Single-Subject Analysis

In a single-subject fMRI study, one collects, at each voxel, a time series of responses (intensities of the BOLD signal) to a stimulus, e.g. an ON (condition B) and OFF (condition A) experiment (box-car design). Let $y_t$, be the time series observed from $t=1\cdots T$, at a given voxel. Among the $T$ values observed at this voxel, $T_A$ of them were recorded while under condition A (OFF or rest condition) and $T_B$ of them were recorded while under condition B (ON or stimulation condition) according to the paradigm. The observations are assumed to be independent and identically distributed (i.i.d.)⁴ and to come from a Normal distribution with the same variance $\sigma^2$ and different means $\mu_A$ and $\mu_B$ for each sample. To decide if there was an activation (at this voxel) during the experiment, one has to compare the means in the two conditions. If the difference of the means is big enough relatively to its dispersion, one will assume activation. For that purpose is used the following statistic (where the denominator is an estimate of the standard deviation of the numerator estimating the difference of the means) with its derived distribution (see appendix A) under the null hypothesis (of no activation $\mu_A=\mu_B$):

$$t_o=\frac{\bar{y_B}-\bar{y_A}}{\sqrt{ \hat{\sigma}^2 (1/T_A + 1/T_B) } }\sim t_{dist}(T_A +T_B -2)$$ (1)

which means that $t_o$ comes from (or follows) a Student distribution ($t$) with $(T_A +T_B -2)$ degrees of freedom (df). $\bar{y_A}$ and $\bar{y_B}$ are the sample means (i.e. $\bar{y_A}=1/T_A \sum_{t\in A} y_t$) and $\hat{\sigma}^2$ is an estimate of the common variance, which is usually the pooled estimate $\hat{\sigma}^2=s_p^2=\frac{(T_A -1)s_A^2 +(T_B -1)s_B^2}{(T_A+T_B -2)}$ with $s_A^2=1/(T_A -1)\sum_{t\in A}( y_t-\bar{y_A})^2$. In fMRI the data is often balanced, i.e. $T_A=T_B=T/2$; thus:

$$t_o=\frac{\bar{y_B}-\bar{y_A}}{2 \hat{\sigma}/ \sqrt{T} } \sim t_{dist}(T -2)$$ (2)

Now following classical hypothesis testing, if the probability $p(t_{dist}(T_A +T_B -2)\geq t_o)<\alpha$ for a chosen level $\alpha$ ($e.g.$ 0.05 or 0.01 or 0.001) one rejects the null hypothesis and decides that the voxel was activated at the level $\alpha$ (level of decision not of activation)⁵. Remark: If you decide (test) that the variances are unequal, the denominator has to be estimated by $\sqrt{ (s_A^2/T_A + s_B^2/T_B)}$ and the degrees of freedom are approximated by the Satterthwaite formula[10]:

$$\frac{[s_A^2/T_A + s_B^2/T_B]^2} {(s_A^2/T_A)^2/(T_A -1)+(s_B^2/T_B)^2/(T_B -1) }$$ (3)

but if the data is balanced and the sample size not too small the departure from equal variances is usually negligible.