(*^
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:[font = subsection; inactive; preserveAspect]
Johnson-Neyman 3 Covariate Case Ancova; acvjn3c[]
Steve Hunka, University of Alberta. 403-492-7276;steve.hunka@ualberta.ca
Last Revision: July 19/96
:[font = subsubsection; inactive; preserveAspect]
This program carries out the Johnson-Neyman one-way Analysis of Covariance for the case when the regression parameters are not homogeneous among the groups. The number of covariates is restricted to 3; the number of groups must be two or more. Using contour plots, regions of statistical significance are identified as a function of covariate values to which the adjustment in group effect is made and the alpha level employed. The user may also determine the significance of group differences for a selected point set {p,q,s} of the covariates. The general linear model using a mu-model (cell-means model) design matrix is employed in the OLS estimate of the parameters ordered as (m1,a11,a12,a13,m2,a21,...) in which the terms in m are adjusted group intercepts and the terms in a the regression coefficients.
A polynomial relating the sum of squares for a specific contrast matrix is formed by
(K'B)'Inverse[K'Inverse[X'X]K]K'B-(dfk MSe F)=0
where K is the contrast matrix; X the design matrix; B the parameter estimates, dfk the degrees of freedom for the contrast matrix, MSe the mean square error term and F the value of the F distribution for a given level of alpha. The regions of significance are obtained by contour plotting the polynomial. Any covariate point set {p,q,s} which falls on the surface of the representation, or within its enclosing volume, will be significant at the alpha level selected or less.
Input to the program is a data matrix of order (N,4) where N represents the number of observations and 4 the number of input variables. The observations are ordered by group. Each row is an observation and contains in order, the dependent variable followed by the 3 covariates.
For the 1 and 2 covariate case see MathSource document 0207-953.
References
1. Johnson, P.O. & Hoyt, C. On determining three dimensional regions of significance. Journal of Experimental Education, Vol. 15, No. 4, 1947, 342-353.
2. Hunka, S. Using Mathematica to Solve Johnson-Neyman Problems, Mathematica in Education, Vol. 3, No. 3, 1994.
3. Hunka S., Identifying Regions of Signficance in ANCOVA Problems Having Nonhomogeneous Regressions, Br. Jr. Math. & Stat. Psych., 1995, 48, 161-188.
4. Hunka, S. & Leighton, J. Defining Johnson-Neyman Regions of Significance in the Three Covariate ANCOVA Using Mathematica, Jr. of Educational and Behavioural Statistics, In press 1996.
;[s]
9:0,0;1782,1;1792,0;1999,1;2010,0;2045,1;2056,0;2355,1;2366,0;2433,-1;
2:5,13,9,Times,1,12,0,0,0;4,13,9,Times,3,12,0,0,0;
:[font = subsubsection; inactive; preserveAspect]
Executing the program
The program is executed by acvjn3c[d_,ng_], where d_ is the data matrix of order observations by variables. Observations are ordered by groups, and the first column is taken as the dependent variable. ng_ is a vector of the number of observations in each group. The polynomial expression representing the relationship between the covariates and SSmcfm is returned in case the researcher wishes to further explore regions (volumes) of significance.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Mathematica Packages required:
Statistics`ContinuousDistributions`
Graphics`Graphics3D`
Graphics`ContourPlot3D`
;[s]
2:0,1;11,0;112,-1;
2:1,13,9,Times,1,12,0,0,0;1,13,9,Times,3,12,0,0,0;
:[font = input; preserveAspect; endGroup]
<{3,0},
TableDirections->{Row},TableHeadings->Automatic]];
Return[cont];
];
:[font = subsubsection; inactive; preserveAspect; startGroup]
pmat[m,pc] print a matrix m in units of pc columns
:[font = input; preserveAspect; startGroup]
pmat[m_,pc_]:=
Module[{m1,a,c,indx},
m1=Transpose[m];
c=Part[Dimensions[m1],1];
a=Floor[c/pc];
indx=Join[Table[pc,{a}],{Mod[c,pc]}];
If[0==Part[indx,-1],
indx=Drop[indx,-1]];
Do[Print[""];
Print["Table Part=",j,"/",Length[indx]];
Print[TableForm[Transpose[Take[m1,indx[[j]]]],
TableHeadings->Automatic,
TableSpacing->{0,4}]];
m1=Drop[m1,indx[[j]]];
,{j,1,Length[indx]}];
];
:[font = message; inactive; preserveAspect; endGroup; endGroup]
General::spell1:
Possible spelling error: new symbol name "indx"
is similar to existing symbol "inx".
