(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 365450, 9846]*) (*NotebookOutlinePosition[ 367046, 9896]*) (* CellTagsIndexPosition[ 367002, 9892]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Analysis of Variance GLM Version 3.0 Using ", StyleBox["Mathematica", FontSlant->"Italic"], " V4.0\nSteve Hunka, University of Alberta,\nEdmonton, Alberta, Canada, T6G \ 2G5\nsteve.hunka@ualberta.ca\nRevision Date: July 20/02" }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "The ", StyleBox["readMe.nb", FontWeight->"Bold"], " notebook contains the following information about this file:\n1) General \ Description of anova procedure used here\n2) Program Modification to V2.0 \ Incorporated in V3.0 for ", StyleBox["Mathematica", FontSlant->"Italic"], " V4.0\n3) Description of Procedures and Cautions\n4) General Description \ of Interactive User Input\n5) Data Input\n6) Example Data Sets (a \ description and source of the data sets in this notebook)\n7) About the \ Output" }], "Text"], Cell["Example Data Sets", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dat1", FontWeight->"Bold"], " is in ungrouped format with 4 cell indices; dependent variable is in Real \ form. N=120. Example used in a {2x3x2} equal n design." }], "Text"], Cell[BoxData[ \(\(dat1 = {{{1, 1, 1, 1}, 0. }, {{1, 1, 1, 1}, 1. }, {{1, 1, 1, 1}, 1. }, {{1, 1, 1, 1}, 2. }, {{1, 1, 1, 1}, 2. }, {{1, 1, 1, 2}, 2. }, {{1, 1, 1, 2}, 2. }, {{1, 1, 1, 2}, 3. }, {{1, 1, 1, 2}, 3. }, {{1, 1, 1, 2}, 4. }, {{1, 1, 2, 1}, 0. }, {{1, 1, 2, 1}, 0.5}, {{1, 1, 2, 1}, 0.5}, {{1, 1, 2, 1}, 1. }, {{1, 1, 2, 1}, 2. }, {{1, 1, 2, 2}, 2. }, {{1, 1, 2, 2}, 2.5}, {{1, 1, 2, 2}, 2.5}, {{1, 1, 2, 2}, 3. }, {{1, 1, 2, 2}, 2. }, {{1, 2, 1, 1}, 2. }, {{1, 2, 1, 1}, 1. }, {{1, 2, 1, 1}, 1. }, {{1, 2, 1, 1}, 2. }, {{1, 2, 1, 1}, 3. }, {{1, 2, 1, 2}, 4. }, {{1, 2, 1, 2}, 4. }, {{1, 2, 1, 2}, 4. }, {{1, 2, 1, 2}, 4. }, {{1, 2, 1, 2}, 5. }, {{1, 2, 2, 1}, 2. }, {{1, 2, 2, 1}, 2. }, {{1, 2, 2, 1}, 1. }, {{1, 2, 2, 1}, 3. }, {{1, 2, 2, 1}, 3. }, {{1, 2, 2, 2}, 4. }, {{1, 2, 2, 2}, 4. }, {{1, 2, 2, 2}, 5. }, {{1, 2, 2, 2}, 5. }, {{1, 2, 2, 2}, 4. }, {{1, 3, 1, 1}, 0.5}, {{1, 3, 1, 1}, 0.5}, {{1, 3, 1, 1}, 0.5}, {{1, 3, 1, 1}, 1. }, {{1, 3, 1, 1}, 1. }, {{1, 3, 1, 2}, 1.5}, {{1, 3, 1, 2}, 1. }, {{1, 3, 1, 2}, 2. }, {{1, 3, 1, 2}, 2. }, {{1, 3, 1, 2}, 2. }, {{1, 3, 2, 1}, 0. }, {{1, 3, 2, 1}, 0. }, {{1, 3, 2, 1}, 0.5}, {{1, 3, 2, 1}, 0.5}, {{1, 3, 2, 1}, 0.5}, {{1, 3, 2, 2}, 1. }, {{1, 3, 2, 2}, 1. }, {{1, 3, 2, 2}, 1.5}, {{1, 3, 2, 2}, 1. }, {{1, 3, 2, 2}, 2. }, {{2, 1, 1, 1}, 3. }, {{2, 1, 1, 1}, 3. }, {{2, 1, 1, 1}, 2. }, {{2, 1, 1, 1}, 2. }, {{2, 1, 1, 1}, 4. }, {{2, 1, 1, 2}, 4. }, {{2, 1, 1, 2}, 5. }, {{2, 1, 1, 2}, 5. }, {{2, 1, 1, 2}, 4. }, {{2, 1, 1, 2}, 4. }, {{2, 1, 2, 1}, 3. }, {{2, 1, 2, 1}, 3. }, {{2, 1, 2, 1}, 3. }, {{2, 1, 2, 1}, 3. }, {{2, 1, 2, 1}, 4. }, {{2, 1, 2, 2}, 4. }, {{2, 1, 2, 2}, 5. }, {{2, 1, 2, 2}, 5. }, {{2, 1, 2, 2}, 5. }, {{2, 1, 2, 2}, 5. }, {{2, 2, 1, 1}, 2. }, {{2, 2, 1, 1}, 2. }, {{2, 2, 1, 1}, 3. }, {{2, 2, 1, 1}, 3. }, {{2, 2, 1, 1}, 4. }, {{2, 2, 1, 2}, 4. }, {{2, 2, 1, 2}, 5. }, {{2, 2, 1, 2}, 5. }, {{2, 2, 1, 2}, 5. }, {{2, 2, 1, 2}, 5. }, {{2, 2, 2, 1}, 4. }, {{2, 2, 2, 1}, 4. }, {{2, 2, 2, 1}, 5. }, {{2, 2, 2, 1}, 5. }, {{2, 2, 2, 1}, 4. }, {{2, 2, 2, 2}, 4. }, {{2, 2, 2, 2}, 5. }, {{2, 2, 2, 2}, 5. }, {{2, 2, 2, 2}, 4. }, {{2, 2, 2, 2}, 4. }, {{2, 3, 1, 1}, 3. }, {{2, 3, 1, 1}, 3. }, {{2, 3, 1, 1}, 4. }, {{2, 3, 1, 1}, 4. }, {{2, 3, 1, 1}, 4. }, {{2, 3, 1, 2}, 4. }, {{2, 3, 1, 2}, 5. }, {{2, 3, 1, 2}, 5. }, {{2, 3, 1, 2}, 4. }, {{2, 3, 1, 2}, 4. }, {{2, 3, 2, 1}, 3. }, {{2, 3, 2, 1}, 3. }, {{2, 3, 2, 1}, 3. }, {{2, 3, 2, 1}, 4. }, {{2, 3, 2, 1}, 4. }, {{2, 3, 2, 2}, 4. }, {{2, 3, 2, 2}, 5. }, {{2, 3, 2, 2}, 5. }, {{2, 3, 2, 2}, 6. }, {{2, 3, 2, 2}, 5. }};\)\)], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dat2", FontWeight->"Bold"], " is the same dependent variable data as dat1 but in grouped format with 3 \ cell indices;" }], "Text"], Cell[BoxData[ \(\(dat2 = {{{1, 1, 1}, {0. , 1. , 1. , 2. , 2. , 2. , 2. , 3. , 3. , 4. }}, {{2, 3, 1}, {3. , 3. , 4. , 4. , 4. , 4. , 5. , 5. , 4. , 4. }}, {{1, 1, 2}, {0. , .5, .5, 1. , 2. , 2. , 2.5, 2.5, 3. , 2. }}, {{1, 2, 1}, {2. , 1. , 1. , 2. , 3. , 4. , 4. , 4. , 4. , 5. }}, {{1, 2, 2}, {2. , 2. , 1. , 3. , 3. , 4. , 4. , 5. , 5. , 4. }}, {{1, 3, 1}, { .5, .5, .5, 1. , 1. , 1.5, 1. , 2. , 2. , 2. }}, {{1, 3, 2}, {0. , 0. , .5, .5, .5, 1. , 1. , 1.5, 1. , 2. }}, {{2, 1, 1}, {3. , 3. , 2. , 2. , 4. , 4. , 5. , 5. , 4. , 4. }}, {{2, 1, 2}, {3. , 3. , 3. , 3. , 4. , 4. , 5. , 5. , 5. , 5. }}, {{2, 2, 1}, {2. , 2. , 3. , 3. , 4. , 4. , 5. , 5. , 5. , 5. }}, {{2, 2, 2}, {4. , 4. , 5. , 5. , 4. , 4. , 5. , 5. , 4. , 4. }}, {{2, 3, 2}, {3. , 3. , 3. , 4. , 4. , 4. , 5. , 5. , 6. , 5. }}};\)\)], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dat3", FontWeight->"Bold"], " is for a 2-factor anova; data is ungrouped and in Integer format. N=33, \ unequal n run as {2x4} design." }], "Text"], Cell[BoxData[ \(\(dat3 = {{{1, 1}, 3}, {{1, 1}, 4}, {{1, 1}, 6}, {{1, 1}, 7}, {{1, 2}, 5}, {{1, 2}, 6}, {{1, 2}, 6}, {{1, 2}, 7}, {{1, 2}, 7}, {{1, 3}, 4}, {{1, 3}, 6}, {{1, 3}, 8}, {{1, 3}, 8}, {{1, 4}, 8}, {{1, 4}, 10}, {{1, 4}, 10}, {{1, 4}, 7}, {{1, 4}, 11}, {{2, 1}, 2}, {{2, 1}, 3}, {{2, 1}, 4}, {{2, 2}, 3}, {{2, 2}, 5}, {{2, 2}, 6}, {{2, 2}, 3}, {{2, 3}, 9}, {{2, 3}, 12}, {{2, 3}, 12}, {{2, 3}, 8}, {{2, 4}, 9}, {{2, 4}, 7}, {{2, 4}, 12}, {{2, 4}, 11}};\)\)], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dat4", FontWeight->"Bold"], " is for a 3-factor anova with 1 observation per cell; df(error)=0 unless \ an interaction term is dropped; data is ungrouped and in Integer form.N=60 in \ a {6x2x5} design" }], "Text"], Cell[BoxData[ \(\(dat4 = {{{1, 1, 1}, 5}, {{1, 1, 2}, 5}, {{1, 1, 3}, 4}, {{1, 1, 4}, 4}, {{1, 1, 5}, 3}, {{1, 2, 1}, 5}, {{1, 2, 2}, 5}, {{1, 2, 3}, 4}, {{1, 2, 4}, 4}, {{1, 2, 5}, 4}, {{2, 1, 1}, 3}, {{2, 1, 2}, 4}, {{2, 1, 3}, 2}, {{2, 1, 4}, 3}, {{2, 1, 5}, 2}, {{2, 2, 1}, 5}, {{2, 2, 2}, 5}, {{2, 2, 3}, 2}, {{2, 2, 4}, 2}, {{2, 2, 5}, 4}, {{3, 1, 1}, 1}, {{3, 1, 2}, 4}, {{3, 1, 3}, 5}, {{3, 1, 4}, 2}, {{3, 1, 5}, 4}, {{3, 2, 1}, 2}, {{3, 2, 2}, 3}, {{3, 2, 3}, 5}, {{3, 2, 4}, 2}, {{3, 2, 5}, 3}, {{4, 1, 1}, 3}, {{4, 1, 2}, 3}, {{4, 1, 3}, 2}, {{4, 1, 4}, 2}, {{4, 1, 5}, 4}, {{4, 2, 1}, 4}, {{4, 2, 2}, 4}, {{4, 2, 3}, 2}, {{4, 2, 4}, 2}, {{4, 2, 5}, 2}, {{5, 1, 1}, 1}, {{5, 1, 2}, 4}, {{5, 1, 3}, 4}, {{5, 1, 4}, 2}, {{5, 1, 5}, 2}, {{5, 2, 1}, 2}, {{5, 2, 2}, 4}, {{5, 2, 3}, 4}, {{5, 2, 4}, 3}, {{5, 2, 5}, 5}, {{6, 1, 1}, 2}, {{6, 1, 2}, 2}, {{6, 1, 3}, 2}, {{6, 1, 4}, 1}, {{6, 1, 5}, 2}, {{6, 2, 1}, 1}, {{6, 2, 2}, 3}, {{6, 2, 3}, 1}, {{6, 2, 4}, 1}, {{6, 2, 5}, 2}};\)\)], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dat5", FontWeight->"Bold"], " is for a 1-factor anova having 3 groups of unequal size; data is grouped \ and in Integer form; N=21" }], "Text"], Cell[BoxData[ \(\(dat5 = {{{1}, {3, 5, 2, 4, 8, 4, 3, 9}}, {{2}, {4, 4, 3, 8, 7, 4, 2}}, {{3}, {6, 7, 8, 6, 7, 9}}};\)\)], "Input"] }, Closed]], Cell[TextData[{ StyleBox["Note", FontWeight->"Bold"], ":Data sets for the repeated measures designs and nested designs, as well \ as the examples from Wolfram are included in cells at the end of the \ notebook." }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Generating the NIST Simon-Lesage Data Sets", FontWeight->"Bold"]], "Text"], Cell["\<\ Test data sets for assessing the accuracy of calculating sum of \ squares are provided by the web site www.nist.gov/itl/div898/strd/anova/anova.html. The functions below can be used to generate the data sets Simon-Lesage 7, 8, \ and 9. The \"certified values\" for each data set provided by the www site \ are as follows: Simon-Lesage7: SStreatment=1.6800....0E+00; SSerror=1.80...0E+00; \ F=2.10....0E+01 Simon-Lesage8: SStreatment=1.6080....0E+01; SSerror=1.80...0E+01; \ F=2.010..0E+02 Simon-Lesage9: SStreatment=1.60080..0E+02; SSerror=1.80...0E+02; \ F=2.0010.0E+03 These data sets are provided by the National Institute of Standards & \ Technology (NIST), Information Technology Laboratory.The www site provides \ information on how extended real-value precision was obtained. For each data \ set, there are 9 cells with an equal number of observation in each \ cell.\ \>", "Text"], Cell["Infinite Precision Results for Simon-Lesage Data Sets", "Text", FontWeight->"Bold"], Cell["\<\ Mathematica allows computation to proceed in rational form \ providing for infinite precision. The SS and R-sqd values which are obtained \ for these data sets using the notebook anova.nb are as follows: Simon7: SStcfm=87/25; SSerr=9/5; SSmcfm=42/25; R-sqd=14/29; (N=189) Simon8: SStcfm=852/25; SSerr=18; SSmcfm=402/25; R-sqd=67/142; (N=1809) Simon9: SStcfm=8502/25; SSerr=180; SSmcfm=4002/25; R-sqd=667/1417; (N=18009) Function getrn which brings Real input data into a rational form, takes about \ 45 seconds on a Macintosh 7200/120 when used with Simon 9.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["The function ", FontVariations->{"CompatibilityType"->0}], StyleBox["gensimon", FontWeight->"Bold"], " generates the data set of Simon-Lesage7 used by NIST for testing SS \ accuracy. The data set returned is in grouped format. By changing the \ iterator {10} to {100} or {1000} the Simon-Lesage data sets 8 (N=1809) and 9 \ (N=18009), respectively, can be generated." }], "Text"], Cell[BoxData[ \(\(gensimon := Module[{d1, d2, d3, s1, s2}, \n d1 = Flatten[Table[{1000000000000.3, 1000000000000.5}, {10}]]; \n d1 = Prepend[d1, 1000000000000.4]; \n d2 = Flatten[Table[{1000000000000.2, 1000000000000.4}, {10}]]; \n d2 = Prepend[d2, 1000000000000.3]; \n d3 = Flatten[Table[{1000000000000.4, 1000000000000.6}, {10}]]; \n d3 = Prepend[d3, 1000000000000.5]; \n s1 = {d1, d2, d3, d2, d3, d2, d3, d2, d3}; \n s2 = Map[Append[{{#}}, s1[\([#]\)]] &, Range[9]]; \n Return[s2];\n\t\t];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(simon7 = gensimon;\)\), "\n", \(Length[simon7]\)}], "Input"], Cell[BoxData[ \(9\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["The function", FontVariations->{"CompatibilityType"->0}], StyleBox[" genunsimon", FontWeight->"Bold"], " generates the Simon-Lesage7 data in an ungrouped format. By changing the \ iterator {10} to {100} or {1000} the Simon-Lesage data sets 8 and 9, \ respectively, can be generated. " }], "Text"], Cell[BoxData[ \(\(genunsimon := Module[{d1, d1c, d2, d2c, d3, d3c, d4c, \n\t\td5c, d6c, d7c, d8c, d9c, s7c}, \n d1 = Flatten[Table[{1000000000000.3, 1000000000000.5}, {10}]]; \n d1 = Prepend[d1, 1000000000000.4]; \t\n d1c = Map[Append[{{1}}, #] &, d1]; \n d2 = Flatten[Table[{1000000000000.2, 1000000000000.4}, {10}]]; \n d2 = Prepend[d2, 1000000000000.3]; \n d2c = Map[Append[{{2}}, #] &, d2]; \t\n d3 = Flatten[Table[{1000000000000.4, 1000000000000.6}, {10}]]; \n d3 = Prepend[d3, 1000000000000.5]; \n d3c = Map[Append[{{3}}, #] &, d3]; \n d4c = Map[Append[{{4}}, #] &, d2]; \n d5c = Map[Append[{{5}}, #] &, d3]; \n d6c = Map[Append[{{6}}, #] &, d2]; \n d7c = Map[Append[{{7}}, #] &, d3]; \n d8c = Map[Append[{{8}}, #] &, d2]; \n d9c = Map[Append[{{9}}, #] &, d3]; \n s7c = {d1c, d2c, d3c, d4c, d5c, d6c, d7c, d8c, d9c}; \n s7c = Flatten[s7c, 1]; \n Return[s7c];\t\t\t\t\t\t\t\t\t\t\n\t\t];\)\)], "Input"] }, Open ]], Cell[BoxData[ \(\(simon7u = genunsimon;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[simon7u]\)], "Input"], Cell[BoxData[ \(189\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Rationalizing Input Data to Obtain High Accuracy", "Subsubsection"], Cell["\<\ The NIST data sets are designed to check the adequacy of \ computational algorithms and do not represent data likely to be encountered \ in a real experimental situation. Even if such a data set was encountered it \ is likely that the researcher would apply a linear transformation to the \ data, e.g., subtracting a constant, so that the data contained only the \ digits to the right of the decimal point. Any linear transformation of the \ form y=a+bx with b\[NotEqual]0, will change the sum of squares and mean \ squares, however, the ratios (F tests) would not be affected. A high degree of accuracy is obtained when the experimental data is in \ integer or rational form. When the data is in real form, it can be \ rationalized using the function getrn called by the functions srtcnt and \ srtgrp. The function getrn is defined in two different forms, one form using \ RealDigits and FromDigits, and the other using only Rationalize[x,0]. The \ user can initialize either form of the function. The differences resulting in \ the use of Rationalize[x,0] and the function using RealDigits and FromDigits \ are illustrated below. For purposes of illustration, the function getrn6 as \ given below is used. This latter function differs from getrn only in that \ getrn operates on a matrix of data, while getrn6 operates on a vector of \ data.\ \>", "Text"], Cell[BoxData[ \(getrn6[d_] := Sign[d]*Map[FromDigits[#] &, RealDigits[d]]\)], "Input"], Cell["\<\ Some large example values are given as the variable tdr shown \ below.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tdr = {100000000000000.3, 1000000000000000.3, 1000000000000000.30, 10000000000000000.3}\)], "Input"], Cell[BoxData[ \({1.000000000000003`*^14, 1.0000000000000002`*^15, 1.0000000000000003`17.301*^15, 1.00000000000000003`17.301*^16}\)], "Output"] }, Open ]], Cell["\<\ In the variable tdr the first value has 14 zeroes, the second has \ 15 zeroes, the third value has 15 zeroes but an additional zero following the \ decimal digit 3, the fourth value has 16 zeroes. The results of \ rationalizing these values using getrn6 are as follows:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(getrn6[tdr]\)], "Input"], Cell[BoxData[ \({1000000000000003\/10, 1000000000000000, 10000000000000003\/10, 100000000000000003\/10}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%3, 20]\)], "Input"], Cell[BoxData[ \({1.000000000000003000000000000007105`20*^14, 1.`20*^15, 1.0000000000000003000000000000007105`20*^15, 1.00000000000000003`20*^16}\)], "Output"] }, Open ]], Cell["\<\ Notice that the second value was not rationalized to the same value \ as the third value. Examining the rationalization further,\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\($MachinePrecision\)\( (*for\ Macintosh\ G4*) \)\)\)], "Input"], Cell[BoxData[ \(16\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Precision[#] &, tdr]\)], "Input"], Cell[BoxData[ \({16, 16, 17, 17}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RealDigits[tdr]\)], "Input"], Cell[BoxData[ \({{{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3}, 15}, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 16}, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0}, 16}, {{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3}, 17}}\)], "Output"] }, Open ]], Cell[TextData[{ "When the value is less than or equal to machine precision, machine \ precision is used, thus the decimal portion of the second value is lost. \ When the value is greater than machine precision, arbitrary precision is \ used. (See sections 3.1.3 and 3.1.4 of the ", StyleBox["Mathematica", FontSlant->"Italic"], " Book.). If we use the ", StyleBox["Mathematica", FontSlant->"Italic"], " function Rationalize, the results are as follows:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Rationalize[#, 0] &, tdr]\)], "Input"], Cell[BoxData[ \({700000000000002\/7, 4000000000000001\/4, 10000000000000003\/10, 30000000000000001\/3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%, 20]\)], "Input"], Cell[BoxData[ \({1.000000000000002857142857142846992`20*^14, 1.00000000000000025`20*^15, 1.0000000000000003000000000000007105`20*^15, 1.000000000000000033333333333333`20*^16}\)], "Output"] }, Open ]], Cell[TextData[{ "In the above results, only the third value is without error, all others \ contain an error in the representation of digits to the right of the decimal \ point. The NIST data is of the order 1.000000000000n x ", Cell[BoxData[ \(TraditionalForm\`10\^12\)]], "(where n is a single digit) so no error is encountered using the built in \ function Rationalize[x,0] or the function getrn. (Note: in the functions that \ require initialization, two forms of getrn are given. One is of the form of \ getrn6 defined above, and the other uses the built-in Rationalize function. \ Only one of these needs to be intialized.)\n\nA comparison similar to that \ given above is given below for very small values." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(tdr2 = { .00000000000014, .000000000000015, \ .0000000000000016, \ .00000000000000017}\)], "Input"], Cell[BoxData[ \({1.4`*^-13, 1.5`*^-14, 1.6`*^-15, 1.7`*^-16}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Precision[#] &, tdr2]\)], "Input"], Cell[BoxData[ \({16, 16, 16, 16}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(getrn6[tdr2]\)], "Input"], Cell[BoxData[ \({7\/50000000000000, 3\/200000000000000, 1\/625000000000000, 17\/100000000000000000}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%, 20]\)], "Input"], Cell[BoxData[ \({1.399999999999999999999999999997`20*^-13, 1.4999999999999999999999999999999998`20*^-14, 1.5999999999999999999999999999999993009`20*^-15, 1.6999999999999999999999999999999913`20*^-16}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Rationalize[#, 0] &, tdr2]\)], "Input"], Cell[BoxData[ \({7\/50000000000000, 3\/200000000000000, 1\/625000000000000, 1\/5882352941176470}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%, 20]\)], "Input"], Cell[BoxData[ \({1.399999999999999999999999999997`20*^-13, 1.4999999999999999999999999999999998`20*^-14, 1.5999999999999999999999999999999993009`20*^-15, 1.700000000000000170000000000000012205`20*^-16}\)], "Output"] }, Open ]], Cell[TextData[{ "If one assumes that typical research data is in the range 0-1000 and \ containing 3 decimal digits, the user could define the input data in the \ following ways:\n\n1) real values which will invoke the function getrn. The \ error in the numerator of the rationalized value may be 1x", Cell[BoxData[ \(TraditionalForm\`10\^\(-16\)\)]], " if the function getrn involves the operations Sign, RealDigits, and \ FromDigits, i.e., the first definition of getrn in the List of Functions to \ Initialize.\n\n2) real values, but redefine the function getrn to be \ getrn[d_]:=Rationalize[d,0], i.e., use the second definition of getrn in the \ List of Functions to Initialize. Examples tried showed no error in the \ numerator of the rationalized values.\n\n3) define the data as integers by \ multiplying by 1000. The calculated means will be increased by ", Cell[BoxData[ \(TraditionalForm\`10\^3\)]], " and sum of squares, mean squares, and variances by ", Cell[BoxData[ \(TraditionalForm\`10\^6\)]], ". The F ratios will be unaffected. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Function to Check Cell Indices", "Subsubsection"], Cell["\<\ The function chkidx can be used to check the adequacy of cell \ indices required of the data used for an anova. The function checks (a) the \ length of the vectors holding the indices, (b) for a Null included as part of \ the indices, and (c) missing cell indices. Output includes number of cells \ found, minimum and maximum length of cell index vectors, location of Null \ indices, cells indices which are missing, and frequency of each cell index \ vector.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["chkidx[d]", FontWeight->"Bold"], " is used to check the adequacy of the cell index vectors in the data set \ d.\nCalls: ndim (located in the next section)\nMod. Date: July 5/02" }], "Text"], Cell[BoxData[ \(\(chkidx[d_] := Module[{indc, ilen, minl, maxl, np, flg = 0, dim, nc, sum, psn0, alli, fm, fm2}, \[IndentingNewLine]Print["\"]; \ \[IndentingNewLine]indc = Map[Part[#, 1] &, d]; \[IndentingNewLine]Print["\", Length[indc]]; \[IndentingNewLine] (*check\ length\ of\ indices\ *) \[IndentingNewLine]ilen = Map[Length[#] &, indc]; \[IndentingNewLine]minl = Min[ilen]; \[IndentingNewLine]maxl = Max[ilen]; \[IndentingNewLine]If[minl\ \[NotEqual] \ maxl, Print["\<**ERROR:Length of index vectors are not same\>"]; flg = 1; \[IndentingNewLine]Print["\", minl, "\< Max length=\>", maxl], \ Print["\"]]; \[IndentingNewLine] \ (*Check\ for\ Null\ in\ index\ vector*) \[IndentingNewLine]np = Position[indc, Null]; \[IndentingNewLine]If[ Length[np] \[NotEqual] \ 0, flg = 1; \[IndentingNewLine]Print["\", np]; \[IndentingNewLine]Print["\"]]; \[IndentingNewLine]If[flg \[Equal] 1, \ Return[]]; \[IndentingNewLine] (*check\ for\ missing\ cell\ \ indices*) \[IndentingNewLine]Print["\"]; \ \[IndentingNewLine]dim = Map[Max[#] &, Transpose[ indc]]; \[IndentingNewLine]Print["\", dim]; \[IndentingNewLine]nc = Apply[Times, dim]; \[IndentingNewLine]\[IndentingNewLine] (*create\ sum\ box\ *) \[IndentingNewLine]fm = Table[0, {nc}]; \[IndentingNewLine]fm = ndim[fm, dim]; \[IndentingNewLine] (*count\ freq\ of\ indices\ *) \[IndentingNewLine]Do[\[IndentingNewLine]sum = 1 + Extract[fm, indc[\([j]\)]]; \[IndentingNewLine]fm = ReplacePart[fm, sum, indc[\([j]\)]]\[IndentingNewLine], {j, 1, Length[indc]}]; \[IndentingNewLine]\[IndentingNewLine]psn0 = Position[fm, 0]; \[IndentingNewLine]Print["\", psn0]; \[IndentingNewLine]fm2 = ndim[Range[nc], dim]; \[IndentingNewLine]alli = Map[Position[fm2, #] &, Range[nc]]; \[IndentingNewLine]alli = Flatten[alli, 1]; \[IndentingNewLine]fm = Transpose[{Flatten[fm]}]; \[IndentingNewLine]Print[ TableForm[Transpose[{alli, fm}], TableDepth \[Rule] 2, TableHeadings \[Rule] {Automatic, {"\", \ "\< Frequency\ \>"}}]];\[IndentingNewLine]\[IndentingNewLine]];\)\)], "Input"] }, Open ]], Cell["\<\ Data for this example is from dat2 with the cell indices {2,3,1} and {1,3,1} \ and their associated data removed.