Disaggregation of Effects With No Growth in the TVC and Time is Balanced

We begin by examining an artificial data set that was created to correspond to conditions under which the person-mean centering approach is expected to properly disaggregate within- and between-person effects. More specifically, we assume that

Equations 13 and

16 hold in the population. For our initial data set we generated

*n* = 500 simulated cases, each with

*T* = 9 repeated measures. We scaled time so that the mean of time was zero (i.e.,

*t* = −4, −3, −2, −1, 0, 1, 2, 3, 4), although given the absence of growth in this condition, the scaling of time has no impact in the current model. Finally, because this design is balanced on time, all individuals are the same age, are assessed at the same points in time, and there are no missing data.

We can first consider the characteristics of the TVC itself prior to examining the simulated outcome variable. We generated the TVC to be independent of the passage of time; in other words, there is no systematic growth process that underlies

*z*_{ti}, consistent with

Equation 16. This might be reflective of daily measures of anxiety in which anxiety varied both within and between individuals, but it did not systematically increase or decrease over time. This can be seen in the conditional distribution of the TVC as a function of time presented in , in which the distribution of the TVC at each specific time point is nearly identical; that is, the mean of the TVC is independent of time.

The box plots in show the distributions of the TVC pooling over individuals within each time point. However, we can also examine the individual trajectories of the TVC over time. shows the model-implied trajectories of the TVC for 50 randomly selected observations. Two characteristics are particularly important. First, because there is no time trend in the population model, the estimated trajectories are perfectly flat with respect to time. That is, there is no systematic change in the TVC as a function of time. Second, there is substantial individual variability in the relative heights of the individual trajectories. That is, some observations reflect higher levels of the TVC, and others report lower levels. This between-person variability is captured in the random intercept term in

Equation 15 from above. Extending our hypothetical example, this figure shows that although anxiety does not change systematically as a function of time, some people are reporting higher overall levels of anxiety, whereas others are not.

It is also helpful to consider the set of observations for just one individual plotted over all the time points; this highlights the within-person variability around each individual trajectory. For example, we could consider the nine repeated measures of anxiety taken on just one individual. The data for a single randomly chosen individual is presented in , in which the observed TVC values are plotted against time. The points are the time-specific measures of the TVC, and the horizontal line demarcates the sample mean for the person, pooling over the set of TVCs. The horizontal line thus shows the overall level of anxiety for this individual, and the points show the time-specific values of anxiety relative to the overall level. We can see that the TVC does not appear to be related to time and that the time-specific measures of the TVC vary randomly around the person-specific mean. This is precisely what allows us to deviate each time-specific measure of the TVC from the person-mean to disaggregate the between-person and within-person effects.

Thus far we have considered only the overtime characteristics of the TVC itself. Next we turn to our simulated outcome,

*y*_{ti}, which was generated to be consistent with

Equation 13; in words, this is a random intercept-only model for a continuously and normally distributed outcome variable with both a within-person and between-person effect of the single TVC

*z*_{ti}. In our hypothetical example, the outcome could represent daily alcohol use that varies both within and between individuals but does not systematically change over time. The overall intercept of the model for

*y*_{ti} was defined to be γ

_{00} = 5.0, the within-person effect was γ

_{10} = −1.0, and the between-person effect was γ

_{01} = 1.5. Thus, higher time-specific deviations of the TVC from the overall person-mean are associated with lower values of the outcome, whereas higher overall person-means are associated with higher values of the outcome.

We chose these values to reflect the hypothetical relation that might be found between daily anxiety symptoms and daily alcohol use. More specifically, the positive between-person effect reflects that, on average, people who are more anxious tend to drink more alcohol; this might be attributable to a self-medication process, where alcohol is consumed to modulate anxiety symptoms (e.g.,

Kassel et al. 2010). In contrast, the negative within-person effect reflects that, on average, people tend to drink less alcohol on days when their anxiety is elevated relative to their typical stable level; this might be attributable to an individual avoiding alcohol-related social contexts on days when anxiety is particularly pronounced (e.g.,

Kaplow et al. 2001). Note that although theory is predictive of these relations, for our purposes here we consider these strictly hypothetical (although we would sure like to see this study done).

