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Profile Analysis. Definition Let X 1 , X 2 , … , X p denote p jointly distributed variables under study Let m 1 , m 2 , … , m p denote the means of these variables s denote the means these variables The profile of these variables is a plot of m i vs i. m i. i.
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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:
