Student mathematics achievement across schools: a multilevel model of motivation and teacher ratio

Multilevel Regression (lme4) for Education and Learning

Field: Education and LearningLevel: Doctoral dissertation (S3)Method: Multilevel Regression (lme4)

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Separating within and between group variation in student mathematics score and testing learning motivation and students per teacher with a multilevel model, illustrated with synthetic data. Multilevel regression (a mixed-effects model) handles nested data, such as students within schools or patients within clinics, with random effects for each group. It suits theses and dissertations whose observations are clustered and therefore correlated within groups.

Student mathematics achievement across schools: a multilevel model of motivation and teacher ratio: result sheet

Project info

Field of studyEducation and Learning
LevelDoctoral dissertation (S3)
Method typeAdvanced Models
Estimated build time± 7 working days
Toolslme4::lmer, lme4::VarCorr, lme4::ranef, lme4::refitML, ggplot2
PriceUS$31 - US$194

Research goal

Separating within and between group variation in student mathematics score and testing learning motivation and students per teacher with a multilevel model, illustrated with synthetic data.

Variables

Role	Variable	Unit
Main	Student mathematics score	score
Supporting 1	Learning motivation	score
Supporting 2	Students per teacher	ratio

Example result

0.494ICC (null model)
0.589Coef. Learning motiva.
<0.001p-value Learning motiva.
9.974Coef. Students
0.001p-value Students
<0.001LR vs OLS p-value

The null-model ICC is 0.494: 49.4% of the variance in Student mathematics score lies at the School level (33 groups). The random intercept + random slope model gives a coefficient of 0.589 for Learning motivation (p < 0.001) and 9.974 for Students per teacher (p = 0.001). The LR test against OLS (p < 0.001) supports the random effects.

Computed from illustrative synthetic data, not from a real study.

About Multilevel Regression (lme4)

Multilevel regression (a mixed-effects model) handles nested data, such as students within schools or patients within clinics, with random effects for each group. It suits theses and dissertations whose observations are clustered and therefore correlated within groups.

Steps

1Arrange the data with a group column and centre the individual-level predictor
2Fit the null model lme4::lmer(y ~ 1 + (1 | group)) to compute the ICC
3Fit a random intercept model with individual-level and group-level predictors
4Try a random slope and keep it if it is not singular and the LR test is significant
5Test the random effects with a likelihood ratio test against plain OLS and read the fixed effects
6Plot group-specific lines, a caterpillar plot of random effects and the group means

Assumptions to check

Data are nested and groups are independent
Random effects and residuals are normal with mean zero
Predictors relate linearly to the outcome
There are enough groups (ideally 20 or more)

Reading and reporting the results

The ICC is the share of variance at the group level; a sizeable ICC justifies a multilevel model. Report the fixed effects with p-values, the random-effect variance components and the likelihood ratio test against a model without random effects.

Example R code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

Show code

# Contoh kode R: Multilevel Regression (lme4)
suppressPackageStartupMessages({
  library(ggplot2)
  library(patchwork)
  library(lme4)
})
set.seed(42)
# ---- Gaya grafik ----
pal <- c("#1f6feb", "#e8590c", "#2f9e44", "#ae3ec9", "#f59f00")
warna <- list(bg = "#f4f6fa", panel = "#ffffff", teks = "#1f2933", redup = "#5f6b78", grid = "#e3e8ee", bar = "#1d3557")
tata <- 1
theme_set(theme_minimal(base_size = 11) + theme(legend.position = "bottom"))

