# To install a package, run the following ONCE (and only once on your computer)
# install.packages("psych")
library(here) # makes reading data more consistent
library(tidyverse) # for data manipulation and plotting
library(haven) # for importing SPSS/SAS/Stata data
library(lme4) # for multilevel analysis
library(sjPlot) # for plotting effects
library(MuMIn) # for computing r-squared
library(broom.mixed) # for augmenting results
library(modelsummary) # for making tables
library(interactions) # for plotting interactionsWeek 4 (R): Lv-1 Predictors
Load Packages and Import Data
Import Data
# Read in the data (pay attention to the directory)
hsball <- read_sav(here("data_files", "hsball.sav"))Between-Within Decomposition
Cluster-mean centering
To separate the effects of a lv-1 predictor into different levels, one needs to first center the predictor on the cluster means:
hsball <- hsball %>%
group_by(id) %>% # operate within schools
mutate(ses_cm = mean(ses), # create cluster means (the same as `meanses`)
ses_cmc = ses - ses_cm) %>% # cluster-mean centered
ungroup() # exit the "editing within groups" mode
# The meanses variable already exists in the original data, but it's slightly
# different from computing it by hand
hsball %>%
select(id, ses, meanses, ses_cm, ses_cmc)Model Equation
Lv-1:
\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses\_cmc}_{ij} + e_{ij}\]
Lv-2:
\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} \end{aligned} \] where
- \(\gamma_{10}\) = regression coefficient of student-level
sesrepresenting the expected difference in student achievement between two students in the same school with one unit difference inses,
- \(\gamma_{01}\) = between-school effect, which is the expected difference in mean achievement between two schools with one unit difference in
meanses.
Running in R
We can specify the model as:
m_bw <- lmer(mathach ~ meanses + ses_cmc + (1 | id), data = hsball)As an alternative to creating the cluster means and centered variables in the data, you can directly create in the formula:
m_bw2 <- lmer(mathach ~ I(ave(ses, id)) + I(ses - ave(ses, id)) + (1 | id),
data = hsball)You can summarize the results using
summary(m_bw)># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + ses_cmc + (1 | id)
># Data: hsball
>#
># REML criterion at convergence: 46568.6
>#
># Scaled residuals:
># Min 1Q Median 3Q Max
># -3.1666 -0.7253 0.0174 0.7557 2.9453
>#
># Random effects:
># Groups Name Variance Std.Dev.
># id (Intercept) 2.692 1.641
># Residual 37.019 6.084
># Number of obs: 7185, groups: id, 160
>#
># Fixed effects:
># Estimate Std. Error t value
># (Intercept) 12.6481 0.1494 84.68
># meanses 5.8662 0.3617 16.22
># ses_cmc 2.1912 0.1087 20.16
>#
># Correlation of Fixed Effects:
># (Intr) meanss
># meanses -0.004
># ses_cmc 0.000 0.000Contextual Effect
With a level-1 predictor like ses, which has both student-level and school-level variance, we can include both the level-1 and the school mean variables as predictors. When the level-1 predictor is present, the coefficient for the group means variable becomes the contextual effect.
Model Equation
Lv-1:
\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses}_{ij} + e_{ij}\]
Lv-2:
\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} \end{aligned} \] where
- \(\gamma_{10}\) = regression coefficient of student-level
sesrepresenting the expected difference in student achievement between two students in the same school with one unit difference inses,
- \(\gamma_{01}\) = contextual effect, which is the expected difference in student achievement between two students with the same
sesbut from two schools with one unit difference inmeanses.
Running in R
We can specify the model as:
contextual <- lmer(mathach ~ meanses + ses + (1 | id), data = hsball)You can summarize the results using
summary(contextual)># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + ses + (1 | id)
># Data: hsball
>#
># REML criterion at convergence: 46568.6
>#
># Scaled residuals:
># Min 1Q Median 3Q Max
># -3.1666 -0.7253 0.0174 0.7557 2.9453
>#
># Random effects:
># Groups Name Variance Std.Dev.
># id (Intercept) 2.692 1.641
># Residual 37.019 6.084
># Number of obs: 7185, groups: id, 160
>#
># Fixed effects:
># Estimate Std. Error t value
># (Intercept) 12.6613 0.1494 84.763
># meanses 3.6750 0.3777 9.731
># ses 2.1912 0.1087 20.164
>#
># Correlation of Fixed Effects:
># (Intr) meanss
># meanses -0.005
># ses 0.004 -0.288If you compare the REML criterion at convergence number, you can see this is the same as the between-within model. The estimated contextual effect is the coefficient of meanses minus the coefficient of ses_cmc, which is the same as what you will get in the contextual effect model.
