# How to choose the correct sample and hypothesis test for spatiotemporal observations?

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

Consider the following scenario:

1. 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.
2. 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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