Hotelling's T**2 & Profile Analysis

 

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*****************************************
**                                     **
** Hotelling's T**2 & Profile Analysis **
**                                     **
*****************************************
* Input matrix of raw scores for group X.

MATRIX.
COMPUTE X = {   7,      5,      9,      8;
		8,      8,      5,      6;
		16,     18,     11,     9;
		8,      3,      7,      9;
		6,      3,      13,     9;
		11,     8,      10,     10;
		12,     7,      9,      8;
		8,      11,     9,      3;
		14,     12,     11,     4;
		13,     13,     13,     6;
		13,     9,      9,      9;
		13,     10,     15,     7;
		14,     11,     12,     8;
		15,     11,     11,     10;
		13,     10,     15,     9;
		10,     5,      8,      6;
		10,     3,      7,      7;
		17,     13,     13,     7;
		10,     6,      10,     7;
		10,     10,     15,     8;
		14,     7,      11,     5;
		16,     11,     12,     11;
		10,     7,      14,     6;
		10,     10,     9,      6;
		10,     7,      10,     10;
		7,      6,      5,      9;
		15,     12,     10,     6;
		17,     15,     15,     8;
		16,     13,     16,     9;
		13,     10,     17,     8;
		13,     10,     17,     10;
		19,     12,     16,     10;
		19,     15,     17,     11;
		13,     10,     7,      8;
		15,     11,     12,     8;
		16,     9,      11,     11;
		14,     13,     14,     9 }.
COMPUTE NX = NROW(X).
COMPUTE KX = NCOL(X).
PRINT {NX, KX} /TITLE "No. of Cases, No. of Vars. in X".
* Input matrix of raw scores for group Y.
COMPUTE Y =   { 9,      5,      10,     8;
		10,     0,      6,      2;
		8,      9,      11,     1;
		13,     7,      14,     9;
		4,      0,      4,      0;
		4,      0,      6,      0;
		11,     9,      9,      8;
		5,      3,      3,      6;
		9,      7,      8,      6;
		7,      2,      6,      4;
		12,     10,     14,     3;
		13,     12,     11,     10 }.
COMPUTE NY = NROW(Y).
COMPUTE KY = NCOL(Y).
PRINT {NY, KY} /TITLE "No. of Cases, No. of Vars. in Y".
* Calculate sample means.                              
COMPUTE UNITX = MAKE (NX, 1, 1.0).                     
COMPUTE MEANSX = T(UNITX) * X &* (1.0 / NX).          
PRINT MEANSX /TITLE "Sample means for group X".
COMPUTE UNITY = MAKE (NY, 1, 1.0).                     
COMPUTE MEANSY = T(UNITY) * Y &* (1.0 / NY).          
PRINT MEANSY /TITLE "Sample means for group Y".
COMPUTE DIFF = MEANSX - MEANSY.
PRINT DIFF /TITLE "Differences".
* Calculate mean-corrected SSCP matrices.                                
COMPUTE BXX = SSCP(X).                                          
COMPUTE SX = BXX - ((1/NX) &* (T(X) * UNITX) * (T(UNITX) * X)).
COMPUTE BYY = SSCP(Y).                                          
COMPUTE SY = BYY - ((1/NY) &* (T(Y) * UNITY) * (T(UNITY) * Y)).
* Calculate covariance-variance matrices.
COMPUTE CX = (1/(NX-1.0)) &* SX.
COMPUTE CY = (1/(NY-1.0)) &* SY.
* Pooled covariance matrix.                                     
COMPUTE COV = (((NX-1) &* CX) + ((NY-1) &* CY)) &/ (NX+NY-2). 
COMPUTE ICOV = INV(COV).
* "Hotelling's T**2 statistic".
COMPUTE T2 = (NX * NY * DIFF * ICOV * T(DIFF)) / (NX+NY).
PRINT /TITLE " ========== Hotelling's Two-Sample T**2 Test of Means ========== ".
PRINT T2 /TITLE "Hotelling's T**2 Statistic".                           
COMPUTE F = T2 * (NX+NY-KX-1) / ((NX+NY-2)*KX).
PRINT { F, KX, NX+NY-KX-1 } /Title "Equivalent F & d.f.".
COMPUTE P = 1.0 - FCDF(F, KX, NX+NY-KX-1).