\ \>", "Text"], Cell[BoxData[ \(\(\({{2, 3, 1}, {3. , 3. , 4. , 4. , 4. , 4. , 5. , 5. , 4. , 4. }}\)\(,\)\)\)], "Input"], Cell[BoxData[ \(\(\({{1, 3, 1}, { .5, .5, .5, 1. , 1. , 1.5, 1. , 2. , 2. , 2. }}\)\(,\)\)\)], "Input"], Cell[BoxData[ \(\(dat21 = {{{1, 1, 1}, {0. , 1. , 1. , 2. , 2. , 2. , 2. , 3. , 3. , 4. }}, {{1, 1, 2}, {0. , .5, .5, 1. , 2. , 2. , 2.5, 2.5, 3. , 2. }}, {{1, 2, 1}, {2. , 1. , 1. , 2. , 3. , 4. , 4. , 4. , 4. , 5. }}, {{1, 2, 2}, {2. , 2. , 1. , 3. , 3. , 4. , 4. , 5. , 5. , 4. }}, {{1, 3, 2}, {0. , 0. , .5, .5, .5, 1. , 1. , 1.5, 1. , 2. }}, {{2, 1, 1}, {3. , 3. , 2. , 2. , 4. , 4. , 5. , 5. , 4. , 4. }}, {{2, 1, 2}, {3. , 3. , 3. , 3. , 4. , 4. , 5. , 5. , 5. , 5. }}, {{2, 2, 1}, {2. , 2. , 3. , 3. , 4. , 4. , 5. , 5. , 5. , 5. }}, {{2, 2, 2}, {4. , 4. , 5. , 5. , 4. , 4. , 5. , 5. , 4. , 4. }}, {{2, 3, 2}, {3. , 3. , 3. , 4. , 4. , 4. , 5. , 5. , 6. , 5. }}};\)\)], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(chkidx[dat21]\)], "Input"], Cell[BoxData[ \("Checking Cell Indices"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Number of vectors of cell \ indices="\[InvisibleSpace]10\), SequenceForm[ "Number of vectors of cell indices=", 10], Editable->False]], "Print"], Cell[BoxData[ \("Index vector lengths all equal"\)], "Print"], Cell[BoxData[ \("Checking for missing cell indices"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Dimensions of data="\[InvisibleSpace]{2, 3, 2}\), SequenceForm[ "Dimensions of data=", {2, 3, 2}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Missing indices="\[InvisibleSpace]{{1, 3, 1}, {2, 3, 1}}\), SequenceForm[ "Missing indices=", {{1, 3, 1}, {2, 3, 1}}], Editable->False]], "Print"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"Index\"\>", "\<\" Frequency\"\>"}, {"1", \({1, 1, 1}\), \({1}\)}, {"2", \({1, 1, 2}\), \({1}\)}, {"3", \({1, 2, 1}\), \({1}\)}, {"4", \({1, 2, 2}\), \({1}\)}, {"5", \({1, 3, 1}\), \({0}\)}, {"6", \({1, 3, 2}\), \({1}\)}, {"7", \({2, 1, 1}\), \({1}\)}, {"8", \({2, 1, 2}\), \({1}\)}, {"9", \({2, 2, 1}\), \({1}\)}, {"10", \({2, 2, 2}\), \({1}\)}, {"11", \({2, 3, 1}\), \({0}\)}, {"12", \({2, 3, 2}\), \({1}\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableDepth -> 2, TableHeadings -> {Automatic, {"Index", " Frequency"}}]&)]], "Print"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Example of Interactive User Input for Data Set dat2", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ dat2 is used in a straight-forward 3-way anova. User queries are in \ quotations. 1. anovanw[dat2] 2. \"Type the number of factors in the design\" 3 3. \"Type vector of Factor indices to use\" {1,2,3} 4. \"Display cell data as Real (1) or Rational (0)?\" 1 5. \"Do you wish to proceed? y/n\" y 6. \"Remove interaction terms? y/n\" n 7. \"Plot interactions? y/n\" y 7.1 \"Type an interaction term to plot\" ab 7.2 \"More plots? y/n\" y 7.3 \"Type an interaction term to plot\" abc 7.4 \"More plots? y/n\" n 8. \"Contrasts on main effects? y/n\" y 8.1 \"Type contrast term\" b 8.2 \"How many rows in contrast? 2 8.3 \"Type contrast matrix\" {{1,0,-1},{0,1,-1}} 8.4 \"More contrasts\" n \ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Required Packages (Requires Initialization)", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(<< "\"\)], "Input"], Cell[BoxData[ \(<< "\"\)], "Input"], Cell[BoxData[ \(<< "\"\)], "Input"], Cell[BoxData[ \(<< "\"\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["List of Functions Used", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ anovanw: main anova starting function anvT2: displays general Anova table of SS; calculates MSe and dfe binselect: selects elements in list corresponding to positions of 1 in binary \ vector bin2alf: generates character representation from binary vector chkidx: checks adequacy of cell index vector (not part of anova procedure) comdta: to handle grouped data by combining data belonging to common cells cutintr: allows user to remove designated interaction vectors in design \ matrix dmpar: generates character and binary representation of all parameters dsgnint: generates interaction vectors for design matrix dsgnmef: generates vectors for main effects design matrix examd: determines whether input data is all Real, all Integers, or mixed findc: locates cols of design matrix used to form interaction vectors findintr: locates 2 & 3 term interactions from binary designation gencon: generates contrast matrices and displays SS table for all terms genrch: generates row/col headings for pmat2 getrn: gets rational values for real valued input data (2 definitions) getss: gets deviated sum of squares and means getucn: obtain user constrast matrix hytst: display hypothesis tested for user contrasts intrplot: entry to plotting of interaction terms kbin3: keyboard input of numerics kbinstrg: keyboard input of characters lctmev: locate starting/ending position of vectors representing anova terms \ in X'X matsum: calculates a matrix of means for plotting interactions ndim: restructures vector of means to a matrix of given dimensions plotnLists: plots 2-term interactions plt3fintr: plots planes of 3-term interactions pmat2: displays a matrix prntimm: to display means matrix used to plot interaction probF: calculate probability of F redspc: sums over specific dimensions of means matrix restrv: restructures a vector rndoff: round off real values srtcnt: sort ungrouped data by factor indices; returns cell sizes, means, \ and variances srtgp: using grouped data, returns cell sizes, means, and variances ssk: calculates SS for contrast matrix K Each function is preceded with documentation including other functions which \ are called. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Error Messages", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Two types of error messages may appear. On input of user parameters or \ options, an error message will be given when the structure of the input is \ inappropriate or not acceptable. In such cases the message will appear on the \ screen, but may be partially covered by the input window in which a re-entry \ is requested. The main function anovanw will Abort if it is presented with \ data which contains both Real and Integer values with a message presented by \ Message[] asking the user to present either all Real or all Integer data. The \ program will also abort if the data presented is of Length[]==0, usually \ caused by the user forgetting to have ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluate the data set." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Functions to Initialize", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["kbin3[q_,p_]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" function to accept numeric input from keyboard posing question \ in q. If p is any integer then an integer is to be input; if p is any real \ value then a real value is to be input; if p={n} then a n element vector is \ expected; if p={r,c} then a matrix with r rows and c columns is expected. If \ the input is inappropriate the question is posed again. If the input is \ appropriate it is returned.\nMod. Date: Mar 24/96", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(kbin3[q_, p_] := Module[{c, ans, vrp, vrp2}, c = 0; vrp = "\< Input Error. Type a vector with # elements=\>"; vrp2 = "\< Input Error. Type a matrix of order \>"; While[c == 0, ans = Input[q]; Which[Head[ans] === Symbol, \(Print[ ans, "\< Input Error; Characters not allowed\>"];\), Head[p] === Integer || Head[p] === Real, \(If[ Head[p] =!= Head[ans], \(Print[ ans, "\< Input Error. Input must be \>", Head[p]];\), c = 1];\), Head[p] === List && Length[p] == 1, \(If[ VectorQ[ans] == False, \(Print[ans, vrp <> ToString[p]];\), If[Length[ans] \[NotEqual] First[p], \(Print[ ans, "\< Error: Vector requires # of elements=\>", First[p]];\), c = 1]];\), Head[p] === List && Length[p] == 2, \(If[ MatrixQ[ans] == False, \(Print[ans, vrp2 <> ToString[p]];\), If[Dimensions[ans] \[NotEqual] p, \(Print[ ans, "\< Error; Matrix dimensions must be \>", p];\), c = 1]];\)];]; Return[ans];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["kbinstrg[q,p]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" used to input a string in response to question in q. p contains \ acceptable answers in the form of a list, eg., p={{ans1},{ans2},....}. If the \ input is appropriate it is returned. If the input is inappropriate the \ question is posed again.\nMod. Date: Mar 24/96", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(kbinstrg[q_, p_] := Module[{c, ans}, c = 0; While[c == 0, ans = InputString[q]; Do[\(If[{ans} == p\[LeftDoubleBracket]j\[RightDoubleBracket], c = 1; Break[]];\), {j, 1, Length[p]}]; If[c \[NotEqual] 1, Print[ans, "\< Input Error. Type Again\>"]];]; Return[ans];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["pmat2[m,pc,rch]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" prints a 2D matrix with either numbered rows and cols or with \ labelled rows and columns, in units of pc columns \nm is the matrix to \ print\npc is the number of columns to use\nrch is optional; if present \ it contains the row and column headings in the \n following format: \ {{\"a\",\"b\",\"c\",...},{\"A\",\"B\",\"C\",\"D\",....}} where\n Part 1 \ gives the row headings, and Part 2 gives the column headings. \n Part 1 \ must have as many headings as rows of the matrix, and Part 2\n must \ have as many headings as the columns of the matrix.\n If the argument \ rch is not used, the row and column labels\n will be the integer \ sequence starting at 1, and the col digits for any 2nd\n part of the \ matrix printed will continue the digit sequence.\n Mod Date: Jan 27/96", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(pmat2[m_, pc_, rch___] := Module[{m1, a, c, indx, col1, row}, m1 = Transpose[m]; c = \(Dimensions[m1]\)\[LeftDoubleBracket]1\[RightDoubleBracket]; a = Floor[c\/pc]; indx = Join[Table[pc, {a}], {Mod[c, pc]}]; If[0 == indx\[LeftDoubleBracket]\(-1\)\[RightDoubleBracket], indx = Drop[indx, \(-1\)]]; If[Length[{rch}] == 0, row = \((ToString[#1] &)\) /@ Range[\(Dimensions[ m]\)\[LeftDoubleBracket]1\[RightDoubleBracket]]; \(col1 \ = \((ToString[#1] &)\) /@ Range[\(Dimensions[ m]\)\[LeftDoubleBracket]2\[RightDoubleBracket]];\), row = rch\[LeftDoubleBracket]1\[RightDoubleBracket]; col1 = rch\[LeftDoubleBracket]2\[RightDoubleBracket];]; Do[Print["\<\>"]; Print[TableForm[ Transpose[ Take[m1, indx\[LeftDoubleBracket]j\[RightDoubleBracket]]], TableHeadings \[Rule] Append[{row}, Take[col1, indx\[LeftDoubleBracket]j\[RightDoubleBracket]]], TableSpacing \[Rule] {0, 4}]]; m1 = Drop[m1, indx\[LeftDoubleBracket]j\[RightDoubleBracket]]; \(col1 = Drop[col1, indx\[LeftDoubleBracket]j\[RightDoubleBracket]];\), {j, 1, Length[indx]}];];\)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(row\)\" is similar to \ existing symbol \"\!\(Row\)\"."\)], "Message"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["genrch[rh,ch,rc]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" generates row and column headings for printing a 2D matrix\nrh \ is the string prefix for row headings\nch is the string prefix for the \ column headings\nrc is the Dimensions of the matrix as a vector\ni1 is \ the starting index for rows\ni2 is the starting index for columns\nThe \ prefix for row and columns will be followed by digits starting at i1 and i2. \ i1 and i2 are optional, but the order of not including is from the right. \ When not present, these are set to 1. The function can be used to generate \ the 3rd argument for pmat2.