This document is licensed under the Creative Commons attribution-noncommercial license. Please share & remix noncommercially, mentioning its origin. Sourcecode and data are available here.

The violation of model-assumptions in RE-models for panel data

This example shows how to address the issue when group factors (random effects) and (time-constant) predictors correlate for mixed models, especially in panel data. Models, where predictors and group factors correlate, may have compromised estimates of uncertainty as well as possible bias. In particular in econometrics, fixed-effects models are considered the gold-standard to address such issues. However, it often makes no sense to consider group-effects as “fixed” over a long time period. Apart from this, there are more shortcomings of FE-models as well, see (Bell, Fairbrother, and Jones 2018), (Bell, Jones, and Fairbrother 2018) and (Bafumi and Gelman 2006).

The following equations and discussion on FE vs. RE are based on (Bell, Fairbrother, and Jones 2018). Further discussion on FE vs. RE, also at the end of this document, refer to (Gelman and Hill 2007) and (Bafumi and Gelman 2006).

Adding group meaned predictors to solve this issue

The solution to the critics from “FE-modelers” is simple: If you include a group-mean of your variables in a random effects model (that is, calculating the mean of the predictor at each group-level and including it as a group-level predictor), it will give the same answer as a fixed effects model (see table 3 very below, and (Bell, Jones, and Fairbrother 2018) as reference). This is why FE-modelers often call this type of RE-model also a “kind of” FE-model, i.e. they define a RE model as a model where predictors are assumed uncorrelated with the residuals. However,

“Calling it a FE model is not just inaccurate. It also does down its potential. Eg FE models don’t usually include random slopes, and failing to do so can lead to incorrect SEs as well as being a less interesting and realistic model.”

“A random effects model is such because it has random effects (that is, higher-level entities treated as a distribution) in it rather than fixed effects (higher-level entities treated as dummy variables) in it.”

source: Twitter-Discussion 1, Twitter-Discussion 2

Problems of ignoring random slopes in Fixed Effects models

(Heisig, Schaeffer, and Giesecke 2017) demonstrate how ignoring random slopes, i.e. neglecting “cross-cluster differences in the effects of lower-level controls reduces the precision of estimated context effects, resulting in unnecessarily wide confidence intervals and low statistical power”. You may refer to this paper to justify a mixed model with random slopes over “simple” FE-models.

Examples

The following code snippets show how to translate the Equations from (Bell, Fairbrother, and Jones 2018) into R-code, using lmer() from the lme4-package.

library(lme4)
library(sjPlot)
library(parameters)
library(lfe)

load("example.RData")

Sourcecode and data are available here.

Description of the data

Variables:
- x_tv : time-varying variable
- z1_ti : first time-invariant variable, co-variate
- z2_ti : second time-invariant variable, co-variate
- QoL : Response (quality of life of patient)
- ID : patient ID
- time : time-point of measurement

“Classical” growth-model for longitudinal data

model_re <- lmer(
  QoL ~ time + age + x_tv + z1_ti + z2_ti + (1 + time | ID),
  data = d
)

Computing the de-meaned and group-meaned variables

Next is a model from Eq. 10, which includes the “de-meaned” time-varying variable as well as the “group-meaned” time-varying variable.

# compute mean of "x_tv" for each subject (ID) and
# then "de-mean" x_tv
d <- cbind(
  d,
  demean(d, select = c("x_tv", "QoL"), group = "ID") # from package "parameters"
)

Now we have:

x_tv_between : time-varying variable with the mean of x_tv accross all time-points, for each patient (ID).
x_tv_within : the de-meaned time-varying variable x_tv

QoL_between and QoL_within are used to test different FE-models, which are described later. In those models, I also use a “de-meaned” response variable without the group-variable (ID) as fixed effect (see Equation 6 in the paper).

The complex random-effect-within-between model (REWB)

Eq. 10 suggests allowing the “within-effect” (de-meaned) vary across individuals, that’s why x_tv_within is added as random slope as well.

Here, the estimate of x_tv_within indicates the within-subject effect, while the estimate of x_tv_between indicates the between-subject effect. This model also allows for heterogenity across level-2 units, that’s why x_tv_within also appears in the random effects. The estimates of z1_ti and z2_ti also indicate a between-subject effect, as this is a level-2 variable, which cannot have a within-subject effect.