Random-Coefficients Model
Explore Variations in Slopes
hsball %>%
# randomly sample 16 schools
filter(id %in% sample(unique(id), 16)) %>%
# ses on x-axis and mathach on y-axis
ggplot(aes(x = ses, y = mathach)) +
# add points (half the original size)
geom_point(size = 0.5) +
# add a linear smoother
geom_smooth(method = "lm") +
# present data of different ids in different panels
facet_wrap(~id)># `geom_smooth()` using formula = 'y ~ x'
Model Equation
Lv-1:
\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses\_cmc}_{ij} + e_{ij}\]
Lv-2:
\[ \begin{aligned} \beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j + u_{0j} \\ \beta_{1j} & = \gamma_{10} + u_{1j} \end{aligned} \] The additional term is \(u_{1j}\), which represents the deviation of the slope of school \(j\) from the average slope (i.e., \(\gamma_{10}\)).
Running in R
We have to put ses in the random part:
ran_slp <- lmer(mathach ~ meanses + ses_cmc + (ses_cmc | id), data = hsball)You can summarize the results using
summary(ran_slp)># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + ses_cmc + (ses_cmc | id)
># Data: hsball
>#
># REML criterion at convergence: 46557.7
>#
># Scaled residuals:
># Min 1Q Median 3Q Max
># -3.1768 -0.7269 0.0160 0.7560 2.9406
>#
># Random effects:
># Groups Name Variance Std.Dev. Corr
># id (Intercept) 2.6931 1.6411
># ses_cmc 0.6858 0.8281 -0.19
># Residual 36.7132 6.0591
># Number of obs: 7185, groups: id, 160
>#
># Fixed effects:
># Estimate Std. Error t value
># (Intercept) 12.6454 0.1492 84.74
># meanses 5.8963 0.3600 16.38
># ses_cmc 2.1913 0.1280 17.11
>#
># Correlation of Fixed Effects:
># (Intr) meanss
># meanses -0.004
># ses_cmc -0.088 0.004Showing the random intercepts and slopes
head(coef(ran_slp)$id) # `head()` shows only the first six schoolsPlotting Random Slopes
# The `augment` function adds predicted values to the data
augment(ran_slp, data = hsball) %>%
ggplot(aes(x = ses, y = .fitted, color = factor(id))) +
geom_smooth(method = "lm", se = FALSE, size = 0.5) +
labs(y = "Predicted mathach") +
guides(color = "none")># Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
># ℹ Please use `linewidth` instead.># `geom_smooth()` using formula = 'y ~ x'
Or using sjPlot::plot_model() (with ses_cmc, not ses, on the x-axis)
plot_model(ran_slp, type = "pred", pred.type = "re", terms = c("ses_cmc", "id"),
# suppress confidence band
ci.lvl = NA, show.legend = FALSE, colors = "gs")
Including the effect of meanses
# augmented data (adding EB estimates)
augment(ran_slp, data = hsball) %>%
ggplot(aes(x = ses, y = mathach, color = factor(id))) +
# Add points
geom_point(size = 0.2, alpha = 0.2) +
# Add within-cluster lines
geom_smooth(aes(y = .fitted),
method = "lm", se = FALSE, size = 0.5) +
# Add group means with `stat_summary`
stat_summary(aes(x = meanses, y = .fitted,
fill = factor(id)),
fun = mean,
geom = "point",
color = "red", # add border
shape = 24, # use triangles
size = 2.5) +
# Add between coefficient
geom_smooth(aes(x = meanses, y = .fitted),
method = "lm", se = FALSE,
color = "black") +
labs(y = "Math Achievement") +
# Suppress legend as there are too many schools
guides(color = "none", fill = "none")># `geom_smooth()` using formula = 'y ~ x'
># `geom_smooth()` using formula = 'y ~ x'
Or on separate graphs:
# Create a common base graph
pbase <- augment(ran_slp, data = hsball) %>%
ggplot(aes(x = ses, y = mathach, color = factor(id))) +
# Add points
geom_point(size = 0.2, alpha = 0.2) +
labs(y = "Math Achievement") +
# Suppress legend
guides(color = "none")
# Lv-1 effect
p1 <- pbase +
# Add within-cluster lines
geom_smooth(aes(y = .fitted),
method = "lm", se = FALSE, size = 0.5)
# Lv-2 effect
p2 <- pbase +
# Add group means
stat_summary(aes(x = meanses, y = .fitted),
fun = mean,
geom = "point",
shape = 17, # use triangles
size = 2.5) +
# Add between coefficient
geom_smooth(aes(x = meanses, y = .fitted),
method = "lm", se = FALSE,
color = "black")
# Put the two graphs together (need the gridExtra package)
gridExtra::grid.arrange(p1, p2, ncol = 2) # in two columns># `geom_smooth()` using formula = 'y ~ x'
># `geom_smooth()` using formula = 'y ~ x'
Effect Size
As discussed in the previous week, we can obtain the effect size (\(R^2\)) for the model:
(r2_ran_slp <- r.squaredGLMM(ran_slp))># Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.># R2m R2c
># [1,] 0.1684161 0.2310862Analyzing Cross-Level Interactions
sector is added to the level-2 intercept and slope equations.
