# Expressing the effect size relative to controls

The apparent Fibre Density (FD) and Fibre Density and Cross-section (FDC) are relative measures and have arbitrary units. Therefore the units of `abs_effect.mif`

output from fixelcfestats are not directly interpretable. In a patient-control group comparison t-test, one way to present results is to express the absolute effect size as a percentage relative to the control group mean.

To compute FD and FDC percentage decrease effect size use:

```
mrcalc stats_fd/abs_effect.mif control_mean.mif -div 100 -mult stats_fd/percentage_effect.mif
```

The mean value in the control group can be obtained in one of two ways:

Use the relevant beta coefficient image from fixelcfestats

If there is a column in the design matrix that contains the value 1 for all subjects in the control group and 0 for all other subjects,

*and*if any & all nuisance regressors were de-meaned prior to inserting them into the design matrix, then the relevant beta coefficient image provided by fixelcfestats (e.g.`stats_fd/beta0.mif`

) can be interpreted directly as the control group mean.Explicitly calculate the control group mean

Using the smoothed data from fixelfilter, the mrmath command with the

`sum`

operation can be used to explicitly compute the mean across the set of images corresponding to the control group.

Because the Fibre Cross-section (FC) measure is a scale factor it is slightly more complicated to compute the percentage decrease. The FC ratio between two subjects (or groups) tells us the direct scale factor between them.

For example, for a given fixel if the patient group mean FC is 0.7, and control mean is 1.4, then this implies encompassing fibre tract in the patients is half as big as the controls: 0.7/1.4 = 0.5. I.e. this is a 50% reduction wrt to the controls: 1 - (FC_patients/FC_controls)

Because we peform FBA of log(FC), the abs_effect that is output from fixelcfestats is: abs_effect = log(FC_controls) - log(FC_patients) = log(FC_controls/FC_patients). Therefore to get the percentage effect we need to perform 1 - 1/exp(abs_effect):

```
mrcalc 1 1 stats_fc/abs_effect.mif -exp -div -sub stats_log_fc/percentage_effect.mif
```