:[font = subsection; inactive; preserveAspect; fontSize = 12; startGroup]
ssk[k_,xx_,pb_]
Function ssk calculates sum of squares for contrast matrix k, X'X, and parameters bp.
:[font = input; preserveAspect; endGroup]
ssk[k_,xx_,pb_]:=
(k.pb).(Inverse[k.Inverse[xx].Transpose[k]]).
(k.pb);
:[font = subsection; inactive; preserveAspect; fontSize = 12; startGroup]
knTrst[k_,ng_]
creates a contrast matrix for SSmcfm; k_ holds pattern e.g., for 3 groups {1,p,q,s} generates {{1,p,q,s,-1,-p,-q,-s,0,0,0,0}, {0,0,0,0,1,p,q,s,-1,-p,-q,-s}}. ng is the number of groups.
:[font = input; preserveAspect]
knTrst[k_,ng_]:=
Module[{t},
t=Table[
Join[k,-k,Table[0,{Length[k] (ng-2)}]],
{ng-1}];(*end Table*)
Do[
t[[j]]=RotateRight[t[[j]],{(j-1) (Length[k])}],
{j,1,ng-1}];
Return[t];
];
:[font = subsubsection; inactive; preserveAspect; startGroup]
cradd[a_,b_]
Carries out circle + operation using matrix a and b. Used to form a linear model design matrix where a and b appear as block diagonals in the result.
:[font = input; preserveAspect; endGroup; endGroup]
cradd[a_,b_]:=
Join[
Transpose[
Join[
Transpose[a],
Transpose[Table[i j 0,
{i,Dimensions[a][[1]]},
{Dimensions[b][[2]]}]]]],
Transpose[
Join[
Table[i j 0,
{i,Dimensions[a][[2]]},
{Dimensions[b][[1]]}],
Transpose[b] ]]]
:[font = subsubsection; inactive; preserveAspect; startGroup]
probF[ss,dfs,mse,dfe] 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.
Requires: <{ycntr[[j]]},MaxRecursion->1,
Axes->True,
DisplayFunction->Identity]],
{j,1,Length[eq]}]; (*end Do *)
title=StringInsert["Regression Planes at ",
ToString[ycntr],-1];
Show[pobj,DisplayFunction->$DisplayFunction,
Axes->True,AxesLabel->{"p","q","s"},
PlotLabel->title];
ans=kbinstrg["Replot at other contour values? y/n",
{{"y"},{"n"}}];
]; (*end While *)
]; (*end Module *)
:[font = subsection; inactive; preserveAspect; fontSize = 12; startGroup]
in43dcp[exp_,ms_,d1_,d2_]; 3D Contour and Shadow Plotting function.
The function carries out a 3D contour and shadow plot of a polynomial of the form z=f(p,q,s) for a specific value of z. It is used in the Johnson-Neyman Ancova analysis where p,q, and s are the three covariates. The argument exp is the polynomial for sum of squares expressed in terms of p,q, and s; ms is the mean square error; d1 is df1; d2 is dfe. The user is asked to input the plot range for each covariate in the form {start,end}, and the alpha level for significance. The MaxRecursion parameter is set to 1 and generally will result in a plot of sufficient resolution within a reasonable time and memory requirements. For a considerable increase in resolution, but with a considerable increase in time and memory requirements, the user can change the option MaxRecursion to 2. Calls kbinstrg, kbin3.
