Hi Ana,

Iâ€™ll give this question its own topic instead, since itâ€™s slightly different to the â€śtricksâ€ť that are currently present in that Wiki page; but itâ€™s a *great* question, involving data manipulations that Iâ€™ve had to do myself for one of my own experiments, and something that should ideally appear if I ever find the time to write a more general GLM wiki page.

What the beta coefficients provided by the GLM encode is the *gradient* between the predicted values of the exploratory variable (i.e. the imaging metric data you provide as input) and the explanatory variable (the column of values in the GLM).

Imagine that you have some nuisance regressor *X*, and that the value of the beta coefficient corresponding to the *X* column is 2.0. And letâ€™s say youâ€™re testing FDC for simplicity. What that means is that the linear model predicts that if, for some hypothetical subject, the value in the *X* column were to increase by 1.0, the value of FDC would increase by 2.0.

In your case, *X* is actually: `((age - mean_age) / std_age)`

.

So, if the second beta coefficient is 2.0, that means that if `((age - mean_age) / std_age)`

were to increase by 1.0, FDC would increase by 2.0.

Therefore, if you *multiply* the value of that beta coefficient by `(std_age)`

, what that gives you is the modelâ€™s prediction of the rate of change of FDC with respect to age!

(The subtraction disappears; weâ€™re only interested in rate of change, and the age offset gets gobbled up as part of the global intercept term; but you can use the same logic to derive the complete estimated linear relationship with respect to age)

Now Iâ€™ve explained all of this in terms of beta coefficients. But the absolute effect is simply the inner product between the beta coefficients and the contrast matrix. Since your contrast is `[0 -1 0]`

, what file `abs_effect_size.mif`

is actually providing you with is: `-((age - mean_age) / std_age)`

(note the negative!). So you would want to multiply by `(-std_age)`

to get the rate of change of FDC with respect to age, as opposed to the negative of such.

This *isnâ€™t actually any different* to the interpretation of these values in a group difference context!

For instance, letâ€™s say that youâ€™re also interested in the magnitude of the sex effect in your cohort. Internally, the GLM doesnâ€™t care that the values in that third column are all +1 or -1; it still treats it as a continuous distribution. What the third beta coefficient provides you with is the rate of change of FDC with respect to *Y*, where *Y* is a dummy variable â€śencodingâ€ť sex. But a unitary change in *Y* is only *half* of the difference between female and male; i.e. the â€śdistanceâ€ť between female and male in the dimension of variable *Y* is 2.0. Therefore, if you actually wanted to know what the model predicts to be the difference in FDC between an exemplar male and an exemplar female (additionally regressing for all other variables), you would need to *double* the corresponding beta coefficient!

(Iâ€™ve actually wondered about advocating the use of `-0.5`

and `0.5`

for categorical dummy variables for this reasonâ€¦)

Cheers

Rob