\nMod. Date: Feb 14/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(genrch[rh_, ch_, rc_, i1_: 1, i2_: 1] := Module[{a, b, ab, ri, ci}, fn[a_, b_] := StringInsert[a, ToString[b], \(-1\)]; a = Table[rh, {rc\[LeftDoubleBracket]1\[RightDoubleBracket]}]; ri = i1 - 1; ci = i2 - 1; a = \((fn[a\[LeftDoubleBracket]#1\[RightDoubleBracket], ri + #1] &)\) /@ Range[rc\[LeftDoubleBracket]1\[RightDoubleBracket]]; b = Table[ch, {rc\[LeftDoubleBracket]2\[RightDoubleBracket]}]; b = \((fn[b\[LeftDoubleBracket]#1\[RightDoubleBracket], ci + #1] &)\) /@ Range[rc\[LeftDoubleBracket]2\[RightDoubleBracket]]; ab = Join[{a}, {b}]; Return[ab];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["rndoff[n,d]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" rounds of real n to d decimal digits", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(rndoff[n_, d_] := N[10\^\(-d\)\ Round[n\ 10\^d]];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["getrn[d]", FontWeight->"Bold"], " is a pure function which converts the data in the structure d containing \ all Real values and returns rational values or integers maintaining the sign \ of each value.\nReturns: a data structure of the same form as the input d, \ but with rational or Integer values.\nMod. Date: Nov 29/97" }], "Text"], Cell[TextData[StyleBox["Note: the first or the second definition of getrn \ requires initialization (not both). See the section entitled Rationalizing \ Input Data to Obtain High Accuracy.", FontWeight->"Bold"]], "Text"], Cell[BoxData[ \(\(getrn[d_] := Map[Map[Sign[#]*FromDigits[RealDigits[#]] &, #] &, d];\)\)], "Input"] }, Open ]], Cell[BoxData[ \(\(getrn[d_] := Rationalize[d, 0];\)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["examd[d]", FontWeight->"Bold"], " determines whether the data structure d is all in the form of Integers, \ Real values, or a mix of Integers and Real values. \nReturns: 0 if all values \ in d are Integers; 1 if all are Real, and 2 if there is a mixture\nMod. Date: \ Jun 26/02: removed duplicate output of number of observations " }], "Text"], Cell[BoxData[ \(\(examd[d_] := Module[{ni, nt, fl}, \n ni = Apply[Plus, Map[IntegerQ[#] /. {True -> 1, False -> 0} &, Flatten[d]]]; \t\n\t\t\tnt = Length[Flatten[d]]; \n\t\tPrint["\", nt]; \t\n\t Which[\n\t\t\tni == nt, Print["\"]; fl = 0, \n\t\t\tni == 0, \ Print["\"\ ]; fl = 1, \n\t\t\ \ ni < \ nt, Print["\", ni]; fl = 2]; \n\t\tReturn[ fl];\n\t\t];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["srtcnt[d,nf]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" using ungrouped data, requests user to input the indices to be \ used for grouping the dependent variable (y), sorts the data according to \ indices and size of y, gets cell frequencies; examd is called to determine \ the type of input data. If all input data is Real calls getrn to bring data \ to rational form. Calls getss to calculate means, variances, and sum of \ squares.\nd: data matrix; {{i11,i12,...i1m},d1},{i21,i22,...i2m},d2}....}, \ where the m factor indices\n for each observation are given by i and the \ associated dependent variable by d. \nnf: number of factors to use\nUser \ input: a vector of nf elements of the factor indices to use, e.g., {1,2,3}, \ {2,1,3}, {1}\nOutput: number of observations; number of levels in each factor\ \nReturns: {sst,ssw,ord,freq,mv,gm}\nsst: total sum of squares deviated about \ grand mean\nssw: pooled sum of squares within group\nord: a vector of the \ number of levels in each factor, \nfreq: the number of observations in each \ cell ordered with last subscript varying the fastest, \nmv: the cell means \ ordered with last subscript varying the fastest\ngm: total group mean\nNote: \ A cell which has zero observations is not directly identified. \nCalls: \ kbin3,getss,restrv,examd, getrn\nRequired: Statistics`DataManipulation`\nMod. \ Date: Dec 1/97\n", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(srtcnt[d_, nf_] := Module[{nob, dn, si, obgrp, mv, frq, gm, ord, sst, ssw, typd, inx}, \[IndentingNewLine]\[IndentingNewLine]nob = Length[d]; (*\(Print["\", nob];\)*) si = kbin3["\", {nf}]; Print["\", si]; dn = Sort[\((ReplacePart[ d\[LeftDoubleBracket]#1\[RightDoubleBracket], \(\(d\ \[LeftDoubleBracket]#1\[RightDoubleBracket]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket]\)\[LeftDoubleBracket]si\[RightDoubleBracket], 1] &)\) /@ Range[nob]]; obgrp = Frequencies[\((\(dn\[LeftDoubleBracket]#1\ \[RightDoubleBracket]\)\[LeftDoubleBracket]1\[RightDoubleBracket] &)\) /@ Range[nob]]; obgrp = \((Reverse[ obgrp\[LeftDoubleBracket]#1\[RightDoubleBracket]] &)\) /@ Range[Length[obgrp]]; obgrp = Partition[Flatten[obgrp], nf + 1]; \n\t\tord = \((Max[\(Transpose[ obgrp]\)\[LeftDoubleBracket]#1\[RightDoubleBracket]] \ &)\) /@ Range[nf + 1]; ord = Take[ord, nf]; Print["\", ord]; \n\t\tdn = Flatten[Rest[Transpose[dn]]]; \n\t\tfrq = Flatten[Take[Transpose[obgrp], \(-1\)]]; \n\t\t\tinx = Map[Take[obgrp[\([#]\)], nf] &, Range[Length[ obgrp]]]; \n\t\t (*next\ restructure\ data\ into\ cell\ \ units\ of\ length\ frq*) \n\t\tdn = restrv[dn, frq]; \n\t\ttypd = examd[dn]; \ (*check\ type\ of\ data*) \[IndentingNewLine]\n\t\t\ Which[\n\t\t\ttypd == 0, {sst, ssw, mv, gm} = getss[dn, inx, frq], \n\t\t\ttypd == 1, {sst, ssw, mv, gm} = getss[getrn[dn], inx, frq], \n\t\t\ttypd == 2, Message[srtcnt::mixed]; Abort[];\n\t\t\t]; \n\t\t\tReturn[{sst, ssw, ord, frq, mv, gm}];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[BoxData[ \(\(srtcnt::mixed = "\";\)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["srtgp[dg,nf]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" takes data grouped by cell index, and for nf factors and a user \ selection of nf cell\nindices, combines the data of cells having a common set \ of indices, then calls getss and calculates the following:\n1) cell means, \ SSdeviated about cell mean and variances, cell sample sizes \n2) total group \ mean and SS deviated about grand mean,\n3) SSdeviated about cell mean is \ summed to form SSwithin group\ndg: data matrix in the form of {{{cell \ indices},{Y for each observation in the cell}},.....}\nnf: number of factors \ in the design\nBefore calculating SS examd is called to assess the type of \ data input in order to achieve maximum accuracy in the results. If all the \ data is in Real Form, getrn is called to convert the data to rational form.\n\ User Input: a vector of cell indices to be used, eg., there may be 3 cell \ indices, but user\n may select only two of them {1,3} \ collapsing the data over index 2.\nReturns:{sst,ssw, orf,obgrp,mv,gm}, where \ \na) sst is SStotal group deviated about grand mean,\nb) ssw SSwithin group\n\ c) orf is a vector of the number of levels in each factor\n d) obgrp is a \ vector of the cell frequencies, \n e) mv a vector of the cell means,\n f) gm \ is the grand mean\nCalls: kbin3,comdta, examd,getss,getrn\nMod. Date: Dec \ 1/97\n ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(\(srtgp[dg_, nf_] := Module[{si, dgri, sd, orf, obgrp, ind, typd, sst, ssw, mv, gm}, \[IndentingNewLine]\n\t\tsi = kbin3["\", {nf}]; Print["\", si]; \n\t\t (*get\ the\ selected\ indices\ in\ place*) \n\t\tdgri \ = \((ReplacePart[ dg\[LeftDoubleBracket]#1\[RightDoubleBracket], \(\(dg\ \[LeftDoubleBracket]#1\[RightDoubleBracket]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket]\)\[LeftDoubleBracket]si\[RightDoubleBracket], 1] &)\) /@ Range[Length[dg]]; sd = Sort[dgri]; \n\t\torf = \ Map[Part[#, 1] &, sd]; \n\t\torf = Map[Max[#] &, Transpose[ orf]]; \n\t\tPrint["\", orf]; \n\t\t (*combine\ cells\ having\ common\ indices*) \n\t\t\ {ind, sd} = comdta[sd]; \n\t\tobgrp = \((Length[#1] &)\) /@ sd; (*cell\ sizes*) \n\t\ttypd = examd[sd]; (*check\ type\ of\ data*) \n\t\tWhich[\n\t\ttypd == 0, {sst, ssw, mv, gm} = getss[sd, ind, obgrp], \n\t\ttypd == 1, \ (*data\ input\ all\ Real\ so\ convert*) \t{sst, ssw, mv, gm} = getss[getrn[sd], ind, obgrp], \n\t\ttypd == 2, Message[srtgp::mixed]; \ Abort[];\n\t\t]; \n\t\tReturn[{sst, ssw, orf, obgrp, mv, gm}];\n\t\t];\)\( (*end\ module*) \)\)\)], "Input"] }, Open ]], Cell[BoxData[ \(\(srtgp::mixed = "\";\)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["getss[sd,ind,obgrp]", FontWeight->"Bold"], " using the data in sd, indices ind, and cell frequencies obgrp, displays a \ table of cell indices, cell frequencies, cell means, and cell variances. The \ cell variances are ML estimates. SS deviated about the group mean and SS for \ each group deviated about the group mean are calculated by multiplying by n \ the variances provided by the function VarianceMLE[] in the package \ Statistics`DescrptiveStatistics`.\nUser Input: user is asked whether cell \ means and variances should be output in real or rational form.\nReturns: \ {sst,ssw,mv,gm} where sst is SStotal deviated, ssw is SSwithin group \ deviated, mv a vector of group means, and gm the grand mean.\nCalls: kbin3\n\ Requires: Statistics`DescriptiveStatistics`\nMod. Date: June 25/02; Modified \ output" }], "Text"], Cell[BoxData[ \(\(getss[sd_, ind_, obgrp_] := Module[{m, v, ror, vt, stuf, rch, sst, gm, ssw, fm}, m = Map[Mean[#] &, sd]; \ (*grp\ means*) \[IndentingNewLine]v = Map[VarianceMLE[#] &, sd]; \[IndentingNewLine]ror = kbin3["\", 1]; \[IndentingNewLine]If[ ror \[Equal] 1, \[IndentingNewLine]stuf = Transpose[{ind, obgrp, N[m, 10], N[v, 10]}], \[IndentingNewLine]stuf = Transpose[{ind, obgrp, m, v}]]; \[IndentingNewLine]\[IndentingNewLine]Print[ TableForm[stuf, TableDepth \[Rule] 2, TableHeadings \[Rule] {Automatic, {"\", \ "\", "\", "\"}}]]; \[IndentingNewLine]\ \[IndentingNewLine] (*grand\ m\ and\ sstcfm*) \[IndentingNewLine]gm = Mean[Flatten[sd]]; \[IndentingNewLine]vt = VarianceMLE[ Flatten[sd]]; \[IndentingNewLine]Print["\", gm, "\<(=\>", N[gm, 16], "\<)\>"]; \[IndentingNewLine]Print["\", vt, "\<(=\>", N[vt, 16], "\<)\>"]; \[IndentingNewLine]sst = Apply[Plus, obgrp]* vt; \ (*get\ ss\ only*) \[IndentingNewLine]ssw = obgrp*v; \ (*ss\ by\ grp*) \[IndentingNewLine]ssw = Apply[Plus, ssw]; \ (*sum\ within\ grp\ \ ss*) \[IndentingNewLine]Print["\", sst, "\<(=\>", N[sst, 16], "\<)\>"]; \[IndentingNewLine]Print["\", ssw, "\<(=\>", N[ssw, 16], "\<)\>"]; \[IndentingNewLine]fm = sst - ssw; \[IndentingNewLine]Print["\", fm, "\< (=\>", N[fm, 16], "\<)\>"]; \[IndentingNewLine]fm = fm/sst; \[IndentingNewLine]Print["\", fm, "\< (=\>", N[fm, 16], "\<)\>"]; \[IndentingNewLine] (*return\ ssotal, sswithin, cell\ means, and\ grand\ mean*) \[IndentingNewLine]Return[{sst, ssw, m, gm}];];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["comdta[d]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " combines data (d) of cells having the same indices; finds unique cell \ indices\nd: data matrix in the form of {{{indices},{data vector}},....}. Each \ row of data is \nexpected to be in order sorted on {indices}.