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I have two categorical, independent groups, and a continuous dependent variable.

The data violates homogeneity of variances, and the dependent variable is not normally distributed for each of the two groups. I applied a log transformation and ran an independent t-test. It yields the same results as performing a Welch t-test on the non-transformed data.

Can I go ahead and report the Welch t-test? Or do I need some other procedure?

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    $\begingroup$ 1. What is the underlying question of interest? 2. What are your sample-sizes? 3. (a) Given you've already looked at the data (which impacts the properties of your inference) ... what kind of non-normality do you mean and how far from normality was it? (b) Were the distributions similar in general shape? $\endgroup$ – Glen_b Apr 26 '16 at 1:33
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    $\begingroup$ I'm looking at whether the means of two groups differ on a self-report measure. Sample size of 60 each. Positive skew (z-score is 4.95). Very similar distribution shapes. $\endgroup$ – mrchimpbanana Apr 26 '16 at 11:59
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Yes, you can go ahead and report the Welch t-test, unless your sample is very small. As long as you have independent observations and your dependent variable has a finite second moment, then $$ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} \leadsto_d N(0,1)$$ where $\bar{x}_1$ and $\bar{x}_2$ are the group sample means, $s_1^2$ and $s_2^2$ are the group sample variances, and $n_1$ and $n_2$ are the number of observations in each group. This means that in large samples, $t$ will be approximately normal distributed (and since with high degrees of freedom, the t-distribution is close to normal, also approximately t-distributed). How big your sample must be for this approximation to give accurate results depends how the exact distribution of your data. Different people have various rules of thumb for a minimal acceptable sample size, but I would say that anything more than 50 and you're okay.

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