Model Equation
Lv-1:
\[\text{mathach}_{ij} = \beta_{0j} + \beta_{1j} \text{ses\_cmc}_{ij} + e_{ij}\]
Lv-2:
\[
\begin{aligned}
\beta_{0j} & = \gamma_{00} + \gamma_{01} \text{meanses}_j +
\gamma_{02} \text{sector}_j + u_{0j} \\
\beta_{1j} & = \gamma_{10} + \gamma_{11} \text{sector}_j + u_{1j}
\end{aligned}
\] where - \(\gamma_{02}\) = regression coefficient of school-level sector variable representing the expected difference in achievement between two students with the same SES level and from two schools with the same school-level SES, but one is a catholic school and the other a private school. - \(\gamma_{11}\) = cross-level interaction coefficient of the expected slope difference between a catholic and a private school with the same school-level SES.
Running in R
We have to put the sector * ses interaction to the fixed part:
# The first step is optional, but let's recode sector into a factor variable
hsball <- hsball %>%
mutate(sector = factor(sector, levels = c(0, 1),
labels = c("Public", "Catholic")))
crlv_int <- lmer(mathach ~ meanses + sector * ses_cmc + (ses_cmc | id),
data = hsball)You can summarize the results using
summary(crlv_int)># Linear mixed model fit by REML ['lmerMod']
># Formula: mathach ~ meanses + sector * ses_cmc + (ses_cmc | id)
># Data: hsball
>#
># REML criterion at convergence: 46514.5
>#
># Scaled residuals:
># Min 1Q Median 3Q Max
># -3.11560 -0.72905 0.01592 0.75280 3.01543
>#
># Random effects:
># Groups Name Variance Std.Dev. Corr
># id (Intercept) 2.3807 1.5430
># ses_cmc 0.2716 0.5212 0.24
># Residual 36.7077 6.0587
># Number of obs: 7185, groups: id, 160
>#
># Fixed effects:
># Estimate Std. Error t value
># (Intercept) 12.0846 0.1987 60.81
># meanses 5.2450 0.3682 14.24
># sectorCatholic 1.2523 0.3062 4.09
># ses_cmc 2.7877 0.1559 17.89
># sectorCatholic:ses_cmc -1.3478 0.2348 -5.74
>#
># Correlation of Fixed Effects:
># (Intr) meanss sctrCt ss_cmc
># meanses 0.245
># sectorCthlc -0.697 -0.355
># ses_cmc 0.071 -0.002 -0.046
># sctrCthlc:_ -0.048 -0.001 0.071 -0.664Effect Size
By comparing the \(R^2\) of this model (with interaction) and the previous model (without the interaction), we can obtain the proportion of variance accounted for by the cross-level interaction:
(r2_crlv_int <- r.squaredGLMM(crlv_int))># R2m R2c
># [1,] 0.1759122 0.2284431# Change
r2_crlv_int - r2_ran_slp># R2m R2c
># [1,] 0.007496143 -0.002643047Therefore, the cross-level interaction between sector and SES accounted for an effect size of \(\Delta R^2\) = 0.7%. This is usually considered a small effect size (i.e., < 1%), but interpreting the magnitude of an effect size needs to take into account the area of research, the importance of the outcome, and the effect sizes when using other predictors to predict the same outcome.
Simple Slopes
With a cross-level interaction, the slope between ses_cmc and mathach depends on a moderator, sector. To find out what the model suggests to be the slope at a particular level of sector, which is also called a simple slope in the literature, you just need to look at the equation for \(\beta_{1j}\), which shows that the predicted slope is \[\hat \beta_{1} = \hat \gamma_{10} + \hat \gamma_{11} \text{sector}.\] So when sector = 0 (public school), the simple slope is \[\hat \beta_{1} = \hat \gamma_{10} + \hat \gamma_{11} (0),\] or 2.7877417. When sector = 1 (private school), the simple slope is \[\hat \beta_{1} = \hat \gamma_{10} + \hat \gamma_{11} (1),\] or 2.7877417 + (-1.3477638)(1) = 1.4399779.
Plotting the Interactions
crlv_int %>%
augment(data = hsball) %>%
ggplot(aes(
x = ses, y = .fitted, group = factor(id),
color = factor(sector) # use `sector` for coloring lines
)) +
geom_smooth(method = "lm", se = FALSE, size = 0.5) +
labs(y = "Predicted mathach", color = "sector")># `geom_smooth()` using formula = 'y ~ x'
You can also use the sjPlot::plot_model() function for just the fixed effects:
plot_model(crlv_int, type = "pred", pred.type = "re",
terms = c("ses_cmc", "id", "sector"),
# suppress confidence band
ci.lvl = NA, show.legend = FALSE, colors = "gs",
# suppress title
title = "",
axis.title = c("SES (cluster-mean centered)", "Math Achievement"),
show.data = TRUE, dot.size = 0.5, dot.alpha = 0.5)
Summaizing Analyses in a Table
You can again use the msummary() function from the modelsummary package to quickly summarize the results in a table.