To begin, consider the simple bivariate scatter plot in , where the TVC is plotted on the *x*-axis and the outcome on the *y*-axis. Although we see a generally positive trend, this is an inextricable aggregation of the between-person effect (which is positive) and the within-person effect (which is negative). Following our hypothetical example, we would conclude from the aggregate analysis that there is a positive relation between anxiety and alcohol use that is modest in size and holds across all individuals in the sample. However, we know the true relation to be patently different. To recover the more complex relation that truly exists, we must disaggregate the TVC into the between-person component (*zb*_{i}) and the within-person component (*zw*_{ti}).

One way to get a better visual sense of these two effects is to plot the relationships observed at each level of analysis. Note that we are only using these plots to visually examine potential differences in levels of effect, and we will formally test these disaggregated effects through the parameterization of the multilevel model. To see the within-person effect, we can plot outcome

*y*_{ti} against the person-mean centered

*ż*_{ti}; to see the between-person effect, we can plot the person-means

_{i} against the person-means

_{i}. presents the person-mean centered TVC plotted against the outcome, and presents the person-mean of the TVC plotted against the person-mean of the outcome.

These plots clearly reflect the strong negative within-person relation between the time-specific measure of the TVC and the outcome () and the strong positive between-person relation between the mean of the TVC and the mean of the outcome (). This is of course precisely how we generated these data. We now use the techniques described above to obtain estimates of the between- and within-person effects via the multilevel model, in which

*zb*_{i} =

_{i} and

*zw*_{ti} =

*ż*_{ti} are included as separate predictors of

*y*_{ti}.

To do this, we fitted a multilevel model consistent with

Equation 13 to formally test the between- and within-person influences of the TVC. Recall that 500 individuals were each assessed nine times, resulting in a total of 4500 person-time observations. We fitted a two-level model under full information maximum likelihood and obtained an estimate of the within-person effect of

_{10} = −0.99 (

*se* = 0.008) and of the between-person effect of

_{01} = 1.51 (

*se* = 0.022). Recall that the corresponding population values were γ

_{10} = −1.0 and γ

_{01} = 1.5, respectively; thus, as expected, we closely replicated these values in our artificial sample.

^{6} Continuing with our hypothetical example, these results would reflect that, on average, people reporting higher overall levels of anxiety tended to drink more alcohol; but at the very same time, on average, people tended to drink less alcohol on days when they reported higher levels of anxiety. This nicely highlights that the first conclusion made with respect to

*between* individual differences, and the second conclusion is made with respect to

*within* individual differences.

As we fully expected based on prior analytic theory, the person-mean centering approach accurately recovered the known population-generating values. However, although comforting, this is at best a modest victory. That is, we generated a population model consistent with

Equations 13 and

16, and then we fit a sample model that corresponded to these same generating equations. Had we found anything other than these results, you would do well to suspect that we made an error in our computer programming. However, we view this as an important endeavor in that it demonstrates that the existing methods work properly when the underlying assumptions are met. Further, it gooses us to think more carefully about the specific conditions under which person-mean centering is a valid method for disaggregating multiple levels of effect.

Disaggregation of Effects with Growth in the TVC and Time is Balanced

The second situation we consider is when there are both fixed and random effects of growth underlying the TVC and the design is balanced on time (i.e.,

Equation 30). Extending our hypothetical example, we remain interested in studying the relation between anxiety and alcohol use. However, we now want to consider the situation in which anxiety is not only increasing over time, but there are also individual differences in both starting point and rate of change. We thus defined a linear growth model to underlie the TVC itself based on the same sample size (

*N* = 500) and same number of time points (

*T* = 9) as before. The TVC in this second data set was defined to have an intercept equal to 25.0 and a linear slope equal to 1.0; these are arbitrary values, but they define a linear growth trajectory for the TVC. Further, we coded time so that the middle point was equal to zero, meaning that that the intercept is defined as the mean of the outcome at the mean of time, and the TVC increased in value by one unit with each unit increase in time. Finally, we allowed for individual variability (that is, random effects) in both the starting point (τ

_{00} = 4) and rate of change over time (τ

_{11} = 1) and a level-1 residual equal to σ

^{2} = 1.

