Advanced Two Level Hierarchical Models
This is a sample hierarchical linear model research. This model begins with the necessary nitty-gritty of data analyses in R and culminates in an advanced 2-level mixed linear model (MLM) which contains fixed slopes, random slopes and even moderators. It took quite a while for me to figure out ways to conduct these analyses, especially the MLMs. Thus, this report is for my own resource. Thus, it doesn’t have much description. I have some explanation especially of the one’s I think I may forget. If you happen to visit this page, you are on luck!!
Highlights
- Null Model Using
lmer()function - Random Intercept, Fixed Slope Models
- Random Intercept, Random Slope Models
- Best Fitting Model
- Model Comparison
- Plotting Models (Random effects and Fixed effects)
- APA Tables
- Calculating R-Squared, Omega-Squared values etc.
Select Working Directory
setwd("C:/Users/nirma/Desktop/FJER Copies")Loading Required Libraries
When you become an advance R user, you refrain loading all the
libraries. We only load the libraries that we use all the times like
{tidyverse} or {ggplot}, and we call other libraries on the go as we
need using {library_name::function()}. R stores all of
these objects in RAM (Random Access Memory), thus, loading all the
libraries may seriously compromise the speed of our machine.
library(tidyverse)
library(ggplot2)
library(sjlabelled)
library(sjPlot)
library(stringr)
library(r2mlm)
library(lme4)
library(readr)
library(here)
library(papaja)
library(geomtextpath)
library(janitor)
library(lmerTest)
library(sjstats)
# library(QuantPsyc)Uploading My Data
final_data <- read_csv("EDF7006_final_Study_TWS_Data_Spring_2018_1.csv")
# Checking the dimension of the data set
dim(final_data)[1] 5561 25# Checking the summary of the data
summary(final_data) T_ID STUDENT TPROGRAM Majors TGRADE
Min. : 1 Min. : 1 Min. :0.000 Min. :0.000 Min. : 0.00
1st Qu.: 68 1st Qu.:1391 1st Qu.:0.000 1st Qu.:0.000 1st Qu.: 2.00
Median :132 Median :2781 Median :0.000 Median :0.000 Median : 4.00
Mean :132 Mean :2781 Mean :0.308 Mean :0.028 Mean : 4.29
3rd Qu.:199 3rd Qu.:4171 3rd Qu.:0.000 3rd Qu.:0.000 3rd Qu.: 5.00
Max. :236 Max. :5561 Max. :2.000 Max. :1.000 Max. :12.00
SGrd_Lvl MALE ETHNICITY FRLUNCH EL
Min. :0.000 Min. :0.000 Min. :0.00 Min. :0.00 Min. :0.000
1st Qu.:0.000 1st Qu.:0.000 1st Qu.:0.00 1st Qu.:0.00 1st Qu.:0.000
Median :0.000 Median :0.000 Median :1.00 Median :1.00 Median :0.000
Mean :0.307 Mean :0.489 Mean :1.11 Mean :0.51 Mean :0.105
3rd Qu.:0.000 3rd Qu.:1.000 3rd Qu.:1.00 3rd Qu.:1.00 3rd Qu.:0.000
Max. :2.000 Max. :1.000 Max. :5.00 Max. :1.00 Max. :1.000
NA's :1 NA's :1 NA's :1
PRELG1 PRELG2 PRELG3 POSTLG1 POSTLG2
Min. : 0 Min. : 0.0 Min. : 0.0 Min. : 0.0 Min. : 0.0
1st Qu.: 1 1st Qu.: 1.0 1st Qu.: 1.0 1st Qu.: 3.0 1st Qu.: 3.0
Median : 3 Median : 2.0 Median : 2.0 Median : 5.0 Median : 4.0
Mean : 7 Mean : 5.1 Mean : 5.1 Mean : 11.7 Mean : 9.4
3rd Qu.: 6 3rd Qu.: 5.0 3rd Qu.: 5.0 3rd Qu.: 10.0 3rd Qu.: 10.0
Max. :100 Max. :100.0 Max. :100.0 Max. :100.0 Max. :100.0
NA's :58 NA's :124 NA's :121 NA's :55 NA's :109
POSTLG3 PREPERCENT POSTPERCENT PPG
Min. : 0.0 Min. : 0.0 Min. : 0.0 Min. :-45.8
1st Qu.: 2.0 1st Qu.: 26.7 1st Qu.: 73.8 1st Qu.: 20.0
Median : 5.0 Median : 43.8 Median : 86.7 Median : 36.8
Mean : 9.6 Mean : 45.2 Mean : 82.5 Mean : 37.3
3rd Qu.: 9.0 3rd Qu.: 62.5 3rd Qu.: 95.8 3rd Qu.: 53.4
Max. :100.0 Max. :100.0 Max. :100.0 Max. :100.0
NA's :96 NA's :90 NA's :90 NA's :90
White Elementary Math Taught_ELEM
Min. :0.000 Min. :-1.000 Min. :-1.000 Min. :-1.000
1st Qu.:0.000 1st Qu.: 1.000 1st Qu.: 0.000 1st Qu.: 0.000
Median :0.000 Median : 1.000 Median : 0.000 Median : 0.000
Mean :0.463 Mean : 0.692 Mean :-0.113 Mean :-0.062
3rd Qu.:1.000 3rd Qu.: 1.000 3rd Qu.: 0.000 3rd Qu.: 0.000
Max. :1.000 Max. : 1.000 Max. : 1.000 Max. : 1.000
Taught_Middle filter_$
Min. :-1.00 Min. :0.000
1st Qu.: 0.00 1st Qu.:0.000
Median : 0.00 Median :0.000
Mean :-0.04 Mean :0.406
3rd Qu.: 0.00 3rd Qu.:1.000
Max. : 1.00 Max. :1.000
Checking the Column Names
names(final_data) [1] "T_ID" "STUDENT" "TPROGRAM" "Majors"
[5] "TGRADE" "SGrd_Lvl" "MALE" "ETHNICITY"
[9] "FRLUNCH" "EL" "PRELG1" "PRELG2"
[13] "PRELG3" "POSTLG1" "POSTLG2" "POSTLG3"
[17] "PREPERCENT" "POSTPERCENT" "PPG" "White"
[21] "Elementary" "Math" "Taught_ELEM" "Taught_Middle"
[25] "filter_$" Subsetting the data by selecting the required columns
# Selecting only the Required Columns
final_data <- final_data |>
select(
T_ID, STUDENT, TPROGRAM,
SGrd_Lvl, MALE, ETHNICITY, FRLUNCH,
EL, PREPERCENT, POSTPERCENT, White,
Elementary, Math, Taught_ELEM, Taught_Middle
)
# Renaming the desired Column
final_data <- final_data |>
rename(TGRADE = SGrd_Lvl)
# Checking the Structure of the Entire Dataset
str(final_data)tibble [5,561 x 15] (S3: tbl_df/tbl/data.frame)
$ T_ID : num [1:5561] 1 1 1 1 1 1 1 1 1 1 ...
