General Linear Model

Second-level analyses allow researchers to make inferences about properties of groups or populations, by generalizing from the observations of only a subset of subjects in a study. CONN uses the General Linear Model (GLM) for all second-level analyses of functional connectivity data. This section describes the mathematics behind the General Linear Model, including model definition, parameter estimation,  and hypothesis testing framework. It also includes several practical examples and general guidelines aimed at helping researchers use this method to answer their specific research questions

The (generally unknown) effective association in the  population between each of the explanatory measures in X and each of the outcome measures in Y is characterized by the matrix B in the GLM equation. The vector epsilon represents the cumulative contribution on the outcome measure Y of all other unspecified factors beyond those that can be predicted from the knowledge of X. (i.e. the model error term). Based on the Central Limit Theorem, it is often reasonable to model this cumulative contribution as a Normally distributed random term. GLM assumes this term is independent across subjects and follows a multivariate normal distribution with mean zero and an arbitrary variance-covariance structure across outcome measures. 

For example, from N subjects' data, we would typically construct the data and design matrices, respectively, as X = [x1' x2' x3' ... xN']' and Y = [y1' y2' y3' ... yN']' , resulting from vertically concatenating the corresponding x[n] and y[n] row vectors across all relevant subjects, and then use the OLS equation above to compute the best linear unbiased estimator of the unknown matrix B from the observed X and Y data.

Choosing different forms of the matrix C allows us to construct hypotheses that address specific combinations of explanatory measures X, as each column in the vector/matrix C is paired with the same column of the design matrix X. Similarly, choosing different forms of the matrix M allows us to construct hypotheses that address specific combinations of outcome measures Y, as each column in the contrast vector/matrix M is paired with the same column of the data matrix Y.  Last, the choice of contrast matrix D determines the hypothesized net effect of the selected combination of explanatory measures on the selected combination of outcome measures (e.g. D is set to zero in many standard null-hypothesis scenarios). 

Wilks' Lambda distributions have three parameters: the number of dimensions a, the error degrees of freedom b, and the hypothesis degrees of freedom c, which are fully determined from the dimensionality and rank of the original data and choice of contrast matrices. Unfortunately, Wilks' Lambda distributions are only tabulated for a limited number of scenarios/dimensions, so CONN GLM implementation uses the following transformations in order to derive appropriate statistics and p-values for any tested hypothesis, depending on the specific values of a, b, and c:

note-1: it could be argued that two-sided hypotheses of the form "CBM'=0 " are almost surely false in real world data, where an effect may be arbitrarily small but almost never precisely zero. Because of this, failure to reject a hypothesis of this form often simply indicates that the effect being evaluated (e.g. difference in connectivity between two groups) is too small to be detectable with the current experimental setup (e.g. with the current acquisition parameters and number of subjects) rather than truly non-existent. In this context, it is generally recommended, and not a bad idea at all, to attempt to always quantify and report the effects measured (e.g. report the estimated B values) instead of only relying on and reporting the significance of the hypotheses being evaluated.  This can help build increasingly better model-based estimates of these effects, going beyond the initial but limited question of whether they "exist" (i.e. are they non-zero) or not. Last, testing one-tailed hypotheses of the form "CBM'>D" (these are available in CONN for univariate effects using LRT case 1) can also be used as a tool to combine "practical" significance (is an effect large?) with "statistical" significance (how confident are we of this, given the available data?) 

Another common alternative in these scenarios is to apply first a linear dimensionality reduction step by projecting the original data Y over the subspace spanned by the first few singular vectors from a model-agnostic Singular Value Decomposition (Strang 2007). As long as the dimensionality of this subspace is chosen to be smaller than the error degrees of freedom b, the full GLM error covariance over this subspace can be properly estimated, allowing the evaluation of Likelihood Ratio Tests using the standard Wilks' Lambda statistics (LRT case 2 or case 4). 

 1. Subject-effects: what is the list of explanatory/independent measures that we would like to include in this analysis? (i.e. what are the columns of X?)

