Leverage for HLMs

This function calculates the leverage of a hierarchical linear model fit by lmer.

# Default S3 method
leverage(model, ...)

# S3 method for class 'mer'
leverage(model, level = 1, ...)

# S3 method for class 'lmerMod'
leverage(model, level = 1, ...)

# S3 method for class 'lme'
leverage(model, level = 1, ...)

Arguments

model: fitted model object of class mer of lmerMod
...: do not use
level: the level at which the leverage should be calculated: either 1 for observation level leverage (default) or the name of the grouping factor (as defined in flist of the mer object) for group level leverage. leverage assumes that the grouping factors are unique; thus, if IDs are repeated within each unit, unique IDs must be generated by the user prior to use of leverage.

Value

leverage returns a data frame with the following columns:

overall: The overall leverage, i.e. $H = H_1 + H_2$.
fixef: The leverage corresponding to the fixed effects.
ranef: The leverage corresponding to the random effects proposed by Demidenko and Stukel (2005).
ranef.uc: The (unconfounded) leverage corresponding to the random effects proposed by Nobre and Singer (2011).

Details

Demidenko and Stukel (2005) describe leverage for mixed (hierarchical) linear models as being the sum of two components, a leverage associated with the fixed ($H_1$) and a leverage associated with the random effects ($H_2$) where $$H_1 = X (X^\prime V^{-1} X)^{-1} X^\prime V^{-1}$$ and $$H_2 = ZDZ^{\prime} V^{-1} (I - H_1)$$ Nobre and Singer (2011) propose using $$H_2^* = ZDZ^{\prime}$$ as the random effects leverage as it does not rely on the fixed effects.

For individual observations leverage uses the diagonal elements of the above matrices as the measure of leverage. For higher-level units, leverage uses the mean trace of the above matrices associated with each higher-level unit.

References

Demidenko, E., & Stukel, T. A. (2005) Influence analysis for linear mixed-effects models. Statistics in Medicine, 24(6), 893–909.

Nobre, J. S., & Singer, J. M. (2011) Leverage analysis for linear mixed models. Journal of Applied Statistics, 38(5), 1063–1072.

Author

Adam Loy loyad01@gmail.com

Examples

data(sleepstudy, package = 'lme4')
fm <- lme4::lmer(Reaction ~ Days + (Days | Subject), sleepstudy)

# Observation level leverage
lev1 <- leverage(fm, level = 1)
head(lev1)
#>      overall       fixef      ranef  ranef.uc
#> 1 0.22930404 0.019191919 0.21011212 0.9345897
#> 2 0.16972999 0.013804714 0.15592528 1.0174683
#> 3 0.12682372 0.009764310 0.11705941 1.2074459
#> 4 0.10058520 0.007070707 0.09351449 1.5045226
#> 5 0.09101445 0.005723906 0.08529055 1.9086983
#> 6 0.09811147 0.005723906 0.09238756 2.4199730

# Group level leverage
lev2 <- leverage(fm, level = "Subject")
head(lev2)
#>    overall      fixef     ranef ranef.uc
#> 1 0.161234 0.01111111 0.1501229 2.592732
#> 2 0.161234 0.01111111 0.1501229 2.592732
#> 3 0.161234 0.01111111 0.1501229 2.592732
#> 4 0.161234 0.01111111 0.1501229 2.592732
#> 5 0.161234 0.01111111 0.1501229 2.592732
#> 6 0.161234 0.01111111 0.1501229 2.592732