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Suppose we have two samples (A and B) of 200 people each. There are 100 men and 100 women in each population.

The income difference between men and women in A is 1000 dollars (per month). The difference in income between men and women in B is 500 dollars. (both differences are statistically significant)

Under what conditions (and via what kind of tests) can we say that the difference in income between men and women in A is significantly different from the difference in income in group B?

Sorry for a very naive question. Any hint/point to the right direction will be welcome.

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  • $\begingroup$ Do you have the full populations? If you do, there there is no notion of statistical significance; either the values are the same or different, and you know that with certainty. $\endgroup$
    – Dave
    Aug 16, 2021 at 14:11
  • $\begingroup$ thanks for your comment. Nope, these are samples drawn from two different populations (A* and B* correspondingly) $\endgroup$
    – Philipp Chapkovski
    Aug 16, 2021 at 14:12
  • $\begingroup$ Perhaps you should change your phrasing in the question to reflect the fact that you have samples, not populations. $\endgroup$
    – Dave
    Aug 16, 2021 at 14:13
  • $\begingroup$ thanks! I just did it $\endgroup$
    – Philipp Chapkovski
    Aug 16, 2021 at 14:14

1 Answer 1

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Say that we denote salaries as $x$, and assume that in each population the indices 1-100 are for women and 101-200 are for men. That is, for each population we can define 100 variables of the form $g_i=x_{100+i}-x_{i}$, so the wage gap for population A as $\bar{g}_A=1000$ and for population B as $\bar{g}_B=500$. From here on it's straightforward two-sample t-test:

  1. Calculate $s^2_A=Var(g_A)$ and $s^2_B=Var(g_B)$, the wage gap variance for each population
  2. Find the pooled variance, $s_p^2=(s_1^2+s_2^2)/2$
  3. The test statistic is

$$t=\frac{\bar{g}_A-\bar{g}_B}{s_p\sqrt{1/n_A+1/n_B}}=\frac{500}{s_p\sqrt{0.02}}$$

the significance of this result depends on your choice of $\alpha$, note that you should use 100+100-2=198 degrees of freedom.

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  • $\begingroup$ "result depends on your choice of 𝛼, " - what do you mean by 𝛼 in this case (you haven't referred to it before that). $\endgroup$
    – Philipp Chapkovski
    Aug 16, 2021 at 14:53
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    $\begingroup$ If I understand correctly, this method assumes that men and women are paired. For example, you assume value #2 is paired to value #102, when computing g. There is nothing in the problem setup to suggest such pairing. $\endgroup$
    – Harvey Motulsky
    Aug 16, 2021 at 15:05
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    $\begingroup$ @HarveyMotulsky No, there's no such pairing at all. Pairing means two treatments for the same sample, while where I have suggested dividing to pairs just to make it easier to understand the concept of 100 differences out of population of 200 (this still stands, as $\bar{x}_m-\bar{x}_f$ would yield the exact same result as $\bar{g}$). Not all pairs are pairings. $\endgroup$
    – Spätzle
    Aug 16, 2021 at 20:49
  • $\begingroup$ @PhilippChapkovskievery the result of statistical significance test depends on the significance level chosen $\alpha$. A different $\alpha$ value results in a different critical value, which in turn might change the binary outcome of the test. $\endgroup$
    – Spätzle
    Aug 16, 2021 at 20:52
  • $\begingroup$ @Spätzle the follow-up question: would it be enough just to test for the significance of interaction term in ANOVA income~group*gender? $\endgroup$
    – Philipp Chapkovski
    Aug 18, 2021 at 16:50

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