If it is not a problem, can you please help me to understand why? Thank you for this beautiful explanation. I was struggling to understand how a centered and uncentered quadratic model differ and why the linear interaction terms become insignificant. Now I am quite clear. Thanks again. Should you also centre variables when appropriate if using a mixed model as opposed to a regression analysis?
I might not be grasping this correctly. How would you interpret this intercept, and could it be statistically significant? Or is there any way to move the Y-axis to the center of the graph so that in this case the mean of Y would be where the mean of X is i.
All predicted values in a regression line are conditional means: the mean of Y at a certain value of X. I have output from a polynomial regression and only the second order term seems centered on its mean.
Do you have an easy explanation? When I teach regression I generally use the centered-regressor form, because that form takes away much of the desire for the intercept to mean something. Pun intended. It is true of models that do not fit a mean term only when the data centroid is the origin.
That is, when we center all the variables involved in the model. Thanks for your comments. The regression line will go through the data centroid by definition, whether we center X or not. Nicely explained.
This gives the appearance of being a meaningful parameter, but generally is not. And in the case where we are doing empirical modeling there is no justification for the assumption. It is now clear what effect the so-called intercept has: it forces the regression line through the data centroid x-bar, y-bar. Your email address will not be published. Skip to primary navigation Skip to main content Skip to primary sidebar Centering predictor variables is one of those simple but extremely useful practices that is easily overlooked.
Learn the approach for understanding coefficients in that regression as we walk through output of a model that includes numerical and categorical predictors and an interaction. The notion of centering and scaling continuous variables in regression is the source of constant debate and questioning Example 1 , Example 2 and often answers are given in terms statistical properties.
For the epidemiologist, what are the practical implications? Centering a variable moves its mean to 0 which is done by subtracting the mean from the variable , standardizing adjusts the scales of magnitude by dividing the centered variable by its standard deviation. Here's a concise list, recognizing that one or more may apply at any given time:.
Users of R can refer to the built-in scale function, which allows both mean centering and standardization of a continuous variables. Other statistical software may have similar features, or require manually centering or scaling the variable s.
One may face an unresolvable challenge in including age or IQ as a covariate in analysis. For instance, suppose the average age is 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. 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, 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. 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 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.
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, 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.
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. 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.
The risk-seeking group is usually younger 20 - 40 years old than the risk-averse group 50 — 70 years old. As Neter et al. However, it is not unreasonable to control for age variability within each group and center each group around a meaningful age e. 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.
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.
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, ; Moore et al.
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.
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. Chen, G.
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