Package {MLMES}

Type: Package
Title: Model-Based Effect Sizes for Multilevel Models
Version: 0.1.2
Description: Computes model-based effect sizes for fixed-effect coefficients in multilevel (hierarchical) models. The coefficient effect sizes are standardized mean differences from zero (d) and unique variance-explained measures (squared semi-partial correlations, sr2). The package also reports variance components and level-specific and total R-squared values. It supports 2-level and 3-level linear and binary logistic models fitted with 'lme4' (Bates et al., 2015) <doi:10.18637/jss.v067.i01>, and 2-level Gaussian and Bernoulli models fitted with 'brms' (Bürkner, 2017) <doi:10.18637/jss.v080.i01>. Sanders, Konold, and Cheng (in press), "Model-based effect sizes for multilevel linear regression coefficients," Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, describe the 2-level linear-model methods.
License: GPL-3
Encoding: UTF-8
Depends: R (≥ 3.5.0)
Imports: lme4 (≥ 1.1.34), Matrix (≥ 1.6-5), dplyr, stats
Suggests: brms, bayestestR, lmerTest (≥ 3.1-0), testthat (≥ 3.0.0)
Config/testthat/edition: 3
NeedsCompilation: no
Config/roxygen2/version: 8.0.0
Packaged: 2026-08-06 14:14:22 UTC; zigzag
Author: Yijun Cheng [aut, cre], Elizabeth A. Sanders [aut], Timothy R. Konold [aut], Zhigang Zhang [aut], Zixie Zheng [aut]
Maintainer: Yijun Cheng <chengxb@uw.edu>
Repository: CRAN
Date/Publication: 2026-08-20 15:52:04 UTC

MLMES: Model-Based Effect Sizes for Multilevel Models

Description

Computes model-based effect sizes for fixed-effect coefficients in multilevel models. The coefficient effect sizes are standardized mean differences from zero (d) and unique variance-explained measures (squared semi-partial correlations, sr^2). The package also reports variance components and level-specific and total R-squared values.

Details

The 2-level linear-model functions implement the methods described by Sanders, Konold, and Cheng (in press). The remaining functions extend the same variance-decomposition calculations to 3-level linear models, 2-level and 3-level binary logistic models, and 2-level Bayesian Gaussian and Bernoulli models. These extensions are not covered in the cited manuscript.

Functions are organized by model type:

2-level linear models (lme4::lmer): lmerES, lmerVarComp, lmerRsq

3-level linear models (lme4::lmer): lmerES_3L, lmerVarComp_3L, lmerRsq_3L

2-level logistic models (lme4::glmer): glmerES, glmerVarComp, glmerRsq

3-level logistic models (lme4::glmer): glmerES_3L, glmerVarComp_3L, glmerRsq_3L

2-level Bayesian models (brms::brm): brms_lmerES, brms_lmerVarComp, brms_lmerRsq, brms_lmerES_ci, brms_lmerVarComp_ci, brms_lmerRsq_ci

Author(s)

Maintainer: Yijun Cheng chengxb@uw.edu

Authors:

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

Bürkner, P.-C. (2017). brms: An R package for Bayesian multilevel models using Stan. Journal of Statistical Software, 80(1), 1–28. doi:10.18637/jss.v080.i01

See Also

hsb_data, readgrowth_data, and achieve_3L_data for example datasets.

Effect Sizes from 2-Level Bayesian Multilevel Models

Description

Computes model-based coefficient effect sizes, variance components, and R-squared values from posterior summaries of a 2-level model fitted with brms::brm.

Usage

brms_lmerVarComp(model, robust = TRUE, probs = 0.95, verbose = TRUE)

brms_lmerRsq(model, robust = TRUE, probs = 0.95, verbose = TRUE)

brms_lmerES(model, robust = TRUE, method = "HDI", probs = 0.95, verbose = TRUE)

Arguments

model

A 2-level Gaussian or Bernoulli model fitted with brms::brm.

robust

Logical; if TRUE (default), posterior locations are summarized by the median and posterior scales by the median absolute deviation.

probs

Numeric scalar giving the posterior interval probability. Default is 0.95.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

method

Character; method for computing credible intervals, passed to bayestestR::ci. Used for the coefficient intervals returned by brms_lmerES. Default is "HDI".

Details

These functions apply the 2-level variance-decomposition calculations to posterior point summaries. Use brms_lmerES_ci, brms_lmerVarComp_ci, or brms_lmerRsq_ci for intervals calculated from posterior draws.

For Bernoulli models, the effect sizes and variance components are on the latent scale. Observation-level variance is based on the mean response probability approximation described by McCulloch, Searle, and Neuhaus (2008).

Predictors should be centered, z-scored, and effect-coded as described in lmerES. All functions require brms; brms_lmerES also requires bayestestR.

