Update stats plots, add longitudinal sample size calculation

[PaulBautin]

This PR intends to homogenize notations and conventions between the graphs presented in the manuscript and the "csa_atrophy" repo.

Done:

• Update plots: sample_size, error_function_of_csa
• Update legends: boxplot_atrophy, boxplot_csa
• Draw bissection automatically on the boxplot_atrophy graph
• Add graph showing error in function of CSA
• Add longitudinal study sample size computation and update ref for sample size computation
• Update README

FIX #95, FIX #92, FIX #100, FIX #80, FIX #103

[PaulBautin]

Up to now the computed longitudinal sample sizes were small (< 1). After investigation, we observed that the SD of the difference of measured CSA across subjects were computed using mean CSA across transformations ex: diff(sI, rX) = Mean[CSA(sI, r1, :)] - Mean[CSA(sI, rX, :)] (1) This had for consequence that the SD of differences did not take into account the variability due to transformations.

Implemented in 7874439, contrary to formula (1) program does not mean CSA across transformations but randomly samples a CSA value for each subject. ex: diff(sI, rX) = CSA(sI, r1, tY) - CSA(sI, rX, tZ) (2) The results with this method are much closer to what was previously found in literature. However, it also puts in evidence a large variability in longitudinal sample sizes due to transformations. This is surprising because the mean intra-subject SD is relatively small (hence, I would not expect such an influence of the transformation-related variability).

[PaulBautin]

With commit 05d7fcd using difference formula (tY and tZ are two different transforms):

diff(sI, rX) = CSA(sI, r1, tY) - CSA(sI, rX, tZ) (2)

longitudinal sample size variability is relatively important

Also note that, when looking at between-group differences (vs. paired differences as described above), the formula was also updated as follows:

CSA(sI, rX) = CSA(sI, rX, tZ) and not CSA(sI, rX) = MEAN[CSA(sI, rX, :)]

Results seem to vary much less: SD of sample size no more than 3% of sample size (between groups).

Therefore, my best guess is that the important variability found computing longitudinal sample sizes are mostly due to the variability of CSA measures between scalings (which has already been shown in article).

@jcohenadad, should we continue with these results? My idea is now to keep the Monte Carlo simulations for both sample size computations.

[jcohenadad]

@PaulBautin This is an interesting investigation but I need more guidance to understand the formula described in https://github.com/sct-pipeline/csa-atrophy/pull/98#issuecomment-845227755. Without the context of the code I cannot advise on what is the most appropriate solution. I suggest we discuss it in a meeting.

[PaulBautin]

@jcohenadad, could you review? I think this PR is ready to be merged (plots in PR match plots in article).

[PaulBautin]

@jcohenadad, could you review? This PR should be merged into master because plots and stats for the article are based on this PR.

[jcohenadad]

sorry-- realistically i will not have time to review