda]\)/2\)\), " ", \(n1\^\(n1/2\)\), " ", \(n2\^\(n2/2\)\), " ", \(\((\(n2\ x\)\/n1)\)\^\(\(n1 - 2\)\/2\)\), " ", \(\((x\ n2 + n2)\)\^\(\(-\(1\/2\)\)\ \((n1 + n2)\)\)\), " ", TagBox[ TagBox[ RowBox[{ RowBox[{ SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", "TraditionalForm"]], SubscriptBox["F", FormBox["1", "TraditionalForm"]]}], "\[InvisibleApplication]", RowBox[{"(", RowBox[{ TagBox[ TagBox[ TagBox[\(\(n1 + n2\)\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[ TagBox[ TagBox[\(n1\/2\), Hypergeometric1F1, Editable->True], InterpretTemplate[ { SlotSequence[ 1]}&]], Hypergeometric1F1, Editable->False], ";", TagBox[\(\(n2\ x\ \[Lambda]\)\/\(2\ \((x\ n2 + n2)\)\)\), Hypergeometric1F1, Editable->True]}], ")"}]}], InterpretTemplate[ HypergeometricPFQ[ #, #2, #3]&], Editable->False], Hypergeometric1F1]}], RowBox[{ InterpretationBox["\[CapitalBeta]", Beta, Editable->False, Selectable->False], "(", \(n1\/2, n2\/2\), ")"}]], ",", \({x, 0, \[Infinity]}\), ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{\(n1 > 0\), "\[And]", \(n2 > 2\), "\[And]", RowBox[{\((n1 | n2)\), "\[Element]", TagBox["\[DoubleStruckCapitalZ]", Function[ {}, Integers]]}], "\[And]", RowBox[{"\[Lambda]", "\[Element]", TagBox["\[DoubleStruckCapitalR]", Function[ {}, Reals]]}], "\[And]", \(\[Lambda] \[GreaterEqual] 0\)}]}]}], "]"}], TraditionalForm]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Decomposition of the expected value", "Subsection"], Cell["\<\ The following takes the different multiplicative components of the expected \ value, so that we can inspect their contribution.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ ev = RSquaredExpectation[Phi, Identity, n, k, \[CurlyPhi]] \ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \ \[CurlyPhi]\)\/2\)\)\ \((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma](n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ decomp = {2^(n/2 - 1), E^((n*\[CurlyPhi])/2), ((-n)*\[CurlyPhi])^(n/2), (k - \ n)*Gamma[n/2] + 2*Gamma[n/2 + 1, -((n*\[CurlyPhi])/2)] - k*Gamma[n/2, -((n*\ \[CurlyPhi])/2)] }\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`{2\^\(n\/2 - 1\), \[ExponentialE]\^\(\(n\ \ \[CurlyPhi]\)\/2\), \((\(-n\)\ \[CurlyPhi])\)\^\(n/2\), \((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma](n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ decomprho2 = decomp /. {n -> 15, k -> 3, \[CurlyPhi] -> rho2}\ \>", "Input"], Cell[BoxData[ \(TraditionalForm\`{64\ \@2, \[ExponentialE]\^\(\(15\ rho2\)/2\), 170859375\ \@15\ \((\(-rho2\))\)\^\(15/2\), \(-3\)\ \(\[CapitalGamma]( 15\/2, \(-\(\(15\ rho2\)\/2\)\))\) + 2\ \(\[CapitalGamma]( 17\/2, \(-\(\(15\ rho2\)\/2\)\))\) - \(405405\ \@\[Pi]\)\/32}\)], \ "Output"] }, Open ]], Cell["\<\ A conclusion of these graphs below is that large values compensate for small \ values, and that the imaginary values cannot be neglected (though the Re[...] \ of the total remains the most sensible selection). PM. 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distribution \ of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], "." }], "Text"], Cell[TextData[{ "To better understand the situation, we also consider the LHS of SSRadj = \ (", StyleBox["b'X'X.b", FontSlant->"Italic"], " - ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ")", StyleBox[" = ", FontSlant->"Italic"], "(", StyleBox["y'.y", FontSlant->"Italic"], " - ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ")." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(b' X' X . b\), FontSlant->"Italic"], \(\[Sigma]\^2\)], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]'\/\[Sigma]\)]], ". ", StyleBox["P. ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], StyleBox[" + ", FontSlant->"Italic"], "2 ", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' . ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], " + ", StyleBox["B / ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox[" ", FontSlant->"Italic"], "with", " \[Theta] = ", StyleBox["B / ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], "." }], "Text"], Cell[TextData[{ "The part ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]'\/\[Sigma]\)]], ". ", StyleBox["P. ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], StyleBox[" ", FontSlant->"Italic"], " is a sum of squares of standard normal variables P \[Epsilon] / \[Sigma], \ and thus has the ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], "distribution." }], "Text"], Cell[TextData[{ "The part ", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' . ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], " is a linear sum (", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' ", FontSlant->"Italic"], ") of the standard normal distibution ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], "~ N[0, ", StyleBox["I ", FontSlant->"Italic"], "]. Thus ", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' . ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], " ~ N[0, \[Theta] ]. But we have two of those, thus 2", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' . ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], " ~ N[0, 2 \[Theta] ]. The constant can be included as the mean of this \ distribution, thus ", StyleBox[" ", FontSlant->"Italic"], "2 ", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' . ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], " + ", StyleBox["B / ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ N[\[Theta], 2 \[Theta] ]. Note that ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[p]\)\(\ \)\)\)]], "has mean ", StyleBox["p", FontSlant->"Italic"], " and variance 2 ", StyleBox["p, ", FontSlant->"Italic"], "so that N[\[Theta], 2 \[Theta] ] approximates a ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[\[Theta]]\)\(\ \)\)\)]], "distribution." }], "Text"], Cell[TextData[{ "Thus ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(b' X' X . b\), FontSlant->"Italic"], \(\[Sigma]\^2\)], TraditionalForm]]], " ~ ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], " + N[\[Theta], 2 \[Theta] ], in a shorthand notation as the sum of two \ random variates. " }], "Text"], Cell[TextData[{ "If we sum the distributions of the two variates we must check whether they \ are dependent or not. A sufficient condition for independence of the linear \ form ", StyleBox["L. ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], " and the quadratic form ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]'\/\[Sigma]\)]], ". ", StyleBox["P. ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\/\[Sigma]\)]], StyleBox[" ", FontSlant->"Italic"], "is that ", StyleBox["L P", FontSlant->"Italic"], " = 0. Substituting ", StyleBox["L", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X' ", FontSlant->"Italic"], "we find ", StyleBox["L P", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[Beta]'\/\[Sigma]\)]], StyleBox["X'", FontSlant->"Italic"], " ", StyleBox["X.", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((X' . X)\)\^\(-1\)\)]], ".", StyleBox["X' ", FontSlant->"Italic"], " ", StyleBox[" = L \[NotEqual] ", FontSlant->"Italic"], "0, which indicates dependence (which is consistent with the notion that a \ sum of squares would always generate nonnegative values)." }], "Text"], Cell[TextData[{ "Denote the \"independent remainder\" as D[\[Theta]]. Thus, as we had some \ variables ", StyleBox["a ~ ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], "and ", StyleBox["b ~ ", FontSlant->"Italic"], "N[\[Theta], 2 \[Theta] ],", StyleBox[" ", FontSlant->"Italic"], "mutually dependent, we now consider the variables ", StyleBox["a ~ ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], " and ", StyleBox["f ~ ", FontSlant->"Italic"], " D[\[Theta]], now independent, such that ", StyleBox["a + b \[DoubleLeftRightArrow] a + f. ", FontSlant->"Italic"], "(Above we had the total as ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k, \ \[Theta]]\)\(\ \)\)\)]], ", thus ", StyleBox["f ~ ", FontSlant->"Italic"], " D[\[Theta]] = ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k, \ \[Theta]]\)\(\ \)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], " can be used.)" }], "Text"], Cell[TextData[{ "The numerator of ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "also contains a term (", StyleBox["- k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ") and we know that", StyleBox[" e'e", FontSlant->"Italic"], " / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ~ ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]], " " }], "Text"], Cell[TextData[{ "In summary, we find for the sample distribution of ", " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " ", StyleBox["/ (", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " + ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], ")", StyleBox[" ", FontSlant->"Italic"], "that:" }], "Text"], Cell[TextData[{ "(a) numerator ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(b' X' X . b\), FontSlant->"Italic"], \(\[Sigma]\^2\)], TraditionalForm]]], " - ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ~ ( ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], " + D[\[Theta]] ) - ", Cell[BoxData[ \(TraditionalForm\`k\/\(n - k\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]] }], "Text"], Cell[TextData[{ " ~ ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[k]\)\(\ \)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`k\/\(n - k\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]], " + D[\[Theta]] " }], "Text"], Cell[TextData[{ " ~ ", Cell[BoxData[ \(TraditionalForm\`\(\[Chi]\^2\)[ k\ - \ k\/\(n - k\)\ \((n - k)\)]\)]], " + D[\[Theta]] " }], "Text"], Cell["\<\ ~ D[\[Theta]] \ \>", "Text"], Cell[TextData[{ "(b) denominator (", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(b' X' X . b\), FontSlant->"Italic"], \(\[Sigma]\^2\)], TraditionalForm]]], " - ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ) + (", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ") = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(b' X' X . b\), FontSlant->"Italic"], \(\[Sigma]\^2\)], TraditionalForm]]], " + (", StyleBox["n - k) ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " = (", StyleBox["b'X'X.b", FontSlant->"Italic"], " + ", StyleBox["e'e", FontSlant->"Italic"], ") / ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]] }], "Text"], Cell[TextData[{ " (", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(b' X' X . b\), FontSlant->"Italic"], \(\[Sigma]\^2\)], TraditionalForm]]], " - ", StyleBox["k ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], " ) + (", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "/ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\)]], ") ~ D[\[Theta], ", StyleBox["k", FontSlant->"Italic"], "] + ", Cell[BoxData[ \(TraditionalForm\`n\/\(n - k\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n - k]\)\(\ \)\)\)]] }], "Text"], Cell[TextData[{ " \ ~ D[\[Theta]] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]] }], "Text"], Cell[TextData[{ "Hence ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ D[\[Theta]] / (D[\[Theta]] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], ")." }], "Text"], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " = \[Theta] / (\[Theta] + n) or \[Theta] = ", StyleBox["n", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], "), thus D[\[Theta]] = \[CapitalDelta][", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", ", StyleBox["n", FontSlant->"Italic"], "]. Note that \[Theta] and \[CapitalDelta], though using Greek letters, are \ still sample functions, but restricted by population parameters." }], "Text"], Cell[TextData[{ "Hence ", " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ \[CapitalDelta][", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", ", StyleBox["n", FontSlant->"Italic"], "] / (\[CapitalDelta][", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", ", StyleBox["n", FontSlant->"Italic"], "] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], ") where it may be noted that there is no explicit term for ", StyleBox["k", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "We would like E[ ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "] = ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ". If it is possible to so, which is the big