by chai90
Last Updated September 20, 2018 22:19 PM

Consider the following scenario:

- There exist $N=10$ participants in a study. Each participant $i$ is monitored under baseline conditions and later under experiment conditions. That is, the baseline and experiment responses are paired in participant space by design.
- The baseline and experiment conditions each spanned $D=10$ days in time, respectively. During each baseline day $d_b$ or experiment day $d_e$, each participant's response was monitored at regular instants in time $t_k \in [0,86400)$ seconds.

*Notation*: Let the baseline and experiment responses for the $i^{th}$ participant on a day $d$ at time instant $t_k$ be represented by $y_b(i,d_b,t_k)$ and $y_e(i,d_e,t_k)$, respectively.

**Question**: Given these spatiotemporal samples which are paired spatially, I need to determine whether or not the experiment and baseline responses are different without completing averaging out the temporality? After several days of exploration of parametric and nonparametric tests, I am stuck with data that doesn't quite fit into the assumptions of any test [see assumption notes below]. I would appreciate any help on constructing the examination sample and hypothesis test, and estimating confidence interval and effect size.

*Constraints*: I do not want to average out the temporality across days (since the spatial sample size is only 10 and I would like my claims generalize not only across the participant population but also across days) but I can average out the temporality across times within a day to obtain daily average for each participant ($Y_b(i,d_b)$, $Y_e(i,d_e)$), defined across several days and participants.

*Independence assumption (of samples)*: Can I treat such daily average differences ($Y_b(i,d_b)-Y_e(i,d_e)$) for a given participant to be independent sampled across days? And given the independence of these differences across participants, can I consider the differential responses $Y_b(i,d_b)-Y_e(i,d_e)$ to be independently sampled across space and time (days)?

*Symmetry/normality assumptions (of sample distribution)*: Neither the sample $Y_b(i,d_b)-Y_e(i,d_e)\ \forall\ i,days$ nor the sample $y_b(i,d_b,t_k)-y_e(i,d_e,t_k)\ \forall\ i,days,k$ seem to be symmetrical (so, not normally distributed either).

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