Paired $t$-tests

Let us assume that for each of $N$ subject there exist two measurements obtained under different conditions $s_1$, $s_2$ and that we are interested in the significance of the mean group difference $b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}=\sum_kb_k/N=\sum_k(\beta_{ks_1}-\beta_{ks_2})/N$. We will assume that the between-subject variance and the between-session variance is equal across subjects and conditions. Note that this is a notational simplification within this framework which might or might not become a necessary condition once we try to estimate the associated group-level parameters. Similar to the previous sections, we model this as

$$\beta_{ks_i}\sim{\cal N}(\mu_i,\sigma^2_b+\sigma^2_s),$$   and$$\quad\textrm{Cov}(\beta_{ks_1},\beta_{ks_2}) = \sigma^2_s.$$

Let

$$V=\left[ \begin{array}{ccc} V_1& & 0 \\ & \ddots & \\ 0... ... & & & \! 1 \\ \!\!-\!1 & & & & \! 1 \\ \end{array}\right],$$

where, again, $\sigma^2_c=\sigma^2_b+\sigma^2_s$ and where the group design matrix, $X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}$, de-means the first level estimates for each subject. Assume for simplicity that $V_k=\sigma^2_w{\mathbf {I}}$, $X_k^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}X_k = 1$ and define $u^2=\sigma^2_w+\sigma^2_c$. Then $V_{\mbox{\tiny\textit{\sffamily {$\!$G2}}}}^{\mbox{\scriptsize\textit{\sffamily {-1}}}}$ will be block diagonal with blocks of

$$\widetilde{U}^{\mbox{\scriptsize\textit{\sffamily {-1}}}}=\frac{1... ...}=\left[\begin{array}{cc} u^2&\sigma_s^2\\ \sigma_s^2&u^2 \end{array}\right].$$

Furthermore, let $c_1 = u^2+\sigma_s^2 =\sigma^2_w+\sigma^2_b +2\sigma_s^2$ and $c_2=u^2-\sigma^2_s = \sigma^2_w+\sigma^2_b$. Then

$$X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsiz... .../N &&&\\ &c_1 && \\ &&\ddots &\\ &&&c_1 \end{array}\right],$$

so that

$$\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}... ...\widehat{\beta}_{Ns_2} \right)^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}.$$

Using $C_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}=(1,0, \dots, 0)^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}$, the group parameter estimate writes as

$$\widehat{b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}}=C_{_{\mbo... ...m{Var}(\widehat{b_{_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}}})=\frac{c_2}{2N}.$$

As expected, the variance of the group level estimate no longer depends on the between-subject variance $\sigma_s^2$. Note that this approach is equivalent to using a three level approach with an unpaired $t$-test of de-meaned repeated measures.