IF, we select the ", StyleBox["f ", FontSlant->"Italic"], " above such that we can also take the expectation of the numerator \ independently from the denominator, so that E[ ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], "] = E[", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " ] ", StyleBox["/ E", FontSlant->"Italic"], "[", Cell[BoxData[ \(TraditionalForm\`B\&^\)]], " + ", StyleBox["n ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`s\^2\)]], "] = ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\[Rho]\^2\), "TraditionalForm"]}], TraditionalForm]]], " and in particular such that D[\[Theta]] = \[CapitalDelta][", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ", ", StyleBox["n", FontSlant->"Italic"], "] = ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], " / (1 - ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\^2\)]], ") = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Chi]\^2\), "[", RowBox[{"n", " ", RowBox[{ FormBox[\(\[Rho]\^2\), "TraditionalForm"], " ", "/", RowBox[{"(", RowBox[{"1", "-", FormBox[\(\[Rho]\^2\), "TraditionalForm"]}], ")"}]}]}], "]"}], " "}], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[\[Theta]]\)\(\ \)\)\)]], "." }], "Text"], Cell[TextData[{ "In that case ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta]] / (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta]] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], "). " }], "Text"], Cell[TextData[{ "Subsequently, we use the property that ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[p]\)\(\ \)\)\)]], " = \[Gamma][2, ", StyleBox["p", FontSlant->"Italic"], " / 2] and that for two variates ", StyleBox["x", FontSlant->"Italic"], " ~ \[Gamma][2, ", StyleBox["n", FontSlant->"Italic"], " / 2] and ", StyleBox["y", FontSlant->"Italic"], " ~ \[Gamma][2, ", StyleBox["m", FontSlant->"Italic"], " / 2] then ", StyleBox["x", FontSlant->"Italic"], " / (", StyleBox["x + y", FontSlant->"Italic"], ") ~ Beta[", StyleBox["n, m", FontSlant->"Italic"], "]. " }], "Text"], Cell[TextData[{ "Thus: ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ Beta[\[Theta], ", StyleBox["n", FontSlant->"Italic"], " ] . That is, if above assumptions on ", StyleBox["f ", FontSlant->"Italic"], " can be made. (The following shows that this runs into problems.)" }], "Text"], Cell[TextData[{ "An estimator is ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], " = ", StyleBox["n ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" / (1 - ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], "), for convenience rounded down to an integer (\"Floor[", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], " ]\"), while negative ", Cell[BoxData[ \(TraditionalForm\`R\&_\^2\)]], " can be set at a small positive value. It is a leap of faith to replace \ the sample parameter \[Theta] by its unbiased ", "estimator", " ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], ", in particular to describe the sample distribution of the ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], " that defines it, but the result is consistent. " }], "Text"], Cell[TextData[{ "Thus: ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ Beta[ ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], ", ", StyleBox["n", FontSlant->"Italic"], " ] = Beta[Max[epsilon, Floor[", StyleBox["n ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" / (1 - ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], ")]], ", StyleBox["n", FontSlant->"Italic"], " ]." }], "Text"], Cell[TextData[{ "Rather though, such a substitution causes an inaccuracy, and we rather \ have (re-introducing ", StyleBox["k", FontSlant->"Italic"], "):" }], "Text"], Cell[TextData[{ "Thus: ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ Beta[ ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\&^\)]], ", ", StyleBox["n-k", FontSlant->"Italic"], " ] = Beta[Max[epsilon, Floor[", StyleBox["n ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" / (1 - ", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], ")]], ", StyleBox["n-k", FontSlant->"Italic"], " ]." }], "Text"], Cell["But