\nCalls: \ binselect\nReturns: {{unique indices}, {data for each cell}}\nMod. Date: Nov \ 27/97 (corrected to properly handle unequal n case)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(comdta[d_] := Module[{di, dd, newc, bs, dn, sd, inx}, di = Flatten[Take[Transpose[d], 1], 1]; (*indices*) \n\t\tdd = \(Transpose[ d]\)\[LeftDoubleBracket]2\[RightDoubleBracket]; (*cell\ data\ \ as\ row\ vectors*) \n\t\tnewc = {}; sd = {}; While[Length[di] > 0, bs = \((di\[LeftDoubleBracket]#1\[RightDoubleBracket] == di\[LeftDoubleBracket]1\[RightDoubleBracket] &)\) /@ Range[Length[di]] /. {True \[Rule] 1, False \[Rule] 0}; \n\t\t\tinx = Flatten[binselect[bs, {Range[Length[bs]]}]]; \n\t\t\tsd = Append[sd, dd[\([inx]\)]]; (*combine\ cells\ data*) dn = Plus @@ bs; AppendTo[newc, Take[di, 1]]; di = Drop[di, dn]; dd = Drop[dd, dn];]; newc = Flatten[newc, 1]; sd = \((Flatten[#1] &)\) /@ sd; \n\t\tReturn[{newc, sd}];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["anvT2[stc,sse,df]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" displays a general anova table containing SStcfm, SSmcfm, \ SSerror, df, MS, and F calculated for SSmcfm only. If df(error) is 0, \ MS(error) is set to 0, and no F tests are carried out.\nstc: SS total cfm \ (SStcfm)\nsse: SS error\nsmd: SS model cfm (SSmcfm); calculated by \ SStcfm-SSerror\ndf: vector of degrees of freedom for SStcfm and SSmcfm\n\ Probability of F is rounded to 4 decimal digits for display; P(F)=0 is <.0001\ \nReturns: {MSerror,dferror}\nCalls: rndoff\nPackages: \ Statistics`ContinuousDistributions`\nNote: SStcfm is SS total corrected for \ the mean; SSmcfm is SS model corrected for the mean. To\ndisplay SS and MS as \ rational values, remove the operator N[....]\nMod. Date: Nov 26/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(anvT2[stc_, sse_, df_] := Module[{ct, dft, dfmd, dfe, mser, ssmcfm, fr, fd, p}, Print["\<\nGeneral Anova Table\>"]; ct = {}; dft = df\[LeftDoubleBracket]1\[RightDoubleBracket]; dfmd = df\[LeftDoubleBracket]2\[RightDoubleBracket]; dfe = dft - dfmd; AppendTo[ ct, {"\", N[stc, 6], dft - 1. , N[stc\/\(dft - 1\), 6]}]; ssmcfm = stc - sse; If[dfe > 0, mser = N[sse\/dfe]; fr = ssmcfm\/\(\((dfmd - 1)\)\ mser\); fd = FRatioDistribution[dfmd - 1, dft - dfmd]; p = rndoff[1. - CDF[fd, fr], 4], fr = "\<*\>"; p = "\<*\>"; mser = 0]; \n\t\tAppendTo[ ct, {"\", N[ssmcfm, 6], dfmd - 1, N[ssmcfm\/\(dfmd - 1\), 6], fr, p}]; If[dfe > 0, AppendTo[ct, {"\", rndoff[sse, 6], dfe, N[mser, 6]}], AppendTo[ct, {"\", Chop[sse], 0, "\<*\>"}]]; Print[TableForm[ct, TableHeadings \[Rule] {Automatic, {"\", "\", \ "\", "\", "\", "\"}}, TableSpacing \[Rule] {0, 2}]]; Return[{mser, dfe}];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["probF[ss,dfs,mse,dfe] ", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox["calculates probability of F. ss is the sum of squares for the \ hypothesis being tested, dfs is the degree of freedom for ss, mse is the mean \ square error term having dfe degrees of freedom.\nCalls: rndoff\nRequires: \ <False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(probF[ss_, dfs_, mse_, dfe_] := Module[{ms, f, fd, p, ct}, ms = ss\/dfs; f = ms\/mse; fd = FRatioDistribution[dfs, dfe]; p = rndoff[1 - CDF[fd, f], 4]; ct = {{"\", N[ss, 6], dfs, N[ms, 6], f, p}}; Print[TableForm[ct, TableHeadings \[Rule] {Automatic, {"\", "\", \ "\", "\", "\", "\"}}, TableSpacing \[Rule] {0, 2}]];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dsgnmef[nl]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" creates a design matrix for main effects only\nnl: vector of \ number of levels in each factor\nAlgorithm: Over each of j elements of nl,\n\ a) a matrix having nl[j]+1 cols is created having nl[j] cols of 0s and the \ last column of ones\nb) indices (the number of shifts required) for left \ shifting elements of the last column of ones into the leading cols are \ generated and replicated as required. Using these indices\nc) the left shifts \ are mapped over the matrix and the last column which now has 0s is dropped\n\ d) the sum reduction adjustment is done by subtracting the last column from \ the leading columns, and the last column dropped.\nReturns: main effects \ design matrix in the form of{{{a1},{a2},...}},{{b1},{b2}...},...}\nMod. Date: \ Dec 8/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(dsgnmef[ln_] := Module[{kln, dm, e1, e2, e3, r1, r2, r3, r4}, kln = Join[{1}, ln]; dm = {}; Do[e1 = Table[ Join[Table[ 0, {ln\[LeftDoubleBracket] j\[RightDoubleBracket]}], {1}], {Times @@ ln}]; e2 = Transpose[ Table[Reverse[ Range[ln\[LeftDoubleBracket] j\[RightDoubleBracket]]], {Times @@ Drop[ln, j]}]]; e3 = Flatten[Table[Flatten[e2], {Times @@ Take[kln, j]}]]; r1 = \((RotateLeft[ e1\[LeftDoubleBracket]#1\[RightDoubleBracket], #1] &)\) \ /@ e3; r1 = Drop[Transpose[r1], \(-1\)]; r2 = Take[r1, ln\[LeftDoubleBracket]j\[RightDoubleBracket] - 1]; r3 = Flatten[Take[r1, \(-1\)]]; r4 = MapThread[Flatten[#1] - r3 &, {r2}]; \(AppendTo[dm, r4];\), {j, 1, Length[ln]}]; Return[dm];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["bin2alf[x]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" returns the character in alph corresponding to the position of a \ 1 in x, e.g., bin2alf[{1,1,1}] ->\"abc\" Currently set for characters \"a\" \ to \"g\" which sets the maximum number of levels in the anova to 7. To \ increase the dimensionality of the design, simply place the additional \ characters in the vector, e.g., {\"a\",......\"g\",\"h\",\"i\"} for a 9-way \ anova.\nMod. Date: June 7/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(bin2alf[x_] := Module[{res, alph}, res = "\<\>"; alph = Reverse[ Take[{"\", "\", "\", "\", "\", "\", \ "\"}, Length[x]]]; \((If[ 1 == x\[LeftDoubleBracket]#1\[RightDoubleBracket], \(res = StringInsert[res, alph\[LeftDoubleBracket]#1\[RightDoubleBracket], \(-1\ \)];\)] &)\) /@ Reverse[Range[Length[x]]]; Return[res];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dmpar[nf]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" returns the character and binary designation for all possible \ parameters in the anova design having nf factors. \nCalls: bin2alf to get \ character designations.\nReturns: { {character, {binary \ representation}},......} in increasing order of the binary representation\n\ Mod. Date: June 7/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(dmpar[nf_] := Module[{bin, bin2, bin3, bin4}, bin = IntegerDigits[Range[\(-1\) + 2\^nf], 2]; bin2 = MapThread[ Join[Table[ 0, {nf - Length[ bin\[LeftDoubleBracket]#1\[RightDoubleBracket]]}], bin\[LeftDoubleBracket]#1\[RightDoubleBracket]] &, \ {Range[\(-1\) + 2\^nf]}]; bin3 = MapThread[ bin2alf[bin2\[LeftDoubleBracket]#1\[RightDoubleBracket]] &, \ {Range[\(-1\) + 2\^nf]}]; bin4 = Transpose[Append[{bin3}, bin2]]; Return[bin4];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["dsgnint[apc,avc,cdm]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" generates interaction vectors\napc: all possible component terms \ for the design {{a,{0,0,1}},{b,{0,1,0}},....} in binary order\navc: \ components available; {{a,{1,0,0},1},{b,{0,1,0},2},{c,{1,0,0},3}...}; on \ entry only the\n main effect components are available\ncdm: main effects \ design matrix; {{{column vectors of component a}},\n \ {{column vectors of component b}}, ....}\n\ Algorithm: \n1) for each binary vector in apc an interaction term is sought, \ i.e., more than one 1\n2) for the term found in (1), and doing backward \ search over binary vectors in avc, subtract\n each avc binary term until \ the result is a binary vector with only one 1; the term last subtracted\n \ and the term resulting from the subtraction identify the vectors of cdm which \ are to be\n multiplied to form the interaction vectors; multiply the \ vectors\n3) the interaction vectors of (2) are appended to cdm2 and avc2 is \ updated indicating which\n vectors are now available to form other \ interaction vectors\nCalls: findc to find location of vectors to form product \ for interaction vectors. fvp is a function internal to dsgnint which \ multiplies the vectors element by element.\nReturns: \navc2: all components \ for complete design matrix in the form of\n \ {{a,{0,0,1},1},{b,{0,1,0},2},{c,{1,0,0},3}...{abc,{1,1,1},7}} giving the \ character\n representation, the corresponding binary vector, and the \ location in the design matrix \tof the vectors. For example, Part[cdm2,7] \ contains all vectors for abc.\ncdm2: the design matrix for all components in \ the form of\n {{{vectors for a}},{{vectors for b}}, ...{{vectors for \ abc}}}\nMod. Date: July 16/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(dsgnint[apc_, avc_, cdm_] := Module[{avc2, cdm2, bin1, dif, lcn1, lcn2, prd, mat1, mat2, newterm}, fvp[a_, b_] := \((a\ b\[LeftDoubleBracket]#1\[RightDoubleBracket] \ &)\) /@ Range[Length[b]]; avc2 = avc; cdm2 = cdm; Do[\(If[1 \[NotEqual] Plus @@ \(apc\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket], bin1 = \(apc\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket]; Do[dif = bin1 - \(avc2\[LeftDoubleBracket] k\[RightDoubleBracket]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket]; \(If[1 == Plus @@ \(dif\^2\), lcn1 = avc2\[LeftDoubleBracket]k\[RightDoubleBracket]; lcn2 = findc[dif, avc2]; mat1 = cdm2\[LeftDoubleBracket] lcn1\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[RightDoubleBracket]; mat2 = cdm2\[LeftDoubleBracket] lcn2\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[RightDoubleBracket]; prd = \((fvp[ mat1\[LeftDoubleBracket]#1\ \[RightDoubleBracket], mat2] &)\) /@ Range[Length[mat1]]; prd = Flatten[prd, 1]; AppendTo[cdm2, prd]; newterm = bin2alf[bin1]; AppendTo[avc2, {newterm, bin1, Length[cdm2]}]; Break[];];\), {k, Length[avc2], 1, \(-1\)}];];\), {j, 1, Length[apc]}]; Return[{avc2, cdm2}];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["findc[c,ac]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " find the vector c in the row vectors of ac; used to locate which col. \ vectors of \nthe design matrix are required to generate an interaction term.\n\ ac: of the form {{\"a\",{0,0,1},1},{\"b\",{0,1,0},2},...} and holds currently \ available vectors\n which can be used to generate interaction vectors\nc: \ binary vector to be found in rows of ac, eg., {0,1,0}\nReturns: a row vector \ of ac which matches, e.g., {b,{0,1,0},2}\nMod. Date: July 16/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(findc[c_, ac_] := Module[{res}, Do[\(If[c == \(ac\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket], res = ac\[LeftDoubleBracket]j\[RightDoubleBracket]; Break[];];\), {j, 1, Length[ac]}]; Return[res];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["cutintr[x,it,nf]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " allows user to remove selected interaction vectors from design matrix\nx: \ design matrix in the form of {{{a1},{a2}},{{b1},{b2},{b3}},....}\nit: \ available terms in the form of {{\"a\",{0,0,1},1},....