Model from Equation 10

Here is the equation 10 from Bell et al. 2018:

y_it = β₀ + β_1W (x_it - ͞x_i) + β_2B ͞x_i + β₃ z_i + υ_i0 + υ_i1 (x_it - ͞x_i) + ε_it

x_it - ͞x_i is the de-meaned predictor, x_tv_within
͞x_i is the group-meaned predictor, x_tv_between
β_1W is the coefficient for x_tv_within (within-subject)
β_2B is the coefficient for x_tv_between (bewteen-subject)
β₃ is the coefficient for z1_ti or z2_ti (bewteen-subject)

# This model this leads to an error (number of observations <= number of 
# random effects), the check for nobs vs. re is ignored here.
model_rewb <- lmer(
  QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + (1 + time + x_tv_within | ID),
  data = d,
  control = lmerControl(check.nobs.vs.nRE = "ignore")
)

# An alternative could be to model the random effects as not correlated.
# m2b <- lmer(
#   QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + (1 + time + x_tv_within || ID),
#   data = d
# )

# here we get no error message, model runs fine...
model_complex_rewb <- lmer(
  QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + 
    (1 + time | ID) + (1 + x_tv_within | ID),
  data = d
)

# an alternative would be to assume independence between random slopes
# and no covariance...
model_complex_rewb_2 <- lmer(
  QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + 
    (1 + time | ID) + (0 + x_tv_within | ID),
  data = d
)

We compare all model fits, but we go on with model_complex_rewb (the second model in the table below) for now…

Table 1: Comparison of complex REWB-Models

tab_model(
  model_rewb, model_complex_rewb, model_complex_rewb_2,
  show.ci = FALSE, 
  show.se = TRUE, 
  auto.label = FALSE, 
  string.se = "SE",
  show.icc = FALSE,
  dv.labels = c("Complex REWB (1)", "Complex REWB (2)", "Complex REWB (3)")
)

	Complex REWB (1)			Complex REWB (2)			Complex REWB (3)
Predictors	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p
(Intercept)	61.82	5.82	<0.001	61.69	5.82	<0.001	62.46	5.85	<0.001
time	1.16	0.61	0.056	1.19	0.60	0.048	1.17	0.61	0.053
age	-0.22	0.19	0.258	-0.22	0.19	0.250	-0.19	0.19	0.318
x_tv_within	-4.44	0.57	<0.001	-4.50	0.58	<0.001	-4.54	0.58	<0.001
x_tv_between	-6.29	0.52	<0.001	-6.30	0.52	<0.001	-6.22	0.52	<0.001
z1_ti	5.11	4.34	0.239	5.12	4.34	0.238	4.74	4.37	0.277
z2_ti	0.00	0.00	0.087	0.00	0.00	0.084	0.00	0.00	0.112
Random Effects
σ²	119.21			119.27			119.61
τ₀₀	143.59 _ID			110.58 _ID			141.55 _ID
				28.22 _ID.1			13.91 _ID.1
τ₁₁	1.23 _ID.time			0.59 _ID.time			1.01 _ID.time
	13.95 _{ID.x_tv_within}			14.45 _{ID.1.x_tv_within}
ρ₀₁	-0.75			-1.00 _ID			-0.77 _ID
	0.36			0.50 _ID.1
N	188 _ID			188 _ID			188 _ID
Observations	564			564			564
Marginal R² / Conditional R²	0.395 / 0.713			0.452 / 0.674			0.415 / 0.694

The simple random-effect-within-between model (REWB) and Mundlak model

After email correspondance, the paper’s authors suggest that, depending on the research interest and necessary complexity of the model, a “simple” random-slope might be suitable as well. As stated in the paper, this is useful if homogenity across level-2 units is assumed. This model usually yields the same results as a FE-model, however, we additionally have information about the random effects - and the model can incorporate time-invariant covariates.

Again, the estimate of x_tv_within indicates the within-subject effect, while the estimate of x_tv_between indicates the between-subject effect.

Model from Equation 2

y_it = β₀ + β_1W (x_it - ͞x_i) + β_2B ͞x_i + β₃ z_i + (υ_i + ε_it)

model_simple_rewb <- lmer(
  QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + (1 + time | ID),
  data = d
)

An alternativ would be the Mundlak model. Here, the estimate of x_tv indicates the within-subject effect, while the estimate of x_tv_between indicates the contextual effect.