this still would not allow for negative values.", "Text"], Cell[TextData[{ "Thus the assumption that ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta]] / (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta]] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], ") runs into problems, since ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], " can have negative values." }], "Text"], Cell[TextData[{ "PM. We cannot use ", Cell[BoxData[ \(TraditionalForm\`\[Rho]\&^\^2\)]], StyleBox[" ", FontSlant->"Italic"], " ~ ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta] ] / (", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta] ] + ", Cell[BoxData[ \(TraditionalForm\`\(\(\(\[Chi]\^2\)[n]\)\(\ \)\)\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta] ] / ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\^2\)]], "[\[Theta] + n ] and then use the F-distribution, since the terms in the \ ratio are dependent." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Aspects of the R2 density", "Subsection"], Cell[CellGroupData[{ Cell["On the hypergeometric function", "Subsubsection"], Cell[CellGroupData[{ Cell["?Hypergeometric0F1Regularized", "Input"], Cell[BoxData[ RowBox[{"\<\"Hypergeometric0F1Regularized[a, z] is the regularized \ confluent hypergeometric function Hypergeometric0F1[a, z]/Gamma[a].\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Hypergeometric0F1Regularized", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3358917601-1004150"] }, Open ]], Cell[TextData[{ "PM ", StyleBox["Mathematica", FontSlant->"Italic"], " notes: \"A limiting form of the confluent hypergeometric function which \ often appears is ", StyleBox["Hypergeometric0F1[", "MR"], StyleBox["a", "TI"], StyleBox[",", "MR"], " ", StyleBox["z", "TI"], StyleBox["]", "MR"], ". This function is obtained as the limit ", Cell[BoxData[ FormBox[ RowBox[{\(\[Null]\_0\), SubscriptBox[ StyleBox["F", "TI"], "1"], RowBox[{"(", RowBox[{";", StyleBox["a", "TI"], ";", StyleBox["z", "TI"]}], ")"}], "=", UnderscriptBox["lim", RowBox[{ StyleBox["q", "TI"], "\[Rule]", "\[Infinity]"}], LimitsPositioning->True], \(\[Null]\_1\), SubscriptBox[ StyleBox["F", "TI"], "1"], RowBox[{"(", RowBox[{ StyleBox["q", "TI"], ";", StyleBox["a", "TI"], ";", StyleBox["z", "TI"], "/", StyleBox["q", "TI"]}], ")"}]}], TraditionalForm]], "InlineFormula"], ". \nThe ", Cell[BoxData[ FormBox[ RowBox[{\(\[Null]\_0\), SubscriptBox[ StyleBox["F", "TI"], "1"]}], TraditionalForm]], "InlineFormula"], " function has the series expansion ", Cell[BoxData[ FormBox[ RowBox[{\(\[Null]\_0\), SubscriptBox[ StyleBox["F", "TI"], "1"], RowBox[{"(", RowBox[{";", StyleBox["a", "TI"], ";", StyleBox["z", "TI"]}], ")"}], "=", UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["k", "TI"], "=", "0"}], "\[Infinity]", LimitsPositioning->True], "1", "/", SubscriptBox[ RowBox[{"(", StyleBox["a", "TI"], ")"}], StyleBox["k", "TI"]], " ", SuperscriptBox[ StyleBox["z", "TI"], StyleBox["k", "TI"]], "/", RowBox[{ StyleBox["k", "TI"], "!"}]}], TraditionalForm]], "InlineFormula"], " and satisfies the differential equation ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["z", "TI"], SuperscriptBox[ StyleBox["y", "TI"], "\[Prime]\[Prime]"], "+", StyleBox["a", "TI"], SuperscriptBox[ StyleBox["y", "TI"], "\[Prime]"], "-", StyleBox["y", "TI"], "=", "0"}], TraditionalForm]], "InlineFormula"], ".\"" }], "Text"], Cell[CellGroupData[{ Cell["?Hypergeometric1F1Regularized", "Input"], Cell[BoxData[ RowBox[{"\<\"Hypergeometric1F1Regularized[a, b, z] is the regularized \ confluent hypergeometric function Hypergeometric1F1[a, b, z]/Gamma[b].\"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"Hypergeometric1F1Regularized", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3358872091-3230279"] }, Open ]], Cell[TextData[{ "PM ", StyleBox["Mathematica", FontSlant->"Italic"], " notes: \"The confluent hypergeometric function can be obtained from the \ series expansion ", Cell[BoxData[ FormBox[ RowBox[{\(\[Null]\_1\), SubscriptBox[ StyleBox["F", "TI"], "1"], RowBox[{"(", RowBox[{ StyleBox["a", "TI"], ";", StyleBox["b", "TI"], ";", StyleBox["z", "TI"]}], ")"}], "=", "1", "+", StyleBox[\(a z\), "TI"], "/", StyleBox["b", "TI"], "+", RowBox[{ StyleBox["a", "TI"], RowBox[{"(", RowBox[{ StyleBox["a", "TI"], "+", "1"}], ")"}]}], "/", RowBox[{ StyleBox["b", "TI"], RowBox[{"(", RowBox[{ StyleBox["b", "TI"], "+", "1"}], ")"}]}], " ", SuperscriptBox[ StyleBox["z", "TI"], "2"], "/", \(2!