Profile Analysis

Definition Let X 1, X 2, …, X p denote p jointly distributed variables under study Let  1,  2, …,  p denote the means of these variables  denote the means these variables The profile of these variables is a plot of  i vs i. ii i

The multivariate Test Let denote a sample of n from the p-variate normal distribution with mean vector  and covariance matrix . Suppose we want to test Let denote a sample of m from the p-variate normal distribution with mean vector  and covariance matrix .

Hotelling’s T 2 statistic for the two sample problem if H 0 is true than has an F distribution with 1 = p and 2 = n +m – p - 1

Profile Comparison X variables p 123 … Group A Group B

Hotelling’s T 2 test, tests against

Profile Analysis

Parallelism

123 … Variables not interacting with groups (parallelism) X variables p groups

Variables interacting with groups (lack of parallelism) X variables p 123 … groups

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

Test for parallelism

Let denote a sample of n from the p-variate normal distribution with mean vector  and covariance matrix . Let denote a sample of m from the p-variate normal distribution with mean vector  and covariance matrix .

Let Then

Consider the data This is a sample of n from the (p -1) -variate normal distribution with mean vector  and covariance matrix. The test for parallelism is Also is a sample of m from the (p -1) -variate normal distribution with mean vector  and covariance matrix.

Hotelling’s T 2 test for parallelism if H 0 is true than has an F distribution with 1 = p – 1 and 2 = n +m – p Thus we reject H 0 if F > F   with 1 = p – 1 and 2 = 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 T 2 test.

Test for Equality of Groups (Parallelism assumed)

123 … Groups equal X variables p groups

If parallelism is proven: It is appropriate to test for equality of profiles i.e.

The t test Thus we reject H 0 if |t| > t  /2  with df = = 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 variables i 123 … groups

Let Then

Consider the data This is a sample of n from the p-variate normal distribution with mean vector  and covariance matrix. The test for equality of variables for the first group is:

Hotelling’s T 2 test for equality of variables if H 0 is true than Thus we reject H 0 if F > F   with 1 = p – 1 and 2 = n – p + 1 has an F distribution with 1 = p – 1 and 2 = 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 T 2 test for a zero mean vector A similar test can be performed for the second sample. Both of these tests do not assume parllelism.

Then This is a sample of n + m from the p-variate normal distribution with mean vector  and covariance matrix. If parallelism is assumed then The test for equality of variables is:

Hotelling’s T 2 test for equality of variables if H 0 is true than Thus we reject H 0 if F > F   with 1 = p – 1 and 2 = n + m – p has an F distribution with 1 = p – 1 and 2 = n +m - p

To perform this test for parallelism, 1.Compute differences of successive variables for each case in each group 2.Combine the two samples into a single sample of n + m and 3.Perform the single-sample Hotelling’s T 2 test for a zero mean vector.

Example Two groups of Elderly males Groups 1.Males identified with no senile factor 2.Males identified with a senile factor Variables – Scores on WAIS (intelligence) test 1.Information 2.Similarities 3.Arithmetic 4.Picture completion

Summary Statistics

Hotellings T 2 test (2 sample) H 0 :equal means, is rejected

Profile Analysis

Hotelling’s T 2 test for parallelism Decision: Accept H 0 : parallelism

The t test for equality of groups assuming parallelism Thus we reject H 0 if t > t   with df = = n +m - 2 = 47

Hotelling’s T 2 test for equality of variables Thus we reject H 0 if F > F   with 1 = p – 1= 3 and 2 = n + m – p = 45 F 0.05  = 6.50 if 1 = 3 and 2 = 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:

The data

Summary Statistics