8.2. When and how to center a variable?¶
8.2.1. Major points¶
Centering is crucial for interpretation when group effects are of interest.
Centering is not necessary if only the covariate effect is of interest.
Centering (and sometimes standardization as well) could be important for the numerical schemes to converge.
Centering does not have to be at the mean, and can be any value within the range of the covariate values.
When multiple groups of subjects are involved, centering becomes more complicated. Sometimes overall centering makes sense. However, in contrast to the popular misconception in the field, under some circumstances within-group centering can be meaningful (and even crucial) and may avoid the following problems with overall or grand-mean centering:
loss of the integrity of group comparisons;
invalid extrapolation of linearity.
When multiple groups of subjects are involved, it is recommended that the interactions between groups and the quantitative covariate be modeled unless prior information exists otherwise
To avoid unnecessary complications and misspecifications, categorical variables, regardless of interest or not, are better modeled directly as factors instead of user-defined variables through dummy coding as typically seen in the field. In doing so, centering can be automatically taken care of by the program without any potential mishandling, and potential interactions would be properly considered.
8.2.2. Usage clarifications of “covariate”¶
There are three usages of the word covariate commonly seen in the literature, and they cause some unnecessary confusions. Originally the word was adopted in the 1940s to connote a variable of quantitative nature (e.g., age, IQ) in ANCOVA, replacing the phrase concomitant variable by R. A. Fisher. Such usage has been extended from the ANCOVA context, and sometimes refers to a variable of no interest (extraneous, confounding or nuisance variable) to the investigator (e.g., sex, handedness, scanner). These subtle differences in usage stem from designs where the effects of interest are experimentally manipulable while the effects of no interest are usually difficult to control or even intractable. Occasionally the word covariate means any explanatory variable among others in the model that co-account for data variability. Mathematically these differences do not matter from the modeling perspective. Here we use quantitative covariate (in contrast to its qualitative counterpart, factor) instead of covariate to avoid confusion.
8.2.3. Purpose of modeling a quantitative covariate¶
Ideally all samples, trials or subjects, in an FMRI experiment are drawn from a completely randomized pool in terms of BOLD response, cognition, or other factors that may have effects on BOLD response. However, such randomness is not always practically guaranteed or achievable. In many situations (e.g., patient recruitment) the investigator does not have a set of homogeneous subjects, and the potentially unaccounted variability sources in cognitive capability or BOLD response could distort the analysis if handled improperly, and may lead to compromised statistical power, inaccurate effect estimates, or even inferential failure. For example, direct control of variability due to subject performance (e.g., response time in each trial) or subject characteristics (e.g., age, IQ, brain volume, psychological features, etc.) is most likely unrealistic. Instead, indirect control through statistical means may become crucial, achieved by incorporating one or more concomitant measures in addition to the variables of primary interest. Such concomitant variables or covariates, when incorporated in the model, might provide adjustments to the effect estimate, and increase statistical power by accounting for data variability some of which cannot be explained by other explanatory variables than the covariate. Such adjustment is loosely described in the literature as a process of “regressing out”, “partialling out”, “controlling for” or “correcting for” the variability due to the covariate effect. Typically, a covariate is supposed to have some cause-effect relation with the outcome variable, the BOLD response in the case of FMRI data. Potential covariates include age, personality traits, and behavioral data. They are sometime of direct interest (e.g., personality traits), and other times are not (e.g., age). They are mostly continuous (or quantitative) variables; however, discrete (qualitative or categorical) variables are occasionally treated as covariates in the literature (e.g., sex) if they are not specifically of interest except to be “regressed” out in the analysis.
While stimulus trial-level variability (e.g., reaction time) is usually modeled through amplitude or parametric modulation in single subject analysis, the covariates typically seen in the brain imaging group analysis are task-, condition-level or subject-specific measures such as age, IQ, psychological measures, and brain volumes, or behavioral data at condition- or task-type level. Although amplitude modulation accounts for the trial-to-trial variability, for example, with linear or quadratic fitting of some behavioral measures that accounts for habituation or attenuation, the average value of such behavioral measure from each subject still fluctuates across subjects. Therefore it may still be of importance to run group analysis with the average measure from each subject as a covariate at group level.