Model from Equation 3

y_it = β₀ + β_1W x_it + β_2C ͞x_i + β₃ z_i + (υ_i + ε_it)

model_mundlak <- lmer(
  QoL ~ time + age + x_tv + x_tv_between + z1_ti + z2_ti + (1 + time | ID),
  data = d
)

The contextual effect, i.e. the coefficient for x_tv_between, indicates the effect of an individual (at level 1) that moves from one level-2 group into another one. In the above model, or in general: in case of longitudinal data, the contextual effect is meaningless, as level-2 predictors are individuals (or subjects) themselves, and by definition cannot “move” to another individual. Therefor, the REWB-model is more informative.

Comparison of models

In table 2, we compare the “classical” RE-model, the complex REWB-model, the “simple” REWB-model and the Mundlak-Model.

Table 2: Comparison of RE, REWB and Mundlak Models

tab_model(
  model_re, model_complex_rewb, model_simple_rewb, model_mundlak,
  show.ci = FALSE, 
  show.se = TRUE, 
  auto.label = FALSE, 
  string.se = "SE",
  show.icc = FALSE,
  dv.labels = c("Classical RE", "Complex REWB", "Simple REWB", "Mundlak")
)

	Classical RE			Complex REWB			Simple REWB			Mundlak
Predictors	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p
(Intercept)	60.96	5.96	<0.001	61.69	5.82	<0.001	62.73	5.85	<0.001	62.73	5.85	<0.001
time	1.14	0.67	0.087	1.19	0.60	0.048	1.09	0.66	0.098	1.09	0.66	0.098
age	-0.15	0.20	0.458	-0.22	0.19	0.250	-0.22	0.19	0.261	-0.22	0.19	0.261
x_tv	-4.69	0.33	<0.001							-3.73	0.41	<0.001
z1_ti	4.48	4.44	0.313	5.12	4.34	0.238	4.43	4.34	0.308	4.43	4.34	0.308
z2_ti	0.00	0.00	0.023	0.00	0.00	0.084	0.00	0.00	0.095	0.00	0.00	0.095
x_tv_within				-4.50	0.58	<0.001	-3.73	0.41	<0.001
x_tv_between				-6.30	0.52	<0.001	-6.30	0.52	<0.001	-2.56	0.66	<0.001
Random Effects
σ²	142.88			119.27			142.12			142.12
τ₀₀	208.07 _ID			110.58 _ID			201.95 _ID			201.95 _ID
				28.22 _ID.1
τ₁₁	12.48 _ID.time			0.59 _ID.time			10.82 _ID.time			10.82 _ID.time
				14.45 _{ID.1.x_tv_within}
ρ₀₁	-0.74 _ID			-1.00 _ID			-0.77 _ID			-0.77 _ID
				0.50 _ID.1
N	188 _ID			188 _ID			188 _ID			188 _ID
Observations	564			564			564			564
Marginal R² / Conditional R²	0.311 / 0.619			0.452 / 0.674			0.382 / 0.649			0.382 / 0.649

Check if a REWB- or simple RE-model suitable

If the estimates of the within- and between-effect (x_tv_within and x_tv_between) are (almost) identical, or if the contextual effect (x_tv_between) in the Mundlak-model is zero and doesn’t give a significant improvement for the model, you can also use a simple RE-model.

A simple way to check this is a likelihood-ratio test between the simple RE-model and the Mundlak-model:

anova(model_re, model_mundlak)
#> Data: d
#> Models:
#> model_re: QoL ~ time + age + x_tv + z1_ti + z2_ti + (1 + time | ID)
#> model_mundlak: QoL ~ time + age + x_tv + x_tv_between + z1_ti + z2_ti + (1 + 
#> model_mundlak:     time | ID)
#>               Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
#> model_re      10 4662.6 4705.9 -2321.3   4642.6                             
#> model_mundlak 11 4649.5 4697.2 -2313.8   4627.5 15.081      1   0.000103 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Here we see a significant improvement of the Mundlak-model over the simple RE-model, indicating that it makes sense to model within- and between-subjects effects, i.e. to apply a REWB-model.