Value

brms_lmerES returns coefficient posterior summaries, coefficient interval bounds, R-hat, and the columns d_lev, d_tot, sr2_lev, and sr2_tot. The effect sizes are calculated from posterior point summaries.

brms_lmerVarComp returns rows for L1, L2, and all variance sources, with columns level, L1_FE_Var, L1_Resid_Var, L1_RS_Var, L2_RI_Var, TotVar, and SD.

brms_lmerRsq returns Effect, TotalRsq, L1Rsq, and L2Rsq. R-squared values are proportions calculated from posterior point summaries.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bürkner, P.-C. (2017). brms: An R package for Bayesian multilevel models using Stan. Journal of Statistical Software, 80(1), 1–28. doi:10.18637/jss.v080.i01

Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing effects and their uncertainty, existence and significance within the Bayesian framework. Journal of Open Source Software, 4(40), 1541. doi:10.21105/joss.01541

McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2008). Generalized, linear, and mixed models (2nd ed.). Wiley.

See Also

brms_lmerES_ci, brms_lmerVarComp_ci, brms_lmerRsq_ci for versions with credible intervals; lmerES for frequentist 2-level models.

Examples


if (requireNamespace("brms", quietly = TRUE) &&
    requireNamespace("bayestestR", quietly = TRUE)) {
  data("hsb_data")

  hsb_brm_m1c <- brms::brm(mathach ~
      minority_eff_CMC + female_eff_CMC + z_ses_CMC +
      z_ses_agg_GMC +
      z_sqrt_size_GMC + sector_eff + z_disclim_GMC + himinty_eff +
      (1 + minority_eff_CMC + female_eff_CMC + z_ses_CMC | school),
    data = hsb_data, family = gaussian(),
    chains = 2, iter = 2000, warmup = 1000, cores = 2, seed = 212931)

  brms_lmerVarComp(hsb_brm_m1c)
  brms_lmerES(hsb_brm_m1c)
  brms_lmerRsq(hsb_brm_m1c)
}

Posterior Intervals for Effect Sizes from 2-Level Bayesian Models

Description

Computes posterior intervals for model-based coefficient effect sizes, variance components, and R-squared values from a 2-level model fitted with brms::brm.

Usage

brms_lmerVarComp_ci(
  model,
  robust = TRUE,
  method = "HDI",
  probs = 0.95,
  verbose = TRUE
)

brms_lmerRsq_ci(
  model,
  robust = TRUE,
  method = "HDI",
  probs = 0.95,
  verbose = TRUE
)

brms_lmerES_ci(
  model,
  robust = TRUE,
  method = "HDI",
  probs = 0.95,
  verbose = TRUE
)

Arguments

model

A 2-level Gaussian or Bernoulli model fitted with brms::brm.

robust

Logical; passed to the brms posterior summaries used during model setup. Default is TRUE.

method

Character; method for computing credible intervals, passed to bayestestR::ci. Default is "HDI".

probs

Numeric; probability for credible intervals. Default is 0.95.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

Intervals are calculated from posterior draws of the corresponding quantities; the returned tables do not include point estimates. For Bernoulli models, the quantities are on the latent scale.

Predictors should be centered, z-scored, and effect-coded as described in lmerES. These functions require brms and bayestestR.

Value

brms_lmerES_ci returns Variable, Level, and RS, followed by lower and upper posterior interval bounds for d_lev, d_tot, sr2_lev, and sr2_tot.

brms_lmerVarComp_ci returns rows for L1, L2, and all variance sources, with lower and upper posterior interval bounds for each variance component, TotVar, and SD.

brms_lmerRsq_ci returns Effect and lower and upper posterior interval bounds for TotalRsq, L1Rsq, and L2Rsq. R-squared values are proportions.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bürkner, P.-C. (2017). brms: An R package for Bayesian multilevel models using Stan. Journal of Statistical Software, 80(1), 1–28. doi:10.18637/jss.v080.i01

Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing effects and their uncertainty, existence and significance within the Bayesian framework. Journal of Open Source Software, 4(40), 1541. doi:10.21105/joss.01541

McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2008). Generalized, linear, and mixed models (2nd ed.). Wiley.

See Also

brms_lmerES, brms_lmerVarComp, brms_lmerRsq for versions without credible intervals.