To better illustrate the implications of the inclusion of this time trend, presents the conditional distributions of the TVC as a function of time. It is clear that the time-specific means are (as we intended) increasing as a function of time. Further, note that the variance of the TVC varies as a function of time; this is also consistent with our population-generating model because there is a random slope component that differentially influences time-specific variability over time. In terms of our hypothetical example, both the mean and variance of anxiety are changing as a function of time; the mean is increasing linearly, and the variance is changing quadratically.

To see the influence of the random components on growth, in we present the individual model-implied trajectories of the TVC for 50 randomly drawn cases. This highlights not only the systematic increase in the TVC over time, but also the individual variability in starting point and rate of change. You can consider each of these lines as an individual’s own trajectory of anxiety symptoms unfolding over the period of observation. On a related point, note that each trajectory spans the entire period of time, reflecting that these data are balanced with respect to time. Finally, relevant to later analysis, note that the relative rank ordering of values on the outcome changes over time. To see this, picture drawing a vertical line at each value of time; because the slopes are not parallel, the individual standing on the TVC varies at each vertical line drawn at a given value of time.

However, why would the systematic relation between the TVC and time potentially undermine the validity of the person-mean centering approach? Although we showed this analytically above (i.e.,

Equation 33), this threat to validity can be saliently visualized when examining the distribution of the TVCs over time for an individual case. In , the TVC is plotted on the

*y*-axis, time is plotted on the

*x*-axis, and the horizontal line demarcates the person-specific mean of the set of TVCs. However, the positively sloped line is the regression line of best fit linking the TVC to time. This is consistent with the increasing value of the TVC associated with the passage of time; that is, the hypothetical individual is reporting progressively higher values of anxiety at each time point.

Importantly, note that the person-mean centering strategy deviates each TVC relative to the horizontal line because of the implicit assumption that the value of the TVC is independent of time. Yet it is clear from this plot that person-mean centering fails to differentiate within-person fluctuations around the time trend. Using existing standard methods, all of the values of the TVC falling below the person-mean receive a negative deviated score, and all of the values falling above the person-mean receive a positive deviated score. These values are incorrect for obtaining a sample estimate of the within-person variability of the TVC over time. Instead, we must deviate the time-specific values of the TVC not from the horizontal line but instead from the positively sloped regression line. Only this will properly isolate the within-person component of the TVC.

To demonstrate this, we first applied the standard methods for disaggregating the between- and within-person effects of the TVC on the outcome. Given that the TVC was generated to be related to time yet the standard methods assume no relation to time, we a priori expect these results to be biased. To evaluate this, we fitted precisely the same person-mean centered model to the second data set as we did to the first. Although in the first data set we nearly perfectly recovered the corresponding population parameters, this did not occur here.

The person-mean deviated TVC resulted in a highly biased estimate of the within-person effect. Specifically, the within-person effect was estimated to be

_{01} = −0.07 (

*se* = 0.006), whereas the corresponding population value was γ

_{10} = −1.0. Thus, applying the standard methods of person-mean centering to data in which the TVC varies as a function of time results in a within-person effect that drastically underestimates the known population value. In our hypothetical example, we would conclude that there was indeed a negative within-person effect, yet we would underestimate the magnitude of this effect by 93%. This is a striking amount of bias that occurs even under what are otherwise ideal conditions (e.g., large sample size, large numbers of repeated measures, no missing data).

In contrast to the highly biased within-person effect, we accurately recovered the population between-person effect; our obtained value was

_{10} = 1.49 (

*se* = 0.029), whereas the corresponding population value was γ

_{10} = 1.5. To better understand this accurate recovery, recall that we generated the TVC such that the mean of time was equal to zero (i.e., time was centered around zero). As such, because this condition is balanced,

_{i} =

= 0 for all individuals. Thus the omitted second set of terms in

Equation 32 (i.e., [γ

_{10} +

*û*_{1i}]

_{i}), drops out and the person-mean accurately recovers the between-person effect. Note, however, that this is strictly a function of the balanced design. If time were unbalanced (e.g., if there were missing data or a cohort-sequential design), then the person-mean would not accurately capture the between-person effect in this situation. Indeed, we demonstrate just this point in the next example.

