How do I perform a 2-sample Hotelling's T-square test in JMP?

[#1]

I'm trying to compare multiple different product specs or variables (k = 8) between 2 different groups. I can't find a way to compare those multiple factors between the 2 groups easily in JMP right now using the 2-sample Hotelling's T-square test. I'd also like to know if JMP can then create the CIs if we reject the Hotelling's T2 test statistic and also review the profile plots? 

[#2]

I found some SAS code on the internet that performed the 2 Sample Hotelling T2 calculation.  I converted the code to JSL.  I tested it against some sample data and it checked out ok.  It is a bit clunking but it works.  Take a look at the annotation in the JSL to see what needs to be set to work with your data.

Names Default To Here( 1 );

// Open the data table
dt = open("$SAMPLE_DATA/Prostate Cancer.jmp");

groupVar = "Status"; // Enter the grouping column name
groupList = {"CCD", "Nor"}; // Set the values for the 2 groups

startColumn = 660;  // Set the starting column number for the data to be included

// Create the group 1 and group 2 matrices for analysis
gp1Rows = dt << get rows where( as column( groupVar) == groupList[1] );
group1 = dt[gp1Rows, startColumn :: N Cols( dt )];

gp1Rows = dt << get rows where( as column( groupVar ) == groupList[2] );
group2 = dt[gp1Rows, startColumn :: N Cols( dt )];

// Get the dimensions of the matricies
nRows1 = N Row( group1 );
nRows2 = N Row( group2 );
numCols = N Col( group1 );

// Create a filled vector for the analysis
j1 = J( nRows1, 1, 1 );
j2 = J( nRows2, 1, 1 );

// Create an identity matrix for each group
identity1 = Identity( nRows1 );
identity2 = Identity( nRows2 );

// Calculate the means for each of the columns by each group
MeanY1 = group1` * j1 / nRows1;
MeanY2 = group2` * j2 / nRows2;

// Create an output window to display the results in
nw = New Window("Hotelling T2", outlinebox("Hotelling 2 Sample T2 Analysis",spacerbox(size(1,15)),lub = vlistbox()));

// Add the means table to the output
obMean = Outline Box("Group Means",
	mTB = Table Box(
		string Col box("Groups", groupList),
		number col box("# Rows", matrix(nRows1)||matrix(nRows2))
	)
);
for(i=startColumn, i<=NCols(dt),i++,
	mTB << append(number col box(column(dt,i)<<get name, MeanY1[i-startColumn+1] || MeanY2[i-startColumn+1]))
		);
lub<<append(obMean);
lub<<append(spacerbox(size(1,15)));

// Create the Sample Variance-Covariance matrix for group 1
sampVarCovarGp1 = group1` * (identity1 - j1 * j1` / nRows1) * group1 / (nRows1 - 1.0);

// Add the matrix to the output window
obSVCG1 = Outline Box("Sample Variance-Covariance for Group = " || groupList[1],
	SVCG1 = TableBox()
);
for(i=1,i<=ncols(sampVarCovarGp1),i++,
	SVCG1 << append(numbercolbox("", sampVarCovarGp1[0,i]))
);
lub<<append(obsvcg1);
lub<<append(spacerbox(size(1,15)));
obSVCG1 << close;

// Create the Sample Variance-Covariance matrix for group 2
sampVarCovarGp2 = group2` * (identity2 - j2 * j2` / nRows2) * group2 / (nRows2 - 1.0);

// Add the matrix to the output window
obSVCG2 = Outline Box("Sample Variance-Covariance for Group = " || groupList[2],
	SVCG2 = TableBox()
);
for(i=1,i<=ncols(sampVarCovarGp2),i++,
	SVCG2 << append(numbercolbox("", sampVarCovarGp2[0,i]))
);
lub<<append(obsvcg2);
lub<<append(spacerbox(size(1,15)));
obSVCG2 << close;

// Create the Pooled Sample Variance-Covariance matrix
sampPool = ((nRows1 - 1.0) * sampVarCovarGp1 + (nRows2 - 1.0) * sampVarCovarGp2) / (nRows1 + nRows2 - 2.0);

// Add the matrix to the output window
obSampPool = Outline Box("Pooled Variance-Covariance Matrix",
	sampPooltb = TableBox()
);
for(i=1,i<=ncols(sampPool),i++,
	sampPooltb << append(numbercolbox("", sampPool[0,i]))
);
lub<<append(obSampPool);
lub<<append(spacerbox(size(1,15)));
obSampPool << close;

// Calculate the result statistics
t2 = (MeanY1 - MeanY2)` * Inv( sampPool * (1 / nRows1 + 1 / nRows2) ) * (MeanY1 - MeanY2);
f = (nRows1 + nRows2 - numCols - 1) * t2 / numCols / (nRows1 + nRows2 - 2);
degFreedomGp1 = numCols;
degFreedomGp2 = nRows1 + nRows2 - numCols - 1;
pValue = 1 - F Distribution( f, degFreedomGp1, degFreedomGp2 );

// Add the results to the display window
obFinal = Outline Box("Results",
	tablebox(
		stringcolbox("Group", groupList),
		numbercolbox("DF", matrix(degFreedomGp1)||matrix(degFreedomGp2)),
		numbercolbox("T2", matrix(t2)),
		numbercolbox("F", matrix(f)),
		pv=numbercolbox("pValue",matrix(pValue))
	)
);
pv<<set format("pvalue");
lub<<append(obFinal);

[#3]

Thank you for the response. I didn't think to try finding a solution using SAS.  Now that I think about it I may have SAS code for doing a multivariate comparison between 2 groups that may also work. That said, I was hoping someone knew if: analyze > fit model menu > Manova , could work multivariate comparison of just 2 groups instead of multiple ones? 

[#4]

Dear Jim

Could I persuade you to post the SAS code as well? That would be most helpful.

Thanks

[#5]

Here is a link to a SAS example.  I believe this is the example I used, but I am not sure.

7.1.15 - The Two-Sample Hotelling's T-Square Test Statistic | STAT 505 (psu.edu)

[#6]

From JMP Help:

[#7]

Thanks Mark. I saw that 1-sample T2 test option under the multivariate menu, but couldn't find anything for a 2-sample test - which is what I'm looking for to compare product specs between 2 groups. I think I may playing around w/ the MANOVA option but use 2 groups instead of the 3+ groups it usually calls for & see if I can get the correct outputs. 

[#8]

For those that are interested - I THINK (someone may need to verify) you can use the MANOVA option to perform a 2-sample Hotelling's T2 analysis as follows: 

• Analyze > Fit Model > select all of your response variables you are trying compare between your 2 groups & move them to "Y", select the column containing your labels for your 2 groups select "Add" under 'Construct Model Effects' > select "Manova" from the 'Personality' dropdown menu > Run > under 'Choose Response' menu, select "Identity" > Run. 
• This should output your F-value and probability for your test. 
• I'm not sure how to do the simultaneous or Bonferroni adjusted intervals from here.