lmm() explicitly evaluates a profiled ML or REML criterion for
V = Z G Z' + R. It can combine random effects with an independent,
compound-symmetric, AR(1), full or fixed-band Toeplitz, or unstructured
residual covariance.
Missing values are handled according to na.action.
Usage
lmm(
data,
formula,
random = NULL,
repeated = NULL,
structure = c("id", "cs", "ar1", "toep", "un"),
method = c("REML", "ML"),
ddf = c("satterthwaite", "residual", "kenward-roger"),
weights = NULL,
offset = NULL,
contrasts = NULL,
control = lmm_control(),
na.action = stats::na.omit
)Arguments
- data
A data frame or tibble. Placing
datafirst makes the function pipe-friendly.- formula
A two-sided fixed-effects formula.
- random
Optional one-sided random-effects formula such as
~ 1 | subject, or a list of such formulas.- repeated
Optional one-sided repeated-measures formula such as
~ time | subject.- structure
Residual covariance structure:
"id","cs","ar1","toep","toep(k)", or"un". The"toep"form estimates every available Toeplitz band. The"toep(k)"form estimateskbands, including the main diagonal, and fixes longer-lag covariances to zero.- method
Estimation method: restricted maximum likelihood (
"REML") or maximum likelihood ("ML").- ddf
Denominator degrees-of-freedom method:
"satterthwaite","kenward-roger", or"residual".- weights
Optional known relative precision weights. The residual covariance is scaled on both sides by
diag(1 / sqrt(weights)).- offset
Optional numeric offset added to the fixed-effect predictor. An
offset()term informulais also supported.- contrasts
Optional contrasts passed to
stats::model.matrix().- control
An object returned by
lmm_control().- na.action
Missing-value action:
stats::na.omit(),stats::na.exclude(), orstats::na.fail().
Value
An object of class lmm. The object stores fixed and random-effect
estimates, covariance components, fitted values, residuals, optimization
diagnostics, the analysis data, and the model matrices.
Details
The random argument accepts a grouping formula or a list of independent
grouping formulas. The repeated argument
accepts one ordering variable and one grouping expression. Each repeated
group may have at most one observation per ordering value. The likelihood
factors independent connected covariance components separately. The fitted
object retains the complete marginal covariance for model methods.
References
Harville, D. A. (1977). Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association, 72(358), 320-338. doi:10.1080/01621459.1977.10480998
LaMotte, L. R. (2007). A direct derivation of the REML likelihood function. Statistical Papers, 48(2), 321-327. doi:10.1007/s00362-006-0335-6
Pinheiro, J. C., and Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer. doi:10.1007/b98882
Kenward, M. G., and Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53(3), 983-997. doi:10.2307/2533558
Examples
fit <- lmm(
data = nlme::Orthodont,
formula = distance ~ age + Sex,
random = ~ 1 | Subject
)
summary(fit)
#> Linear mixed model
#> Estimation: REML
#> Denominator df: satterthwaite
#> Residual covariance: ID
#>
#> Fixed effects
#> # A tibble: 3 × 8
#> Term Estimate `Std Error` Statistic DF `p value` `Conf Low` `Conf High`
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 (Interc… 17.7 0.834 21.2 99.4 1.04e-38 16.1 19.4
#> 2 age 0.660 0.0616 10.7 80.0 3.95e-17 0.538 0.783
#> 3 SexFema… -2.32 0.761 -3.05 25.0 5.38e- 3 -3.89 -0.753
#> Type III tests
#> # A tibble: 2 × 5
#> Term `Num DF` `Den DF` Statistic `p value`
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 age 1 80.0 115. 3.95e-17
#> 2 Sex 1 25.0 9.29 5.38e- 3
#> Covariance parameters
#> # A tibble: 2 × 5
#> Group Term Component Estimate `Std Error`
#> <chr> <chr> <chr> <dbl> <dbl>
#> 1 Subject (Intercept) var 3.27 1.07
#> 2 Residual id var 2.05 0.324
#> Information criteria
#> logLik AIC BIC deviance
#> -218.7563 447.5125 460.9232 437.5125