\), "+", "\[CenterEllipsis]", "=", UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["k", "TI"], "=", "0"}], "\[Infinity]", LimitsPositioning->True], SubscriptBox[ RowBox[{"(", StyleBox["a", "TI"], ")"}], StyleBox["k", "TI"]], "/", SubscriptBox[ RowBox[{"(", StyleBox["b", "TI"], ")"}], StyleBox["k", "TI"]], " ", SuperscriptBox[ StyleBox["z", "TI"], StyleBox["k", "TI"]], "/", RowBox[{ StyleBox["k", "TI"], "!"}], " "}], TraditionalForm]], "InlineFormula"], ". Some special results are obtained when ", Cell[BoxData[ FormBox[ StyleBox["a", "TI"], TraditionalForm]], "InlineFormula"], " and ", Cell[BoxData[ FormBox[ StyleBox["b", "TI"], TraditionalForm]], "InlineFormula"], " are both integers. 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\(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\)\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " makes life complex. We are only interested in real values, but ", StyleBox["Mathematica", FontSlant->"Italic"], " makes sure that we don't forget about the complex plane, and they present \ us with generalizations which we might not need: \"", StyleBox["Gamma[", "MR"], StyleBox["z", "TI"], StyleBox["]", "MR"], " is defined by the integral ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalGamma]", RowBox[{"(", StyleBox["z", "TI"], ")"}], "=", \(\[Integral]\_0\%\[Infinity]\), SuperscriptBox[ StyleBox["t", "TI"], RowBox[{ StyleBox["z", "TI"], "-", "1"}]], SuperscriptBox[ StyleBox["e", "TI"], RowBox[{"-", StyleBox["t", "TI"]}]], StyleBox["d", "TI"], StyleBox["t", "TI"]}], TraditionalForm]], "InlineFormula"], ". For positive integer ", Cell[BoxData[ FormBox[ StyleBox["n", "TI"], TraditionalForm]], "InlineFormula"], ", ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalGamma]", RowBox[{"(", StyleBox["n", "TI"], ")"}], "=", RowBox[{"(", RowBox[{ StyleBox["n", "TI"], "-", "1"}], ")"}], \(\(!\)\(\ \)\)}], TraditionalForm]], "InlineFormula"], ". ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalGamma]", RowBox[{"(", StyleBox["z", "TI"], ")"}]}], TraditionalForm]], "InlineFormula"], " can be viewed as a generalization of the factorial function, valid for \ complex arguments ", Cell[BoxData[ FormBox[ StyleBox["z", "TI"], TraditionalForm]], "InlineFormula"], ".\" Also, ", StyleBox["Gamma[", "MR"], StyleBox["a", "TI"], StyleBox[",", "MR"], " ", StyleBox["z", "TI"], StyleBox["]", "MR"], " is defined by the integral ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalGamma]", RowBox[{"(", RowBox[{ StyleBox["a", "TI"], ",", StyleBox["z", "TI"]}], ")"}], "=", SubsuperscriptBox["\[Integral]", StyleBox["z", "TI"], "\[Infinity]"], SuperscriptBox[ StyleBox["t", "TI"], RowBox[{" ", RowBox[{ StyleBox["a", "TI"], "-", "1"}]}]], SuperscriptBox[ StyleBox["e", "TI"], RowBox[{"-", StyleBox["t", "TI"]}]], StyleBox["d", "TI"], StyleBox["t", "TI"]}], TraditionalForm]], "InlineFormula"], ". Note that the ", StyleBox["z ", FontSlant->"Italic"], "in \[CapitalGamma][", StyleBox["z", FontSlant->"Italic"], "] determines the power whereas the ", StyleBox["a ", FontSlant->"Italic"], "does in \[CapitalGamma][", StyleBox["a, z", FontSlant->"Italic"], "], which doesn't look as consistent notation. Note that the incomplete \ \[CapitalGamma] above uses negative values, and the documentation of ", StyleBox["Mathematica", FontSlant->"Italic"], " states: \"", StyleBox["Gamma[", "MR"], StyleBox["a", "TI"], StyleBox[",", "MR"], " ", StyleBox["z", "TI"], StyleBox["]", "MR"], " has a branch cut discontinuity in the complex ", StyleBox["z", "TI"], " plane running from ", Cell[BoxData[ \(TraditionalForm\`\(-\[Infinity]\)\)], "InlineFormula"], " to ", Cell[BoxData[ \(TraditionalForm\`0\)], "InlineFormula"], "\". But we are only interested in real values, as our assumptions clearly \ stated that Im[\[Theta]] == 0." }], "Text"], Cell[BoxData[ \(MyGamma[z_, \ a_]\ := \ Gamma[a, \ z]\)], "Input"], Cell["\<\ For us, \[CurlyPhi] \[GreaterEqual] 0 and hence only the negative axis for \ -zz in \[CapitalGamma][z, -zz] is relevant.