Requires: Statistics`ContinuousDistributions`
Graphics`ContourPlot3D`
Graphics`Graphics3D`
Mod. Date: Apr 30/96
:[font = input; preserveAspect; startGroup]
in43dcp[exp_,ms_,d1_,d2_]:=
Module[{rpq,par,sv,quest,q1,siglev,alf,gobj,
labelalf},
rpq="y";
q1="For covariate type plotting range values eg.{1,10}";
sv={"p ","q ","s "};
While[rpq=="y", par=List[];
Do[
quest=StringInsert[q1,sv[[j]],15];
AppendTo[par,
kbin3[quest,{2}]],{j,1,3}];
par=Flatten[par];
Print["Current plot ranges=",par];
alf=kbin3["Type alpha level.",0.0];
siglev=ms d1 Quantile[FRatioDistribution[d1,d2],1-alf];
Print["Mse=",ms," df1=",d1," df2=",d2," alpha=",alf];
Print["SS Point of Significance=",siglev];
labelalf=StringJoin["Region of Signficance for alpha=",
ToString[alf]];
gobj=ContourPlot3D[exp,
{p,par[[1]],par[[2]]},{q,par[[3]],par[[4]]},
{s,par[[5]],par[[6]]},MaxRecursion->1,
Contours->{siglev},DisplayFunction->Identity,
Axes->True];
Shadow[gobj,Axes->True,AxesLabel->{"p","q","s"},
DisplayFunction->$DisplayFunction,
PlotLabel->labelalf,
AxesEdge->{{-1,-1},{1,-1},{-1,-1}},
FaceGrids->{{-1,0,0},{0,1,0},{0,0,-1}}];
rpq=kbinstrg["Do you want to replot? y/n",
{{"y"},{"n"}}];
]; (*end While *)
]; (*end Module*)
:[font = message; inactive; preserveAspect; endGroup; endGroup]
General::spell1:
Possible spelling error: new symbol name "gobj"
is similar to existing symbol "pobj".
:[font = subsubsection; inactive; preserveAspect; startGroup]
chkPts[x_,b_,mse_,dfe_,ng_] calculates the probability in testing SS of a contrast associated with a particular set of covariate values input by user. x is X'X where X is the design matrix; b is a vector of parameter estimates; mse is the mean square error; dfe is the degree of freedom for mse; ng is the number of groups. First the symbolic contrast is constructed, then user values of the covariate given as {p,q,s} are substituted into the symbolic contrast. ssk is called to get the sum of squares, and probF to carry out the F test.
Calls kbin3, kbinstrg, getCon, probF, ssk.
Mod. Date: April 27/96
:[font = input; preserveAspect; endGroup]
chkPts[x_,b_,mse_,dfe_,ng_]:=
Module[{rpq,con,ss,dfc,pqs,rpcon,consub},
rpcon="y";
Print["\nTest of adjustment to user (p,q,s)"];
While[rpcon=="y",
rpq="y";
con=getCon[ng]; (*generates symbolic contrast*)
While[rpq=="y",
pqs=kbin3["Type specific covariate values as vector {p,q,s}",
{3}];
Print["User input {p,q,s}:",pqs];
(*now substitute specific values of p,q,s *)
consub=con /. {p->pqs[[1]],q->pqs[[2]],s->pqs[[3]]};
Print["Numerical Contrast matrix:"];
Print[TableForm[consub,TableSpacing->{3,0},
TableDirections->{Row},TableHeadings->Automatic]];
dfc=Length[consub];
ss=ssk[consub,x,b]; (*get SS *)
probF[ss,dfc,mse,dfe];
rpq=kbinstrg["For current contrast do you have more covariate values? y/n",
{{"y"},{"n"}}];
];(*end While rpq==y*)
rpcon=kbinstrg["Do you have another contrast? y/n",
{{"y"},{"n"}}];
]; (*end While rpcon==y*)
];(*end Module*)
:[font = subsubsection; inactive; preserveAspect; startGroup]
acvjn3c is the main Mathematica program to do the Johnson-Neyman 1-way ancova analysis. d_ is a data matrix of order (N,4) where N is the number of observations ordered by group membership; ng is a vector of the group sample sizes.
Calls: cradd, getMp, kbinstrg, kbin3, pmat, knTrst, ssk, regcntr3d, in43dcp, getCon, chkPts, probF.