Whereas in the balanced case the person-specific mean of time (

_{i}) is constant over individual, the deviation of the individual value of time from the mean (

*x*_{ti} −

_{i}) is not. Thus the traditional method neglects the term (γ

_{10} +

*û*_{1i})(

*x*_{ti} −

_{i}) from

Equation 33 in the calculation of the time-specific deviation of

*z*_{ti} from the person-mean. This is why our sample estimate of the within-person effect was equal to −0.07 when the corresponding population value was equal to −1.0. Fortunately, though, we can draw on our prior developments to obtain an unbiased estimate of this known population effect.

To do this, we need a person-specific estimate of γ_{10} + *u*_{1i} to use in the calculation of *zw*_{ti}. More specifically, instead of deviating the time-specific TVC measures with respect to the person-mean, we can deviate the TVCs with respect to the individual-specific regression line linking the TVC and time. This strategy can be more clearly understood by reconsidering . Here we plotted the TVCs against time for a single individual, and we superimposed both a horizontal line representing the person-mean and the best-fitting regression line estimating the positive relation between time and the TVC. Whereas the traditional person-mean centering approach deviates the TVC with respect to the horizontal line, we can instead deviate the TVC with respect to the regression line. We refer to this strategy as detrending.

The general concept of detrending is far from novel, and it has been used in various forms in time-series analysis for decades (e.g.,

Chatfield 1996). However, to our knowledge there has been no prior discussion of applying these techniques in the multilevel model in order to disaggregate between- and within-person effects of a TVC on the outcome when the TVC itself is related to time. Our proposed approach for detrending is simple. We first regress the TVC on time separately for each individual using ordinary least squares (OLS). We then deviate each time-specific TVC not from the overall person-mean (as is done in the traditional approach) but instead from the model-implied value of the TVC specific to that particular unit of time. In other words, our deviated TVC measure is simply the residual (i.e., the observed minus expected value) from the regression of the TVC on time computed separately for each individual case.

We can present this more formally as a one-predictor regression equation estimated separately (case by case) for each individual in the sample. This is given as

where

*z*_{ti} is the time-specific measure of the TVC,

*x*_{ti} is the measure of time,

*b*_{0i} and

*b*_{1i} are sample estimates of the intercept and the slope of the regression of the TVC on time, respectively, and

*e*_{ti} is the time-specific residual.

^{7} A trivial rearrangement of this equation shows that

where

*e*_{ti} is the detrended rescaling of the TVC. In other words, the residual

*e*_{ti} is computed by deviating the time-specific TVC from the model-implied value of the TVC that includes information about the specific value of time. Thus the TVC is deviated not relative to the horizontal line but instead relative to the regression line. We now define

*zw*_{ti} as

*e*_{ti}.

An interesting generalization can be seen here as well. We could fit the OLS regression of the TVC on time defined in

Equation 36 to our initial artificial data set in which the TVC was unrelated to the passage of time. Given the structure of the data, there would be no

*b*_{1}*x*_{ti} term in

Equation 36, and this would simplify to

and the deviation of the TVC would be

which is precisely equal to the traditional person-mean centering approach we first described (because

*b*_{0i} =

_{i} when there are no predictors in the regression equation). However, the more general conclusion is that the person-mean centering approach is equivalent to detrending but under the implicit assumption that there is no relation between the TVC and time, and thus

*b*_{1i} is zero for all cases. Here we simply extend this approach to allow

*b*_{1i} to take on some nonzero value from the data.

To examine the utility of this approach, we detrended the TVC in the second data set with respect to the regression line fitted to each case individually.

^{8} Once detrended, we then used this rescaling of the TVC in precisely the same way as before; namely, we included the detrended TVC as the level-1 predictor (

*zw*_{ti}), and we retained the OLS intercept from

Equation 36 as the level-2 predictor (

*zb*_{i}). Because in this balanced condition the OLS intercept is equal to the person-mean used in our initial model that we fitted to these data, we get the same estimate of the between-person effect as we did before:

=1.49 (

*se* = 0.029). However, whereas our prior estimate of the within-person effect was highly biased when using the person-mean centered TVC, we recover this with near-precision using the detrended TVC:

= −0.99 (

*se* = 0.018). These results demonstrate that when the TVC is systematically related to the passage of time, it is critical that the TVC be deviated not with respect to the person-mean but instead with respect to the individual-specific regression linking the TVC and time.

