You are asking the wrong question...

You are talking about the t-test, but the hypothesis you state is "that there is no difference of salary between gender groups". The t-test test the hypothesis that the expected salaries are the same (or that the expected difference in salaries is zero). It can be that the expectations are the same but the salary distributions are not. The hypothesis that the salary distributions are the same is tested by the Kolmogorow-Smirnow-test. The hypothesis that there is an equal probability of a higher salary in both groups is tested by Wilcoxon-Mann-Whitney-test.

Another question you should ask is if a mean difference in salaries is really a relevant quentity. I suspect that the salary ratio is the more interesting statistic (the hypothesis is: the expected ratio is 1; this can be tested with a t-test as the expected difference of log(salary)).

Good explanation by Dr.Jochen. Regards

Just to note that the t-test is, in fact, a linear regression with a single predictor measured on a 0/1 scale. The coefficient and its 95% CI are none other than the difference between the means and its CI.

The advantage to using regression is that you can add other variables to the model. The t-test on its own is a dead end.

When I tell people I'm running t-tests, I actually use a regression command. It's easier to build the model further, and I can use robust variance estimation where needed.

@Jochen

"There is no difference in salary" is not equal to "The salaries are the same"? The t-test is for the difference in means. "No difference" equals "are the same."

@Sabri

A difference in means is not sufficient to show salaries are different if the salary structure is different,

Perform a quantile plot of the salaries to see the salary structure. You cannot rely on tests to evaluate data.

Dear all, thank you for your responses to my question but I see that you are focusing on a salary instead of main question here. Assume that there are two independent groups and we are going to test whether their means are equal or not. If I were you I would Imagine any other scale variable such as 5-point likert which is derived interval and I would create a null hypothesis such that:

H0: Both group means in populations are equal.

Please focus on the questions in part a and b, not the salary!

It is important that the assumptions make sense, particularily for small samples. Hence:

a) A formal test is nonsense. However, it may be good to confirm that the data is not strongly contradicting the assumptions.

b) Same as for a. It would need some intellectual work to argue why the variance should assumed different when the assumption of normality is considered sensible. Often either both or none of these two assumptions make sense.

Sabri – 'you say thank you for your responses to my question but I see that you are focusing on a salary instead of main question here'.

I have a very different take as I think you cannot talk in data analysis about models purely in the abstract but in terms of the real variables of interest and that you need to think especially about the specifics of the data generating process. So, the fact that it is salaries and you are interested in some aspect of the gender gap, and that data is (presumably) observational and (presumably) cross-sectional is important. From reading Joan Fisher-Box’s book on her father (the progenitor of the t test) he argued that way, and was not impressed when the things he had developed were used out of context. Departments of theoretical statistics were and an anathema to Fisher. I regularly see t tests being applied to autocorrelated time and spatial series without any account of the dependency that is likely to be found in such data.

I would conceive of raises in salary not as an absolute amount but as a proportional rise – so not a $2k increment but 2.5%. That sort of compound interest growth suggest that a log of salary may be an appropriate transform; it will probably help with potential positive skewness (few have high earnings) and potential differential heteroscedasticity between the genders.

Having graphed up the two distributions before and after transforms I would probably fit a random-coefficients model with a dummy variable for gender in the fixed part and two variances in the random part (something that was not available to Fisher). I would not presume homoscedasticity. This would allow the evaluation the difference in the means and in variability around those means for different genders. I would then perform some diagnostic plots – I am particularly worried about the presence of a millionaire really affecting the results.

I would also know about other studies and would be concerned about the sort of differences that are found between USA and Canada. If we take the arithmetic mean, Canadians earn $2,000 dollars less than Americans, which may suggest a policy of lower taxes and cutbacks is needed. But if you calculate what the average Canadian earns, the median is $2,000 higher than in America, because the US is a much more unequal society with the arithmetic mean being pulled upwards by relatively few, big earners – different techniques, different answers, different implications.

So, I would be interested in seeing the differences for different ‘cuts’ of the distribution – median, quartiles, highest 10 percent say. I would use quantile regression to do that. In Keonker’s chapter 1 of Quantile Regression he uses the approach to look at salaries (of American statisticians!) conditioning on experience, and to look at the effect of years of schooling on the wages of identical twins – and that potentially highly dependent structure I would undoubtedly need to understand and take account of when applying the modelling.

My point is you need technique and substantive understanding, and that you often need to do things in more than one way to see how robust the findings are, Having got that of my chest, back to serious striking and getting no salary today!

Sabri Erdem, you have been provided with some interesting and inevitably valuable advice above, and by now you have probably resolved your questions.

First up, apparently the t-test is quite robust despite departures from normality

As I see it, your second question is simple and it can be easily addressed. If you are using SPSS, the output for an independent-samples t-test contains the Levene test for homogeneity of variances. If that test (i.e., the Levene test) is significant, there is a difference in variance between the two groups and you simply report the t value and degrees of freedom associated with the output that relates to that finding (the relevant information is in the second row of the t-test output).

Slightly unrelated to your query, however, I would check out the possibility that your data contain outliers. They could contaminate your results badly and therefore might need to be attended to one way or another.

I hope my contribution is helpful, perhaps for others even if not for you any more.