{\"abc\",{1,1,1},9}}\n\ nf: number of factors in the design\nUser Input: interaction term to be \ removed, e.g., abc\nCalls: kbinstrg,binselect\nReturns: {new design matrix, \ available terms updated}\nMod. Date: July 21/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(cutintr[x_, it_, nf_] := Module[{xc, itc, can, bs, flg = 0, strm, indx}, can = Transpose[{Drop[Flatten[Take[Transpose[it], 1]], nf]}]; bs = Table[0, {Length[it]}]; While[flg == 0, strm = kbinstrg["\", can]; Print["\", strm]; bs = bs + \((strm == \(it\[LeftDoubleBracket]#1\ \[RightDoubleBracket]\)\[LeftDoubleBracket]1\[RightDoubleBracket] &)\) /@ Range[Length[it]] /. {False \[Rule] 0, True \[Rule] 1}; If["\" == kbinstrg["\", {{"\"}, {"\"}}], flg = 1];]; bs = \((bs\[LeftDoubleBracket]#1\[RightDoubleBracket] === 0 &)\) /@ Range[Length[bs]] /. {False \[Rule] 1, True \[Rule] 0}; bs = Table[1, {Length[it]}] - bs; indx = binselect[bs, {Range[Length[it]]}]; indx = Flatten[indx]; xc = x\[LeftDoubleBracket]indx\[RightDoubleBracket]; itc = it\[LeftDoubleBracket]indx\[RightDoubleBracket]; itc = \((ReplacePart[ itc\[LeftDoubleBracket]#1\[RightDoubleBracket], #1, \ \(-1\)] &)\) /@ Range[Length[itc]]; Return[{xc, itc}];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["ssk[k,xi,pb]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " Function ssk calculates sum of squares for contrast\nmatrix k, with xi as \ Inverse[X'X], and pb the parameter estimates\nMod. Date: June 21/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(ssk[k_, xi_, pb_] := Flatten[{k . pb} . Inverse[k . xi . Transpose[k]] . Transpose[{k . pb}]]\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["ndim[vc,di]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" restructures vector vc into matrix of order given by vector di. \ Used to restructure\ncell means into a matrix having dimensions equal to the \ number of levels in each factor.\nReturns: matrix of Dimensions[]={di}\nMod. \ Date: June 27/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(ndim[vc_, di_] := Module[{res, indx}, res = vc; indx = Drop[Reverse[di], \(-1\)]; Do[res = Partition[res, indx\[LeftDoubleBracket]j\[RightDoubleBracket]], {j, 1, Length[indx]}]; Return[res]];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["fndintr[sl]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ "; finds and returns all 2 & 3 term interactions in list sl\nsl: list of \ all available anova terms; {\"mu\",{0,0,0},1},{\"a\",{0,0,1},2}...}\nReturns: \ interaction term of the form {\"ab\",{0,1,1},5}; if no interaction terms are \ found {}is returned\nMod. Date: July 24/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(fndintr[sl_] := Module[{sl2, cr}, sl2 = {}; Do[cr = Plus @@ \(sl\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket]; \(If[ 2 == cr || 3 == cr, \(AppendTo[sl2, sl\[LeftDoubleBracket]j\[RightDoubleBracket]];\)];\), {j, 1, Length[sl]}]; Return[sl2];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["plotnLists[m,tl,xu,ps,pm,pn]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " plots a colored line for each column of m. The values in the column are \ y coordinates and the x coordinates are taken to be from 1 to the number of \ rows of the matrix.\ntl is a title for the plot, \nxu is the label for the \ x-axis\nps set to 1 for individual plots for each col; 0 for no individual \ plots\npm set to 1 for one plot holding multiple lines; 0 for no plot\npn \ (Optional) set to 1 for line numbers at end of lines when pm=1; else 0 \nFor \ single plots (ps=1), the title is appended to a plot sequence number.\n The \ colors are rotated through the following color sequence:(1) black, (2) blue \ (3) green, (4) cyan, (5) red, (6) magenta, (7) yellow, (8) dark blue, (9) \ dark green (10) turquoise, (11)brown, and (12) purple. (Colors numbered 8 to \ 12 may vary with different monitors.) To get a simple example of the colors \ used, do plotnLists[Table[Range[0,11],{2}],\"Legend\",\"units\",0,1,1] \nMod. \ Date: July 11/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(plotnLists[m_, tl_, xu_, ps_, pm_, pn_: 0] := Module[{pobj, colr, ipt, cntr = 0}, If[ps == 0 && pm == 0, Print["\"]; Return[]]; pobj = {}; colr = { .5, .0, .5, .0, .0, .0, .0, .0, 1. , .0, 1. , .0, .0, 1. , 1. , 1. , .0, .0, 1. , .0, 1. , 1. , 1. , .0, .0, .0, .5, .0, .5, .0, .0, .5, .5, .5, \ .0, .0}; If[ps == 1, ipt = $DisplayFunction, ipt = Identity]; pobj = MapThread[ ListPlot[#1, AxesLabel \[Rule] {xu, "\"}, PlotJoined \[Rule] True, PlotRange \[Rule] All, PlotLabel \[Rule] StringInsert[tl, ToString[cntr = cntr + 1], 1], PlotStyle \[Rule] {RGBColor[\((colr = RotateLeft[colr, 3])\)\[LeftDoubleBracket]1\ \[RightDoubleBracket], colr\[LeftDoubleBracket]2\[RightDoubleBracket], colr\[LeftDoubleBracket]3\[RightDoubleBracket]]}, DisplayFunction \[Rule] ipt] &, {Transpose[m]}]; If[pn == 1, \(MapThread[ AppendTo[pobj\[LeftDoubleBracket]#1, 1\[RightDoubleBracket], Text[#1, Last[pobj\[LeftDoubleBracket]#1, 1, 2, 1\[RightDoubleBracket]]]] &, {Range[ Length[Transpose[m]]]}];\)]; If[pm == 1, Show[pobj, PlotLabel \[Rule] tl, DisplayFunction \[Rule] $DisplayFunction]];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["binselect[b_,a_]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " columns of the matrix a corresponding to 1 in b are selected and \ returned as a new matrix. b must have as many elements as columns of a. If a \ is a vector, enclose it in {} and then Flatten the result returned. \ Equivalent to APL reduction operation.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(binselect[b_, a_] := Module[{indx, z}, indx = DeleteCases[ b\ Range[\(Dimensions[ a]\)\[LeftDoubleBracket]2\[RightDoubleBracket]], 0]; z = Transpose[\(Transpose[a]\)\[LeftDoubleBracket] indx\[RightDoubleBracket]]; Return[z];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["matsum[nlv,ptr,m]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" forms means over different dimensions of matrix m specified \ indirectly by the binary terms in ptr; used to calculate means for plotting \ of interactions\nnlv: number of levels in each factor\nptr: interaction term; \ {\"ab\",{0,1,1},psn}\nm: matrix of means having dimensions defined by those \ terms in nlv \nThe summation is made over those dimensions having 0 in the \ corresponding position of binary vector ptr.\nReturns: a matrix of means\n\ Calls: binselect, redspc\nMod. Date: July 11/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(matsum[nlv_, ptr_, m_] := Module[{inx, dv, ms}, inx = binselect[ Table[1, {Length[nlv]}] - ptr\[LeftDoubleBracket]2\[RightDoubleBracket], {Reverse[ nlv]}]; dv = Times @@ Flatten[inx]; Print["\", dv]; If[0 == dv, dv = 1]; ms = redspc[m, ptr\[LeftDoubleBracket]2\[RightDoubleBracket]]; ms = N[ms\/dv]; Return[ms];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["redspc[m,bv]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" reduces the dimensionality of the matrix m by summing over \ dimensions \nidentified by the position of 0 in the binary vector bv; used to \ find the sums (and later the means) for plotting 2d and 3d interaction terms. \ \nNote: The binary vector bv low-order bit represents \"a\"; the next \"b\", \ etc.\nMod. Date: Jun 30/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(redspc[m_, bv_] := Module[{rm, spr = 1, c}, rm = m; c = Reverse[Table[1, {Length[bv]}] - bv]\ Range[Length[bv]]; Do[\(If[0 \[NotEqual] c\[LeftDoubleBracket]j\[RightDoubleBracket], rm = Apply[Plus, rm, {c\[LeftDoubleBracket]j\[RightDoubleBracket] - spr}]; spr = spr + 1];\), {j, 1, Length[c]}]; Return[rm];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["plt3fintr[mnm,int] ", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ "plots 3D matrix of means;each plane represents 1st term of the interaction\ \nmnm: matrix of means\nint: interaction term, e.g., \"abc\"\nMod. Date: July \ 11/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(plt3fintr[mnm_, int_] := Module[{pobj}, pobj = MapThread[ ListPlot3D[#1, DisplayFunction \[Rule] Identity] &, {mnm}]; Show[pobj, PlotLabel \[Rule] StringInsert["\", int, \(-1\)], AxesLabel \[Rule] Join[Take[Reverse[Characters[int]], 2], {"\"}], DisplayFunction \[Rule] $DisplayFunction];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["intrplot[m,st,nl]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " plots 2 and 3 term interactions based on cell means\nm: vector of cell \ means\nst: anova terms available; \ {{\"mu\",{0,0,0},1},{\"a\",{0,0,1},2},.....}\nnl: vector of the number of \ levels in each factor\nSteps:\n1) fndintr finds the interaction terms, and \ allows user selection of 2 & 3 term interactions and\n returns \ {\"ab\",{binary vector representation}, posn}\n2) ndim restructures m (means) \ to dimensions given by nl (number of levels in each factor) \n3) matsum \ uses the results returned by (1) and sums over the appropriate\n \ dimensions of m to get the means for plotting\n4) plot 2 term interaction as \ line plots; 3 term interaction as planes\nCalls: \ ndim,findintr,kbinstrg,matsum,plotnLists,plt3fintr,prntimm\nMod. Date: Dec \ 8/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(intrplot[m_, st_, nl_] := Module[{mm, trm, ans, rp, pltrm, msrd, flag = 0}, trm = fndintr[st]; If[0 == Length[trm], Print["\"]; Return[]]; mm = ndim[m, nl]; ans = Transpose[{\((#1\[LeftDoubleBracket]1\[RightDoubleBracket] &)\ \) /@ trm}]; Print["\<\nInteraction terms available=\>", ans]; While[flag == 0, rp = kbinstrg["\", ans]; Print["\<\nPlot for interaction=\>", rp]; Do[\(If[{rp} == ans\[LeftDoubleBracket]j\[RightDoubleBracket], pltrm = trm\[LeftDoubleBracket]j\[RightDoubleBracket]; Break[]];\), {j, 1, Length[ans]}]; If[StringLength[rp] == 2, If[Length[nl] == 2, prntimm[mm, 6, rp]; plotnLists[Transpose[mm], StringInsert["\", rp, \(-1\)], \(Reverse[ Characters[ rp]]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 0, 1, 1], msrd = matsum[nl, pltrm, mm]; prntimm[msrd, 6, rp]; plotnLists[Transpose[msrd], StringInsert["\", rp, \(-1\)], \(Reverse[ Characters[ rp]]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 0, 1, 1]], \(If[Length[nl] == 3, prntimm[mm, 6, rp]; plt3fintr[mm, rp], msrd = matsum[nl, pltrm, mm]; prntimm[msrd, 6, rp]; plt3fintr[msrd, rp]];\)]; If["\" == kbinstrg["\", {{"\"}, {"\"}}], flag = 1];];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["prntimm[m,pc,is]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" displays 2d or 3d interaction means matrix\nm: a 2d or 3d matrix \ of means\npc: max number of columns to use in display\nis: 2 or 3 character \ anova interaction term; eg., \"ac\", \"abd\"\nCalls: pmat2, genrch\nMod. \ Date: Dec. 2/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(prntimm[m_, pc_, is_] := Module[{lv, rwch}, Print[StringInsert["\", is, \(-1\)]]; If[2 == Length[Dimensions[m]], rwch = genrch[StringTake[is, {1}], StringTake[is, {2}], Dimensions[m]]; pmat2[N[m], pc, rwch], rwch = genrch[StringTake[is, {2}], StringTake[is, {3}], Rest[Dimensions[m]]]; Do[lv = StringInsert[StringTake[is, 1], ToString[j], \(-1\)]; Print["\<\nLevel=\>", lv]; pmat2[N[m\[LeftDoubleBracket]j\[RightDoubleBracket]], pc, rwch], {j, 1, \(Dimensions[ m]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}];];];\)\)]\ , "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["restrv[v,p]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ " restructures elements in vector v into units of length p\nv: a vector to \ restructure\np: length of each unit \nReturns: new structure of \ Length=Length[p]\nMod. Date: July 11/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(restrv[v_, p_] := Module[{rv, temp}, temp = v; rv = {}; Do[AppendTo[rv, Take[temp, p\[LeftDoubleBracket]j\[RightDoubleBracket]]]; temp = RotateLeft[temp, p\[LeftDoubleBracket]j\[RightDoubleBracket]], {j, 1, Length[p]}]; Return[rv];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["getucn[at,df,xi,inxi,ems,edf,mefb]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" obtains a user contrast matrix in non-sum reduced form for main \ effect terms in at; calculates and displays sum-reduced and row echelon form \ \nat: available terms; string vector of the form {\"a\",\"b\",\"c\",...