Examples


if (requireNamespace("brms", quietly = TRUE) &&
    requireNamespace("bayestestR", quietly = TRUE)) {
  data("hsb_data")

  hsb_brm_m1c <- brms::brm(mathach ~
      minority_eff_CMC + female_eff_CMC + z_ses_CMC +
      z_ses_agg_GMC +
      z_sqrt_size_GMC + sector_eff + z_disclim_GMC + himinty_eff +
      (1 + minority_eff_CMC + female_eff_CMC + z_ses_CMC | school),
    data = hsb_data, family = gaussian(),
    chains = 2, iter = 2000, warmup = 1000, cores = 2, seed = 212931)

  brms_lmerVarComp_ci(hsb_brm_m1c)
  brms_lmerES_ci(hsb_brm_m1c)
  brms_lmerRsq_ci(hsb_brm_m1c)
}

Effect Sizes for 2-Level Binary Logistic Multilevel Models

Description

Computes model-based coefficient effect sizes, variance components, and R-squared values for a 2-level binary logistic model fitted with lme4::glmer. The calculations use a latent-scale variance decomposition.

Usage

glmerES(model, verbose = TRUE)

glmerVarComp(model, verbose = TRUE)

glmerRsq(model, verbose = TRUE)

Arguments

model

A fitted model object from lme4::glmer with 2-level structure and binomial family with logit link.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

The standardized mean difference and squared semi-partial correlation follow the definitions used for linear models, but their denominators are based on latent variance. The mean response probability is approximated from the model intercept and random-effects variance following McCulloch, Searle, and Neuhaus (2008). Observation-level variance is then calculated as 1 / \{\bar p(1 - \bar p)\}.

L1 predictors should be cluster-mean-centered. Continuous L1 and L2 predictors should be z-scored after centering, and categorical predictors should be effect-coded.

Value

glmerES returns a data frame with the following columns:

Variable

Predictor name.

Level

Predictor level: 1 or 2.

RS

Random-slope status: 1 = yes, 0 = no.

Coeff

Unstandardized logit coefficient.

SE

Standard error.

z

z statistic.

p

p value.

d_lev, d_tot

Standardized mean differences using level-specific and total latent variance.

sr2_lev, sr2_tot

Unique proportions of level-specific and total latent variance explained.

glmerVarComp returns rows for L1, L2, and all variance sources, with columns level, L1_FE_Var, L1_Resid_Var, L1_RS_Var, L2_RI_Var, TotVar, and SD.

glmerRsq returns Effect, TotalRsq, L1Rsq, and L2Rsq. R-squared values are proportions.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2008). Generalized, linear, and mixed models (2nd ed.). Wiley.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

glmerES_3L, glmerVarComp_3L, glmerRsq_3L for 3-level models; lmerES, lmerVarComp, lmerRsq for linear models.

Examples


data("hsb_data")

cutvalue_math <- quantile(hsb_data$mathach, probs = 0.40)
hsb_data$mathach_pass <- ifelse(hsb_data$mathach >= cutvalue_math, 1, 0)

hsb_glm_m1c <- lme4::glmer(mathach_pass ~
    minority_eff_CMC + female_eff_CMC + z_ses_CMC +
    z_ses_agg_GMC +
    z_sqrt_size_GMC + sector_eff + z_disclim_GMC + himinty_eff +
    (1 + minority_eff_CMC + female_eff_CMC + z_ses_CMC | school),
  data = hsb_data,
  family = binomial,
  control = lme4::glmerControl(optimizer = "bobyqa",
                               optCtrl = list(maxfun = 2e5)))

glmerES(hsb_glm_m1c)
glmerVarComp(hsb_glm_m1c)
glmerRsq(hsb_glm_m1c)

Three-Level Student Achievement Dataset

Description

A processed subset of the Achieve data used in Chapter 4, "Level-3 and Higher Models," of Finch, Bolin, and Kelley (2024). The data contain 10,320 students (L1) nested within 568 classrooms (L2), which are nested within 160 schools (L3).

Usage

data("achieve_3L_data")

Format

A data frame with 10,320 observations on 9 variables:

geread

General reading achievement score (outcome)

z_gevocab_CMC

General vocabulary score, classroom-mean-centered and standardized (L1)

gender_eff

Effect-coded gender: original code 1 = -1 and original code 2 = 1 (L1)

z_gemath_agg_CMC

Classroom mean mathematics achievement, school-mean-centered and standardized (L2)

z_ptratio_CMC

Pupil-teacher ratio, school-mean-centered and standardized (L2)

z_ses_GMC

School socioeconomic status, grand-mean-centered and standardized (L3)

school

School identifier (L3 unit)

class

Classroom code within school

school_class

Unique classroom identifier (L2 unit)

Details

Continuous predictors were centered according to their model level and then standardized. The student vocabulary score was centered within classrooms; classroom mean mathematics achievement and pupil-teacher ratio were centered within schools; and school socioeconomic status was centered at the grand mean. Only the variables used in the package examples are retained.

Source

https://www.mlminr.com/data-sets

References

Finch, W. H., Bolin, J. E., & Kelley, K. (2024). Multilevel modeling using R (3rd ed.). Chapman & Hall/CRC. doi:10.1201/b23166

Coefficient Effect Sizes for 3-Level Binary Logistic Multilevel Models

Description

Computes the standardized mean difference from zero (d) and unique variance-explained (sr^2) for each fixed-effect coefficient in a 3-level binary logistic model fitted with lme4::glmer. Effect sizes are based on latent variance.