$ STUDENT : num [1:5561] 1 2 3 4 5 6 7 8 9 10 ...
$ TPROGRAM : num [1:5561] 0 0 0 0 0 0 0 0 0 0 ...
$ TGRADE : num [1:5561] 0 0 0 0 0 0 0 0 0 0 ...
$ MALE : num [1:5561] 0 1 0 1 1 1 0 0 1 0 ...
$ ETHNICITY : num [1:5561] 1 0 0 3 3 0 1 1 0 1 ...
$ FRLUNCH : num [1:5561] 1 1 0 1 1 1 1 1 1 1 ...
$ EL : num [1:5561] 1 0 1 0 1 0 1 1 0 1 ...
$ PREPERCENT : num [1:5561] 50 80 70 60 70 NA 50 40 70 60 ...
$ POSTPERCENT : num [1:5561] 90 80 70 80 80 NA 70 60 70 60 ...
$ White : num [1:5561] 0 1 1 0 0 1 0 0 1 0 ...
$ Elementary : num [1:5561] 1 1 1 1 1 1 1 1 1 1 ...
$ Math : num [1:5561] 0 0 0 0 0 0 0 0 0 0 ...
$ Taught_ELEM : num [1:5561] 0 0 0 0 0 0 0 0 0 0 ...
$ Taught_Middle: num [1:5561] 0 0 0 0 0 0 0 0 0 0 ...Cleaning the Column Names for Consistency
# Using clean columns names for consistency
final_data <- final_data |>
clean_names()
# Checking the column names
names(final_data) [1] "t_id" "student" "tprogram" "tgrade"
[5] "male" "ethnicity" "frlunch" "el"
[9] "prepercent" "postpercent" "white" "elementary"
[13] "math" "taught_elem" "taught_middle"# Calculating Pretest and Posttest Means
pretest_mean <- mean(final_data$prepercent, na.rm = TRUE)
posttest_mean <- mean(final_data$postpercent, na.rm = TRUE)
# Displaying the means together as a data frame
data.frame(pretest_mean, posttest_mean) pretest_mean posttest_mean
1 45.2 82.5Changing the labels and class of the columns
Sometimes, we have to change the dummy codes to the actual categories
for better understanding and interpretation of the output. The codes
below display the action where I am changing lavels into
the desired labels. They were dummy coded on the SPSS
files. The names did not get carried over to R but just the codes.
# Setting dummy codes to real names and changing the columns to factor
final_data$el <- factor(final_data$el, levels = c(0, 1), labels = c("non_ELs", "ELs"))
final_data$male <- factor(final_data$male, levels = c(0, 1), labels = c("Female", "Male"))
final_data$tgrade <- factor(final_data$tgrade, levels = c(0, 1, 2), labels = c("Elementary Grades", "Middle School", "High School"))
final_data$ethnicity <- factor(final_data$ethnicity, levels = c(0, 1, 2, 3, 4, 5), labels = c("White", "Black", "Hispanics", "Asian & Pacific Islanders", "Alaskan Natives or American Indians", "Other or Multiracial"))
final_data$frlunch <- factor(final_data$frlunch, levels = c(0, 1), labels = c("high_SES", "low_SES"))
final_data$tprogram <- factor(final_data$tprogram, levels = c(0, 1, 2), labels = c("Elementary Education", "Math Education", "English Language Arts"))
final_data$white <- factor(final_data$white, levels = c(0, 1), labels = c("Minority", "non_Minority"))Creating Long Data (Person Period Data) out of Wide Data (Person Level Data)
There are many instances when we need our data to be in a long format
(person period data - one row for each measurement/time having
multiple rows per person based on the number or times they were
measured) rather than in a wide one (person level data
- one row of data per participant, and one column per
variable), for example, while conducting ANOVA, or other
advanced regression. We can change the wide data to long data using the
pivot_longer() function in base R. However, its scope is
limited for me to use here. I want to be able to transfer some
columns as they are, while stacking some of the columns for combined
measurement. Use of `data.frame() & stack() function would allow me
to do so.