# ---- Data sintetis: individu (level 1) tersarang dalam kelompok (level 2) ----
n <- 200
J <- sample(20:36, 1)                                         # banyak kelompok
kode <- sprintf("%s %02d", "Group", 1:J)
uk <- as.numeric(rmultinom(1, n - 3 * J, rep(1, J))) + 3      # ukuran kelompok tidak sama (minimal 3)
kelompok <- rep(kode, uk)
idx <- rep(1:J, uk)
z2g <- rnorm(J)                                               # prediktor level kelompok
u0 <- rnorm(J, 0, runif(1, 0.4, 0.7))                        # efek acak intersep
u1 <- rnorm(J, 0, runif(1, 0.12, 0.3))                        # efek acak slope X1
z1 <- rnorm(n, 0.3 * z2g[idx], 1)
b1 <- runif(1, 0.35, 0.6)
b2 <- runif(1, 0.45, 0.75) * sample(c(-1, 1), 1)
y <- b1 * z1 + b2 * z2g[idx] + u0[idx] + u1[idx] * z1 + rnorm(n, 0, runif(1, 0.55, 0.8))
d <- data.frame(kelompok = kelompok,
                learning_motivation = round(50 + 10 * z1, 3),
                students_per_teacher = round(30 + 6 * z2g[idx], 3),
                student_mathematics_score = round(100 + 15 * y, 3))
write.csv(d, "data_contoh.csv", row.names = FALSE)

# ---- Analisis: regresi multilevel (lme4::lmer) ----
d$kelompok <- factor(d$kelompok)
d$x1_pusat <- d$learning_motivation - mean(d$learning_motivation)                       # X1 dipusatkan pada rata-rata keseluruhan
# 1) Model null untuk ICC
m0 <- lmer(student_mathematics_score ~ 1 + (1 | kelompok), data = d, REML = TRUE)
vc0 <- as.data.frame(VarCorr(m0))
icc <- vc0$vcov[1] / sum(vc0$vcov)
cat(sprintf("ICC model null = %.3f\n", icc))
# 2) Random intercept, lalu coba random slope X1 (cadangan bila singular/gagal)
m1 <- lmer(student_mathematics_score ~ x1_pusat + students_per_teacher + (1 | kelompok), data = d, REML = TRUE)
m2 <- tryCatch(lmer(student_mathematics_score ~ x1_pusat + students_per_teacher + (1 + x1_pusat | kelompok), data = d, REML = TRUE,
                    control = lmerControl(calc.derivs = FALSE)), error = function(e) NULL)
p_slope <- NA
if (!is.null(m2) && !isSingular(m2)) {
  a12 <- suppressMessages(anova(m1, m2, refit = FALSE))          # LR uji varians slope acak
  print(a12)
  p_slope <- a12[2, "Pr(>Chisq)"]
}
fit <- if (is.finite(p_slope) && p_slope < 0.05) m2 else m1
label_model <- if (identical(fit, m2)) "random intercept + random slope" else "random intercept"
print(summary(fit))
ct <- coef(summary(fit))
pval <- unname(2 * pnorm(abs(ct[, "t value"]), lower.tail = FALSE))  # p-value aproksimasi normal (Wald z)
print(cbind(ct, "p (Wald z)" = round(pval, 4)))
# 3) Uji LR efek acak: model terpilih vs OLS biasa (ML), p dibagi dua (uji di batas)
ols <- lm(student_mathematics_score ~ x1_pusat + students_per_teacher, data = d)
fit_ml <- refitML(fit)
lr_stat <- max(0, as.numeric(2 * (logLik(fit_ml) - logLik(ols))))
df_lr <- attr(logLik(fit_ml), "df") - attr(logLik(ols), "df")
p_lr <- 0.5 * pchisq(lr_stat, df_lr - 1, lower.tail = FALSE) + 0.5 * pchisq(lr_stat, df_lr, lower.tail = FALSE)
cat(sprintf("Model terpilih: %s; LR vs OLS = %.2f, p = %.4g\n", label_model, lr_stat, p_lr))