Incorporating a quantitative covariate in a model at the group level may serve two purposes, increasing statistical power by accounting for data variability and estimating the magnitude (and significance) of the confounding effect. However, two modeling issues deserve more attention in practice, covariate centering and its interactions with other effects, due to their consequences on result interpretability and inferences. And these two issues are a source of frequent inquiries, confusions, model misspecifications and misinterpretations across analysis platforms, and not even limited to neuroimaging community. Centering a covariate is crucial for interpretation if inference on group effect is of interest, but is not if only the covariate effect is of interest. And in contrast to the popular conception, centering does not have to hinge around the mean, and can be any value that is meaningful and when linearity holds. This is the reason we prefer the generic term “centering” instead of the popular description “demeaning” or “mean-centering” in the field.
Centering with one group of subjects¶
Two parameters in a linear system are of potential research interest, the intercept and the slope. The former reveals the group mean effect when the covariate is at the value of zero, and the slope shows the covariate effect accounting for the subject variability in the covariate. In other words, the slope is the marginal (or differential) effect of the covariate, the amount of change in the response variable when the covariate increases by one unit. For example, in the case of IQ as a covariate, the slope shows the average amount of BOLD response change when the IQ score of a subject increases by one. Depending on the specific scenario, either the intercept or the slope, or both, are of interest to the investigator. However, one would not be interested in the group or population effect with an IQ of 0. Instead the investigator would more likely want to estimate the average effect at the sample mean (e.g., 104.7) of the subject IQ scores or the population mean (e.g., 100). If the group average effect is of research interest, a practical technique, centering, not usually highlighted in formal discussions, becomes crucial because the effect corresponding to the covariate at the raw value of zero is not necessarily interpretable or interesting.
Centering typically is performed around the mean value from the sampled subjects, and such a convention was originated from and confounded by regression analysis and ANOVA/ANCOVA framework in which sums of squared deviation relative to the mean (and sums of products) are computed. In most cases the average value of the covariate is a valid estimate for an underlying or hypothetical population, providing a pivotal point for substantive interpretation. However, the centering value does not have to be the mean of the covariate, and should be based on the expediency in interpretation. Suppose the IQ mean in a group of 20 subjects is 104.7. By subtracting each subject’s IQ score by 104.7, one provides the centered IQ value in the model (1), and the estimate of intercept α0 is the group average effect corresponding to the group mean IQ of 104.7. On the other hand, suppose that the group of 20 subjects recruited from a college town has an IQ mean of 115.0, which is not well aligned with the population mean, 100. Through the manual transformation of centering (subtracting the raw covariate values by the center), one may analyze the data with centering on the population mean instead of the group mean so that one can make inferences about the whole population, assuming the linear fit of IQ holds reasonably well within the typical IQ range in the population. Another example is that one may center the covariate with the same value as a previous study so that cross-study comparison can be achieved. Similarly, centering around a fixed value other than the mean is typically seen in growth curve modeling for longitudinal studies (Biesanz et al., 2004) in which the average time in one experiment is usually not generalizable to others. For instance, in a study of child development (Shaw et al., 2006) the inferences on the correlation between cortical thickness and IQ required that centering of the age be around, not the mean, but each integer within a sampled age range (from 8 up to 18). In general, centering artificially shifts the values of a covariate by a value that is of specific interest (e.g., IQ of 100) to the investigator so that the new intercept corresponds to the effect when the covariate is at the center value. In other words, by offsetting the covariate to a center value c the x-axis shift transforms the effect corresponding to the covariate at c to a new intercept in a new system.
In addition to the distribution assumption (usually Gaussian) of the residuals (e.g., di in the model (1)), the following two assumptions are typically mentioned in traditional analysis with a covariate (e.g., ANCOVA): exact measurement of the covariate, and linearity between the covariate and the dependent variable. Regarding the first assumption, the explanatory variables in a regression model such as (1) should be idealized predictors (e.g., presumed hemodynamic response function), or they have been measured exactly and/or observed without error. This assumption is unlikely to be valid in behavioral data, and significant unaccounted-for estimation errors in the covariates can lead to inconsistent results and potential underestimation of the association between the covariate and the response variable—the attenuation bias or regression dilution (Greene, 2003). In regard to the linearity assumption, the linear fit of the covariate effect may predict well for a subject within the covariate range, but does not necessarily hold if extrapolated beyond the range that the sampled subjects represent as extrapolation is not always reliable or even meaningful. The assumption of linearity in the traditional ANCOVA framework is due to the limitations in modeling interactions in general, as we will see more such limitations later. Nonlinearity, although unwieldy to handle, are not necessarily prohibitive, if there are enough data to fit the model adequately. And nonlinear relationships become trivial in the context of general linear model (GLM), and, for example, quadratic or polynomial relationship can be interpreted as self-interaction.
