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Explore profile analysis methods in multivariate testing, including comparison between groups, testing for parallelism, and examining equality across variables. Learn how to conduct Hotelling’s T2 tests for various hypotheses and interpret the results.
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Definition • Let X1, X2, … , Xp denote p jointly distributed variables under study • Let m1, m2, … , mpdenote the means of these variables s denote the means these variables • The profile of these variables is a plot of mi vs i. • mi • i
The multivariate Test Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test
Hotelling’s T2 statisticfor the two sample problem if H0 is true than has an F distribution with n1= p and n2= n +m – p - 1
Profile Comparison X Group A Group B p … 1 2 3 variables
Hotelling’s T2 test, tests against
Variables not interacting with groups(parallelism) X groups … p 1 2 3 variables
Variables interacting with groups(lack of parallelism) X groups p … 1 2 3 variables
Parallelism • Group differences are constant across variables Lack of Parallelism • Group differences are variable dependent • The differences between groups is not the same for each variable
Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix S.
Let Then
The test for parallelism is Consider the data This is a sample of n from the (p -1) -variate normal distribution with mean vector and covariance matrix . Also is a sample of m from the (p -1) -variate normal distribution with mean vector and covariance matrix .
Hotelling’s T2 test for parallelism if H0 is true than has an F distribution with n1= p – 1 and n2= n +m – p Thus we reject H0 if F > Fawith n1= p – 1 and n2= n +m – p
To perform the test for parallelism, compute differences of successive variables for each case in each group and perform the two-sample Hotelling’s T2 test.
Test for Equality of Groups (Parallelism assumed)
Groups equal X groups … p 1 2 3 variables
If parallelism is proven: It is appropriate to test for equality of profiles i.e.
The t test Thus we reject H0 if |t|> ta/2with df = n= n +m - 2 To perform this test, average all the variables for each case in each group and perform the two-sample t-test.
Test for equality of variables (Parallelism Assumed)
Variables equal X groups i … 1 2 3 variables
Let Then
The test for equality of variables for the first group is: Consider the data This is a sample of n from the p-variate normal distribution with mean vector and covariance matrix .
Hotelling’s T2 test for equality of variables if H0 is true than has an F distribution with n1= p – 1 and n2= n - p + 1 Thus we reject H0 if F > Fawith n1= p – 1 and n2= n – p + 1
To perform the test, compute differences of successive variables for each case in the group and perform the one-sample Hotelling’s T2 test for a zero mean vector A similar test can be performed for the second sample. Both of these tests do not assume parllelism.
If parallelism is assumed then Then This is a sample of n + m from the p-variate normal distribution with mean vector and covariance matrix . The test for equality of variables is:
Hotelling’s T2 test for equality of variables if H0 is true than has an F distribution with n1= p – 1 and n2= n +m - p Thus we reject H0 if F > Fawith n1= p – 1 and n2= n + m – p
To perform this test for parallelism, • Compute differences of successive variables for each case in each group • Combine the two samples into a single sample of n + m and • Perform the single-sample Hotelling’s T2 test for a zero mean vector.
Example • Two groups of Elderly males • Groups • Males identified with no senile factor • Males identified with a senile factor • Variables – Scores on WAIS (intelligence) test • Information • Similarities • Arithmetic • Picture completion
Hotellings T2 test (2 sample) H0 :equal means, is rejected
Hotelling’s T2 test for parallelism Decision: Accept H0 :parallelism
The t test for equality of groups assuming parallelism Thus we reject H0 if t > tawith df = n= n +m - 2 = 47
Hotelling’s T2 test for equality of variables Thus we reject H0 if F > Fawith n1= p – 1= 3 and n2= n + m – p = 45 F0.05= 6.50 if n1 = 3 and n2= 45
Example 2: Profile Analysis for Manova In the following study, n = 15 first year university students from three different School regions (A, B and C) who were each taking the following four courses (Math, biology, English and Sociology) were observed: The marks on these courses is tabulated on the following slide:
