Profile Analysis

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Profile Analysis
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• Definition
• Let X1, X2, , Xp denote p jointly distributed
  variables under study
• Let m1, m2, , mp denote 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

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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
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Hotellings T2 statistic for the two sample
problem
if H0 is true than
has an F distribution with n1 p and n2 n m
p - 1
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Profile Comparison
X
Group A
Group B
p

1
2
3
variables
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Hotellings T2 test, tests
against
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Profile Analysis
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Parallelism
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Variables not interacting with groups(parallelism
)
X
groups

p
1
2
3
variables
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Variables interacting with groups(lack of
parallelism)
X
groups
p

1
2
3
variables
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• 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

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Test for parallelism
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• 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.
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• Let

Then
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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 .
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Hotellings 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 gt Fa with n1 p 1 and
n2 n m p
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To perform the test for parallelism, compute
differences of successive variables for each case
in each group and perform the two-sample
Hotellings T2 test.
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Test for Equality of Groups

• (Parallelism assumed)

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Groups equal
X
groups

p
1
2
3
variables
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• If parallelism is proven
• It is appropriate to test for equality of profiles

i.e.
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The t test
Thus we reject H0 if t gt ta/2 with 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.
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Test for equality of variables

• (Parallelism Assumed)

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Variables equal
X
groups
i

1
2
3
variables
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• Let

Then
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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 .

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Hotellings 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 gt Fa with n1 p 1 and
n2 n p 1
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To perform the test, compute differences of
successive variables for each case in the group
and perform the one-sample Hotellings 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.
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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
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Hotellings 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 gt Fa with n1 p 1 and
n2 n m p
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• 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 Hotellings T2 test for
  a zero mean vector.

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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

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Summary Statistics
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Hotellings T2 test (2 sample)
H0 equal means, is rejected
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Profile Analysis
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Hotellings T2 test for parallelism
Decision Accept H0 parallelism
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The t test for equality of groups assuming
parallelism
Thus we reject H0 if t gt ta with df n n m -
2 47
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Hotellings T2 test for equality of variables
Thus we reject H0 if F gt Fa with n1 p 1 3
and n2 n m p 45
F0.05 6.50 if n1 3 and n2 45
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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

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The data
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Summary Statistics