To reiterate the case of modeling a covariate with one group of subjects, the inclusion of a covariate is usually motivated by the more accurate group effect (or adjusted effect) estimate and improved power than the unadjusted group mean and the corresponding significance testing obtained through the conventional one-sample Student’s t-test. Centering the covariate may be essential in interpreting the group effect (or intercept) while controlling for the variability in the covariate, and it is unnecessary only if the covariate effect (or slope) is of interest in the simple regression model. The center value can be the sample mean of the covariate or any other value of interest in the context.
Centering with more than one group of subjects¶
When multiple groups are involved, four scenarios exist regarding centering and interaction across the groups: same center and same slope; same center with different slope; same slope with different center; and different center and different slope. None of the four scenarios is prohibited in modeling as long as a meaningful hypothesis can be framed. However, presuming the same slope across groups could be problematic unless strong prior knowledge exists. We suggest that researchers report their centering strategy and justifications of interaction modeling or the lack thereof. Extra caution should be exercised if a categorical variable is considered as an effect of no interest because of its coding complications on interpretation and the consequence from potential model misspecifications.
Is within-group centering meaningful?¶
When more than one group of subjects are involved, even though within-group centering is generally considered inappropriate (e.g., Poldrack et al., 2011), it not only can improve interpretability under some circumstances, but also can reduce collinearity that may occur when the groups differ significantly in group average. More specifically, within-group centering makes it possible in one model
to compare the group difference while accounting for within-group age differences, and at the same time, and
to examine the age effect and its interaction with the groups.
If the groups differ significantly regarding the quantitative covariate, cross-group centering may encounter three issues: collinearity between the subject-grouping variable and the quantitative covariate, invalid extrapolation of linearity to the overall mean where little data are available, and loss of the integrity of group comparison. Not only may centering around the overall mean nullify the effect of interest (group difference), but it could also lead to either uninterpretable or unintended results such as Lord’s paradox (Lord, 1967; Lord, 1969). In contrast, within-group centering, even though rarely performed, offers a unique modeling strategy that should be seriously considered when appropriate (e.g., see Chen et al. (2014)).
When to center within- or across-groups?¶
Suppose that one wants to compare the response difference between the two sexes to face relative to building images. Other than the conventional two-sample Student’s t-test, the investigator may consider the age (or IQ) effect in the analysis even though the two groups of subjects were roughly matched up in age (or IQ) distribution when they were recruited. Further suppose that the average ages from the two sexes are 36.2 and 35.3, very close to the overall mean age of 35.7. One may center all subjects’ ages around the overall mean of 35.7 or (for comparison purpose) an average age of 35.0 from a previous study. However, one extra complication here than the case with one group of subject discussed in the previous section is that the investigator has to decide whether to model the sexes with the same of different age effect (slope). However, unless one has prior knowledge of same age effect across the two sexes, it would make more sense to adopt a model with different slopes, and, if the interaction between age and sex turns out to be statistically insignificant, one may tune up the original model by dropping the interaction term and reduce to a model with same slope.
However, if the age (or IQ) distribution is substantially different across the two sexes, systematic bias in age exists across the two groups; that is, age as a variable is highly confounded (or highly correlated) with the grouping variable. One may face an unresolvable challenge in including age (or IQ) as a covariate in analysis. For instance, suppose the average age is 22.4 years old for males and 57.8 for females, and the overall mean is 40.1 years old. Even without explicitly considering the age effect in analysis, a two-sample Student t-test is problematic because sex difference, if significant, might be partially or even totally attributed to the effect of age difference, leading to a compromised or spurious inference. If one includes age as a covariate in the model through centering around a constant or overall mean, one wants to “control” or “correct” for the age variability across all subjects in the two groups, but the risk is that, with few or no subjects in either or both groups around the center value (or, overall average age of 40.1 years old), inferences on individual group effects and group difference based on extrapolation are not reliable as the linearity assumption about the age effect may break down. Another issue with a common center for the covariate is that the inference on group difference may partially be an artifact of measurement errors in the covariate (Keppel and Wickens, 2004). On the other hand, one may model the age effect by centering around each group’s respective constant or mean. Even though the age effect is controlled within each group and the risk of within-group linearity breakdown is not severe, the difficulty now lies in the same result interpretability as the corresponding two-sample Student t-test: the sex difference may be compounded with the effect of age difference across the groups.