This is typically defined simply by listing a series of subject-level covariates (e.g. age, IQ). In CONN these variables are defined in Setup.Covariates (2nd-level). In addition to continuous variables, such as age, or IQ, dummy-coded group variables are often useful to identify groups of subjects in our studies (e.g. a Patients covariate may take a value of 1 for patients, and 0 for controls). One such dichotomous variable identifying the entire group of subjects in our study is often used in simple designs where constant effects across all subjects are needed (in CONN this variable is automatically created and named AllSubjects, containing the value 1 for every subject)  

 2. Between-subjects contrasts: among these explanatory/independent measures, which one(s) or which combination of them do we want to evaluate/test?  (i.e. what is the C vector/matrix?)

In the simplest scenario this contrast is just a vector, with as many elements as explanatory/independent measures, and having a value of 1 for the individual effect that we would like to evaluate/test and 0's for all other elements (e.g. if we have entered two subject-effects, characterizing Patient and Control subjects, respectively, a contrast vector with values [1, 0] would specify that we would like to evaluate/test the effect in Patients only. Other simple scenarios involve a contrast that acts to compare two effects, which is defined simply by entering a 1 and a -1 in the two elements that we would like to compare, and 0's in all other elements -if there are any-  (e.g. a contrast vector with values [-1, 1] in the previous example would compare the effect in Controls to that in Patients). More complex contrast vectors can be specified simply as the weights of any desired linear combination of our model subject-effects (e.g. a contrast vector with values [0.5, 0.5] in the previous example would estimate the average effect across both Patients and Control subjects). Last, contrast matrices can be used to evaluate/test multiple effects jointly, where each individual effect is defined in the regular manner as an contrast vector, and those vectors are simply concatenated into a matrix (e.g. in a model with three groups instead of two, a contrast matrix [1, -1, 0; 0, 1, -1; 1, 0, -1] evaluates/tests the presence of any differences between the three groups)

 3. Conditions (also known as measures or outcomes): what is the list of outcome/dependent measures that we would like to include in this analysis? (i.e. what are the columns of Y?)

This is typically defined by listing the individual outcome variables that we would like to investigate (e.g. SBC maps during a single rest condition in a standard resting state analysis, or SBC maps during Pre and Post conditions in an intervention design). In CONN this is defined by a combination of choosing which particular first-level functional connectivity measures, and which experimental conditions (if applicable), we would like to evaluate (e.g. SBC maps with one or several seeds during rest)

 4. Between-conditions contrasts (also known as between-measures or within-subjects contrast): among these outcome/dependent measures, which one(s) or which combination of them do we want to evaluate/test ? (i.e. what is the M vector/matrix?)

These contrasts are defined in the same way as the between-subject contrasts above, now spanning across conditions/sources instead of across subject-effects (e.g. if we have selected two conditions, characterizing SBC maps pre- and post- intervention, a contrast vector with values [-1, 1] would compare the connectivity values across these two conditions)

Behind the apparent simplicity of these choices, a perhaps surprising array of different analyses can be specified using this framework. Some examples are shown in the table below 

These results indicate that the functional connectivity trajectories before and after intervention in the two groups are significantly different (with higher connectivity in clinic #1 compared to clinic #2; 0.15 higher pre-intervention and 0.26 higher post-intervention). Note that GLM analyses in this context are exactly equivalent to those from a mixed-model two-way ANCOVA evaluating potential main effects of clinic (a between-subjects factor).

These results indicate that there are significant functional connectivity changes post- vs. pre- intervention in our subjects (with general increases in connectivity post- intervention; a 0.23 increase in clinic #1 and a 0.12 increase in clinic #2). Note that GLM analyses in this context are exactly equivalent to those from a mixed-model two-way ANCOVA evaluating potential main effects of treatment (a within-subjects factor).

These results indicate that, if there is a clinic by treatment interaction, the effect is relatively small and cannot be detected with this study sample size. Note that, as before, GLM analyses in this context are exactly equivalent to those from a mixed-model two-way ANCOVA evaluating potential interactions between clinic (a between-subjects factor) and treatment (a within-subjects factor).

References

Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979). Multivariate analysis. Probability and Mathematical Statistics, London: Academic Press, 1979.

Rao, C. R. (1951). An asymptotic expansion of the distribution of Wilks’ criterion. Bulletin of the International Statistical Institute, 33(2), 177-180.

Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics bulletin, 2(6), 110-114.

see cluster-level inferences for additional information on how to make inferences and interpret second-level GLM results