Requires: Statistics`ContinuousDistributions`, Graphics`ContourPlot3D`,
Graphics`Graphics3D`
Mod. Date: May 1/96
;[s]
3:0,0;20,1;31,0;461,-1;
2:2,13,9,Times,1,12,0,0,0;1,14,10,Times,3,12,0,0,0;
:[font = input; preserveAspect; startGroup]
acvjn3c[d_,ng_]:= (*Johnson-Neyman Ancova*)
Module[{ans,nt,ngrp,x,y,ncv,xtx,dm,b,dfe,mse,con,ssmcfm,
concv,ss1,f1,fdist,prob,ss2,ss3,mnx,dt,d1,dfm,
bm,meanxy,ymns},
If[4!=Part[Dimensions[d],2],
Print["Error:Input data must have 4 cols. Dimensions=",
Dimensions[d]];
Return[]];
nt=Length[d];(*N total sample size*)
If[Apply[Plus,ng]!=nt,
Print["Error:Sum group sizes not equal to ",nt];
Return[]];
y=Flatten[Take[Transpose[d],1]];
x=Drop[Transpose[d],1];
ngrp=Length[ng];
x=Transpose[Join[{Table[1,{nt}]},x]];
(*now set up the design matrix*)
dm=cradd[getMp[1,x,ng],getMp[2,x,ng]];
Do[dm=cradd[dm,getMp[j,x,ng]],{j,3,ngrp}];
xtx=Transpose[dm].dm;
ans=kbinstrg["Display X'X? y/n",{{"y"},{"n"}}];
If[ans=="y",pmat[xtx,6]];
b=N[Inverse[xtx].Transpose[dm].y];
Print["Mu-Model Parameter Estimates"];
Print[TableForm[b,TableSpacing->{0,4},
TableHeadings->Automatic]];
ssmcfm=(b.xtx.b)-(Apply[Plus,y]^2)/nt;
dfm=Length[xtx]-1;
Print["\nSSmodelcfm=",ssmcfm," dfm=",dfm];
dfe=Dimensions[dm].{1,-1};
mse=((y.y)-b.Transpose[dm].y)/dfe;
probF[ssmcfm,dfm,mse,dfe];
(*now test each covariate regression coeff*)
Print["\nTest of Regression Homogeneity"];
con={1,0,0,0};
Do[Print["\nCovariate ",j];
con=RotateRight[con,1];
concv=knTrst[con,ngrp];
Print[TableForm[concv,TableSpacing->{3,0},
TableDirections->{Row},TableHeadings->Automatic]];
ss1=ssk[concv,xtx,b];
d1=Length[concv];
probF[ss1,d1,mse,dfe];
,{j,1,3}];(*end do*)
bm=Partition[b,4];
mnx=Table[Join[{Min[Transpose[d][[j]]]},
{Max[Transpose[d][[j]]]}],{j,1,4}];
Print["\nVariable by Min/Max of data:"];
Print[TableForm[mnx,TableSpacing->{0,4},
TableHeadings->Automatic]];
meanxy=List[];
Do[ (*do over number of groups*)
AppendTo[meanxy,
N[Apply[Plus,getMp[j,d,ng]]/ng[[j]]]];
,{j,1,Length[ng]}]; (*end do *)
Print["\nVariable means Y, P, Q, S:"];
Print[TableForm[meanxy,TableSpacing->{0,4},
TableHeadings->Automatic]];
ymns=Flatten[Take[Transpose[meanxy],1]];
ans=kbinstrg["Plot Regression planes? y/n",{{"y"},{"n"}}];
Print["\nRegression plots: ",ans];
If[ans=="y",
regcntr3d[bm,mnx]];
(*Now do the symbolic processing*)
Print["\nProcessing Plot for SSmcfm"];
concv=knTrst[{1,p,q,s},ngrp]; (*for SSmcfm*)
Print["SSmcfm Symbolic Contrast Matrix"];
Print[TableForm[concv,TableSpacing->{3,0},
TableDirections->{Row},TableHeadings->Automatic]];
ss2=Simplify[ssk[concv,xtx,b]];
(*Now do the 3D contour plot of region of significance*)