In the above example of two groups with different covariate distribution, age (or IQ) strongly correlates with the grouping variable, and it violates an assumption in conventional ANCOVA, the covariate is independent of the subject-grouping variable. Regardless the centering options (different or same), covariate modeling has been discouraged or strongly criticized in the literature (e.g., Neter et al., 1996; Miller and Chapman, 2001; Keppel and Wickens, 2004; Sheskin, 2004). The moral here is that this kind of modeling difficulty is due to imprudent design in subject recruitment, and can and should be prevented. If a subject-related variable might have impact on the experiment, the variable distribution should be kept approximately the same across groups when recruiting subjects.
A different situation from the above scenario of modeling difficulty is the following, which is not formally covered in literature. Suppose that one wishes to compare two groups of subjects, adolescents and seniors, with their ages ranging from 10 to 19 in the adolescent group and from 65 to 100 in the senior group. Again age (or IQ) is strongly correlated with the grouping variable, and violates the assumption in conventional ANCOVA, the covariate is independent of the subject-grouping factor. Although not a desirable analysis, one might center all subjects’ ages around a constant or overall mean and ask the following trivial or even uninteresting question: would the two groups differ in BOLD response if adolescents and seniors were no different in age (e.g., centering around the overall mean of age for all subjects, for instance, 43.7 years old)? In addition to the interpretation difficulty, when the common center value is beyond the covariate range of each group, the linearity does not necessarily hold well when extrapolated to a region where the covariate has no or only few data points available. A third issue surrounding a common center is that the inference on group difference may partially be an artifact of measurement errors in the covariate (Keppel and Wickens, 2004). However, what is essentially different from the previous example is that the problem in this case lies in posing a sensible question in the substantive context, but not in modeling with a covariate per se that is correlated with a subject-grouping factor in general. In addition, the independence assumption in the conventional ANCOVA is not needed in this case. More specifically, we can reasonably test whether the two groups have the same BOLD response while controlling for the within-group variability in age. When the groups differ significantly on the within-group mean of a covariate, the model could be formulated and interpreted in terms of the effect on the response variable relative to what is expected from the difference across the groups on their respective covariate centers (controlling for within-group variability), not if the two groups had no difference in the covariate (controlling for variability across all subjects). That is, if the covariate values of each group are offset by the within-group center (mean or a specific value of the covariate for that group), one can compare the effect difference between the two subpopulations, assuming that the two groups have same or different age effect. Again unless prior information is available, a model with different age effect between the two groups (Fig. 2D) is more favorable as a starting point.
We have discussed two examples involving multiple groups, and both examples consider age effect, but one includes sex groups while the other has young and old. The common thread between the two examples is that the covariate distribution is substantially different across groups, and the subject-specific values of the covariate is highly confounded with another effect (group) in the model. However, unlike the situation in the former example, the age distribution difference in the two groups of young and old is not attributed to a poor design, but to the intrinsic nature of subject grouping. Such an intrinsic difference of covariate distribution across groups is not rare. A similar example is the comparison between children with autism and ones with normal development while IQ is considered as a covariate. Again comparing the average effect between the two groups if they had the same IQ is not particularly appealing. Instead one is usually interested in the group contrast when each group is centered around the within-group IQ center while controlling for the within-group IQ effects. A third case is to compare a group of subjects who are averse to risks and those who seek risks (Neter et al., 1996). The risk-seeking group is usually younger (20 - 40 years old) than the risk-averse group (50 – 70 years old). As Neter et al. (1996) argued, comparing the two groups at the overall mean (e.g., 45 years old) is inappropriate and hard to interpret, and therefore they discouraged considering age as a controlling variable in the analysis. However, it is not unreasonable to control for age variability within each group and center each group around a meaningful age (e.g. group mean). A fourth scenario is reaction time or anxiety rating as a covariate in comparing the control group and an anxiety group where the groups have preexisting mean difference in the covariate values. All these examples show that proper centering not only improves interpretability and allows for testing meaningful hypotheses, but also may help in resolving the confusions and controversies surrounding some unnecessary assumptions about covariate modeling.