# Changing the wide data into the long data needed in some analyses
long_data <- data.frame(t_id = final_data$t_id, student = final_data$student, el = final_data$el, white = final_data$white, frlunch = final_data$frlunch, stack(final_data, select = prepercent:postpercent)) |>
mutate(
test_scores = as.numeric(values),
test_types = as.factor(ind)
) |>
select(t_id, student, el, white, frlunch, test_scores, test_types)
# changing the labels of the newly minted variable neamed test_types
long_data$test_types <- factor(long_data$test_types, labels = c("prepercent", "postpercent"))
# Checking if that worked
head(long_data) t_id student el white frlunch test_scores test_types
1 1 1 ELs Minority low_SES 50 prepercent
2 1 2 non_ELs non_Minority low_SES 80 prepercent
3 1 3 ELs non_Minority high_SES 70 prepercent
4 1 4 non_ELs Minority low_SES 60 prepercent
5 1 5 ELs Minority low_SES 70 prepercent
6 1 6 non_ELs non_Minority low_SES NA prepercentRunning some Exploratory Data Analyses using Visualization Techniques
Visualizing the distribution of pretest and posttest scores
pre_post_density <- long_data |>
na.omit() |>
ggplot(aes(test_scores, fill = test_types, label = test_types), alpha = 0.3) +
geom_textdensity(size = 4, fontface = 4, hjust = 0.2, vjust = 0.3) +
geom_textvline(xintercept = pretest_mean, label = "pretest mean score = 45.2", hjust = 0.8, vjust = 1.3) +
geom_textvline(xintercept = posttest_mean, label = "posttest mean score = 82.5", hjust = 0.8, vjust = 1.3) +
xlab("Test Scores") +
theme(legend.position = "none")
# APA Plot
pre_post_density + theme_apa()
Comparing the Pretest and Posttest Scores as Function of EL Status
# Creating EL pretest and posttest score plot
b_bar <- long_data |>
na.omit() |>
group_by(el, test_types) |>
summarize(
mean_value = mean(test_scores, na.rm = TRUE),
sd_value = sd(test_scores, na.rm = TRUE)
) |>
ggplot(aes(x = factor(test_types), mean_value, fill = test_types)) +
geom_bar(stat = "identity", position = position_dodge()) +
geom_errorbar(aes(ymin = mean_value - sd_value, ymax = mean_value + sd_value), width = .2, position = position_dodge(0.9)) +
scale_fill_manual(values = c("darkgrey", "lightgray")) +
labs(y = "Test Scores", x = "") +
facet_wrap(~el)
# printing the plot in apa theme
b_bar + theme_apa()
Checking out various necessary statistics
# Calculating total students by teacher program
distinct_t_subject <- final_data |>
select(t_id, tprogram) |>
group_by(t_id, tprogram) |>
count() |>
print()# A tibble: 236 x 3
# Groups: t_id, tprogram [236]
t_id tprogram n
<dbl> <fct> <int>
1 1 Elementary Education 21
2 2 Elementary Education 18
3 3 Elementary Education 18
4 4 Elementary Education 22
5 5 Elementary Education 14
6 6 Elementary Education 16
7 7 Elementary Education 19
8 8 Elementary Education 15
9 9 Elementary Education 4
10 10 Elementary Education 17
# ... with 226 more rows# Calculating total teachers by teacher program
teacher_count_program <- distinct_t_subject |>
select(tprogram) |>
group_by(tprogram) |>
count() |>
mutate(pct = n / 236) |>
print()# A tibble: 3 x 3
# Groups: tprogram [3]
tprogram n pct
<fct> <int> <dbl>
1 Elementary Education 220 0.932
2 Math Education 5 0.0212
3 English Language Arts 11 0.0466# Calculating total students by teacher grades
distinct_grade <- final_data |>
select(tgrade, t_id) |>
group_by(t_id, tgrade) |>
count() |>
print()# A tibble: 236 x 3
# Groups: t_id, tgrade [236]
t_id tgrade n
<dbl> <fct> <int>
1 1 Elementary Grades 21
2 2 Elementary Grades 18
3 3 Elementary Grades 18
4 4 Elementary Grades 22
5 5 Elementary Grades 14
6 6 Elementary Grades 16
7 7 Elementary Grades 19
8 8 Elementary Grades 15
9 9 Elementary Grades 4
10 10 Elementary Grades 17
# ... with 226 more rows# Calculating total teacher by the grades they taught
teacher_count_grade <- distinct_grade |>
select(tgrade) |>
group_by(tgrade) |>
count() |>
mutate(pct = n / 236) |>
print()# A tibble: 3 x 3
# Groups: tgrade [3]
tgrade n pct
<fct> <int> <dbl>
1 Elementary Grades 218 0.924
2 Middle School 7 0.0297
3 High School 11 0.0466Regressions
Getting Rid of Missing Values for easy calculations
final_data <- na.omit(final_data)A. Pretest Null Model
# Null Model
pretest_null_model <- lmer(prepercent ~ 1 + (1 | t_id), data = final_data)
# Printing Summary Table
# summary(pretest_null_model)
# R-squared
r2mlm(pretest_null_model)
$Decompositions
total within between
fixed, within 0 0 NA
fixed, between 0 NA 0
slope variation 0 0 NA
mean variation 0.480392530283301 NA 1
sigma2 0.519607469716699 1 NA
$R2s
total within between
f1 0 0 NA
f2 0 NA 0
v 0 0 NA
m 0.480392530283301 NA 1
f 0 NA NA
fv 0 0 NA
fvm 0.480392530283301 NA NA # sjPlot::tab_model(pretest_null_model)B. Posttest Null Model
# Null Model
posttest_null_model <- lmer(postpercent ~ 1 + (1 | t_id), data = final_data)
# Printing Summary Table
summary(posttest_null_model)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: postpercent ~ 1 + (1 | t_id)
Data: final_data
REML criterion at convergence: 45393
Scaled residuals:
Min 1Q Median 3Q Max
-5.975 -0.490 0.201 0.655 2.810
Random effects:
Groups Name Variance Std.Dev.