in43dcp[ss2,mse,d1,dfe];
(*using concv do row contrasts separately*)
Print["\nProcessing SSmcfm row contrasts"];
If[ngrp>=3,
ans=kbinstrg["Display plot of row contrasts of SSmcfm? y/n",
{{"y"},{"n"}}];
Print["Display plots of row contrasts of SSmcfm: ",ans];
If[ans=="y",
Do[Print["Row Contrast Matrix is:"];
Print[TableForm[concv[[j]],TableSpacing->{0,3},
TableDirections->{Column},TableHeadings->Automatic]];
ss3=Simplify[ssk[{concv[[j]]},xtx,b]];
in43dcp[ss3,mse,1,dfe];
,{j,1,ngrp-1}]; (*end Do*)
]]; (*end if ans==y;end if ngrp>=3 *)
(*User contrast wt matrix input next*)
Print["\nProcessing user weights for contrasts"];
ans="y";
While[ans=="y",
ans=kbinstrg["User input of wts for symbolic contrast? y/n",
{{"y"},{"n"}}];
Print["User wts for symbolic contrast: ",ans];
If[ans=="n",Break[]];
concv=getCon[ngrp];
ss3=Simplify[ssk[concv,xtx,b]];
in43dcp[ss3,mse,Length[concv],dfe];
]; (*end while ans==y*)
(*now allow checking sign.of a specific {p,q,s}*)
If["y"==kbinstrg["Test sign.of specific covariate values? y/n",
{{"y"},{"n"}}],
chkPts[xtx,b,mse,dfe,ngrp]];
Return[ss2];
]; (*End Module*)
:[font = message; inactive; preserveAspect; endGroup; endGroup]
General::spell1:
Possible spelling error: new symbol name "prob"
is similar to existing symbol "probF".
:[font = subsubsection; inactive; preserveAspect]
Test data as used in reference 4; the data is taken from: Huitema, B.E. (1980), The Analysis of Covariance and Alternatives, New York; Wiley.
:[font = input; preserveAspect; startGroup]
huitdta={{10,1,9,3},{10,2,7,5},{11,2,9,4},{10,3,9,3},
{11,4,9,3},{11,5,8,4},{10,5,7,4},{11,6,9,4},
{11.5,6,6,3},{12,7,8,5},{12,8,7,4},{11,8,7,5},
{11,9,6,6},{12.5,10,6,6},{12,11,6,6},
{5,1,8,2},{6,1.5,9,4},{6,2.5,9,4},{7,3.5,8,5},
{8,4.5,8,4},{9,4.5,8,4},{9,5,7,3},{9,6,8,3},
{10.5,6,6,4},{11,7,7,3},{12.5,7,5,5},
{12.5,7.5,8,5},{14,8,6,6},{14.5,9,7,5},
{16,10,6,6}}
:[font = output; output; inactive; preserveAspect; endGroup]
{{10, 1, 9, 3}, {10, 2, 7, 5}, {11, 2, 9, 4}, {10, 3, 9, 3},
{11, 4, 9, 3}, {11, 5, 8, 4}, {10, 5, 7, 4}, {11, 6, 9, 4},
{11.5, 6, 6, 3}, {12, 7, 8, 5}, {12, 8, 7, 4}, {11, 8, 7, 5},
{11, 9, 6, 6}, {12.5, 10, 6, 6}, {12, 11, 6, 6},
{5, 1, 8, 2}, {6, 1.5, 9, 4}, {6, 2.5, 9, 4}, {7, 3.5, 8, 5},
{8, 4.5, 8, 4}, {9, 4.5, 8, 4}, {9, 5, 7, 3}, {9, 6, 8, 3},
{10.5, 6, 6, 4}, {11, 7, 7, 3}, {12.5, 7, 5, 5},
{12.5, 7.5, 8, 5}, {14, 8, 6, 6}, {14.5, 9, 7, 5},
{16, 10, 6, 6}}
;[o]
{{10, 1, 9, 3}, {10, 2, 7, 5}, {11, 2, 9, 4}, {10, 3, 9, 3},
{11, 4, 9, 3}, {11, 5, 8, 4}, {10, 5, 7, 4}, {11, 6, 9, 4},
{11.5, 6, 6, 3}, {12, 7, 8, 5}, {12, 8, 7, 4}, {11, 8, 7, 5},
{11, 9, 6, 6}, {12.5, 10, 6, 6}, {12, 11, 6, 6},
