# Unequal sample size t-test, but with group-effect

I am researching how people watch video lectures together. I have two conditions as follows:

(1) 3 groups of 4 learners watch one video lecture on a shared display, but there is only one control (to pause for jump in video)

(2) 3 groups of 4 learners watch one video lecture separately on individual displays, with individual control. They watch the same video on their own pace, but they can of course talk during watching, so there is a group effect.

In the end, I want have a research question which is formulated by the following null hypothesis:

The two conditions have no effect on the frequency of pauses made.

This is a typical question that can usually be answered by doing a t-test (assuming normality and homogeneity). But the problem is that, for those who watch with single control and display, a group has a single log of interactions, however, for those who watch with multiple control and displays, a group has N (number of users) logs of interactions. So the sample size is different, but some samples are not independent, so we cannot directly use Mann-Whitney Test to replace t-test.

I mean, the log files become something like this:

  group    user    numOfPauses  condition
A       A          20          Shared
B       A          32          Shared
C       A          22          Shared
D       D1         10          Individual
D       D2         23          Individual
D       D3         15          Individual
D       D4         3           Individual
E       E1         16          Individual
E       E2         38          Individual
E       E3         11          Individual
E       E4         9           Individual
F       F1         32          Individual
F       F2         21          Individual
F       F3         12          Individual
F       F4         17          Individual


Finally the number of observations from individually watching groups are N times the shared watching groups.

Is there a way to balance the unequal size considering the group-effect? Or it is better to just make an average for individual watching groups, or expanding shared groups by duplicating the same value N times?

• The user in each group A, B, C has name A whatever his group. Is it the same user ? – Stéphane Laurent Jul 7 '15 at 8:55

$$x_{i|j}=\begin{cases} 0 & j=0 \mbox{ (Groups)}\\ 1 & j>0 \mbox{ (Individual)} \end{cases}\\ \eta_{i|j}=\alpha+\beta_{j}x_i\\ \alpha\sim N(0,k_{\alpha})\\ \beta_{j}\sim N(\beta_{\mu},\sigma_{\beta})\\ \sigma_{beta}\sim half\mbox{-}Cauchy(0,k_{\sigma})\\ \beta_{\mu}\sim N(0,k_{\beta})$$
In this case, $\alpha$ would represent the average number of pauses taken at the group level, $\beta_\mu$ is the average effect of individual control on the number of pauses, and $\beta_j$ codifies the assumption of non-independence of individuals within each group. The $k$ values are user-specified constants that control for the scaling of the prior distributions.
You could then fit your linear predictor $\eta$ to an appropriate distribution (e.g., $y\sim Poisson(log(\eta))$, $y\sim N(\eta,\sigma)$, etc.