t_id (Intercept) 85.6 9.25
Residual 214.1 14.63
Number of obs: 5469, groups: t_id, 236
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 82.36 0.64 232.38 129 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# R-squared
r2mlm(posttest_null_model)
$Decompositions
total within between
fixed, within 0 0 NA
fixed, between 0 NA 0
slope variation 0 0 NA
mean variation 0.285498104191483 NA 1
sigma2 0.714501895808518 1 NA
$R2s
total within between
f1 0 0 NA
f2 0 NA 0
v 0 0 NA
m 0.285498104191483 NA 1
f 0 NA NA
fv 0 0 NA
fvm 0.285498104191483 NA NA # sjPlot::tab_model(posttest_null_model)C. EL Pretest Fixed Slope Model
el_pretest_model <- lmer(prepercent ~ el + (1 | t_id), data = final_data)
# summary(el_pretest_model)
# r2mlm(el_pretest_model)
# sjPlot::tab_model(el_pretest_model)
effectsize::standardize_parameters(el_pretest_model, method = "pseudo")# Standardization method: pseudo
Parameter | Coefficient (std.) | 95% CI
-------------------------------------------------
(Intercept) | 0.00 | [ 0.00, 0.00]
elELs | -0.15 | [-0.18, -0.12]D. EL Pretest Random Slope Model
el_pretest_random <- lmer(prepercent ~ el + (el | t_id), data = final_data)
# sjPlot::tab_model(el_pretest_random)
effectsize::standardize_parameters(el_pretest_random, method = "pseudo")# Standardization method: pseudo
Parameter | Coefficient (std.) | 95% CI
-------------------------------------------------
(Intercept) | 0.00 | [ 0.00, 0.00]
elELs | -0.16 | [-0.19, -0.13]E. Putting them Together
sjPlot::tab_model(pretest_null_model, el_pretest_model, el_pretest_random)| prepercent | prepercent | prepercent | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 47.18 | 44.95 – 49.41 | <0.001 | 48.15 | 45.92 – 50.38 | <0.001 | 48.18 | 45.92 – 50.45 | <0.001 |
| el [ELs] | -8.59 | -10.23 – -6.94 | <0.001 | -9.15 | -11.05 – -7.26 | <0.001 | |||
| Random Effects | |||||||||
| σ2 | 313.29 | 307.37 | 304.48 | ||||||
| τ00 | 289.64 t_id | 286.11 t_id | 296.15 t_id | ||||||
| τ11 | 33.78 t_id.elELs | ||||||||
| ρ01 | -0.46 t_id | ||||||||
| ICC | 0.48 | 0.48 | 0.49 | ||||||
| N | 236 t_id | 236 t_id | 236 t_id | ||||||
| Observations | 5469 | 5469 | 5469 | ||||||
| Marginal R2 / Conditional R2 | 0.000 / 0.480 | 0.012 / 0.488 | 0.013 / 0.495 | ||||||
F. EL FRPL Models
el_frpl_model <- lmer(prepercent ~ el + frlunch + (el + frlunch | t_id), data = final_data)
el_frpl_interaction <- lmer(prepercent ~ el + frlunch + el * frlunch + (el + frlunch | t_id), data = final_data)
# sjPlot::tab_model(el_frpl_model, el_frpl_interaction)
plot_model(el_frpl_interaction, type = "pred", terms = c("el", "frlunch"))
G. Level 1 Models
level_1_model <- lmer(prepercent ~ el + frlunch + male + white + (el + frlunch | t_id), data = final_data)
# sjPlot::tab_model(el_frpl_model, el_frpl_interaction, level_1_model)H. Model Comparison
anova(pretest_null_model, el_frpl_model, level_1_model)Data: final_data
Models:
pretest_null_model: prepercent ~ 1 + (1 | t_id)
el_frpl_model: prepercent ~ el + frlunch + (el + frlunch | t_id)
level_1_model: prepercent ~ el + frlunch + male + white + (el + frlunch | t_id)
npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
pretest_null_model 3 47664 47683 -23829 47658
el_frpl_model 10 47461 47527 -23721 47441 216.3 7 < 2e-16 ***
level_1_model 12 47442 47521 -23709 47418 23.2 2 9.1e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1I. Level 1 Level 2 Models
level1_level2_model <- lmer(prepercent ~ el + frlunch + male + ethnicity + (el + frlunch | t_id) + (1 + male | tprogram) + (1 + male | tgrade), data = final_data, control = lmerControl(optimizer = "nloptwrap", optCtrl = list(maxfun = 2e5)))
# Putting Results together
sjPlot::tab_model(pretest_null_model, level_1_model, level1_level2_model)| prepercent | prepercent | prepercent | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 47.18 | 44.95 – 49.41 | <0.001 | 49.02 | 46.48 – 51.56 | <0.001 | 45.58 | 35.79 – 55.38 | <0.001 |