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MyGamma[z_, \ a_]\ = \ Gamma\ [z]\ + \ Integrate[\ t^\((z - 1)\)\ E^\((\(-t\))\), \ {t, \ a, \ 0}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma](z) + If[Re(z) > 0, \[CapitalGamma](z, a) - \[CapitalGamma](z), Integrate[\[ExponentialE]\^\(-t\)\ t\^\(z - 1\), {t, a, 0}, Assumptions \[Rule] Re(z) \[LessEqual] 0]]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " doesn't recognize this formally:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Gamma[z, \ 0]\ \[Equal] \ Gamma[z]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma](z, 0) \[LongEqual] \[CapitalGamma]( z)\)], "Output"] }, Open ]], Cell["But for random numbers it does:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((rand\ = \ Random[]; \ {rand, \ Gamma[rand, \ 0], \ Gamma[rand], \ \ Gamma[rand, \ 0]\ \[Equal] \ Gamma[rand]})\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{0.34845133180429305`, 2.5579177299897324`, 2.5579177299897324`, True}\)], "Output"] }, Open ]], Cell["Note that this doesn't hold for the negative numbers", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((rand\ = \ \(-Random[]\); \ \[IndentingNewLine]{rand, \ Gamma[rand, \ 0], \ Gamma[rand], \ \ Gamma[rand, \ 0]\ \[Equal] \ Gamma[rand]})\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{\(-0.6928982296004675`\), ComplexInfinity, \(-4.212685791605587`\), False}\)], "Output"] }, Open ]], Cell["And neither for complex numbers.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((rand\ = \ I\ Random[]; \ {rand, \ Gamma[rand, \ 0], \ Gamma[rand], \ \ Gamma[rand, \ 0]\ \[Equal] \ Gamma[rand]})\)\)], 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But not in this case, it would seem. Anyway, we can consider the \ series for small values of \[CurlyPhi].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ev2\ = \ RSquaredExpectation[Phi, \ Identity, \ n, \ k, \ \[CurlyPhi]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`2\^\(n\/2 - 1\)\ \[ExponentialE]\^\(-\(\(n\ \ \[CurlyPhi]\)\/2\)\)\ \((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\ \((\((k - n)\)\ \(\[CapitalGamma](n\/2)\) + 2\ \(\[CapitalGamma](n\/2 + 1, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\) - k\ \(\[CapitalGamma]( n\/2, \(-\(\(n\ \[CurlyPhi]\)\/2\)\))\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[ev2, \ {\[CurlyPhi], \ 0, \ 3}]\ // \ FullSimplify\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(\((\(-n\)\ \[CurlyPhi])\)\^\(\(-n\)/2\)\), " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", InterpretationBox[ RowBox[{\(\(k\ \((\(-n\))\)\^\(n/2\)\)\/n\), "+", \(\(\((\(-n\))\)\^\(n/2\)\ \((n - k)\)\ \[CurlyPhi]\)\/\(n + 2\)\), "+", \(\(\((\(-n\))\)\^\(n\/2 + 1\)\ \((n - k)\)\ \[CurlyPhi]\^2\)\/\(n\^2 + 6\ n + 8\)\), "+", \(\(\((\(-n\))\)\^\(n\/2 + 2\)\ \((n - k)\)\ \[CurlyPhi]\^3\)\/\(\((n + 2)\)\ \((n + 4)\)\ \((n + 6)\)\)\), "+", InterpretationBox[\(O(\[CurlyPhi]\^4)\), SeriesData[ \[CurlyPhi], 0, {}, 0, 4, 1], Editable->False]}], SeriesData[ \[CurlyPhi], 0, { Times[ k, Power[ Times[ -1, n], Times[ Rational[ 1, 2], n]], Power[ n, -1]], Times[ Power[ Times[ -1, n], Times[ Rational[ 1, 2], n]], Power[ Plus[ 2, n], -1], Plus[ Times[ -1, k], n]], Times[ Power[ Times[ -1, n], Plus[ 1, Times[ Rational[ 1, 2], n]]], Plus[ Times[ -1, k], n], Power[ Plus[ 8, Times[ 6, n], Power[ n, 2]], -1]], Times[ Power[ Times[ -1, n], Plus[ 2, Times[ Rational[ 1, 2], n]]], Power[ Plus[ 2, n], -1], Power[ Plus[ 4, n], -1], Power[ Plus[ 6, n], -1], Plus[ Times[ -1, k], n]]}, 0, 4, 1], Editable->False], ")"}], " ", \(\[CurlyPhi]\^\(n/2\)\)}], "+", InterpretationBox[\(O(\[CurlyPhi]\^4)\), SeriesData[ \[CurlyPhi], 0, {}, 4, 4, 1], Editable->False]}], ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(%\ /. \ {n\ \[Rule] \ 15, \ k\ \[Rule] \ 3}\)], "Input"], Cell[BoxData[ FormBox[ InterpretationBox[\(O(1\/\[CurlyPhi]\^\(7/2\))\), SeriesData[ \[CurlyPhi], 0, {}, -7, -7, 2], Editable->False], TraditionalForm]], "Output"] }, Open ]], Cell["which neither shows a way to get rid of the complex values.", "Text"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["References", "Section"], Cell["\<\ Texts with a * were not available to the author. Texts with ** were too \ dense. Texts with *** are kept in mind. Note that some papers concern \ normally distributed regressors while we are interested in fixed \ regressors.