PRINT P /TITLE "P-value" /FORMAT = F10.9.                          
PRINT /TITLE " ========== Hotelling's Profile Analysis ========== ".                                                            
* Create Design Matrix C for successive contrasts.                                       
COMPUTE V1 = { 1, -1 }.                                                                  
COMPUTE V2 = MAKE (1, KX-1, 0).
COMPUTE V3 = {V1, V2}.                                                                 
COMPUTE V4 = V3.                                                                         
LOOP I = 1 to KX-1.
COMPUTE V4 = {V4 ; V3}.
END LOOP.
COMPUTE C = RESHAPE (V4, KX+1, KX).
COMPUTE C = C(1:(KX-1), 1:KX).
PRINT C /TITLE "Contrast matrix".
* Differences between adjacent scales for group X.
COMPUTE DIFF1 = MEANSX * T(C).
PRINT DIFF1 /TITLE "Differences between adjacent scales for group X".
* Differences Between adjacent scales for group Y.
COMPUTE DIFF2 = MEANSY * T(C).
PRINT DIFF2 /TITLE "Differences between adjacent scales for group Y".
* Pooled covariance matrix of differences.
COMPUTE POOL = C * COV * T(C).
PRINT POOL /TITLE "Pooled covariance matrix of differences".
COMPUTE IPOOL = INV(POOL).                                                               
PRINT IPOOL /TITLE "Inverted pooled cov. matrix of diff.".
PRINT /TITLE " ----- Hypothesis 1: Parallelism (no group X scale interactions) ----- ".  
COMPUTE PARALLEL = (NX * NY * DIFF * T(C) * IPOOL * C * T(DIFF)) / (NX+NY).
PRINT PARALLEL /TITLE "Hotelling's T**2 statistic for Parallelism".                      
COMPUTE F = PARALLEL * (NX+NY-KX) / ((NX+NY-2)*(KX-1)).
PRINT { F, KX-1, NX+NY-KX } /TITLE "Equivalent F & d.f.".
COMPUTE P = 1.0 - FCDF(F, KX-1, NX+NY-KX).
PRINT P /TITLE "P-value" /FORMAT F10.9.                                                 
PRINT /TITLE " ----- Hypothesis 2: Equal Levels (assuming parallelism) ----- ".          
* Create design vector J for equal levels.                                               
COMPUTE J = MAKE (1, KX, 1.0).                                                           
* Grand total for group x.
COMPUTE GRANDX = MEANSX * T(J).
PRINT GRANDX /TITLE "Grand mean for group X".
* Grand total for group y.
COMPUTE GRANDY = MEANSY * T(J).
PRINT GRANDY /TITLE "Grand mean for group Y".
* Differences in levels.
COMPUTE DIF = J * T(DIFF).
PRINT DIF /TITLE "Difference in levels".                                                               
* Variance of differences.                                               
COMPUTE CV = J * COV * T(J).
PRINT CV /TITLE "Variance of differences".                                                             
COMPUTE EQUAL = J * T(DIFF) / (SQRT ( CV * (1/NX + 1/NY) ) ).
PRINT { EQUAL, NX+NY-2 } /TITLE "Hotelling's t-test for Equal levels (one-tailed)".
COMPUTE P = (1.0 - TCDF(ABS(EQUAL), NX+NY-2 )) * 0.5. 
PRINT P /TITLE "P-value" /FORMAT F10.9.
PRINT /TITLE " ----- Hypothesis 3: Flatness (again assuming parallelism) ----- ".        
* Total Grand Means.
COMPUTE GRAND = ((MEANSX * NX) + (MEANSY * NY)) / (NX + NY).
PRINT GRAND /TITLE "Total Group Means".
COMPUTE FLAT = (NX+NY) * GRAND * T(C) * IPOOL * C * T(GRAND).
PRINT FLAT /TITLE "Hotelling's T**2 statistic for Flat Profiles".
COMPUTE F = FLAT * (NX+NY-KX) / ((NX+NY-2) * (KX-1)).
PRINT { F, KX-1, NX+NY-KX } /TITLE "Equivalent F & d.f.".
COMPUTE P = 1.0 - FCDF( F, KX-1, NX+NY-KX ).
PRINT P /TITLE "P-value" /FORMAT F10.9.
END MATRIX.
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