Comparison FE- and REWB-Model

The function felm() from the package lfe was use to compute the fixed effects regression models. Base R’s lm() gives the same result, however, the output is much longer due to the ID-parameters.

y_it = β₁ (x_it - ͞x_i) + (υ_i + ε_it)

# Model from Equation 5
model_fe_ID <- felm(
  QoL ~ time + x_tv_within | ID,
  data = d
)

# same as this lm-model
# model_fe_ID <- lm(
#   QoL ~ 0 + time + x_tv_within + ID,
#   data = d
# )

Equation 6 describes a fixed effects model with de-meaned dependent variable.

(y_it - ͞y_i) = β₁ (x_it - ͞x_i) + ε_it

# Model from Equation 6
model_fe_y_within <- felm(
  QoL_within ~ time + x_tv_within,
  data = d
)

# or ...
# model_fe_y_within <- lm(
#   QoL_within ~ 0 + time + x_tv_within,
#   data = d
# )

We compare the results from the FE-models with a simple RE-model and the REWB-model.

model_re_2 <- lmer(
  QoL ~ time + x_tv_within + x_tv_between + (1 | ID),
  data = d
)

# Compare with complex REWB-model
model_complex_rewb3 <- lmer(
  QoL ~ time + x_tv_within + x_tv_between +
    (1 + time | ID) + (1 + x_tv_within | ID),
  data = d
)

As we can see, the estimates of the FE-models and the RE-model are identical. However, the estimates from the REWB-model differ. This is because the time-varying predictors, the within-subject effect x_tv_within, is allowed to vary between subjects as well (i.e. it is modelled as random slope).

Table 3: Comparison of FE- and RE-models

tab_model(
  model_fe_ID, model_fe_y_within, model_re_2, model_complex_rewb3,
  show.ci = FALSE, 
  show.se = TRUE, 
  auto.label = FALSE, 
  string.se = "SE",
  show.icc = FALSE,
  dv.labels = c("FE-model with ID", "FE, de-meaned Y (with Intercept)", "RE", "Complex REWB")
)

	FE-model with ID			FE, de-meaned Y (with Intercept)			RE			Complex REWB
Predictors	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p
time	1.09	0.64	0.089	1.09	0.52	0.037	1.09	0.64	0.088	1.19	0.61	0.050
x_tv_within	-3.66	0.41	<0.001	-3.66	0.33	<0.001	-3.66	0.41	<0.001	-4.50	0.58	<0.001
(Intercept)				-2.18	1.13	0.054	71.53	1.56	<0.001	71.33	1.54	<0.001
x_tv_between							-6.28	0.50	<0.001	-6.38	0.50	<0.001
Random Effects
σ²							152.89			119.05
τ₀₀							97.62 _ID			51.73 _ID
										81.70 _ID.1
τ₁₁										0.99 _ID.time
										14.34 _{ID.1.x_tv_within}
ρ₀₁										-1.00 _ID
										0.25 _ID.1
N	188 _ID						188 _ID			188 _ID
Observations	564			564			564			564
R² / R² adjusted	NA			0.180 / 0.177			0.369 / 0.615			0.576 / NA

Comparison with the panelr-package

The panelr-package provides functions to fit models similar to those suggested by Bell et al. 2018, especially the “simple REWB model” (model_simple_rewb) and the Mundlak-model (model_mundlak). For the complex REWB model (model_complex_rewb), I needed some slight modification when using panelr::wbm(), so the following model model_complex_rewb_panelr that mimics the complex REWB model is similar to the above model model_rewb.

Here we compare the results from panelr::wbm() with our previous models model_complex_rewb (complex REWB), model_simple_rewb (simple REWB) and model_mundlak (Mundlak).

library(panelr)

# prepare the data for processing with "panelr"
pd <- panel_data(d, id = ID, wave = time)

# the complex REWB-model
model_complex_rewb_panelr <- wbm(QoL ~ x_tv | age + z1_ti + z2_ti + time  | (time + x_tv | ID), 
           data = pd, 
           control = lmerControl(check.nobs.vs.nRE = "ignore"))