}\n\ df: degrees of freedom for terms in at; {dfa,dfb,dfc,....}\nxi: submatrix of \ Inverse[X'X] formed by dropping 1st row and column (constant term)\ninxi: \ start position and number of vectors in xi for each term in at; \ {{\"a\",s,n\"},{\"b\",s,n},....}\nems: Mean Square error for F test\nedf: \ degrees of freedom for ems\nmefb: parameter estimates for main effects; \ {{a1,a2,..},{b1,b2...},....}\nRqd: LinearAlgebra`MatrixManipulation`\nCalls: \ binselect, probF,hytst \nMod. Date: Dec 8/97\n", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(getucn[at_, df_, xi_, inxi_, ems_, edf_, mefb_] := Module[{iflg = 1, bs, cntrm, dfk, crw, kcon, kct, rwrc, erf, sbinx, si, di, subxi, prr, sumsk}, Print["\<\nUser Defined Contrasts\>"]; While[iflg == 1, cntrm = kbinstrg["\", Transpose[{at}]]; Print["\", cntrm]; bs = \((#1 == cntrm &)\) /@ Flatten[at] /. {True \[Rule] 1, False \[Rule] 0}; dfk = \(Flatten[ binselect[ bs, {df}]]\)\[LeftDoubleBracket]1\[RightDoubleBracket]; erf = 1; While[erf == 1, crw = kbin3["\", 0]; If[crw > dfk, Print["\", dfk], erf = 0];]; Print["\", crw]; Print["\", dfk + 1]; erf = 1; While[erf == 1, kcon = kbin3["\", {crw, dfk + 1}]; Print["\", kcon]; rwrc = RowReduce[kcon]; Print["\", rwrc]; If[False === \((\((Table[ 0, {\(Dimensions[ rwrc]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket]}] == #1 &)\) /@ rwrc == Table[False, {\(Dimensions[ rwrc]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket]}])\), Print["\"], erf = 0];]; hytst[cntrm, kcon]; kct = Transpose[kcon]; kcon = Transpose[ Drop[kct + Table[\(-Last[kct]\), {Length[kct]}], \(-1\)]]; Print["\", kcon]; Print["\", RowReduce[kcon]]; bs = \((cntrm === \(inxi\[LeftDoubleBracket]#1\ \[RightDoubleBracket]\)\[LeftDoubleBracket]1\[RightDoubleBracket] &)\) /@ Range[Length[inxi]] /. {True \[Rule] 1, False \[Rule] 0}; sbinx = binselect[bs, Transpose[inxi]]; si = Flatten[ Table[sbinx\[LeftDoubleBracket]2\[RightDoubleBracket], {2}]]; di = Flatten[ Table[sbinx\[LeftDoubleBracket]3\[RightDoubleBracket], {2}]]; subxi = SubMatrix[xi, si, di]; prr = mefb\[LeftDoubleBracket] Flatten[ binselect[bs, {Range[Length[mefb]]}]]\[RightDoubleBracket]; prr = Flatten[prr]; Print["\", N[prr, 6]]; sumsk = ssk[kcon, subxi, prr]; probF[sumsk, Length[kcon], ems, edf]; If["\" == kbinstrg["\", {{"\"}, {"\"}}], iflg = 0];];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["hytst[t,k]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" displays string expression for hypothesis being tested for term \ \"t\" and contrast matrix k\nt: term in string form, eg., \"a\", \"ab\"\nk: \ contrast matrix\nNote:Chop[N[k]] is used so that \"=0\" is not displayed at \ divisor position when rational values\n are entered in k and Chop to \ ensure a term multiplied by 0. is removed.\nMod. Date: July 17/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(hytst[t_, k_] := Module[{prd, stcf}, Print[StringInsert["\", t, \(-1\)]]; prd = Chop[N[k]] . Array[t, \(Dimensions[ k]\)\[LeftDoubleBracket]2\[RightDoubleBracket]]; stcf = MapThread[ToString[#1] &, {prd}]; MapThread[Print[#1, "\<=0\>"] &, {stcf}];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["lctmev[d,at] ", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ "calc starting and ending position of vectors in X'X \nd: degrees of \ freedom for terms in at (number of row/col for each term)\nat: terms \ ordered,e.g., {\"a\",\"b\",\"c\"...} corresponding to terms represented in \ X'X\nReturns: {{\"a\",begin row/col,#of row/col},{\"b\",begin row/col,#of \ row/col},....}\nMod. Date: July 8/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(lctmev[d_, at_] := Module[{lse, vn, bgn, indxi}, lse = {}; vn = Range[Plus @@ d]; Do[AppendTo[lse, Take[vn, d\[LeftDoubleBracket]j\[RightDoubleBracket]]]; \(vn = RotateLeft[vn, d\[LeftDoubleBracket]j\[RightDoubleBracket]];\), {j, 1, Length[d]}]; bgn = \((Take[#1, 1] &)\) /@ lse; indxi = Transpose[Append[{bgn}, Transpose[{d}]]]; indxi = \((Flatten[#1, 1] &)\) /@ indxi; indxi = \((Prepend[ indxi\[LeftDoubleBracket]#1\[RightDoubleBracket], at\[LeftDoubleBracket]#1\[RightDoubleBracket]] &)\) /@ Range[Length[at]]; Return[indxi];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["gencon[s,d,p,xtxi,mse,dfe]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[ ": calc SS,MS, F, and prob for each main effect and interaction term and \ displays result.Prob is rounded to 4 decimal digits.\ns: contrasts available \ in the form of {{\"mu\",{0,0,0},1},{\"a\",{0,0,1},2},....}\nd: vector of \ degrees of freedom for each term in s\np: vector of parameter estimates\n\ xtxi: Inverse[X'X]\nmse: Mean Square error\ndfe: degrees of freedom for mse\n\ Calls: ssk, rndoff,restrv,lctmev\nRequired: LinearAlgebra`MatrixManipulation\n\ Mod. Date: Dec 5/97", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(gencon[s_, d_, p_, xtxi_, mse_, dfe_] := Module[{dftrm, strm, con, inxt, rxi, par, lcn, si, di, ct, msk, fr, fd, pr, sumsq}, strm = Drop[Flatten[Take[Transpose[s], 1]], 1]; dftrm = Drop[d, 1]; par = restrv[Drop[p, 1], dftrm]; lcn = lctmev[dftrm, strm]; di = Length[xtxi] - 1; rxi = SubMatrix[xtxi, {2, 2}, {di, di}]; ct = {}; Do[con = IdentityMatrix[ dftrm\[LeftDoubleBracket]j\[RightDoubleBracket]]; si = \(lcn\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket]; di = \(lcn\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]3\ \[RightDoubleBracket]; inxt = SubMatrix[rxi, {si, si}, {di, di}]; sumsq = ssk[con, inxt, par\[LeftDoubleBracket]j\[RightDoubleBracket]]; msk = sumsq\/dftrm\[LeftDoubleBracket]j\[RightDoubleBracket]; If[dfe \[GreaterEqual] 1, fr = msk\/mse; fd = FRatioDistribution[ dftrm\[LeftDoubleBracket]j\[RightDoubleBracket], dfe]; pr = rndoff[1. - CDF[fd, fr], 4], fr = "\<*\>"; pr = "\<*\>"]; \(AppendTo[ ct, {strm\[LeftDoubleBracket]j\[RightDoubleBracket], N[sumsq, 6], dftrm\[LeftDoubleBracket]j\[RightDoubleBracket], N[msk, 6], fr, pr}];\), {j, 1, Length[strm]}]; AppendTo[ct, {"\", mse\ dfe, dfe, mse}]; Print["\<\nAnova Table for all Terms (Type III SS)\>"]; Print[TableForm[ct, TableHeadings \[Rule] {Automatic, {"\", "\", \ "\", "\", "\", "\"}}, TableSpacing \[Rule] {0, 2}]];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["anovanw[d]", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Bold"], StyleBox[" does anova using data set d \n Calls Directly: kbin3, srtgrp, \ srtcnt, kbinstrg, dsgnmef, dmpar, dsgnint, cutintr, anvT2, gencon, intrplot, \ lctmev, restrv, getucn \nSpecial Variables:\n xdsm: design matrix; \ starts with main effects; interaction appended by dsgnint\n xdscode: holds \ defn of all terms in the form \ {\"a\",{0,0,1}},{\"b\",{0,1,0}},...{\"abc,{1,1,1}}..\n slctd: holds defn of \ currently available terms in form of xdscode as well as location of the term \ in xdsm. \n Mod. Date: Dec 6/97\n Mod. Date: Jun 4/02: changed local variable \ list", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(anovanw[d_] := Module[{ntot, nf, ntrm, nlev, xdsm, xdscode, slctd, sstot, sse, means, conv, ncs, xtx, xxi, df, gm, bpar, ssmod, temp, mse, dfe, ixr, xtx2, bpar1, cutf = 0, mef, dfa, xindx, bmef}, Print["\"]; \n\t\tIf[0 == Length[d], Message[anovanw::nodat]; Abort[]]; \n\t\tPrint["\", Length[\(d\[LeftDoubleBracket]1\[RightDoubleBracket]\)\ \[LeftDoubleBracket]1\[RightDoubleBracket]]]; nf = kbin3["\", 0]; Print["\", nf]; If[True == VectorQ[\(d\[LeftDoubleBracket]1\[RightDoubleBracket]\)\ \[LeftDoubleBracket]2\[RightDoubleBracket]], Print["\"]; {sstot, sse, nlev, ncs, means, gm} = srtgp[d, nf], Print["\"]; {sstot, sse, nlev, ncs, means, gm} = srtcnt[d, nf];]; ntot = Plus @@ ncs; Print["\"]; ntrm = \(-1\) + 2\^nf; Print["\", ntrm + 1]; Print["\
", 1 + Plus @@ \((nlev - 1)\)]; Print["\", Times @@ nlev - \((1 + Plus @@ \((nlev - 1)\))\)]; If[1 > Plus @@ ncs - Times @@ nlev, Print["\<\nWARNING: No df available for SSerror.\>"]; Print["\"]]; If["\" == kbinstrg["\", {{"\"}, \ {"\"}}], Return[]]; xdsm = dsgnmef[nlev]; xdscode = dmpar[nf]; slctd = xdscode\[LeftDoubleBracket]2\^\(\(-1\) + Range[nf]\)\ \[RightDoubleBracket]; slctd = \((Append[ slctd\[LeftDoubleBracket]#1\[RightDoubleBracket], #1] \ &)\) /@ Range[Length[slctd]]; \[IndentingNewLine] (*Print["\", xdscode]; \[IndentingNewLine]Print["\", slctd]; \[IndentingNewLine]Print["\", xdsm];*) \[IndentingNewLine]\[IndentingNewLine]{slctd, xdsm} = dsgnint[xdscode, slctd, xdsm]; \[IndentingNewLine] (*Print["\", slctd]; \[IndentingNewLine]Print["\", xdsm]; \[IndentingNewLine]Abort[];*) \[IndentingNewLine]If[ nf \[GreaterEqual] 2 && "\" == kbinstrg["\", {{"\"}, \ {"\"}}], cutf = 1; {xdsm, slctd} = cutintr[xdsm, slctd, nf]]; ntrm = Length[slctd]; df = \((Take[ Dimensions[ xdsm\[LeftDoubleBracket]#1\[RightDoubleBracket]], 1] &)\) /@ Range[ntrm]; xdsm = Flatten[xdsm, 1]; conv = Table[1, {Times @@ nlev}]; xdsm = Join[{conv}, xdsm]; df = Prepend[Flatten[df], 1]; slctd = Prepend[slctd, {"\", Table[0, {nf}], 0}]; slctd = \((ReplacePart[ slctd\[LeftDoubleBracket]#1\[RightDoubleBracket], #1, \ \(-1\)] &)\) /@ Range[Length[slctd]]; xtx = \((ncs\ xdsm\[LeftDoubleBracket]#1\[RightDoubleBracket] &)\) /@ Range[Length[xdsm]]; \n\t\txtx = xtx . Transpose[xdsm]; \n\t\ \ xxi = Inverse[xtx]; (*xxi\ is\ in\ rational\ form*) \n\t\tbpar = xxi . xdsm . \((means*\ ncs)\); \n\t\ttemp = N[restrv[bpar, df], 16]; Do[Print["\<\nTerm=\>", \(slctd\[LeftDoubleBracket] j\[RightDoubleBracket]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket], "\< df=\>", df\[LeftDoubleBracket]j\[RightDoubleBracket]]; Print["\", temp\[LeftDoubleBracket]j\[RightDoubleBracket]], {j, 1, Length[df]}]; (*If\ any\ interaction\ term\ was\ dropped, \ calc\ new\ sse*) \n\t\tIf[ cutf == 1, \n\t\tbpar1 = Drop[bpar, 1]; \n\t\txtx2 = SubMatrix[ xtx, {2, 2}, {Length[bpar1], Length[bpar1]}]; \n\t\tssmod = bpar1 . xtx2 . bpar1; \n\t\tPrint["\", ssmod, "\< (=\>", N[ssmod, 16], "\<)\>"]; \n\t\tsse = sstot - ssmod; \n\t\tPrint["\", sse, "\< (=\>", N[sse, 16], "\<)\>"];\n\t\t\t]; (*end\ If\ cutf == 1*) \n\t\t{mse, dfe} = anvT2[sstot, sse, {ntot, Length[bpar]}]; gencon[slctd, df, bpar, xxi, mse, dfe]; If[dfe == 0, Return[]]; If[nf \[GreaterEqual] 2, \(If["\" == kbinstrg["\", {{"\"}, \ {"\"}}], intrplot[means, slctd, nlev]];\)]; If["\" == kbinstrg["\", {{"\"}, \ {"\"}}], mef = Take[ Drop[\(Transpose[ slctd]\)\[LeftDoubleBracket]1\[RightDoubleBracket], 1], nf]; dfa = Take[Rest[df], nf]; xindx = lctmev[dfa, mef]; ixr = SubMatrix[xxi, {2, 2}, Table[\(-1\) + Length[xxi], {2}]]; temp = Take[Rest[bpar], Plus @@ dfa]; bmef = restrv[temp, dfa]; getucn[mef, dfa, ixr, xindx, mse, dfe, bmef];];];\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[BoxData[ \(\(anovanw::nodat = "\";\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Textbook Examples", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(anovanw[dat1]\)], "Input"], Cell[BoxData[ \("GLM Anova V3.0"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Number of indices defining factor levels="\ \[InvisibleSpace]4\), SequenceForm[ "Number of indices defining factor levels=", 4], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Number of factors="\[InvisibleSpace]3\), SequenceForm[ "Number of 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Thus, the 3-way analysis \ contains one observation per cell. An important restriction is that the \ number of data points must be equal in each cell of the original nested \ design. The data re-cast as a 3-way anova with one observation per cell and \ with cell subscripts as required for analysis using functions in this \ notebook are given below.