In reading a paper by Rouder et al (2009, p. 234), I came across the following to explain that each score in each group in a two-sample t-test is distributed as: $$ x_i\stackrel{{\rm iid}}{\sim}{\rm Normal}\big(\mu-\frac \alpha 2, \sigma^2\big),\quad i = 1,\ldots, N_x \\ y_i\stackrel{{\rm iid}}{\sim}{\rm Normal}\big(\mu+\frac \alpha 2, \sigma^2\big),\quad i = 1,\ldots, N_y $$

where $\mu$ is the grand mean, and $\alpha$ is the total effect. For my personal edification (NOT HOMEWORK), I have 3 questions, which are:

  1. How is $\alpha$ obtained?
  2. Why is $\alpha$ divided by 2?
  3. Why for one group, do we subtract $\alpha/2$ from the grand mean, but for the other group, we add $\alpha/2$ to the grand mean?
  • 1
    $\begingroup$ If this is homework you must include the self-study tag and read up on the restrictions on using it. $\endgroup$ – Carl Nov 16 '16 at 21:21
  • $\begingroup$ We have problems with students asking us to do their homework. I changed your question to make it clear that you are asking not for reasons of getting your homework done for you. Welcome to the site, just protecting you here. Newbies can have a rough time of it, chin-up! $\endgroup$ – Carl Nov 16 '16 at 21:29

This is just the standard idea of the data generating process that lies behind a run of the mill t-test.

  1. $\alpha$ is just the difference between the two groups' means in raw units (i.e., not standardized). That is, $\alpha$ isn't "obtained" from somewhere else, it just is. It is a primitive with respect to this setup.
  2. $\alpha$ is divided by two because the groups are balanced around their combined mean. The variances are intended to be equal. I think the sample sizes seem to be intended to be equal as well (although the notation is ambiguous on that), but it could be that they mean that the mean of the samples could diverge from the grand mean by having a greater sample size in one group vs. the other.
  3. The grand mean is halfway between the two means, so one group is above the grand mean and the other is below it.
  • $\begingroup$ gung, thanks so, are you suggesting a better notation? $\endgroup$ – user138773 Nov 16 '16 at 21:47
  • $\begingroup$ @user138773, they use $N_{\bf x}$ & $N_{\bf y}$. W/ 2 different subscripts, you could have 2 different N's, but the shifts of + & - 1/2 the raw gap implies the populations' grand mean is exactly between the means of the two groups. Those two seem to be subtly discordant, IMO. Note that I tweaked my answer to clarify what I meant. $\endgroup$ – gung - Reinstate Monica Nov 16 '16 at 21:52
  • $\begingroup$ You're welcome, @user138773. $\endgroup$ – gung - Reinstate Monica Nov 16 '16 at 21:59

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