# the simple REWB-model
model_rewb_panelr <- wbm(QoL ~ x_tv | age + z1_ti + z2_ti + time | (time | ID), data = pd)

# the Mundlak model
model_mundlak_panelr <- wbm(QoL ~ x_tv | age + z1_ti + z2_ti + time | (time | ID), data = pd, model = "contextual")

tab_model(
  model_complex_rewb_panelr, model_rewb_panelr, model_mundlak_panelr,
  show.ci = FALSE, 
  show.se = TRUE, 
  auto.label = FALSE, 
  string.se = "SE",
  show.icc = FALSE,
  dv.labels = c("Complex REWB", "Simple REWB", "Mundlak")
)

	Complex REWB			Simple REWB			Mundlak
Predictors	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p
x_tv	-4.44	0.57	<0.001	-3.73	0.41	<0.001	-3.73	0.41	<0.001
(Intercept)	61.82	5.82	<0.001	62.73	5.85	<0.001	62.73	5.85	<0.001
imean(x_tv)	-6.29	0.52	<0.001	-6.30	0.52	<0.001	-2.56	0.66	<0.001
age	-0.22	0.19	0.259	-0.22	0.19	0.262	-0.22	0.19	0.262
z1_ti	5.11	4.34	0.241	4.43	4.34	0.309	4.43	4.34	0.309
z2_ti	0.00	0.00	0.089	0.00	0.00	0.097	0.00	0.00	0.097
time	1.16	0.61	0.057	1.09	0.66	0.100	1.09	0.66	0.100
Random Effects
σ²	118.90			142.12			142.12
τ₀₀	145.29 _ID			201.95 _ID			201.95 _ID
τ₁₁	1.60 _ID.time			10.82 _ID.time			10.82 _ID.time
	13.92 _{ID.x_tv}
ρ₀₁	-0.70			-0.77 _ID			-0.77 _ID
	0.36
N	188 _ID			188 _ID			188 _ID
Observations	564			564			564
Marginal R² / Conditional R²	0.395 / 0.714			0.382 / 0.649			0.382 / 0.649

# compare with other models
tab_model(
  model_complex_rewb, model_simple_rewb, model_mundlak,
  show.ci = FALSE, 
  show.se = TRUE, 
  auto.label = FALSE, 
  string.se = "SE",
  show.icc = FALSE,
  dv.labels = c("Complex REWB", "Simple REWB", "Mundlak")
)

	Complex REWB			Simple REWB			Mundlak
Predictors	Estimates	SE	p	Estimates	SE	p	Estimates	SE	p
(Intercept)	61.69	5.82	<0.001	62.73	5.85	<0.001	62.73	5.85	<0.001
time	1.19	0.60	0.048	1.09	0.66	0.098	1.09	0.66	0.098
age	-0.22	0.19	0.250	-0.22	0.19	0.261	-0.22	0.19	0.261
x_tv_within	-4.50	0.58	<0.001	-3.73	0.41	<0.001
x_tv_between	-6.30	0.52	<0.001	-6.30	0.52	<0.001	-2.56	0.66	<0.001
z1_ti	5.12	4.34	0.238	4.43	4.34	0.308	4.43	4.34	0.308
z2_ti	0.00	0.00	0.084	0.00	0.00	0.095	0.00	0.00	0.095
x_tv							-3.73	0.41	<0.001
Random Effects
σ²	119.27			142.12			142.12
τ₀₀	110.58 _ID			201.95 _ID			201.95 _ID
	28.22 _ID.1
τ₁₁	0.59 _ID.time			10.82 _ID.time			10.82 _ID.time
	14.45 _{ID.1.x_tv_within}
ρ₀₁	-1.00 _ID			-0.77 _ID			-0.77 _ID
	0.50 _ID.1
N	188 _ID			188 _ID			188 _ID
Observations	564			564			564
Marginal R² / Conditional R²	0.452 / 0.674			0.382 / 0.649			0.382 / 0.649

As we can see, coefficients, standard errors and p-values of all relevant parameters are identical for the simple REWB and Mundlak models from both packages (panelr::wbm() and lme4::lmer()). This confirms the correct “translation” of the formulae from Bell at al. 2018 into lmer()-syntax.

The complex REWB models are also (almost) identical, the minor variation after the second fractional part is most likely due to the slightly different random effects specification.