# Basic table from `modelsummary`
msummary(list(
"Between-within" = m_bw,
"Contextual" = contextual,
"Random slope" = ran_slp,
"Cross-level interaction" = crlv_int
))| Between-within | Contextual | Random slope | Cross-level interaction | |
|---|---|---|---|---|
| (Intercept) | 12.648 | 12.661 | 12.645 | 12.085 |
| (0.149) | (0.149) | (0.149) | (0.199) | |
| meanses | 5.866 | 3.675 | 5.896 | 5.245 |
| (0.362) | (0.378) | (0.360) | (0.368) | |
| ses_cmc | 2.191 | 2.191 | 2.788 | |
| (0.109) | (0.128) | (0.156) | ||
| ses | 2.191 | |||
| (0.109) | ||||
| sectorCatholic | 1.252 | |||
| (0.306) | ||||
| sectorCatholic × ses_cmc | −1.348 | |||
| (0.235) | ||||
| SD (Intercept id) | 1.641 | 1.641 | 1.641 | 1.543 |
| SD (ses_cmc id) | 0.828 | 0.521 | ||
| Cor (Intercept~ses_cmc id) | −0.195 | 0.244 | ||
| SD (Observations) | 6.084 | 6.084 | 6.059 | 6.059 |
| Num.Obs. | 7185 | 7185 | 7185 | 7185 |
| R2 Marg. | 0.167 | 0.167 | 0.168 | 0.176 |
| R2 Cond. | 0.224 | 0.224 | 0.231 | 0.228 |
| AIC | 46578.6 | 46578.6 | 46571.7 | 46532.5 |
| BIC | 46613.0 | 46613.0 | 46619.8 | 46594.4 |
| ICC | 0.07 | 0.07 | 0.08 | 0.06 |
| RMSE | 6.03 | 6.03 | 5.99 | 6.00 |
Bonus: More “APA-like” Table
Make it look like the table at https://apastyle.apa.org/style-grammar-guidelines/tables-figures/sample-tables#regression. You can see a recent recommendation in this paper published in the Journal of Memory and Language
# Step 1. Create a map for the predictors and the terms in the model.
# Need to use '\\( \\)' to show math.
# Rename and reorder the rows. Need to use '\\( \\)' to
# show math. If this does not work for you, don't worry about it.
cm <- c("(Intercept)" = "Intercept",
"meanses" = "school mean SES",
"ses_cmc" = "SES (cluster-mean centered)",
"sectorCatholic" = "Catholic school",
"sectorCatholicses_cmc" = "Catholic school x SES (cmc)",
"SD (Intercept id)" = "\\(\\tau_0\\)",
"SD (ses_cmc id)" = "\\(\\tau_1\\) (SES)",
"Cor (Intercept~ses_cmc id)" = "\\(\\tau_{01}\\)",
"SD (Observations)" = "\\(\\sigma\\)")
# Step 2. Create a list of models to have one column for coef,
# one for SE, one for CI, and one for p
models <- list(
"Estimate" = crlv_int,
"SE" = crlv_int,
"95% CI" = crlv_int
)
# Step 3. Add rows to say fixed and random effects
# (Need same number of columns as the table)
new_rows <- data.frame(
term = c("Fixed effects", "Random effects"),
est = c("", ""),
se = c("", ""),
ci = c("", "")
)
# Specify which rows to add
attr(new_rows, "position") <- c(1, 7)
# Step 4. Render the table
msummary(models,
estimate = c("estimate", "std.error",
"[{conf.low}, {conf.high}]"),
statistic = NULL, # suppress the extra rows for SEs
# suppress gof indices (e.g., R^2)
gof_omit = ".*",
coef_map = cm,
add_rows = new_rows)| Estimate | SE | 95% CI | |
|---|---|---|---|
| Fixed effects | |||
| Intercept | 12.085 | 0.199 | [11.695, 12.474] |
| school mean SES | 5.245 | 0.368 | [4.523, 5.967] |
| SES (cluster-mean centered) | 2.788 | 0.156 | [2.482, 3.093] |
| Catholic school | 1.252 | 0.306 | [0.652, 1.853] |
| \(\tau_0\) | 1.543 | ||
| Random effects | |||
| \(\tau_1\) (SES) | 0.521 | ||
| \(\tau_{01}\) | 0.244 | ||
| \(\sigma\) | 6.059 |