Inference for the mean vector

Univariate Inference Let x1, x2, … , xn denote a sample of n from the normal distribution with mean m and variance s2. Suppose we want to test H0: m = m0 vs HA: m ≠ m0 The appropriate test is the t test: The test statistic: Reject H0 if |t| > ta/2

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

Roy’s Union- Intersection Principle This is a general procedure for developing a multivariate test from the corresponding univariate test. Convert the multivariate problem to a univariate problem by considering an arbitrary linear combination of the observation vector.

Perform the test for the arbitrary linear combination of the observation vector. Repeat this for all possible choices of Reject the multivariate hypothesis if H0 is rejected for any one of the choices for Accept the multivariate hypothesis if H0 is accepted for all of the choices for Set the type I error rate for the individual tests so that the type I error rate for the multivariate test is a.

Application of Roy’s principle to the following situation Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test Then u1, …. un is a sample of n from the normal distribution with mean and variance .

to test we would use the test statistic:

and

Thus We will reject if

Using Roy’s Union- Intersection principle: We will reject We accept

i.e. We reject We accept

Consider the problem of finding: where

thus

Thus Roy’s Union- Intersection principle states: We reject We accept is called Hotelling’s T2 statistic

Choosing the critical value for Hotelling’s T2 statistic We reject , we need to find the sampling distribution of T2 when H0 is true. It turns out that if H0 is true than has an F distribution with n1 = p and n2 = n - p

Thus Hotelling’s T2 test We reject or if

Another derivation of Hotelling’s T2 statistic Another method of developing statistical tests is the Likelihood ratio method. Suppose that the data vector, , has joint density Suppose that the parameter vector, , belongs to the set W. Let w denote a subset of W. Finally we want to test

The Likelihood ratio test rejects H0 if

The situation Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test

The Likelihood function is: and the Log-likelihood function is:

the Maximum Likelihood estimators of are and

the Maximum Likelihood estimators of when H 0 is true are: and

The Likelihood function is: now

Thus similarly

and

Note: Let

and Now and

Also

Thus

Thus using

Then Thus to reject H0 if l < la This is the same as Hotelling’s T2 test if

Example For n = 10 students we measure scores on Math proficiency test (x1), Science proficiency test (x2), English proficiency test (x3) and French proficiency test (x4) The average score for each of the tests in previous years was 60. Has this changed?

The data

Summary Statistics

Simultaneous Inference for means Recall (Using Roy’s Union Intersection Principle)

Now

Thus and the set of intervals Form a set of (1 – a)100 % simultaneous confidence intervals for

Recall Thus the set of (1 – a)100 % simultaneous confidence intervals for

The two sample problem

Univariate Inference Let x1, x2, … , xn denote a sample of n from the normal distribution with mean mx and variance s2. Let y1, y2, … , ym denote a sample of n from the normal distribution with mean my and variance s2. Suppose we want to test H0: mx = my vs HA: mx ≠ my

The appropriate test is the t test: The test statistic: Reject H0 if |t| > ta/2 d.f. = n + m -2

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 statistic for the two sample problem if H0 is true than has an F distribution with n1 = p and n2 = n +m – p - 1

Thus Hotelling’s T2 test We reject

Simultaneous inference for the two-sample problem Hotelling’s T2 statistic can be shown to have been derived by Roy’s Union-Intersection principle

Thus

Thus

Thus Hence

Thus form 1 – a simultaneous confidence intervals for

A graphical explanation Hotelling’s T2 test A graphical explanation

Hotelling’s T2 statistic for the two sample problem

is the test statistic for testing:

Hotelling’s T2 test X2 Popn A Popn B X1

Univariate test for X1 X2 Popn A Popn B X1

Univariate test for X2 X2 Popn A Popn B X1

Univariate test for a1X1 + a2X2 Popn A Popn B X1

A graphical explanation Mahalanobis distance A graphical explanation

Euclidean distance

Mahalanobis distance: S, a covariance matrix

Hotelling’s T2 statistic for the two sample problem

Case I X2 Popn A Popn B X1

Case II X2 Popn A Popn B X1

In Case I the Mahalanobis distance between the mean vectors is larger than in Case II, even though the Euclidean distance is smaller. In Case I there is more separation between the two bivariate normal distributions