Categorical variables as regressors of no interest¶
It is not rarely seen in literature that a categorical variable such as sex, scanner, or handedness is “partialled” or “regressed” out as a covariate (in the usage of regressor of no interest). Since such a variable is dummy-coded with quantitative values, caution should be taken in centering, because it would have consequences in the interpretation of other effects. Furthermore, if the effect of such a variable is included in the model, examining first its effect and potential interactions with effects of interest might be necessary, regardless whether such an effect – and its interaction with other fixed effects – is of scientific interest. Such a strategy warrants a detailed discussion because of its consequences in interpreting other effects. That is, when one discusses an overall mean effect with a grouping factor (e.g., sex) as an explanatory variable, it is implicitly assumed that interactions or varying average effects occur across groups. Were the average effect the same across all groups, one would model the effects without having to specify which groups are averaged over, and the grouping factor would not be considered in the first place. The interactions usually shed light on the generalizability of main effects because the interpretation of the main effects may be affected or tempered by the presence of a significant interaction (Keppel and Wickens, 2004; Moore et al., 2004; Chow, 2003; Cabrera and McDougall, 2002; Muller and Fetterman, 2002). Simple partialling without considering potential main effects and/or interactions may distort the estimation and significance testing for the effects of interest, and merely including a grouping factor as additive effects of no interest without even an attempt to discuss the group differences or to model the potential interactions invites for potential misinterpretation or misleading conclusions.
We do not recommend that a grouping variable be modeled as a simple additive effect for two reasons: the influence of group difference on interpreting other effects, and the risk of model misspecification in the presence of interactions with other effects. All possible interactions with other effects (continuous or categorical variables) should be considered unless they are statistically insignificant or can be ignored based on prior knowledge. When an overall effect across groups is desirable, one needs to pay attention to centering when adopting a coding strategy, and effect coding is favorable for its immunity to unequal number of subjects across groups. However, such overall effect is not generally appealing: if group differences exist, they deserve more deliberations, and the overall effect may be difficult to interpret in the presence of group differences or with the existence of interactions between groups and other effects; if group differences are not significant, the grouping variable can be dropped through model tuning. Overall, we suggest that a categorical variable (regardless of interest or not) be treated a typical factor. In doing so, one would be able to avoid the complications of dummy coding and the associated centering issues.
So far we have only considered such fixed effects of a continuous variable as well as a categorical variable that separates subjects into multiple groups. Historically ANCOVA was the merging fruit of ANOVA and regression, and we have seen the limitations imposed on the traditional ANCOVA framework. Naturally the GLM provides a further integration beyond ANCOVA. It is worth mentioning that another assumption about the traditional ANCOVA with two or more groups is the homogeneity of variances, same variability across groups. However, it is challenging to model heteroscedasticity, different variances across groups, even under the GLM scheme. Furthermore, of note in the case of a subject-grouping (or between-subjects) factor is that all its levels are independent with each other. When the effects from a within-subject (or repeated-measures) factor are involved, the GLM approach becomes cumbersome. Furthermore, a model with random slope is not possible within the GLM framework. These limitations necessitate the extension of GLM and lead to the multivariate modeling (MVM) (Chen et al., 2013) and linear mixed-effect (LME) modeling (Chen et al., 2014) so that the cross-levels correlations of such a factor and random slopes can be properly modeled.
Chen, G., Adleman, N.E., Saad, Z.S., Leibenluft, E., Cox, R.W. (2014). Applications of Multivariate Modeling to Neuroimaging Group Analysis: A Comprehensive Alternative to Univariate General Linear Model. NeuroImage 99, 571-588. 10.1016/j.neuroimage.2014.06.027 https://afni.nimh.nih.gov/pub/dist/HBM2014/Chen_in_press.pdf
Poldrack, R.A., Mumford, J.A., Nichols, T.E., 2011. Handbook of Functional MRI Data Analysis. Cambridge University Press.