| el [ELs] | -7.50 | -9.37 – -5.62 | <0.001 | -7.90 | -9.83 – -5.96 | <0.001 | |||
| frlunch [low SES] | -5.38 | -6.88 – -3.88 | <0.001 | -4.80 | -6.29 – -3.31 | <0.001 | |||
| male [Male] | 1.51 | 0.58 – 2.44 | 0.001 | 0.84 | -5.71 – 7.39 | 0.802 | |||
| white [non Minority] | 1.95 | 0.89 – 3.00 | <0.001 | ||||||
| ethnicity [Black] | -2.29 | -3.56 – -1.02 | <0.001 | ||||||
| ethnicity [Hispanics] | 13.73 | 4.90 – 22.56 | 0.002 | ||||||
|
ethnicity [Asian & Pacific Islanders] |
-4.35 | -5.86 – -2.84 | <0.001 | ||||||
|
ethnicity [Alaskan Natives or American Indians] |
3.15 | 0.89 – 5.42 | 0.006 | ||||||
|
ethnicity [Other or Multiracial] |
-0.18 | -2.84 – 2.48 | 0.893 | ||||||
| Random Effects | |||||||||
| σ2 | 313.29 | 294.13 | 291.28 | ||||||
| τ00 | 289.64 t_id | 307.66 t_id | 292.54 t_id | ||||||
| 0.00 tprogram | |||||||||
| 55.59 tgrade | |||||||||
| τ11 | 26.88 t_id.elELs | 28.68 t_id.elELs | |||||||
| 34.22 t_id.frlunchlow_SES | 31.92 t_id.frlunchlow_SES | ||||||||
| 21.93 tprogram.maleMale | |||||||||
| 8.58 tgrade.maleMale | |||||||||
| ρ01 | -0.53 | -0.54 t_id.elELs | |||||||
| -0.30 | -0.26 t_id.frlunchlow_SES | ||||||||
| 0.10 tgrade | |||||||||
| ICC | 0.48 | 0.50 | |||||||
| N | 236 t_id | 236 t_id | 236 t_id | ||||||
| 3 tprogram | |||||||||
| 3 tgrade | |||||||||
| Observations | 5469 | 5469 | 5469 | ||||||
| Marginal R2 / Conditional R2 | 0.000 / 0.480 | 0.032 / 0.511 | 0.069 / NA | ||||||
J. Final
# plot(final_data)
# Checking the fixed and random effects of all predictors(Results not populated because of space issue)
# head(coef(level1_level2_model))
# Plotting the Random effects of the mentioned model (only Random effects)
plot(ranef(level1_level2_model))$t_id
$tprogram
$tgrade
# Plotting the whole model(fitted vs. residuals)
plot(level1_level2_model)
K. R-Squared
# Calculating R-Squared for the Mentioned Models
r2(pretest_null_model)# R2 for Mixed Models
Conditional R2: 0.480
Marginal R2: 0.000r2(level_1_model)# R2 for Mixed Models
Conditional R2: 0.511
Marginal R2: 0.032r2(level1_level2_model)Random effect variances not available. Returned R2 does not account for random effects.# R2 for Mixed Models
Conditional R2: NA
Marginal R2: 0.069L. Level 1 Models
level_1_model_int <- lmer(prepercent ~ el + frlunch + male + white + el * frlunch + el * male + el * white + frlunch * male + frlunch * white + male * white + (el + frlunch | t_id), data = final_data)
summary(level_1_model_int)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: prepercent ~ el + frlunch + male + white + el * frlunch + el *
male + el * white + frlunch * male + frlunch * white + male *
white + (el + frlunch | t_id)
Data: final_data
REML criterion at convergence: 47389
Scaled residuals:
Min 1Q Median 3Q Max
-4.243 -0.602 0.009 0.610 4.613
Random effects:
Groups Name Variance Std.Dev. Corr
t_id (Intercept) 307.7 17.54
elELs 27.9 5.28 -0.52
frlunchlow_SES 34.0 5.83 -0.30 0.50
Residual 294.0 17.15
Number of obs: 5469, groups: t_id, 236
Fixed effects:
Estimate Std. Error df t value
(Intercept) 49.2810 1.4090 374.3784 34.98
elELs -8.1092 1.8916 436.5526 -4.29
frlunchlow_SES -6.2253 1.0909 502.9885 -5.71
maleMale 0.7262 0.9374 5156.7984 0.77
whitenon_Minority 2.5124 0.9090 5156.5537 2.76
elELs:frlunchlow_SES 0.8634 1.9826 531.6911 0.44
elELs:maleMale -0.0809 1.6659 3640.8564 -0.05
elELs:whitenon_Minority 0.6052 3.1068 2191.4970 0.19
frlunchlow_SES:maleMale 2.2185 0.9991 5161.7614 2.22
frlunchlow_SES:whitenon_Minority -0.6039 1.0727 4929.4219 -0.56
maleMale:whitenon_Minority -0.6818 1.0273 5165.9737 -0.66
Pr(>|t|)
(Intercept) < 2e-16 ***
elELs 2.2e-05 ***
frlunchlow_SES 2.0e-08 ***
maleMale 0.4386
whitenon_Minority 0.0057 **
elELs:frlunchlow_SES 0.6634
elELs:maleMale 0.9613