Usage

glmerES_3L(model, verbose = TRUE)

Arguments

model

A 3-level model fitted with lme4::glmer, using the binomial family and logit link.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

The standardized mean difference and squared semi-partial correlation follow the definitions used for linear models, but their denominators are based on latent variance. L1 predictors should be centered within L2 units, and L2 predictors should be centered within L3 units. Continuous predictors should be z-scored after centering, and categorical predictors should be effect-coded.

Value

A data frame with coefficient estimates and model-based effect sizes:

Variable

Predictor name.

Level

Predictor level: 1, 2, or 3.

RS_L2

Random-slope status at L2: 1 = yes, 0 = no.

RS_L3

Random-slope status at L3: 1 = yes, 0 = no.

Coeff

Unstandardized logit coefficient.

SE

Standard error.

z

z statistic.

p

p value.

d_lev

Standardized mean difference using level-specific variance.

d_L1_L2

Standardized mean difference using L1 and L2 variance; reported for L1 predictors.

d_L2_L3

Standardized mean difference using L2 and L3 variance; reported for L2 predictors.

d_tot

Standardized mean difference using total variance.

sr2_lev

Unique proportion of level-specific variance explained.

sr2_L1_L2

Unique proportion of L1 and L2 variance explained; reported for L1 predictors.

sr2_L2_L3

Unique proportion of L2 and L3 variance explained; reported for L2 predictors.

sr2_tot

Unique proportion of total variance explained.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2008). Generalized, linear, and mixed models (2nd ed.). Wiley.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

glmerES for 2-level models, glmerVarComp_3L, glmerRsq_3L

Examples

data("achieve_3L_data")

cutvalue_read <- stats::quantile(achieve_3L_data$geread, probs = 0.40)
achieve_binary <- transform(
  achieve_3L_data,
  geread_pass = as.integer(geread >= cutvalue_read)
)

three_level_logit_fit <- lme4::glmer(
  geread_pass ~ z_gevocab_CMC + gender_eff +
    z_gemath_agg_CMC + z_ptratio_CMC + z_ses_GMC +
    (1 | school_class) + (1 | school),
  data = achieve_binary,
  family = binomial,
  control = lme4::glmerControl(
    optimizer = "bobyqa",
    optCtrl = list(maxfun = 2e5)
  )
)

glmerES_3L(three_level_logit_fit)

R-Squared Measures for 3-Level Binary Logistic Multilevel Models

Description

Returns latent-scale R-squared values for the variance sources in a 3-level binary logistic model fitted with lme4::glmer.

Usage

glmerRsq_3L(model, verbose = TRUE)

Arguments

model

A 3-level model fitted with lme4::glmer, using the binomial family and logit link.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

Rows beginning with FE, L1_RS, L2_RS, or RI give R-squared values for the indicated variance source. Rows containing total combine fixed- and random-effects variance sources across levels.

Value

A data frame with R-squared values for each variance source or combination of sources:

Effect

Variance source or combination of sources.

TotalRsq

Proportion of total latent variance explained.

L1Rsq

Proportion of L1 latent variance explained.

L2Rsq

Proportion of L2 latent variance explained.

L3Rsq

Proportion of L3 latent variance explained.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2008). Generalized, linear, and mixed models (2nd ed.). Wiley.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

glmerRsq for 2-level models, glmerVarComp_3L, glmerES_3L

Examples

data("achieve_3L_data")

cutvalue_read <- stats::quantile(achieve_3L_data$geread, probs = 0.40)
achieve_binary <- transform(
  achieve_3L_data,
  geread_pass = as.integer(geread >= cutvalue_read)
)

three_level_logit_fit <- lme4::glmer(
  geread_pass ~ z_gevocab_CMC + gender_eff +
    z_gemath_agg_CMC + z_ptratio_CMC + z_ses_GMC +
    (1 | school_class) + (1 | school),
  data = achieve_binary,
  family = binomial,
  control = lme4::glmerControl(
    optimizer = "bobyqa",
    optCtrl = list(maxfun = 2e5)
  )
)

glmerRsq_3L(three_level_logit_fit)

Variance Components for 3-Level Binary Logistic Multilevel Models

Description

Returns the latent-scale variance components used to calculate model-based effect sizes for a 3-level binary logistic model fitted with lme4::glmer.

Usage

glmerVarComp_3L(model, verbose = TRUE)

Arguments

model

A 3-level model fitted with lme4::glmer, using the binomial family and logit link.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

The mean response probability is approximated from the model intercept and random-effects variance following McCulloch, Searle, and Neuhaus (2008). Observation-level variance is then calculated as 1 / \{\bar p(1 - \bar p)\}.