{5, 1, 8, 2}, {6, 1.5, 9, 4}, {6, 2.5, 9, 4}, {7, 3.5, 8, 5},
{8, 4.5, 8, 4}, {9, 4.5, 8, 4}, {9, 5, 7, 3}, {9, 6, 8, 3},
{10.5, 6, 6, 4}, {11, 7, 7, 3}, {12.5, 7, 5, 5},
{12.5, 7.5, 8, 5}, {14, 8, 6, 6}, {14.5, 9, 7, 5},
{16, 10, 6, 6}}
:[font = subsubsection; inactive; preserveAspect]
User input to program queries for the example given below:
1. Display X'X?; y
2. Plot Regression planes?; y
a) Type vector of 3 contour values; {10.75,9.05,11.8}
b) Replot at other contour values; n
3. For plotting of the region of signficance for SSmcfm;
a) For covariate p type plotting range values; {0,10}
b) For covariate q type plotting range values; {0,10}
c) For covariate s type plotting range values; {0,10}
d) Type alpha level; .01
e) Do you want to replot?; y
(replotted with same ranges but alpha at .001)
4. Display plot of row contrasts of SSmcfm? y
For each row of the contrast, the following queries:
a) For covariate p type plotting range values; {0,20}
b) For covariate q type plotting range values; {0,10}
c) For covariate s type plotting range values; {0,10}
d) Type alpha level; .01
e) Do you want to replot?; n
(same input values for next row of contrast)
5. User input of weights for symbolic contrast?; y
a) How many rows in the contrast?; 1
b) Type contrast matrix weights only;
{{1,1,1,1,-.5,-.5,-.5,-.5,-.5,-.5,-.5,-.5}}
c) For covariate p type plotting range values; {0,15}
d) For covariate q type plotting range values; {0,10}
e) For covariate s type plotting range values; {0,10}
f) Type alpha level; .01
g) Do you want to replot?; n
h) User input of weights for symbolic contrast?; n
6. Test sign. of specific covariate values?; y
a) How many rows in the contrast?; 2
b) Type contrast matrix weights only;
{{1,1,1,1,-1,-1,-1,-1,0,0,0,0},{0,0,0,0,1,1,1,1,-1,-1,-1,-1}}
c) Type specific covariate values as a vector {p,q,s}; {3,4,5}
d) For current contrast do you have more covariate values?; n
e) Do you have another contrast?; n
:[font = subsubsection; inactive; preserveAspect]
Processing Time: Approximately 10 minutes including user input on a Power Macintosh 6100/60.
:[font = input; preserveAspect; startGroup]
acvjn3c[huitdta,{10,10,10}]
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Table Part=1/2
1 2 3 4 5 6
1 10 41 81 38 0 0.
2 41 205 325 159 0 0.
3 81 325 667 306 0 0.
4 38 159 306 150 0 0.
5 0 0 0 0 10 59.
6 0. 0. 0. 0. 59. 472.
7 0 0 0 0 74 400.
8 0 0 0 0 46 305.5
9 0 0 0 0 0 0.
10 0. 0. 0. 0. 0. 0.
11 0 0 0 0 0 0.
12 0 0 0 0 0 0.
Table Part=2/2
1 2 3 4 5 6
1 0 0 0 0. 0 0
2 0 0 0 0. 0 0
3 0 0 0 0. 0 0
4 0 0 0 0. 0 0
5 74 46 0 0. 0 0
6 400. 305.5 0. 0. 0. 0.
