It is well known that between- and within-person effects can be efficiently and unambiguously disaggregated within the multilevel model using the strategy of person-mean centering. Traditionally, the term centering is used to describe the rescaling of a random variable by deviating the observed values around the variable mean (e.g.,Aiken&West 1991 , pp. 28–48). For example, within the standard fixed-effects regression model, a predictor xi is centered via , where x̄ is the observed mean of xi, and is the mean-deviated rescaling of xi (see, e.g., Cohen et al. 2003 , p. 261). By definition, the mean of a centered variable is equal to zero, and this offers both interpretational and sometimes computational advantages in a number of modeling applications.
However, centering becomes more complex when considering TVCs. This is because multiple repeated measures are nested within each individual, and there are thus two means to consider: the grand mean of the TVC pooling over all time points and all individuals, and each person-specific mean pooling over all time points within individual. There are two ways that we can center the TVC.
First, we can deviate the TVC around the grand mean pooling over all individuals. Here, where z̈ti represents the grand mean centered TVC, zti is the observed TVC, and z̄‥ is the grand mean of zti pooling over all individuals and all time points. In other words, we simply compute the grand mean of the TVC and subtract this from each individual- and time-specific TVC score. Second, we can deviate the TVC around the person-specific mean of the TVC unique to each individual. Here, where żti represents the person-mean centered TVC, zti is again the observed TVC, and z̄i is the person-specific mean for individual i. In other words, we subtract just the person-specific mean of the TVC from each of that same person’s time-specific TVC scores. We can use zti, żti, or z̈ti as the level-1 predictor inEquation 8 , and each is associated with a potentially different inference with respect to the disaggregation of effects.
Methods exist that allow for the disaggregation of the between-person and within-person effects using zti, żti, or z̈ti (Kreft et al. 1995 , Raudenbush & Bryk 2002 ). However, direct estimates of these effects can be most easily obtained within the multilevel model by incorporating the person-mean centered TVC at level-1 (i.e., żti) and the person-mean at level-2 (i.e., z̄i) (Raudenbush & Bryk 2002 , equation 5.41). Specifically, where all is defined as above. This requires three steps: We first compute the mean of the time-specific TVCs within each individual to obtain z̄i; we then subtract that person-specific mean from each individual’s time-specific TVC values to obtain żti; finally, we use both z̄i and żti as predictors in our multilevel model.
The reduced form equation for this model is where γ00 is the intercept (or grand mean), γ01 is a direct estimate of the between-person effect, and γ10 is a direct estimate of the within-person effect. Following our earlier hypothetical example, γ01 would capture the relation between average levels of anxiety and average levels of substance use pooling over individuals. In contrast, γ10 would capture the mean relation between a given person’s time-specific deviation in anxiety (relative to the overall level of anxiety) and the individual’s time-specific substance use.
The approach we outline above is currently regarded as best practice for the disaggregation of between-person and within-person effects in multilevel growth models (e.g., Raudenbush & Bryk 2002 , pp. 181-85; Singer & Willett 2003 , pp. 173-77), and there is no question that this is a valid method for accomplishing these goals. As we describe in greater detail below, however, the validity of this approach heavily relies on a set of specific conditions that may or may not be met in practice. Further, we have found that these conditions are rarely, if ever, discussed in either the quantitative or applied literatures. To better define these specific conditions, we next propose a more general framework for defining within-person and between-person effects. This framework both more formally establishes these expressions and allows us to explicate precisely under what conditions standard approaches are and are not valid.
However, centering becomes more complex when considering TVCs. This is because multiple repeated measures are nested within each individual, and there are thus two means to consider: the grand mean of the TVC pooling over all time points and all individuals, and each person-specific mean pooling over all time points within individual. There are two ways that we can center the TVC.
First, we can deviate the TVC around the grand mean pooling over all individuals. Here, where z̈ti represents the grand mean centered TVC, zti is the observed TVC, and z̄‥ is the grand mean of zti pooling over all individuals and all time points. In other words, we simply compute the grand mean of the TVC and subtract this from each individual- and time-specific TVC score. Second, we can deviate the TVC around the person-specific mean of the TVC unique to each individual. Here, where żti represents the person-mean centered TVC, zti is again the observed TVC, and z̄i is the person-specific mean for individual i. In other words, we subtract just the person-specific mean of the TVC from each of that same person’s time-specific TVC scores. We can use zti, żti, or z̈ti as the level-1 predictor in
Methods exist that allow for the disaggregation of the between-person and within-person effects using zti, żti, or z̈ti (Kreft et al. 1995 , Raudenbush & Bryk 2002 ). However, direct estimates of these effects can be most easily obtained within the multilevel model by incorporating the person-mean centered TVC at level-1 (i.e., żti) and the person-mean at level-2 (i.e., z̄i) (Raudenbush & Bryk 2002 , equation 5.41). Specifically, where all is defined as above. This requires three steps: We first compute the mean of the time-specific TVCs within each individual to obtain z̄i; we then subtract that person-specific mean from each individual’s time-specific TVC values to obtain żti; finally, we use both z̄i and żti as predictors in our multilevel model.
The reduced form equation for this model is where γ00 is the intercept (or grand mean), γ01 is a direct estimate of the between-person effect, and γ10 is a direct estimate of the within-person effect. Following our earlier hypothetical example, γ01 would capture the relation between average levels of anxiety and average levels of substance use pooling over individuals. In contrast, γ10 would capture the mean relation between a given person’s time-specific deviation in anxiety (relative to the overall level of anxiety) and the individual’s time-specific substance use.
The approach we outline above is currently regarded as best practice for the disaggregation of between-person and within-person effects in multilevel growth models (e.g., Raudenbush & Bryk 2002 , pp. 181-85; Singer & Willett 2003 , pp. 173-77), and there is no question that this is a valid method for accomplishing these goals. As we describe in greater detail below, however, the validity of this approach heavily relies on a set of specific conditions that may or may not be met in practice. Further, we have found that these conditions are rarely, if ever, discussed in either the quantitative or applied literatures. To better define these specific conditions, we next propose a more general framework for defining within-person and between-person effects. This framework both more formally establishes these expressions and allows us to explicate precisely under what conditions standard approaches are and are not valid.