L1 predictors should be centered within L2 units, and L2 predictors should be centered within L3 units. Continuous predictors should be z-scored after centering, and categorical predictors should be effect-coded.

Value

A data frame with rows for L1, L2, L3, and all variance sources:

levels

Variance level: L1, L2, L3, or All.

FE_Var

Fixed-effects variance.

L1_Resid_Var

Observation-level variance.

L1_RS_L2_Var

Variance from L1 slopes varying across L2 units.

L1_RS_L3_Var

Variance from L1 slopes varying across L3 units. When an L1 and an L2 predictor both have random slopes at L3, their slope contributions covary, and the full L3 slope variance exceeds the sum of the two separate variances by twice that covariance. This component includes one copy of the covariance and L2_RS_Var includes the other, so the two together account for the full L3 slope variance. Under the cluster-mean centering the framework assumes, the covariance is exactly zero and this term is the plain variance.

L2_RI_Var

Conditional L2 random-intercept variance.

L2_RS_Var

Variance from L2 slopes varying across L3 units. Includes the second copy of the covariance described under L1_RS_L3_Var.

L3_RI_Var

Conditional L3 random-intercept variance.

TotVar

Variance for the specified level.

SD

Square root of TotVar.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2008). Generalized, linear, and mixed models (2nd ed.). Wiley.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

glmerVarComp for 2-level models, glmerRsq_3L, glmerES_3L

Examples

data("achieve_3L_data")

cutvalue_read <- stats::quantile(achieve_3L_data$geread, probs = 0.40)
achieve_binary <- transform(
  achieve_3L_data,
  geread_pass = as.integer(geread >= cutvalue_read)
)

three_level_logit_fit <- lme4::glmer(
  geread_pass ~ z_gevocab_CMC + gender_eff +
    z_gemath_agg_CMC + z_ptratio_CMC + z_ses_GMC +
    (1 | school_class) + (1 | school),
  data = achieve_binary,
  family = binomial,
  control = lme4::glmerControl(
    optimizer = "bobyqa",
    optCtrl = list(maxfun = 2e5)
  )
)

glmerVarComp_3L(three_level_logit_fit)

High School and Beyond (HSB) Dataset

Description

Cross-sectional data on math achievement and sociodemographic characteristics for 7,185 students from 160 U.S. high schools. Students (L1) are nested within schools (L2). The data include the original variables and the centered, effect-coded, and z-scored predictors used in the package examples.

Usage

data("hsb_data")

Format

A data frame with 7,185 observations on 27 variables:

school

School identifier (level-2 unit)

size

School size

sector

School sector (1 = Catholic, 0 = public)

disclim

Disciplinary climate

himinty

Indicator for more than 40 percent minority enrollment (1 = yes, 0 = no)

student

Student identifier

minority

Minority status (1 = yes, 0 = no)

female

Female indicator (1 = yes, 0 = no)

ses

Socioeconomic status

mathach

Mathematics achievement score (outcome)

minority_eff

Minority status, effect-coded

female_eff

Female indicator, effect-coded

minority_eff_CMC

Minority status, effect-coded and cluster-mean-centered (L1)

female_eff_CMC

Female indicator, effect-coded and cluster-mean-centered (L1)

ses_CMC

SES, cluster-mean-centered (L1)

z_ses_CMC

SES, cluster-mean-centered and z-scored (L1)

mathach_agg

School-aggregated math achievement

ses_agg

School-aggregated SES

sqrt_size

Square root of school size

sector_eff

School sector, effect-coded (L2)

himinty_eff

High minority enrollment, effect-coded (L2)

ses_agg_GMC

Aggregated SES, grand-mean-centered (L2)

sqrt_size_GMC

Square root of school size, grand-mean-centered (L2)

disclim_GMC

Disciplinary climate, grand-mean-centered (L2)

z_ses_agg_GMC

Aggregated SES, grand-mean-centered and z-scored (L2)

z_sqrt_size_GMC

Square root of school size, grand-mean-centered and z-scored (L2)

z_disclim_GMC

Disciplinary climate, grand-mean-centered and z-scored (L2)

References

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks, CA: Sage Publications.

Coefficient Effect Sizes for 2-Level Linear Multilevel Models

Description

Computes two model-based effect sizes for each fixed-effect coefficient in a 2-level hierarchical linear model fitted with lme4::lmer: the standardized mean difference from zero (d) and unique variance-explained (sr^2). Each effect size is reported relative to level-specific variance and total variance.

Usage

lmerES(model, verbose = TRUE)

Arguments

model

A 2-level linear model fitted with lme4::lmer.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

The standardized mean difference expresses a coefficient relative to the model-implied standard deviation of the outcome. The squared semi-partial correlation gives the proportion of outcome variance uniquely explained by a predictor after accounting for its shared variance with the other predictors. Level-specific effect sizes use variance sources associated with the predictor level; total effect sizes use all L1 and L2 variance sources.