elELs:whitenon_Minority 0.8456
frlunchlow_SES:maleMale 0.0264 *
frlunchlow_SES:whitenon_Minority 0.5735
maleMale:whitenon_Minority 0.5069
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) elELs fr_SES maleMl whtn_M eEL:_S elEL:M eEL:_M f_SES:M
elELs -0.263
frlnchl_SES -0.501 0.205
maleMale -0.340 0.127 0.311
whtnn_Mnrty -0.422 0.248 0.438 0.394
elELs:f_SES 0.146 -0.753 -0.294 -0.027 -0.173
elELs:malMl 0.084 -0.385 0.042 -0.241 -0.133 -0.014
elELs:wht_M 0.028 -0.166 0.030 -0.040 -0.089 0.002 0.085
frlnc_SES:M 0.216 0.016 -0.451 -0.646 -0.145 0.039 -0.105 0.001
frln_SES:_M 0.253 -0.140 -0.525 -0.011 -0.567 0.221 -0.004 -0.086 -0.003
mlMl:whtn_M 0.234 -0.111 -0.128 -0.687 -0.564 0.009 0.238 0.069 0.245
f_SES:_
elELs
frlnchl_SES
maleMale
whtnn_Mnrty
elELs:f_SES
elELs:malMl
elELs:wht_M
frlnc_SES:M
frln_SES:_M
mlMl:whtn_M 0.005 r2(level_1_model_int)# R2 for Mixed Models
Conditional R2: 0.512
Marginal R2: 0.032# Plotting all the interactions in the mentioned models
theme_set(theme_sjplot())
plot_model(level_1_model_int, type = "int")[[1]]
[[2]]
[[3]]
[[4]]
[[5]]
[[6]]
M. Posttest Model
posttest_model <- lmer(postpercent ~ prepercent + el + frlunch + male + ethnicity + (prepercent + el + frlunch | t_id) + (1 + male | tprogram) + (1 + male | tgrade), data = final_data, control = lmerControl(optimizer = "nloptwrap", optCtrl = list(maxfun = 2e5)))
summary(posttest_model)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: postpercent ~ prepercent + el + frlunch + male + ethnicity +
(prepercent + el + frlunch | t_id) + (1 + male | tprogram) +
(1 + male | tgrade)
Data: final_data
Control: lmerControl(optimizer = "nloptwrap", optCtrl = list(maxfun = 2e+05))
REML criterion at convergence: 44050
Scaled residuals:
Min 1Q Median 3Q Max
-6.754 -0.464 0.124 0.610 3.948
Random effects:
Groups Name Variance Std.Dev. Corr
t_id (Intercept) 217.0624 14.733
prepercent 0.0188 0.137 -1.00
elELs 30.2508 5.500 0.28 -0.28
frlunchlow_SES 11.2576 3.355 -0.16 0.16 -0.30
tprogram (Intercept) 129.4065 11.376
maleMale 62.6082 7.913 0.14
tgrade (Intercept) 102.7192 10.135
maleMale 100.3981 10.020 -0.09
Residual 163.1701 12.774
Number of obs: 5469, groups: t_id, 236; tprogram, 3; tgrade, 3
Fixed effects:
Estimate Std. Error df
(Intercept) 65.3276 8.9612 751.9088
prepercent 0.3196 0.0134 145.3309
elELs -5.1100 0.8228 111.0696
frlunchlow_SES -1.4796 0.5117 140.9150
maleMale 1.0306 7.4138 36.0167
ethnicityBlack -0.4624 0.4821 5277.5468
ethnicityHispanics -2.2567 3.3180 4810.2639
ethnicityAsian & Pacific Islanders -1.2604 0.5735 5238.3554
ethnicityAlaskan Natives or American Indians -0.0848 0.8563 4898.7513
ethnicityOther or Multiracial 0.1756 1.0118 5243.3166
t value Pr(>|t|)
(Intercept) 7.29 7.9e-13 ***
prepercent 23.90 < 2e-16 ***
elELs -6.21 9.4e-09 ***
frlunchlow_SES -2.89 0.0044 **
maleMale 0.14 0.8902
ethnicityBlack -0.96 0.3375
ethnicityHispanics -0.68 0.4964
ethnicityAsian & Pacific Islanders -2.20 0.0280 *
ethnicityAlaskan Natives or American Indians -0.10 0.9211
ethnicityOther or Multiracial 0.17 0.8622
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) prprcn elELs fr_SES maleMl ethncB ethncH etA&PI eANoAI
prepercent -0.116
elELs 0.007 -0.014
frlnchl_SES -0.032 0.129 -0.108
maleMale 0.015 0.000 0.003 0.000
ethnctyBlck -0.016 0.034 -0.241 -0.148 -0.001
ethnctyHspn -0.002 -0.025 0.000 -0.010 0.000 0.052
ethnctyA&PI -0.015 0.056 0.020 -0.160 0.000 0.336 0.037
ethnctANoAI -0.009 -0.026 -0.087 0.022 -0.003 0.204 0.026 0.135
ethnctyOtoM -0.007 0.004 -0.035 -0.043 0.001 0.168 0.018 0.141 0.076
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')sjPlot::tab_model(posttest_model)| postpercent | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 65.33 | 47.76 – 82.90 | <0.001 |
| prepercent | 0.32 | 0.29 – 0.35 | <0.001 |
| el [ELs] | -5.11 | -6.72 – -3.50 | <0.001 |
| frlunch [low SES] | -1.48 | -2.48 – -0.48 | 0.004 |
