# How to model the relationship between number of subjects per group, derived using standard power analysis methods, and study specific parameters

I am interested in estimating how many subjects should be included in a brain imaging study. Although the design is a fairly straight forward cross-sectional comparison, there are a number of tweakable image processing steps between the raw image and the processed image in which we carry out pixel-wise comparisons.

I've used the power.t.test function in R to estimate the number of subjects required to reach statistical significance, using hypothetical delta values and sd estimates from a control population. I've run these analyses with varying image processing settings, resulting in a large number of "number of subjects per group" vs "some image processing parameter" plots. So many that it looks kind of messy to provide hundreds of plots.

What I would like is an empirical equation that allows users to estimate the number of subjects per group as a function of effect size, alpha, and a few other image processing specific parameters. That way I only need to report the coefficients for the equation rather than supplying hundreds of graphs. Is this kind of thing possible?

So far if I model N ~ k1/effect.size^2 that looks OK for some values of alpha (for example) but doesn't work so good for others. A log-log plot doesn't model the relationship well either.

I can't seem to find an explicit formula that relates the number of subjects per group with the other factors in a power analysis. Does this exist?

Thanks

## 1 Answer

I understand you have

• many brain imaging datasets
• classified into 2 groups, study and control
• image processing methods
• parameters of the image processing to tune
• a collection of processed images with various parameter settings

and that you will

• run a new study recruiting similar subjects and controls
• pick a single pixel of each dataset
• compare pixels of study subjects with controls
• use a two sample T test

and you want

• a sample size formula that includes the parameters of image processing.

I believe you need to explore and understand how the difference between groups and the standard deviation depends on the parameters of image processing. In a second step you can understand how the requires sample size depends on the parameters. (You mentioned log-log plot: the relationship may only be linear after double log transformation if very special conditions are fulfilled, however a linear approximation may be satisfactory in the parameter range you find practical.)

I suggest to perform visualization of the dependence of effect size and SD on the parameters of image processing, and other explorative statistics. After these you can set up a model that predicts the inputs of the sample size formula using the image processing parameters.

You may find that even your hundreds of parameter settings already evaluated do not give sufficient insight (especially if there are many parameters), in which case you may need to evaluate further parameter combinations. Most image processing methods may be automated, automation saves a lot of time when tweaking the parameters.