Degrees of freedom and p values are taken from summary(model). Fit the model with lmerTest::lmer, or convert an existing lme4::lmer model with lmerTest::as_lmerModLmerTest, to report these values. For an lme4::lmer model, the df and p columns contain NA.

L1 predictors should be cluster-mean-centered. Continuous L1 and L2 predictors should be z-scored after centering, and categorical predictors should be effect-coded. When the model is fitted by maximum likelihood (REML = FALSE), finite-sample bias corrections are applied before the effect sizes are calculated.

Value

A data frame with coefficient estimates and model-based effect sizes:

Variable

Predictor name.

Level

Predictor level: 1 or 2.

RS

Random-slope status: 1 = yes, 0 = no.

Coeff

Unstandardized coefficient estimate.

SE

Standard error.

t

t statistic.

df

Degrees of freedom, when available.

p

p value, when available.

d_lev

Standardized mean difference using level-specific variance.

d_tot

Standardized mean difference using total variance.

sr2_lev

Unique proportion of level-specific variance explained.

sr2_tot

Unique proportion of total variance explained.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

Kuznetsova, A., Brockhoff, P. B., & Christensen, R. H. B. (2017). lmerTest package: Tests in linear mixed effects models. Journal of Statistical Software, 82(13), 1–26. doi:10.18637/jss.v082.i13

See Also

lmerES_3L for 3-level models, lmerVarComp, lmerRsq

Examples

data("hsb_data")

if (requireNamespace("lmerTest", quietly = TRUE)) {
  hsb_m1c <- lmerTest::lmer(mathach ~
      minority_eff_CMC + female_eff_CMC + z_ses_CMC +
      z_ses_agg_GMC +
      z_sqrt_size_GMC + sector_eff + z_disclim_GMC + himinty_eff +
      (1 + minority_eff_CMC + female_eff_CMC + z_ses_CMC | school),
    data = hsb_data, REML = FALSE,
    control = lme4::lmerControl(optimizer = "bobyqa",
                                optCtrl = list(maxfun = 2e5)))

  lmerES(hsb_m1c)
}

Coefficient Effect Sizes for 3-Level Linear Multilevel Models

Description

Computes two model-based effect sizes for each fixed-effect coefficient in a 3-level hierarchical linear model fitted with lme4::lmer: the standardized mean difference from zero (d) and unique variance-explained (sr^2). Effect sizes are reported relative to level-specific, adjacent-level, and total variance where applicable.

Usage

lmerES_3L(model, verbose = TRUE)

Arguments

model

A 3-level linear model fitted with lme4::lmer.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

The standardized mean difference expresses a coefficient relative to the model-implied standard deviation of the outcome. The squared semi-partial correlation gives the proportion of outcome variance uniquely explained by a predictor after accounting for its shared variance with the other predictors.

Degrees of freedom and p values are taken from summary(model). Fit the model with lmerTest::lmer, or convert an existing lme4::lmer model with lmerTest::as_lmerModLmerTest, to report these values. For an lme4::lmer model, the df and p columns contain NA.

L1 predictors should be centered within L2 units, and L2 predictors should be centered within L3 units. Continuous predictors should be z-scored after centering, and categorical predictors should be effect-coded. When the model is fitted by maximum likelihood (REML = FALSE), finite-sample bias corrections are applied before the effect sizes are calculated.

Value

A data frame with coefficient estimates and model-based effect sizes:

Variable

Predictor name.

Level

Predictor level: 1, 2, or 3.

RS_L2

Random-slope status at L2: 1 = yes, 0 = no.

RS_L3

Random-slope status at L3: 1 = yes, 0 = no.

Coeff

Unstandardized coefficient estimate.

SE

Standard error.

t

t statistic.

df

Degrees of freedom, when available.

p

p value, when available.

d_lev

Standardized mean difference using level-specific variance.

d_L1_L2

Standardized mean difference using L1 and L2 variance; reported for L1 predictors.

d_L2_L3

Standardized mean difference using L2 and L3 variance; reported for L2 predictors.

d_tot

Standardized mean difference using total variance.

sr2_lev

Unique proportion of level-specific variance explained.

sr2_L1_L2

Unique proportion of L1 and L2 variance explained; reported for L1 predictors.

sr2_L2_L3

Unique proportion of L2 and L3 variance explained; reported for L2 predictors.

sr2_tot

Unique proportion of total variance explained.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

Kuznetsova, A., Brockhoff, P. B., & Christensen, R. H. B. (2017). lmerTest package: Tests in linear mixed effects models. Journal of Statistical Software, 82(13), 1–26. doi:10.18637/jss.v082.i13

See Also

lmerES for 2-level models, lmerVarComp_3L, lmerRsq_3L

Examples

data("achieve_3L_data")

three_level_fit <- lmerTest::lmer(
  geread ~ z_gevocab_CMC + gender_eff +
    z_gemath_agg_CMC + z_ptratio_CMC + z_ses_GMC +
    (1 | school_class) + (1 | school),
  data = achieve_3L_data,
  REML = FALSE
)

lmerES_3L(three_level_fit)

R-Squared Measures for 2-Level Linear Multilevel Models

Description

Returns R-squared values for the variance sources in a 2-level hierarchical linear model fitted with lme4::lmer. Level-specific values use the variance sources associated with L1 or L2; total values use all L1 and L2 variance sources.