\ \>", "Text"], Cell[BoxData[ \(\(krd = {{{1, 1, 1}, 3}, {{1, 1, 2}, 6}, {{1, 1, 3}, 3}, {{1, 1, 4}, 3}, {{1, 2, 1}, 1}, {{1, 2, 2}, 2}, {{1, 2, 3}, 2}, {{1, 2, 4}, 2}, {{1, 3, 1}, 5}, {{1, 3, 2}, 6}, {{1, 3, 3}, 5}, {{1, 3, 4}, 6}, {{1, 4, 1}, 2}, {{1, 4, 2}, 3}, {{1, 4, 3}, 4}, {{1, 4, 4}, 3}, {{2, 1, 1}, 7}, {{2, 1, 2}, 8}, {{2, 1, 3}, 7}, {{2, 1, 4}, 6}, {{2, 2, 1}, 4}, {{2, 2, 2}, 5}, {{2, 2, 3}, 4}, {{2, 2, 4}, 3}, {{2, 3, 1}, 7}, {{2, 3, 2}, 8}, {{2, 3, 3}, 9}, {{2, 3, 4}, 8}, {{2, 4, 1}, 10}, {{2, 4, 2}, 10}, {{2, 4, 3}, 9}, {{2, 4, 4}, 11}};\)\)], "Input"], Cell["\<\ The results of the anova are given below. Because there is only one \ observation per cell, the SSerror and dfe is 0. 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The output from MGems.nb is given \ below. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(ems3[4]\)\( (*Kirk\ p460 - 2*) \)\)\)], "Input"], Cell[BoxData[ \("Expected Mean Squares"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Factors Defined="\[InvisibleSpace]{"a", "b:a", "c:ab", "r:abc"}\), SequenceForm[ "Factors Defined=", {"a", "b:a", "c:ab", "r:abc"}], Editable->False]], "Print"], Cell[BoxData[ \("Input Terms and Binary Representation"\)], "Print"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"Source\"\>", "\<\"Source Term\"\>", "\<\"Nesting \ Term\"\>"}, {"\<\"\"\>", "\<\"a\"\>", "\<\"{0, 0, 0, 1}\"\>", "\<\"{{0, 0, 0, \ 0}}\"\>"}, {"\<\"\"\>", "\<\"b:a\"\>", "\<\"{0, 0, 1, 0}\"\>", "\<\"{{0, 0, 0, \ 1}}\"\>"}, {"\<\"\"\>", "\<\"c:ab\"\>", "\<\"{0, 1, 0, 0}\"\>", "\<\"{{0, 0, \ 0, 1}, {0, 0, 1, 0}}\"\>"}, {"\<\"\"\>", "\<\"r:abc\"\>", "\<\"{1, 0, 0, 0}\"\>", "\<\"{{0, 0, \ 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}}\"\>"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableHeadings -> {{}, {"Source", "Source Term", "Nesting Term"}}]&)]], "Print"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"Source\"\>", "\<\"Type\"\>", "\<\"Levels\"\>"}, {"\<\"\"\>", "\<\"a\"\>", "\<\"Fixed\"\>", "\<\"2\"\>"}, {"\<\"\"\>", "\<\"b\"\>", "\<\"Random\"\>", "\<\"4\"\>"}, {"\<\"\"\>", "\<\"c\"\>", "\<\"Random\"\>", "\<\"4\"\>"}, {"\<\"\"\>", "\<\"r\"\>", "\<\"Random\"\>", "\<\"1\"\>"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableHeadings -> {{}, {"Source", "Type", "Levels"}}]&)]], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {"\<\"\"\>", "\<\"Source\"\>", "\<\"df\"\>", "\<\"coef\"\>", \ "\<\"EMS\"\>"}, {"\<\"\"\>", "\<\"a:\"\>", "\<\"1\"\>", "\<\"16\"\>", GridBox[{ {"\<\"a:\"\>", "\<\"b:a\"\>", "\<\"c:ab\"\>", \ "\<\"r:abc\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {"\<\"\"\>", "\<\"b:a\"\>", "\<\"6\"\>", "\<\"4\"\>", GridBox[{ {"\<\"b:a\"\>", "\<\"c:ab\"\>", "\<\"r:abc\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {"\<\"\"\>", "\<\"c:ab\"\>", "\<\"24\"\>", "\<\"1\"\>", GridBox[{ {"\<\"c:ab\"\>", "\<\"r:abc\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {"\<\"\"\>", "\<\"r:abc\"\>", "\<\"0\"\>", "\<\"1\"\>", GridBox[{ {"\<\"r:abc\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Center}], TableForm[ {{"a:", "1", "16", {{"a:", "b:a", "c:ab", "r:abc"}}}, {"b:a", "6", "4", {{"b:a", "c:ab", "r:abc"}}}, {"c:ab", "24", "1", {{"c:ab", "r:abc"}}}, {"r:abc", "0", "1", {{"r:abc"}}}}, TableHeadings -> {{}, { "Source", "df", "coef", "EMS"}}]]], "Print"], Cell[BoxData[ \("Deleting terms where df=0"\)], "Print"], Cell[BoxData[ InterpretationBox[GridBox[{ {"\<\"\"\>", "\<\"Source\"\>", "\<\"df\"\>", "\<\"coef\"\>", \ "\<\"EMS\"\>"}, {"\<\"\"\>", "\<\"a:\"\>", "\<\"1\"\>", "\<\"16\"\>", GridBox[{ {"\<\"a:\"\>", "\<\"b:a\"\>", "\<\"c:ab\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {"\<\"\"\>", "\<\"b:a\"\>", "\<\"6\"\>", "\<\"4\"\>", GridBox[{ {"\<\"b:a\"\>", "\<\"c:ab\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {"\<\"\"\>", "\<\"c:ab\"\>", "\<\"24\"\>", "\<\"1\"\>", GridBox[{ {"\<\"c:ab\"\>"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Center}], TableForm[ {{"a:", "1", "16", {{"a:", "b:a", "c:ab"}}}, {"b:a", "6", "4", {{"b:a", "c:ab"}}}, {"c:ab", "24", "1", {{"c:ab"}}}}, TableHeadings -> {{}, {"Source", "df", "coef", "EMS"}}]]], "Print"] }, Open ]], Cell["\<\ The next step is to combine various SS and df from the 3-way \ analysis to form the appropriate SS and df for the nested design. For the \ nested design the sources of variation are given in the last table as A, B:A, \ and C:AB. A heuristic for forming the combinations is as follows: 1) If a term is not nested, the associated SS and df are used without change. 2) If a term is nested, then specific SS and associated df must be summed. \ All the sources of the SS for a given term and the df must be summed. In the \ example above B is nested within A, i.e., b:a, so the SS for b:a is formed by \ summing the SS(b) and all combinations of b with those terms appearing to the \ right of the colon , i.e., SS(ab). If more than one term appears to the right \ of the colon, then SS(b) is summed with each SS term taken one at a time, two \ at a time, etc. (This is illustrated in the next step.) The SS(b:a) now \ becomes 75.25+29.25 = 104.50 and df(b:a)=3+3=6. 3) The SS for the term c:ab now includes SS(c)+SS(ac)+SS(bc)+SS(abc), i.e., \ SS(c:ab)=5.25+0.75+6.00+6.5=18.50. The df(c:ab)=3+3+9+9=24. 4) The mean squares are formed in the usual manner using the new SS and df. \ For example, the mean square for the term A is 112.50/1=112.50; the mean \ square for b:a is 104.50/6=17.417; the mean square for c:ab is \ 18.50/24=0.771. Determining which mean square to use in the denominator to obtain F tests for \ A and B nested within A requires knowing the expected mean squares for each \ term being tested. From column labelled EMS of the last table of the above output, factor A is \ tested using the mean square of B:A in the denominator, and B:A is tested \ using the mean square for C:AB in the denominator. The F test for A is given \ by 112.50/17.417=6.46 with df=(1,6). The F test for B:A is \ 17.417/0.771=22.59 with df=(6,24). \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Winer Repeated Measure p 526.", "Subsubsection"], Cell["\<\ In this example there are 2 factors with repeated measurements on \ the second factor. The characteristics are as follows: 1) Factor A: 2 levels; fixed 2) Factor B: repeated measures factor with 4 levels (4 repetitions); fixed 3) There are a total of 6 subjects, 3 in A1 and 3 in A2\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(w526 = {{0, 0, 5, 3}, {3, 1, 5, 4}, {4, 3, 6, 2}, {4, 2, 7, 8}, {5, 4, 6, 6}, {7, 5, 8, 9}}\)], "Input"], Cell[BoxData[ \({{0, 0, 5, 3}, {3, 1, 5, 4}, {4, 3, 6, 2}, {4, 2, 7, 8}, {5, 4, 6, 6}, {7, 5, 8, 9}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Print[ TableForm[w526, TableSpacing \[Rule] 1, TableHeadings \[Rule] {{"\", "\", "\", \ "\", "\", "\"}, {"\", "\", "\", \ "\"}}]]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"B1\"\>", "\<\"B2\"\>", "\<\"B3\"\>", \ "\<\"B4\"\>"}, {"\<\"A1, s1\"\>", "0", "0", "5", "3"}, {"\<\"A1, s2\"\>", "3", "1", "5", "4"}, {"\<\"A1, s3\"\>", "4", "3", "6", "2"}, {"\<\"A2, s4\"\>", "4", "2", "7", "8"}, {"\<\"A2, s5\"\>", "5", "4", "6", "6"}, {"\<\"A2, s6\"\>", "7", "5", "8", "9"} }, RowSpacings->1, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableSpacing -> 1, TableHeadings -> {{"A1, s1", "A1, s2", "A1, s3", "A2, s4", "A2, s5", "A2, s6"}, {"B1", "B2", "B3", "B4"}}]&)]], "Print"] }, Open ]], Cell["\<\ As in the example taken from Kirk, a new factor C is defined \ containing 3 levels and holding the data points of each subject. Factor C is \ nested under Factor A. This results in a 3-way design having 1 observation \ per cell. The data structured for the anova analysis is shown below.\ \>", \ "Text"], Cell[BoxData[ \(\(\(rmd\)\(=\)\({{{1, 1, 1}, 0}, {{1, 1, 2}, 3}, {{1, 1, 3}, 4}, {{1, 2, 1}, 0}, {{1, 2, 2}, 1}, {{1, 2, 3}, 3}, {{1, 3, 1}, 5}, {{1, 3, 2}, 5}, {{1, 3, 3}, 6}, {{1, 4, 1}, 3}, {{1, 4, 2}, 4}, {{1, 4, 3}, 2}, {{2, 1, 1}, 4}, {{2, 1, 2}, 5}, {{2, 1, 3}, 7}, {{2, 2, 1}, 2}, {{2, 2, 2}, 4}, {{2, 2, 3}, 5}, {{2, 3, 1}, 7}, {{2, 3, 2}, 6}, {{2, 3, 3}, 8}, {{2, 4, 1}, 8}, {{2, 4, 2}, 6}, {{2, 4, 3}, 9}}\)\(\ \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(anovanw[rmd]\)], "Input"], Cell[BoxData[ \("GLM Anova V3.0"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Number of indices defining factor levels="\ \[InvisibleSpace]3\), SequenceForm[ "Number of indices defining factor levels=", 3], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Number of factors="\[InvisibleSpace]3\), SequenceForm[ "Number of factors=", 3], Editable->False]], "Print"], Cell[BoxData[ \("Data input is in 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To obtain the SS for 1) C:A we form SS(c)+SS(ac): 14.5833+2.58333=17.16663 with 2+2=4 df. 2) BC:A we form SS(BC)+SS(ABC): 9.41667+5.41667=14.8334 with 6+6=12 df Also from the output given above (column labelled EMS) the denominator \ mean-squares for F tests can be determined: 1) To test A, the mean-square for C:A is used in the denominator 2) To test B, the mean-square for BC:A is used in the denominator 3) To test AB, the mean square for BC:A is used The F tests are: F(A)=(51.0417/1) / (17.1663/4)=11.8932, df=(1,4) F(B)= (47.4583/3) / (14.8334/12) = 12.7977, df=(3,12) F(AB)=(7.45833/3) / (14.8334/12)= 2.01123, df=(3,12)\ \>", "Text"] }, Closed]] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 640}, {0, 460}}, PrintingStyleEnvironment->"Working", WindowToolbars->{}, CellGrouping->Manual, WindowSize->{491, 368}, WindowMargins->{{74, Automatic}, {Automatic, 17}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Cell[ TextData[ { ValueBox[ "Date"], "/", ValueBox[ "Time"]}], "Header"], Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, Magnification->1, MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002n88o?moogl" ] (*********************************************************************** Cached data follows. 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