Conclusion

When group factors (random effects) and (time-constant) predictors correlate, it’s recommended to fit a complex random-effect-within-between model (REWB) instead of a “simple” mixed effects model. This requires de- and group-meaning the time-varying predictors. Depending on the data structure, random slope and intercept may correlate or not.

The random effects structure, i.e. how to model random slopes and intercepts and allow correlations among them, depends on the nature of the data. The benefits from using mixed effects models over fixed effects models are more precise estimates (in particular when random slopes are included) and the possibility to include between-subjects effects.

In case of convergence problems or singular fits, note that changing the optimizer might help. In this context, some models ran fine in lme4, while other models that had problems being fitted in lme4 ran without any problems in glmmTMB

# compute group-mean of "x_tv" for each subject (ID) and
# then "de-mean" x_tv
d <- cbind(
  d,
  demean(d, select = c("x_tv", "QoL"), group = "ID") # from package "parameters"
)

# fit complex REWB-model
m <- lmer(
  QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + 
    (1 + time | ID) + (1 + x_tv_within | ID),
  data = d
)

# an alternative would be to assume independence between random slopes
# and no covariance...
m <- lmer(
  QoL ~ time + age + x_tv_within + x_tv_between + z1_ti + z2_ti + 
    (1 + time | ID) + (0 + x_tv_within | ID),
  data = d
)

x_tv_within indicates the within-subject effect
x_tv_between indicates the between-subject effect
z1_ti and z2_ti also indicate a between-subject effect

Further critics of the FE-approach

(source: http://andrewgelman.com/2012/04/02/fixed-effects-and-identification/)

“But the so-called fixed effects model does not in general minimize bias. It only minimizes bias under some particular models. As I wrote above, ‘it’s just another model.’ Another way to see this, in the time-series cross-sectional case, is to recognize that there’s no reason to think of group-level coefficients as truly ‘fixed’. One example I remember was a regression on some political outcomes, with 40 years of data for each of 50 states, where the analysis included ‘fixed effects’ for states. I’m sorry but it doesn’t really make sense to think of Vermont from 1960 through 2000 as being ‘fixed’ in any sense.”

“I just don’t see how setting the group-level variance to infinity can be better than estimating it from the data or setting it to a reasonable finite value. That said, the big advantage of multilevel (“random effects”) modeling comes when you are interested in the varying coefficients themselves, or if you’re interested in predictions for new groups, or if you want the treatment effect itself to vary by group. On a slightly different note, I’m unhappy with many of the time-series cross-sectional analyses I’ve seen because I don’t buy the assumption of constancy over time. That is, I don’t really think those effects are “fixed”!"

“I don’t know that there’s anything much that’s time-invariant in what I study. But, in any case, the so-called fixed-effects analysis is mathematically a special case of multilevel modeling in which the group-level variance is set to infinity. I agree that there’s no need to “believe” that model for the method to work; however, I think it works because of some implicit additivity assumptions. I’d prefer to (a) allow the group-level variance to be finite, and (b) work in the relevant assumptions more directly."

References

Bafumi, Joseph, and Andrew Gelman. 2006. “Fitting Multilevel Models When Predictors and Group Effects Correlate.” In. Philadelphia, PA: Annual meeting of the American Political Science Association.

Bell, Andrew, Malcolm Fairbrother, and Kelvyn Jones. 2018. “Fixed and Random Effects Models: Making an Informed Choice.” Quality & Quantity. https://doi.org/10.1007/s11135-018-0802-x.

Bell, Andrew, Kelvyn Jones, and Malcolm Fairbrother. 2018. “Understanding and Misunderstanding Group Mean Centering: A Commentary on Kelley et Al.’s Dangerous Practice.” Quality & Quantity 52 (5): 2031–6. https://doi.org/10.1007/s11135-017-0593-5.

Gelman, Andrew, and Jennifer Hill. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. Analytical Methods for Social Research. Cambridge ; New York: Cambridge University Press.

Heisig, Jan Paul, Merlin Schaeffer, and Johannes Giesecke. 2017. “The Costs of Simplicity: Why Multilevel Models May Benefit from Accounting for Cross-Cluster Differences in the Effects of Controls.” American Sociological Review 82 (4): 796–827. https://doi.org/10.1177/0003122417717901.

Fixed and Random Effects Models

Daniel Lüdecke

22 May 2019