Usage

lmerRsq(model, verbose = TRUE)

Arguments

model

A 2-level linear model fitted with lme4::lmer.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

Rows beginning with FE_L1, FE_L2, RS_L1, or RI_L2 give R-squared values for a single variance source. Rows beginning with FE_total, FE_RS_total, or FE_RS_RI_total combine the indicated fixed- and random-effects variance sources.

Value

A data frame with R-squared values for each variance source or combination of sources:

Effect

Variance source or combination of sources.

TotalRsq

Proportion of total variance explained.

L1Rsq

Proportion of L1 variance explained.

L2Rsq

Proportion of L2 variance explained.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

lmerRsq_3L for 3-level models, lmerVarComp, lmerES

Examples

data("hsb_data")

hsb_m1c <- lme4::lmer(mathach ~
    minority_eff_CMC + female_eff_CMC + z_ses_CMC +
    z_ses_agg_GMC +
    z_sqrt_size_GMC + sector_eff + z_disclim_GMC + himinty_eff +
    (1 + minority_eff_CMC + female_eff_CMC + z_ses_CMC | school),
  data = hsb_data, REML = FALSE,
  control = lme4::lmerControl(optimizer = "bobyqa",
                              optCtrl = list(maxfun = 2e5)))

lmerRsq(hsb_m1c)

R-Squared Measures for 3-Level Linear Multilevel Models

Description

Returns R-squared values for the variance sources in a 3-level hierarchical linear model fitted with lme4::lmer. Level-specific values use the variance sources associated with L1, L2, or L3; total values use all three levels.

Usage

lmerRsq_3L(model, verbose = TRUE)

Arguments

model

A 3-level linear model fitted with lme4::lmer.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

Rows beginning with FE, L1_RS, L2_RS, or RI give R-squared values for the indicated variance source. Rows containing total combine fixed- and random-effects variance sources across levels.

Value

A data frame with R-squared values for each variance source or combination of sources:

Effect

Variance source or combination of sources.

TotalRsq

Proportion of total variance explained.

L1Rsq

Proportion of L1 variance explained.

L2Rsq

Proportion of L2 variance explained.

L3Rsq

Proportion of L3 variance explained.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

lmerRsq for 2-level models, lmerVarComp_3L, lmerES_3L

Examples

data("achieve_3L_data")

three_level_fit <- lme4::lmer(
  geread ~ z_gevocab_CMC + gender_eff +
    z_gemath_agg_CMC + z_ptratio_CMC + z_ses_GMC +
    (1 | school_class) + (1 | school),
  data = achieve_3L_data,
  REML = FALSE
)

lmerRsq_3L(three_level_fit)

Variance Components for 2-Level Linear Multilevel Models

Description

Returns the variance components used to calculate model-based effect sizes for a 2-level hierarchical linear model fitted with lme4::lmer. The components are L1 and L2 fixed-effects variance, L1 random-slopes variance, conditional L2 random-intercept variance, and residual L1 variance.

Usage

lmerVarComp(model, verbose = TRUE)

Arguments

model

A 2-level linear model fitted with lme4::lmer.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

L1 predictors should be cluster-mean-centered. Continuous L1 and L2 predictors should be z-scored after centering, and categorical predictors should be effect-coded. When the model is fitted by maximum likelihood (REML = FALSE), finite-sample bias corrections are applied to the random-effects and residual variance estimates.

Value

A data frame with rows for L1, L2, and all variance sources:

level

Variance level: L1, L2, or All.

FE_Var

Fixed-effects variance.

L1_Resid_Var

Residual L1 variance.

L1_RS_Var

L1 random-slopes variance.

L2_RI_Var

Conditional L2 random-intercept variance.

TotVar

Variance for the specified level.

