Analysis

Instead of testing the null hypothesis for the ratios of the means over their hypothesized means are all equal to one, profile analysis involves testing the null hypothesis that all of these ratios are equal to one another, but not necessarily equal to 1.

After rejecting

\(H_0\colon \dfrac{\mu_1}{\mu^0_1} = \dfrac{\mu_2}{\mu^0_2} = \dots = \dfrac{\mu_p}{\mu^0_p} = 1\)

we may wish to test

\(H_0\colon \dfrac{\mu_1}{\mu^0_1} = \dfrac{\mu_2}{\mu^0_2} = \dots = \dfrac{\mu_p}{\mu^0_p}\)

Profile Analysis can be carried out using the following procedure.

Step 1: Compute the differences between the successive ratios. That is we take the ratio of the j + 1-th variable over its hypothesized mean and subtract this from the ratio of jth variable over its hypothesized mean as shown below:

\(D_{ij} = \dfrac{X_{ij+1}}{\mu^0_{j+1}}-\dfrac{X_{ij}}{\mu^0_j}\)

We call this ratio \(D_{ij}\) for observation i.

\(H_0\colon \boldsymbol{\mu}_D = \mathbf{0}\)

Step 2: Apply Hotelling’s \(T^{2}\) test to the data \(D_{ij}\) to test null hypothesis that the mean of these differences is equal to 0.

\(H_0\colon \boldsymbol{\mu}_D = \mathbf{0}\)

Analysis

The results yield a Hotelling's \(T^{2}\) of 1,030.7953 and an F value of 256.64843, 4 and 733 degrees of freedom and a p-value very close to 0.