| male [Male] | 1.03 | -13.50 – 15.56 | 0.889 |
| ethnicity [Black] | -0.46 | -1.41 – 0.48 | 0.338 |
| ethnicity [Hispanics] | -2.26 | -8.76 – 4.25 | 0.496 |
|
ethnicity [Asian & Pacific Islanders] |
-1.26 | -2.38 – -0.14 | 0.028 |
|
ethnicity [Alaskan Natives or American Indians] |
-0.08 | -1.76 – 1.59 | 0.921 |
|
ethnicity [Other or Multiracial] |
0.18 | -1.81 – 2.16 | 0.862 |
| Random Effects | |||
| σ2 | 163.17 | ||
| τ00 t_id | 217.06 | ||
| τ00 tprogram | 129.41 | ||
| τ00 tgrade | 102.72 | ||
| τ11 t_id.prepercent | 0.02 | ||
| τ11 t_id.elELs | 30.25 | ||
| τ11 t_id.frlunchlow_SES | 11.26 | ||
| τ11 tprogram.maleMale | 62.61 | ||
| τ11 tgrade.maleMale | 100.40 | ||
| ρ01 t_id.prepercent | -1.00 | ||
| ρ01 t_id.elELs | 0.28 | ||
| ρ01 t_id.frlunchlow_SES | -0.16 | ||
| ρ01 tprogram | 0.14 | ||
| ρ01 tgrade | -0.09 | ||
| N t_id | 236 | ||
| N tprogram | 3 | ||
| N tgrade | 3 | ||
| Observations | 5469 | ||
| Marginal R2 / Conditional R2 | 0.300 / NA | ||
r2(posttest_model)Random effect variances not available. Returned R2 does not account for random effects.# R2 for Mixed Models
Conditional R2: NA
Marginal R2: 0.300plot_model(posttest_model, type = "pred", colors = "bw")$prepercent
$el
$frlunch
$male
$ethnicity
effectsize::omega_squared(posttest_model, ci.lvl = .95)# Effect Size for ANOVA (Type III)
Parameter | Omega2 (partial) | 95% CI
--------------------------------------------
prepercent | 0.79 | [0.75, 1.00]
el | 0.25 | [0.14, 1.00]
frlunch | 0.05 | [0.01, 1.00]
male | -0.03 | [0.00, 1.00]
ethnicity | 1.11e-04 | [0.00, 1.00]
- One-sided CIs: upper bound fixed at (1).N. Posttest Final
posttest_model_final <- lmer(postpercent ~ prepercent + el + frlunch + male + white + (prepercent + el + frlunch | t_id) + (1 + male | tprogram) + (1 + male | tgrade), data = final_data, control = lmerControl(optimizer = "nloptwrap", optCtrl = list(maxfun = 2e5)))
summary(posttest_model_final)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula:
postpercent ~ prepercent + el + frlunch + male + white + (prepercent +
el + frlunch | t_id) + (1 + male | tprogram) + (1 + male | tgrade)
Data: final_data
Control: lmerControl(optimizer = "nloptwrap", optCtrl = list(maxfun = 2e+05))
REML criterion at convergence: 44052
Scaled residuals:
Min 1Q Median 3Q Max
-6.691 -0.462 0.125 0.609 3.881
Random effects:
Groups Name Variance Std.Dev. Corr
t_id (Intercept) 191.6689 13.844
prepercent 0.0163 0.128 -1.00
elELs 31.8890 5.647 0.33 -0.33
frlunchlow_SES 11.0141 3.319 -0.08 0.08 -0.35
tprogram (Intercept) 56.1465 7.493
maleMale 2.9007 1.703 0.16
tgrade (Intercept) 40.3482 6.352
maleMale 16.1061 4.013 -0.03
Residual 163.6160 12.791
Number of obs: 5469, groups: t_id, 236; tprogram, 3; tgrade, 3
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 64.6258 5.9049 4.2874 10.94 0.00027 ***
prepercent 0.3201 0.0129 139.8924 24.82 < 2e-16 ***
elELs -4.9208 0.8106 102.6222 -6.07 2.2e-08 ***
frlunchlow_SES -1.5760 0.5062 139.1503 -3.11 0.00224 **
maleMale 1.2611 2.6113 0.0543 0.48 0.92401
whitenon_Minority 0.6154 0.3985 5243.9647 1.54 0.12256
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) prprcn elELs fr_SES maleMl
prepercent -0.164
elELs 0.005 -0.034
frlnchl_SES -0.051 0.111 -0.119
maleMale 0.012 -0.001 0.007 0.001
whtnn_Mnrty -0.037 -0.038 0.166 0.161 0.003
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')sjPlot::tab_model(posttest_model_final)| postpercent | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 64.63 | 53.05 – 76.20 | <0.001 |
| prepercent | 0.32 | 0.29 – 0.35 | <0.001 |
| el [ELs] | -4.92 | -6.51 – -3.33 | <0.001 |
| frlunch [low SES] | -1.58 | -2.57 – -0.58 | 0.002 |
| male [Male] | 1.26 | -3.86 – 6.38 | 0.629 |
| white [non Minority] | 0.62 | -0.17 – 1.40 | 0.123 |
| Random Effects | |||
| σ2 | 163.62 | ||
| τ00 t_id | 191.67 | ||
