Can an independent t-test be used on paired data when the pairing is unknown? Suppose the effectiveness of a training course is examined, and performance of each individual in a group is taken both before and after, and the differences are compared in a paired $t$-test.
Would it be possible to also perform a two-independent-samples $t$-test to investigate the mean difference if the data before and after were mixed as to no longer be paired? Or would the independent condition still not be satisfied, as the two sets of observations are not independent?
 A: A long hint: You could base the comparison between a paired and an unpaired analysis on the following simple model, and doing power calculations based on the model (either theoretically or by simulation).
Let $(Y_{i1}, Y_{i2})$ be independent pairs, each pair with the following model:
$$
    Y_{i1} = \mu +\epsilon_{i1}, \\
    Y_{i2} = \mu + \Delta + \epsilon_{i2}, \quad i=1\dotsc,n,
$$
where the pair $(\epsilon_{i1}, \epsilon_{i2})$ has a bivariate normal distribution with expectation 0, equal variances $\sigma^2$ and covariance $\rho \sigma^2$. Then the paired analysis is based on the differences $D_i= Y_{i2}-Y_{i1}$ and their mean $\bar{D}$. The t-test then is based on $T_D =\sqrt{n}\bar{D}/ s_D$, under the null hypothesis $\Delta=0$ this have a t-distribution with $n-1$ degrees of freedom.
The independent samples t-test is based on $\bar{Y}_2 - \bar{Y}_1$. Now calculate its null mean (0) and variance (will depend on $\rho$), find the t-statistic, and do the comparison.
There are some interesting similar posts:  Paired test, unknown sample identites,  t-test for partially paired and partially unpaired data