\ \>", "Text"], Cell["\<\ Anderson, T. W. (1996), \"R. A. Fisher and Multivariate Analysis\", \ Statistical Science, Vol. 11, No. 1, 20-34, \ http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ss/1032209662\ \>", "Text"], Cell["\<\ * Barten, A.P. (1962), \"Note on the unbiased estimation of the squared \ multiple correlation coefficient\", Statistisca Neerlandica, 16, 151-163\ \>", "Text"], Cell["\<\ *** Campos, J. N. R. Ericsson, and D. F. Hendry (2005), \"General-to-specific \ Modeling: An Overview and Selected Bibliography\", Board of Governors of the Federal \ Reserve System International Finance Discussion Papers Number 838, specifically section 4.5 \ on model selection criteria, \ http://www.nuff.ox.ac.uk/users/hendry/ElgarCEH.pdf\ \>", "Text"], Cell["\<\ Carapeto, M. and W. Holt (2003), \"On the equivalence of the distributions of \ R2 and adjusted R2\", \ http://www.cass.city.ac.uk/faculty/mcarapeto/papers/R2.pdf\ \>", "Text"], Cell["\<\ * Cramer, J.S. (1987), \"Mean and variance of R2 in small and moderate \ samples\", J. Econometrics 35, 253-266.\ \>", "Text"], Cell["\<\ ** Fisher, R. A. (1928), \"The general sampling distribution of the multiple \ correlation coefcient\". Proc. Roy. Soc. London Ser. A 121 654-673. See R.A. \ Fisher digital archive at http://digital.library.adelaide.edu.au/coll/special/fisher/\ \>", "Text"], Cell["Green, W.H. (1993), \"Econometric analysis\", Prentice Hall", "Text"], Cell["\<\ Lohninger, H. (1999), \"Teach/Me data analysis\", \ http://www.vias.org/tmdatanaleng/cc_test_r.html (stochastic regressors)\ \>", "Text"], Cell["\<\ MATH2011 Statistical distribution theory, Notes 8: Beta, F- and \ t-distribution, http://www.maths.soton.ac.uk/teaching/units/math2011/handouts/Notes8.pdf\ \>", "Text"], Cell["\<\ * Ohtani, K. (1994), \"The density functions of R2 and, and their risk \ performance under asymmetric loss in misspecified linear regression models\", \ Economic Modelling, 1994, vol. 11, issue 4, pages 463-471 http://www.sciencedirect.com/science/article/B6VB1-46KC3HK-3/2/\ 79f6781f3559cf10c775fa59ed68cabb\ \>", "Text"], Cell["\<\ * Ohtani, K. (2000), \"Bootstrapping R2 and adjusted R2 in regression \ analysis\", Economic Modelling, 2000, vol. 17, issue 4, pages 473-483 \ http://econpapers.repec.org/scripts/search.asp?ft=Ohtani see also http://www.econ.kobe-u.ac.jp/katudou/publication/paper/paper78-00.\ html\ \>", "Text"], Cell[TextData[{ "Ohtani, K. and H. Tanizaki (2004), \"Exact distribution of ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "and adjusted ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], "in a linear regression model with multivariate t error terms\", J. Japan \ Statist. Soc. Vol 34 No 1 p101-109,\n\ http://www.scipress.org/journals/jjss/pdf/3401/34010101.pdf" }], "Text"], Cell["\<\ * Press, S.J. and A. Zellner (1978), \"Posterior distribution for the \ multiple correlation coefficient with fixed regressors\", Journal of \ Econometrics, 8, 307-321\ \>", "Text"], Cell["* Steiger, http://www.interchange.ubc.ca/steiger/aboutr2.htm", "Text"], Cell["Theil, H. (1971), \"Principles of econometrics\", Wiley", "Text"], Cell["* Weisstein, E., http://www.mathworld.wolfram.com", "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 800}, {0, 527}}, WindowSize->{663, 507}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, ShowSelection->True, ShowCellLabel->False, Magnification->1 ] (******************************************************************* Cached data follows. 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