Within- and Between-Person Variance with variance_components()

Analytic question

variance_components() quantifies whether variation occurs mainly between people or within a person over time. The function takes a repeated-measures data frame, one or more variables, and a person identifier. It returns the within-person variance, between-person variance, total variance, intraclass correlation, and reliability of the person mean.

variance_components() defines the intraclass correlation as the share of total variance attributable to stable differences between people. A variable with an intraclass correlation near zero varies mainly within people. A variable with an intraclass correlation near one differs mainly between people.

Decompose the observed variance

variance_components() uses the analysis-of-variance estimator by default. The optional REML estimator requires lme4. Table 1 reports the default decomposition for three self-regulated-learning variables.

components <- variance_components(analysis_data, vars = c("efficacy", "monitoring", "effort"), id = "name")
components
#> VARIANCE COMPONENTS
#>   Grouping   name
#>   Method     anova
#> 
#>                  Within    Between     ICC   Reliability
#> --------------------------------------------------------
#>   efficacy     435.4422   290.6398   0.400         0.990
#>   monitoring   516.7891   313.4551   0.378         0.990
#>   effort       560.3849   194.4532   0.258         0.982
#> 
#>   ICC         share of variance lying BETWEEN groups
#>   Reliability precision of each group's own mean

variance_components() assigns intraclass correlations of approximately 0.40 to efficacy, 0.38 to monitoring, and 0.26 to effort in Table 1. Most observed variation therefore occurs within learners for all three variables. Stable between-learner differences remain large enough to make a raw pooled slope potentially mix levels of association.

Plot the variance shares

plot_variance() converts each intraclass correlation into complementary between-person and within-person shares. Figure 1 uses blue for between-person variance and orange for within-person variance.

plot_variance(components)

Figure 1. Within-person and between-person shares of observed variance.

Figure 1 shows that the within-person share exceeds the between-person share for each variable. Effort has the largest within-person share among the three. This pattern supports models of daily fluctuation while retaining a separate term for stable person means.

Assumptions and failure checks

variance_components() assumes that person identifiers define independent clusters. The default decomposition uses observed group means and pooled within-group variation. The reliability column depends on both the intraclass correlation and the number of observations per person. High reliability of a person mean does not imply high within-person reliability of individual occasions.

variance_components() requires variation across people to estimate a between-person component. A variable that is constant within each person can support a between-person comparison but cannot support a person-specific time-varying slope. A variable with no between-person variation cannot identify a contextual contrast.

When to use which

variance_components() is appropriate before choosing a level of analysis. describe_persons() is appropriate for person-specific data quality and temporal movement. fit_within_between() is appropriate when the variance decomposition motivates separate within-person and between-person coefficients. pool_coefs() is appropriate after estimating individual slopes and addresses heterogeneity of effects rather than heterogeneity of variable levels.

References