Contents

# Introduction

This page describes the differences and similarities between DODS and DOSS. DODS stands for "Different Offset, Different Slope". DOSS stands for "Different Offset, Same Slope". These refer to methods for **automatically** creating a design matrix from an FSGD file and is used by mri_glmfit and qdec. The terms DODS and DOSS are unique to FreeSurfer (ie, you won't find them in a stats textbook:). For examples of FSGD files see here.

There are only differences between the two in cases where there are more than one group (or class) and at least one continuous covariate (variable).

Consider the simplest case, in which there are two groups (eg, Male and Female) and one continuous covariate (eg, Age), and we are interested in how cortical thickness relates to these variables using a linear model.

A linear model implies two parameters: an offset and a slope. The offset is also called an intercept. The offset/intercept is the thickness at a covariate of 0 (eg, birth) and is measured in the same units as thickness (eg, mm). The slope is the rate of change of the thickness and is measured in mm/years.

# DODS

Generally speaking, we would probably use a separate linear model for each group, meaning that each group would have its own offset and its own slope (thus DODS) for a total of four parameters. The model here would imply that Males and Females could start at different thicknesses and the thicknesses could grow/shrink at different rates. The design matrix would have four columns (one for each parameter). As always, the design matrix has a row for each subject. In the automatic construction, the columns (regressors) would have the following format and interpretation:

Regressor 1: has a value of 1 if the subject (row) is a Male, 0 otherwise. This regressor codes for membership in the Male group.

Regressor 2: has a value of 1 if the subject (row) is a Female, 0 otherwise. This regressor codes for membership in the Female group.

Regressor 3: has a value of the subject's age if the subject is a Male, 0 otherwise. This regressor codes for the slope of the Males' thickness vs age.

Regressor 4: has a value of the subject's age if the subject is a Female, 0 otherwise. This regressor codes for the slope of the Females' thickness vs age.

Generally speaking, the number of DODS regressors will be:

NregressorsDODS = Nclasses*(Nvariables+1) = 2*(1+1) = 4

The order will be intercept, slope for variable 1, slope for variable 2, etc.

# DOSS

In the DOSS model, we still allow for the different groups to start at different places (different offset), but we constrain them to evolve at the same rate (same slope). In a scatter plot of the data and best fit lines, the two lines would be constrained to be parallel. The assumption is that the rate (slope) is the same across groups.

The design matrix would have three columns. As always, the design matrix has a row for each subject. In the automatic construction, the columns (regressors) would have the following format and interpretation:

Regressor 1: has a value of 1 if the subject (row) is a Male, 0 otherwise. This regressor codes for membership in the Male group.

Regressor 2: has a value of 1 if the subject (row) is a Female, 0 otherwise. This regressor codes for membership in the Female group.

Regressor 3: has a value of the subject's age regardless of the group. This regressor codes for the slope of the thickness vs age.

Generally speaking, the number of DODS regressors will be:

NregressorsDOSS = Nclasses + Nvariables = 2+1 = 3

The order will be the Nclasses intercepts followed by the Nvariables slopes.

# DODS or DOSS: which one should I use?

The choice of DODS and DOSS usually comes down to three things:

Truth

Power

Your Test of Interest

"Truth" refers to how you believe the data actually behave. If you truly believe that the groups evolve at different rates, then your model should reflect this, and thus you should use DODS.

"Power" refers to the fact that DOSS has fewer regressors than DODS and so will have a larger degree-of-freedom (DOF). The DOF is the number of subjects minus the number of regressors. In the above example, the DODS only has one fewer DOF than DOSS, which probably won't be that important if you have more than 20 subjects. On the other hand, if you have 4 groups and 3 variables, then DODS will have 16 regressors whereas DOSS will have only 7, meaning that DODS will have 8 fewer DOFs than DOSS, which could be important.

"Your Test of Interest". If you are interested in an interaction between the covariate and group (eg, Gender-x-Age), then you can only test this with a DODS model. If you are interested in the difference between the groups regressing out the effect of your covariate (eg, is there a difference between males and females accounting for age?), then you should first use DODS to test for an interaction. If there is not a significant interaction, then re-run the analysis using DOSS testing for a difference in offset between the groups. If there is a significant interaction, then things get complicated. Some statisticians will say that you cannot/should not even test for a difference between the groups. Having said that, it happens all the time in neuroimaging.

You can, of course, drive yourself crazy deciding how to model your data. I refer you to the words of the famous statistician George Box: "All models are wrong, but some are useful". Make sure your models are useful.

# SODS?

We could have taken the extra step of automatically generating models that have the same offset but allow for different slopes. But we didn't.

# I want more flexibility than DODS and DOSS!

To be sure, there are a lot more ways to model your data than DODS and DOSS, however, it becomes increasingly difficult to **automatically** generate design matrices. The FreeSurfer tools (namely mri_glmfit) allows you as much flexibility as you want in modeling, but you have to construct the design matrix yourself and pass it to mri_glmfit with the --X flag.

*Author: Doug Greve*