| τ00 tprogram | 56.15 | ||
| τ00 tgrade | 40.35 | ||
| τ11 t_id.prepercent | 0.02 | ||
| τ11 t_id.elELs | 31.89 | ||
| τ11 t_id.frlunchlow_SES | 11.01 | ||
| τ11 tprogram.maleMale | 2.90 | ||
| τ11 tgrade.maleMale | 16.11 | ||
| ρ01 t_id.prepercent | -1.00 | ||
| ρ01 t_id.elELs | 0.33 | ||
| ρ01 t_id.frlunchlow_SES | -0.08 | ||
| ρ01 tprogram | 0.16 | ||
| ρ01 tgrade | -0.03 | ||
| N t_id | 236 | ||
| N tprogram | 3 | ||
| N tgrade | 3 | ||
| Observations | 5469 | ||
| Marginal R2 / Conditional R2 | 0.300 / NA | ||
r2(posttest_model_final)Random effect variances not available. Returned R2 does not account for random effects.# R2 for Mixed Models
Conditional R2: NA
Marginal R2: 0.300plot_model(posttest_model_final, type = "pred", colors = "bw")$prepercent
$el
$frlunch
$male
$white
effectsize::omega_squared(posttest_model_final, ci.lvl = .95)# Effect Size for ANOVA (Type III)
Parameter | Omega2 (partial) | 95% CI
--------------------------------------------
prepercent | 0.81 | [0.77, 1.00]
el | 0.26 | [0.14, 1.00]
frlunch | 0.06 | [0.01, 1.00]
male | -0.60 | [0.00, 1.00]
white | 2.64e-04 | [0.00, 1.00]
- One-sided CIs: upper bound fixed at (1).Posttest, pretest*el
posttest_pretest_el <- lmer(postpercent ~ prepercent + el + frlunch + male + white + prepercent * el + prepercent * frlunch + prepercent * male + prepercent * white + el * frlunch + el * white + el * male + frlunch * white + frlunch * male + white * male + (1 | t_id), data = final_data)
r2(posttest_pretest_el)# R2 for Mixed Models
Conditional R2: 0.454
Marginal R2: 0.231summary(posttest_pretest_el)Linear mixed model fit by REML. t-tests use Satterthwaite's method [
lmerModLmerTest]
Formula: postpercent ~ prepercent + el + frlunch + male + white + prepercent *
el + prepercent * frlunch + prepercent * male + prepercent *
white + el * frlunch + el * white + el * male + frlunch *
white + frlunch * male + white * male + (1 | t_id)
Data: final_data
REML criterion at convergence: 44194
Scaled residuals:
Min 1Q Median 3Q Max
-6.146 -0.458 0.089 0.626 3.480
Random effects:
Groups Name Variance Std.Dev.
t_id (Intercept) 70.1 8.37
Residual 171.7 13.10
Number of obs: 5469, groups: t_id, 236
Fixed effects:
Estimate Std. Error df t value
(Intercept) 67.7780 1.1912 2613.2557 56.90
prepercent 0.3190 0.0183 5428.0963 17.38
elELs -7.2096 1.7888 5296.5945 -4.03
frlunchlow_SES -3.1426 1.0606 5395.6418 -2.96
maleMale 1.4159 1.0212 5260.6555 1.39
whitenon_Minority 2.0176 1.0597 5294.6704 1.90
prepercent:elELs 0.0695 0.0279 5320.7234 2.49
prepercent:frlunchlow_SES 0.0461 0.0174 5428.6126 2.65
prepercent:maleMale -0.0237 0.0153 5261.1301 -1.55
prepercent:whitenon_Minority -0.0239 0.0167 5305.0136 -1.43
elELs:frlunchlow_SES -0.6092 1.4521 5298.8825 -0.42
elELs:whitenon_Minority 2.8501 2.2868 5271.9067 1.25
elELs:maleMale 0.6731 1.2550 5264.4600 0.54
frlunchlow_SES:whitenon_Minority -1.2322 0.8082 5311.6098 -1.52
frlunchlow_SES:maleMale 0.5713 0.7659 5270.0186 0.75
maleMale:whitenon_Minority 0.8050 0.7800 5265.9265 1.03
Pr(>|t|)
(Intercept) < 2e-16 ***
prepercent < 2e-16 ***
elELs 5.6e-05 ***
frlunchlow_SES 0.0031 **
maleMale 0.1656
whitenon_Minority 0.0570 .
prepercent:elELs 0.0127 *
prepercent:frlunchlow_SES 0.0081 **
prepercent:maleMale 0.1209
prepercent:whitenon_Minority 0.1515
elELs:frlunchlow_SES 0.6748
elELs:whitenon_Minority 0.2127
elELs:maleMale 0.5917
frlunchlow_SES:whitenon_Minority 0.1274
frlunchlow_SES:maleMale 0.4558
maleMale:whitenon_Minority 0.3021
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1a <- plot_model(posttest_pretest_el, type = "int")
a[[1]] + theme_apa()
a[[2]] + theme_apa()
a[[3]] + theme_apa()
a[[4]] + theme_apa()
a[[5]] + theme_apa()
a[[6]] + theme_apa()
a[[7]] + theme_apa()
a[[8]] + theme_apa()
a[[9]] + theme_apa()
a[[10]] + theme_apa()
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