In sum, this second artificial data set was generated so that there was a random growth process underlying the TVC. However, this was embedded in the unrealistic condition of complete and balanced data. Our third and final data example considers the same growth model for the TVC but embedded in a more realistic condition of unbalanced time.

Disaggregation of Effects with Growth in the TVC and Time is Unbalanced

An important characteristic of the first two artificial data sets is that each simulated subject was followed for precisely the same nine time periods. This is consistent with a birth-cohort design in which an entire cohort of individuals is assessed at the same age at each assessment period and there are no missing data. Because we numerically coded time to range from −4 to 4, the mean value (or midpoint) of time is equal to 0 for each of the 500 individuals. As such, every single person has the same mean of time, equal to zero. The person-mean of the TVC cannot then covary with the person-mean of time because all person-mean values of time are equal for all individuals.

However, as we described above, the time-balanced birth-cohort design is rare in many behavioral science research applications. Instead, multiple cohorts are often considered simultaneously, whether intentionally by design (e.g., one sample of 5-year-olds is recruited, one sample of 6-year-olds is recruited, etc.) or unintentionally by happenstance of the distribution of age within each assessment (e.g., inclusion criteria include children 5 to 9 years of age at first assessment). Further, given that missing data are endemic in longitudinal social science research, even a true birth cohort design will typically be unbalanced.

To simulate this much more realistic situation, we began with precisely the same empirical data as was used in our second example. However, we made one very simple yet critically important modification to this data set: we randomly divided the *N* = 500 individuals into six discrete groups, each representing one distinct cohort (there were 83 individuals in each of five cohorts and 85 in the sixth). Once we created the six groups, we then retained just the first through fourth assessments for the first cohort (i.e., time points −4, −3, −2, −1) and just the second through fifth assessments for the second cohort (i.e., time points −3, −2, −1, 0); we did this for each cohort, ending with the retention of the sixth through ninth assessments for the final cohort. There were thus still 500 individuals with the very same data as before, but here we only retained four assessments from any given individual, the specific four of which depended on the cohort to which the individual belonged.^{9} This design is unbalanced with respect to time.

Whereas in each individual trajectory spans all nine time points, here any given trajectory spans only four time points. Further, which four time points are spanned varies as a function of cohort membership. This can be seen in , in which the trajectories of the TVC and time are shown for 50 random cases. Two implications arise from this unbalanced design.

First, recall that in the balanced case the mean of time (i.e.,

_{i}) was equal to zero across all 500 individuals. However, now the mean of time varies as a function of within which cohort the individuals reside. Specifically, the mean values of time for the six cohorts range from −2.5 to 2.5 by increments of 1 (e.g.,

_{i} = −2.5 for cohort 1;

_{i} = −1.5 for cohort 2; and so on). Because the mean of time now varies over individual, we must account for this additional information in the disaggregation of our between- and within-person effects.

Second, even when the TVC is related to time, in the balanced condition there is just one unique value of the person-specific mean of the TVC pooling over the total period of time. That is, each person is characterized by a mean-value of the TVC pooling over the nine time points. However, when the TVC is related to time in the unbalanced condition, the person-specific mean value of the TVC varies as a function of precisely when in time the individual was assessed. For example, if the TVC is increasing over the nine time points, the person-specific mean of the TVC will also increase as the four-time-point assessment window increases (e.g., the mean of the TVC is directly related to the mean of time).

This can best be seen in the conditional distributions of the person-means of the TVC as a function of cohort membership; this is presented in . To clarify, there were *N* = 83 individuals belonging to cohort 1 who were assessed at the first four time points (coded −4, −3, −2, −1); the first box plot in presents the distribution of the person-specific means of the TVC for these individuals, and this has an overall mean of 22.62. The second box plot presents the distribution of the person-specific means of the TVC for the next *N* = 83 individuals who belong to cohort 2 (and who were thus assessed between times −3 and 0), and this has an overall mean of 23.61; and so on. The horizontal line denotes the grand mean of the TVC, which is equal to 25. Notice that no cohort-specific mean is equal to the grand mean.

