# 7.2.7 - Testing for Equality of Mean Vectors when $$Σ_1 ≠ Σ_2$$

7.2.7 - Testing for Equality of Mean Vectors when $$Σ_1 ≠ Σ_2$$

The following considers a test for equality of the population mean vectors when the variance-covariance matrices are not equal.

Here we will consider the modified Hotelling's T-square test statistic given in the expression below:

$$T^2 = \mathbf{(\bar{x}_1-\bar{x}_2)}'\left\{\dfrac{1}{n_1}\mathbf{S}_1+\dfrac{1}{n_2}\mathbf{S}_2\right\}^{-1}\mathbf{(\bar{x}_1-\bar{x}_2)}$$

Again, this is a function of the differences between the sample means for the two populations. Instead of being a function of the pooled variance-covariance matrix, we can see that the modified test statistic is written as a function of the sample variance-covariance matrix, $$\mathbf{S}_{1}$$, for the first population and the sample variance-covariance matrix, $$\mathbf{S}_{2}$$, for the second population. It is also a function of the sample sizes $$n_{1}$$ and $$n_{2}$$.

For large samples, that is if both samples are large, $$T^{2}$$ is approximately chi-square distributed with p d.f. We will reject $$H_{0}$$ : $$\boldsymbol{\mu}_{1}$$ = $$\boldsymbol{\mu}_{2}$$ at level $$α$$ if $$T^{2}$$ exceeds the critical value from the chi-square table with p d.f. evaluated at level $$α$$.

$$T^2 > \chi^2_{p, \alpha}$$

For small samples, we can calculate an F transformation as before using the formula below.

$$F = \dfrac{n_1+n_2-p-1}{p(n_1+n_2-2)}T^2\textbf{ } \overset{\cdot}{\sim}\textbf{ } F_{p,\nu}$$

This formula is a function of sample sizes $$n_{1}$$ and $$n_{2}$$, and the number of variables p. Under the null hypothesis this will be F-distributed with p and approximately ν degrees of freedom, where 1 divided by ν is given by the formula below:

$$\dfrac{1}{\nu} = \sum_{i=1}^{2}\frac{1}{n_i-1} \left\{ \dfrac{\mathbf{(\bar{x}_1-\bar{x}_2)}'\mathbf{S}_T^{-1}(\dfrac{1}{n_i}\mathbf{S}_i)\mathbf{S}_T^{-1}\mathbf{(\bar{x}_1-\bar{x}_2)}}{T^2} \right\} ^2$$

This involves summing over the two samples of bank notes, a function of the number of observations of each sample, the difference in the sample mean vectors, the sample variance-covariance matrix for each of the individual samples, as well as a new matrix $$\mathbf{S}_{T}$$ which is given by the expression below:

$$\mathbf{S_T} = \dfrac{1}{n_1}\mathbf{S_1} + \dfrac{1}{n_2}\mathbf{S}_2$$

We will reject $$H_{0}$$ \colon $$\mu_{1}$$ = $$\mu_{2}$$ at level $$α$$ if the F-value exceeds the critical value from the F-table with p and ν degrees of freedom evaluated at level $$α$$.

$$F > F_{p,\nu, \alpha}$$

A reference for this particular test is given in: Seber, G.A.F. 1984. Multivariate Observations. Wiley, New York.

#### Using SAS

This modified version of Hotelling's T-square test can be carried out on the Swiss Bank Notes data using the SAS program below:

View the video explanation of the SAS code.

#### Using Minitab

At this time Minitab does not support this procedure.

#### Analysis

As before, we are given the sample sizes for each population, the sample mean vector for each population, followed by the sample variance-covariance matrix for each population.

In the large sample approximation, we find that T-square is 2412.45 with 6 degrees of freedom, (because we have 6 variables), and a p-value that is close to 0.

Note! This value for the Hotelling's T-square is identical to the value that we obtained for our un-modified test. This will always be the case if the sample sizes are equal to one another.
• When $$n_{1}$$ = $$n_{2}$$, the modified values for $$T^{2}$$ and F are identical to the original unmodified values obtained under the assumption of homogeneous variance-covariance matrices.
• Using the large-sample approximation, our conclusions are the same as before. We find that mean dimensions of counterfeit notes do not match the mean dimensions of genuine Swiss bank notes. $$\left( T ^ { 2 } = 2412.45 ; \mathrm { d.f. } = 6 ; p < 0.0001 \right)$$.
• Under the small-sample approximation, we also find that mean dimensions of counterfeit notes do not match the mean dimensions of genuine Swiss bank notes. $$( F = 391.92 ; \mathrm { d } . \mathrm { f } . = 6,193 ; p < 0.0001 )$$.

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