SD

Square root of TotVar.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

lmerVarComp_3L for 3-level models, lmerRsq, lmerES

Examples

data("hsb_data")

hsb_m1c <- lme4::lmer(mathach ~
    minority_eff_CMC + female_eff_CMC + z_ses_CMC +
    z_ses_agg_GMC +
    z_sqrt_size_GMC + sector_eff + z_disclim_GMC + himinty_eff +
    (1 + minority_eff_CMC + female_eff_CMC + z_ses_CMC | school),
  data = hsb_data, REML = FALSE,
  control = lme4::lmerControl(optimizer = "bobyqa",
                              optCtrl = list(maxfun = 2e5)))

lmerVarComp(hsb_m1c)

Variance Components for 3-Level Linear Multilevel Models

Description

Returns the variance components used to calculate model-based effect sizes for a 3-level hierarchical linear model fitted with lme4::lmer. Components are reported for L1, L2, L3, and all variance sources.

Usage

lmerVarComp_3L(model, verbose = TRUE)

Arguments

model

A 3-level linear model fitted with lme4::lmer.

verbose

Logical; if TRUE (default), prints a note explaining column abbreviations.

Details

This function extends the 2-level variance decomposition described by Sanders, Konold, and Cheng (in press) to three levels.

L1 predictors should be centered within L2 units, and L2 predictors should be centered within L3 units. Continuous predictors should be z-scored after centering, and categorical predictors should be effect-coded. When the model is fitted by maximum likelihood (REML = FALSE), finite-sample bias corrections are applied to the random-effects and residual variance estimates.

Value

A data frame with rows for L1, L2, L3, and all variance sources:

levels

Variance level: L1, L2, L3, or All.

FE_Var

Fixed-effects variance.

L1_Resid_Var

Residual L1 variance.

L1_RS_L2_Var

Variance from L1 slopes varying across L2 units.

L1_RS_L3_Var

Variance from L1 slopes varying across L3 units. When an L1 and an L2 predictor both have random slopes at L3, their slope contributions covary, and the full L3 slope variance exceeds the sum of the two separate variances by twice that covariance. This component includes one copy of the covariance and L2_RS_Var includes the other, so the two together account for the full L3 slope variance. Under the cluster-mean centering the framework assumes, the covariance is exactly zero and this term is the plain variance.

L2_RI_Var

Conditional L2 random-intercept variance.

L2_RS_Var

Variance from L2 slopes varying across L3 units. Includes the second copy of the covariance described under L1_RS_L3_Var.

L3_RI_Var

Conditional L3 random-intercept variance.

TotVar

Variance for the specified level.

SD

Square root of TotVar.

References

Sanders, E. A., Konold, T. R., & Cheng, Y. (in press). Model-based effect sizes for multilevel linear regression coefficients. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1–48. doi:10.18637/jss.v067.i01

See Also

lmerVarComp for 2-level models, lmerRsq_3L, lmerES_3L

Examples

data("achieve_3L_data")

three_level_fit <- lme4::lmer(
  geread ~ z_gevocab_CMC + gender_eff +
    z_gemath_agg_CMC + z_ptratio_CMC + z_ses_GMC +
    (1 | school_class) + (1 | school),
  data = achieve_3L_data,
  REML = FALSE
)

lmerVarComp_3L(three_level_fit)

Early Reading Growth Dataset

Description

Longitudinal data from 79 children who received an individual, phonics-based reading intervention in first grade. Children were assessed at four waves over three years, beginning in the fall of first grade. Repeated observations (L1) are nested within children (L2), and the data are in long format.

Usage

data("readgrowth_data")

Format

A data frame with 316 observations on 18 variables:

schnum2

School identifier

subnum2

Child identifier (level-2 unit)

ppvty1_1

Receptive vocabulary, Peabody Picture Vocabulary Test standard score

rlny1_1

Rapid letter naming, Comprehensive Test of Phonological Processing raw score

yoppy1_1

Phonemic awareness, Yopp-Singer Test of Phoneme Segmentation raw score

ppvt_GMC

Receptive vocabulary (PPVT), grand-mean-centered (L2)

rln_GMC

Rapid letter naming, grand-mean-centered (L2)

yopp_GMC

Phonemic awareness (Yopp-Singer), grand-mean-centered (L2)

z_ppvt_GMC

Receptive vocabulary (PPVT), grand-mean-centered and z-scored (L2)

z_rln_GMC

Rapid letter naming, grand-mean-centered and z-scored (L2)

z_yopp_GMC

Phonemic awareness (Yopp-Singer), grand-mean-centered and z-scored (L2)

time

Time point

WR

Woodcock-Johnson Mastery Test Word Identification raw score (outcome)

Decode

Decoding score

Spell

Spelling score

TimeLin

Linear time regressor

Rate1

Piecewise rate regressor for growth during first grade (L1)

Rate2

Piecewise rate regressor for growth during grades 2 and 3 (L1)

References

Vadasy, P. F., Sanders, E. A., & Abbott, R. D. (2008). Effects of supplemental early reading intervention at 2-year follow up: Reading skill growth patterns and predictors. Scientific Studies of Reading, 12(1), 51–89. doi:10.1080/10888430701746906