Returning to our hypothetical example, these data would reflect that earlier (and thus younger) cohorts are reporting less overall anxiety compared to the later cohorts. Interestingly, this is not some strange statistical artifact; this is an accurate reflection of the sample characteristics in that later cohorts do indeed report higher overall levels of anxiety than do earlier cohorts. However, the sole source of this difference is that the later cohorts are assessed at a later age than are the earlier cohorts, and anxiety is increasing with time. Thus person-mean values of anxiety are confounded with time. This is directly analogous to measuring height over time where one cohort was assessed between ages 5 and 10 and a second cohort between ages 9 and 14. Of course the second cohort reports higher values of average height—they are older, and children tend to increase in height with age. But this in no way implies that the second cohort would have been taller than the first had both cohorts been assessed at the same age. This is the crux of the challenge we face: We need to isolate the within-person and between-person differences in the TVC while adjusting for the different values of time at which the assessments were obtained.

clearly reflects that the cohort-specific mean of the person-means of the TVC increases monotonically as a function of the cohort to which individuals belong. Because cohort is directly related to time, the person-mean of the TVC is also unambiguously linked to the passage of time. It is very important to note that this is not a contrived or tortured example; indeed, this situation is almost universally encountered in any cohort-sequential design in which the TVC itself is related to the passage of time.

To examine the implications of this, we first used the standard person-mean centering approach to disaggregate the between-person and within-person influences of the TVC on the outcome. We thus fitted

Equation 12 to the artificial data and (as expected) found significantly biased effects for both the within- and between-person influences. The within-person effect was

_{10} = −0.24 (compared to the population value of −1.0), and the between-person effect was

_{01} = 0.71 (compared to the population value of 1.50). Notice that whereas the person-mean successfully recovered the between effect in the balanced condition, this is now underestimated by more than 50% based on the very same data in the unbalanced condition. Thus under conditions that are likely common in many areas of psychological research, the standard methods for disaggregating effects are highly biased.

We next drew on our expressions for computing

*zb*_{i} and

*zw*_{ti} in the presence of random growth to obtain the necessary disaggregated components of the TVC. In words, we simply regressed the TVC on time within each individual where time is grand-mean centered. We then retained the time-specific residuals as our estimate of

*zw*_{ti} (i.e.,

*e*_{ti} from

Equation 36), and we retained the sample estimate of the regression intercept as our estimate of

*zb*_{i} (i.e.,

*b*_{0i} from

Equation 36). Using these as predictors in the model for our outcome

*y*_{ti}, we obtained an estimate of the within-person effect of

_{10} = −0.95 (

*se* = 0.036) and an estimate of the between-person effect of

_{01} = 1.25 (

*se* = 0.041). Although the within-person effect was underestimated by 5% and the between-person effect by 17% relative to their population counterparts, these estimated values are substantially more accurate than those obtained using traditional methods for disaggregating effects. This is because information about time (via

*x*_{ti} and

_{i}) is explicitly considered in the computation of

*zb*_{i} and

*zw*_{ti}, whereas this is omitted when using standard methods.

There are two related reasons why the between- and within-person effects were recovered with near-perfect precision in the balanced case but with only modest bias in the unbalanced case. First, all cases in the balanced condition had

*T* = 9 repeated measures, and all cases in the unbalanced condition had

*T* = 4 repeated measures. Thus the OLS estimates used as

*zb*_{i} and

*zw*_{ti} are estimated with greater precision, given higher numbers of repeated measures. Second, and more importantly, recall that we are using the person-specific estimate of the intercept term (i.e.,

*b*_{0i}) from the regression of the TVC on time. As in any regression, the intercept reflects the mean of the TVC at the mean of time (i.e., since our coding of time means that

= 0). In the balanced case, all individuals were observed across all time points, so

*b*_{0i} was estimated within the range of observed data (that is, each individual was observed at

*x*_{ti} = 0). In contrast, in the unbalanced case, not every individual was observed when

*x*_{ti} = 0. For example, individuals in cohort 1 were observed at times −4, −3, −2, and −1), yet the estimate of

*b*_{0i} reflects the mean of the TVC when

*x*_{ti} = 0, which is outside of the range of observed data in this cohort. As such, the estimates of

*b*_{0i} were projected beyond the window of observation for many individuals, thus further undermining the precision of estimates beyond simply having fewer repeated measures than in the balanced condition. Nevertheless, our obtained estimate for the between-person effect is still much improved by using

*b*_{0i} compared to

_{i}.