7 560 331 0 0. 0 0
8 331 226 0 0. 0 0
9 0 0 10 70. 68 44
10 0. 0. 70. 516.5 470. 321.5
11 0 0 68 470. 472 294
12 0 0 44 321.5 294 206
Mu-Model Parameter Estimates
1 8.47578
2 0.267507
3 0.115835
4 0.0629428
5 1.12989
6 1.02299
7 0.493876
8 -0.38483
9 2.82128
10 1.00184
11 -0.173829
12 0.715419
SSmodelcfm=171.498 dfm=11
SS=171.498 df=11 MS=15.5907
MSe=0.303821 dfe=18
-11
F=51.3155 Prob=2.75259 10
Test of Regression Homogeneity
Covariate 1
1 2
1 0 0
2 1 0
3 0 0
4 0 0
5 0 0
6 -1 1
7 0 0
8 0 0
9 0 0
10 0 -1
11 0 0
12 0 0
SS=7.19694 df=2 MS=3.59847
MSe=0.303821 dfe=18
F=11.8441 Prob=0.000521607
Covariate 2
1 2
1 0 0
2 0 0
3 1 0
4 0 0
5 0 0
6 0 0
7 -1 1
8 0 0
9 0 0
10 0 0
11 0 -1
12 0 0
SS=0.698468 df=2 MS=0.349234
MSe=0.303821 dfe=18
F=1.14947 Prob=0.338993
Covariate 3
1 2
1 0 0
2 0 0
3 0 0
4 1 0
5 0 0
6 0 0
7 0 0
8 -1 1
9 0 0
10 0 0
11 0 0
12 0 -1
SS=2.97748 df=2 MS=1.48874
MSe=0.303821 dfe=18
F=4.90005 Prob=0.0200003
Variable by Min/Max of data:
1 2
1 5 16
2 1 11
3 5 9
4 2 6
Variable means Y, P, Q, S:
1 2 3 4
1 10.75 4.1 8.1 3.8
2 9.05 5.9 7.4 4.6
3 11.8 7. 6.8 4.4
Regression plots: y
User selected contour values:{10.75, 9.05, 11.8}
Processing Plot for SSmcfm
SSmcfm Symbolic Contrast Matrix
1 2
1 1 0
2 p 0
3 q 0
4 s 0
5 -1 1
6 -p p
7 -q q
8 -s s
9 0 -1
10 0 -p
11 0 -q
12 0 -s
Current plot ranges={0, 10, 0, 10, 0, 10}
Mse=0.303821 df1=2 df2=18 alpha=0.01
SS Point of Significance=3.65369
Current plot ranges={0, 10, 0, 10, 0, 10}
Mse=0.303821 df1=2 df2=18 alpha=0.001
SS Point of Significance=6.31335
Processing SSmcfm row contrasts
Display plots of row contrasts of SSmcfm: y
Row Contrast Matrix is:
1 1
2 p
3 q
4 s
5 -1
6 -p
7 -q
8 -s
9 0
10 0
11 0
12 0
Current plot ranges={0, 20, 0, 10, 0, 10}
Mse=0.303821 df1=1 df2=18 alpha=0.01
SS Point of Significance=2.51729
Row Contrast Matrix is:
1 0
2 0
3 0
4 0
5 1
6 p
7 q
8 s
9 -1
10 -p
11 -q
12 -s
Current plot ranges={0, 20, 0, 10, 0, 10}
Mse=0.303821 df1=1 df2=18 alpha=0.01
SS Point of Significance=2.51729
Processing user weights for contrasts
User wts for symbolic contrast: y
Contrast Matrix Weights Input
{{1, 1, 1, 1, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5}}
Contrast Matrix is:
1
1 1
2 p
3 q
4 s
5 -0.5
6 -0.5 p
7 -0.5 q
8 -0.5 s
9 -0.5
10 -0.5 p
11 -0.5 q
12 -0.5 s
Current plot ranges={0, 15, 0, 10, 0, 10}
Mse=0.303821 df1=1 df2=18 alpha=0.01
SS Point of Significance=2.51729
User wts for symbolic contrast: n
Test of adjustment to user (p,q,s)
Contrast Matrix Weights Input
{{1, 1, 1, 1, -1, -1, -1, -1, 0, 0, 0, 0},
{0, 0, 0, 0, 1, 1, 1, 1, -1, -1, -1, -1}}
Contrast Matrix is:
1 2
1 1 0
2 p 0
3 q 0
4 s 0
5 -1 1
6 -p p
7 -q q
8 -s s
9 0 -1
10 0 -p
11 0 -q
12 0 -s
User input {p,q,s}:{3, 4, 5}
Numerical Contrast matrix:
1 2
1 1 0
2 3 0
3 4 0
4 5 0
5 -1 1
6 -3 3
7 -4 4
8 -5 5
9 0 -1
10 0 -3
11 0 -4
12 0 -5
SS=2.21001 df=2 MS=1.105
MSe=0.303821 dfe=18
F=3.63702 Prob=0.047139
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