I'm trying to determine the effect of taking a training course on various aspects of being a good quality worker. I have pre-test and post-test data on 7 different ways job performance can be viewed. I'm currently performing 7 different paired samples t-tests on the data, to see the effect on each individual aspect. Is there a test to calculate the overall effect on job performance (by comparing all 7 aspects at once)? I'm trying to use SPSS but not sure how to proceed.

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#### Best Answer

You need to first spell out what amounts to "overall improvement". If "overall improvement" means improvement in **each** of the aspects then your test is essentially **AND** of individual tests. i.e.

Suppose your model is:

$ improvement = beta_0 + beta_1cdot x_1 + … + beta_7cdot x_7 + training*(delta_0 + delta_1cdot x_1 + … + delta_7cdot x_7) $

where $training$ is a binary indicator whether training was received or not. The parameters of this model are $vec{theta} = (beta_0, …, beta_7, delta_0, …, delta_7)^T$. Say $vec{theta}_delta = (delta_0, …, delta_7)^T$

Then as you mentioned you can individually test for:

$H_0: delta_i vs H_a: delta_i <= 0, forall i in (1, 7)$.

However if you wish to test for all of them simultaneously, then you are essentially testing for:

$H_0: vec{theta}_delta > vec{0} vs H_a: vec{theta}_delta <= 0$

Now, you can easily find the estimates $vec{hat{theta}_delta}$ of $vec{theta}_delta$ and $hat{sigma}{vec{theta}_delta}$.

Also,

$frac{vec{hat{theta}_delta} – vec{theta}_delta}{hat{sigma}{vec{theta}_delta}} sim t_{df}$ and $frac{vec{hat{theta}_delta}}{hat{sigma}{vec{theta}_delta}} stackrel{H_0}{sim} t_{df}$

Thus, all you need is a simple one-sided t-test.