# After a transformation of data do I carry out a t-test or a non-parametric

For a stats question the data was not normally distributed but the question required a two way ANOVA, a transformation was therefore used and all worked out fine.

Now the next part requires the data set to be split by one of the nominal variables (in SPSS) and a t-test to be run.

Obviously the transformed data is normally distributed and therefore a t-test is applicable, however the original data (not normally distributed) could also be applicable using a non-parametric test. Which one would be the best to use and why?

• When you say 'for a stats question' -- do you mean a question like in a textbook or coursework? – Glen_b Jan 1 '14 at 8:57
• Coursework, although the data is coded from actual dating adverts features in two newspapers. – Pink Teddy Jan 1 '14 at 18:07
• Pink Teddy, I've taken the liberty of adding the self-study tag, for which you should probably read the tag wiki info if you're not already familiar with it. – Glen_b Jan 2 '14 at 0:51
• Could you say more about what the actual dependent variable is here? – Glen_b Jan 2 '14 at 2:42

For a stats question the data was not normally distributed but the question required a two way ANOVA, a transformation was therefore used and all worked out fine.

The raw data isn't assumed to be normally distributed in two way ANOVA.

(Something is assumed to be normal, but it's not the data. At least, not unconditionally. What did you check for normality, and how?)

What transformation has done is made your comparison no longer a comparison of means.

On the other hand, ANOVA isn't particularly sensitive to mild non-normality, and the larger the samples, the more non-normality it can tolerate.

Now the next part requires the data set to be split by one of the nominal variables (in SPSS) and a t-test to be run.

Obviously the transformed data is normally distributed

You have no basis on which to assert that the transformed variable is normal. It might look normal, but that doesn't mean it is. (On the other hand, you can tolerate approximate normality, so this error isn't of much consequence.)

and therefore a t-test is applicable, however the original data (not normally distributed) could also be applicable using a non-parametric test. Which one would be the best to use and why?

Possibly either, possibly neither. What is the required null and alternative? What assumptions are you prepared to make?

The nonparametric test is not a test of equality of means, for example, unless you add some assumptions (such as a location-shift alternative).

If the variances are close to equal, the t-test is reasonably robust to mild/moderate non-normality. (And if you use the Welch approximation, the t-test deals pretty well with unequal variance)

Aside from those two, and a t-test after a transformation, another possibility is to perform a permutation test.

You really need to give more detail about the specific hypotheses you wish to consider.

As a piece of general advice, either the Welch t-test or the Wilcoxon-Mann-Whitney might be reasonable, but there's presently not enough information to suggest leaning toward one or the other, or indeed something else.

• Thanks for the reply but as far as I understand the dependent variable should be approximately normally distributed for each combination of the groups of the two independent variables. Therefore if this assumption is not present (as it was in two of my groups) then a transformation is a valid step (after which normality is achieved in all groups). – Pink Teddy Jan 1 '14 at 18:02
• What sample sizes are involved in each subgroup? How was normality assessed? – Glen_b Jan 2 '14 at 0:49
• The phrase "normality is achieved" is again incorrect (this will be an event with probability zero). You probably mean something nearer to 'the data don't look to be badly non-normal'. – Glen_b Jan 2 '14 at 2:41

In general, parametric tests have more statistical power than non-parametric tests, provided that the underlying parametric model is valid. If your transformed data satisfies assumptions of normality, then the parametric approach will have smaller probability of Type II error.