# ---- Grafik ----
# Garis regresi tiap kelompok (spaghetti) dan garis efek tetap
rentang <- do.call(rbind, lapply(split(d, d$kelompok), function(g)
  data.frame(kelompok = g$kelompok[1], x1_pusat = range(g$x1_pusat), students_per_teacher = g$students_per_teacher[1])))
rentang$prediksi <- predict(fit, newdata = rentang)
rentang$x <- rentang$x1_pusat + mean(d$learning_motivation)
umum <- data.frame(x1_pusat = range(d$x1_pusat), students_per_teacher = mean(d$students_per_teacher))
umum$prediksi <- predict(fit, newdata = umum, re.form = NA)
umum$x <- umum$x1_pusat + mean(d$learning_motivation)
p1 <- ggplot() +
  geom_point(data = d, aes(learning_motivation, student_mathematics_score), colour = warna$redup, alpha = 0.25, size = 1.1) +
  geom_line(data = rentang, aes(x, prediksi, group = kelompok), colour = pal[1], alpha = 0.55, linewidth = 0.6) +
  geom_line(data = umum, aes(x, prediksi), colour = pal[2], linewidth = 1.6) +
  labs(title = sprintf("%s-specific lines and fixed-effect line", substr("Group", 1, 18)), x = "learning_motivation", y = "student_mathematics_score")
# Caterpillar plot intersep acak dengan CI 95%
re <- ranef(fit, condVar = TRUE)$kelompok
se_re <- sqrt(attr(ranef(fit, condVar = TRUE)$kelompok, "postVar")[1, 1, ])
cat_df <- data.frame(kelompok = rownames(re), b = re[, 1], se = se_re)
cat_df <- cat_df[order(cat_df$b), ]
cat_df$urut <- seq_len(nrow(cat_df))
cat_df$sig <- ifelse(cat_df$b - 1.96 * cat_df$se > 0 | cat_df$b + 1.96 * cat_df$se < 0, "CI excludes 0", "CI includes 0")
p2 <- ggplot(cat_df, aes(b, urut, colour = sig)) +
  geom_vline(xintercept = 0, linetype = "dashed", colour = warna$redup) +
  geom_segment(aes(x = b - 1.96 * se, xend = b + 1.96 * se, yend = urut), linewidth = 0.6) +
  geom_point(size = 1.6) +
  scale_colour_manual(values = c("CI excludes 0" = pal[2], "CI includes 0" = pal[3]), name = NULL) +
  scale_y_continuous(breaks = NULL) +
  labs(title = "Random intercepts (caterpillar, 95% CI)", x = "Deviation from fixed intercept",
       y = sprintf("%s (sorted)", substr("Group", 1, 20)))
# Rata-rata kelompok terhadap prediktor level kelompok
rk <- aggregate(cbind(student_mathematics_score, students_per_teacher) ~ kelompok, data = d, FUN = mean)
rk$n <- as.numeric(table(d$kelompok)[as.character(rk$kelompok)])
p3 <- ggplot(rk, aes(students_per_teacher, student_mathematics_score)) +
  geom_hline(yintercept = mean(d$student_mathematics_score), linetype = "dotted", colour = warna$redup) +
  geom_smooth(method = "lm", formula = y ~ x, colour = pal[4], fill = pal[4], alpha = 0.15, linewidth = 0.9) +
  geom_point(aes(size = n), colour = pal[5], alpha = 0.8) +
  scale_size_continuous(range = c(1.2, 4), guide = "none") +
  labs(title = sprintf("%s means vs level-2 predictor", substr("Group", 1, 18)), x = "students_per_teacher", y = "Mean student_mathematics_score")
p_hasil <- if (tata == 1) p1 | (p2 / p3) else p1 / (p2 | p3)

print(p_hasil)

The project file you can download

The original project file is available. It is not just the picture above: you get the working R project. It can be downloaded after payment, and we send it by WhatsApp or email.

ZIP: analisis.R + data_contoh.csv + BACA-DULU.txt

What is inside the ZIP

The complete analysis script (analisis.R), commented step by step, ready to run in RStudio.
A sample dataset (data_contoh.csv) so the script runs straight away.
A short guide (BACA-DULU.txt): which packages to install, how to run it and how to swap in your own data.

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

Package 1

Code and Explanation

US$31

excl. 11% VAT

Duration: 7 working days

Output: R files + result figures

Deliverables

Fully commented R code (.R) for Multilevel Regression (lme4)
Example data and step-by-step run instructions
Ready-to-use result figures and metrics table
Short explanation of the method, its assumptions and how to read the results

Choose Package 1

Package 2 · most chosen

Your Data Analysis + Interpretation

US$88

excl. 11% VAT

Duration: 13 working days

Output: Code + analysis + interpretation notes

Includes all of Package 1, plus

Adapting the code to your own research dataset (CSV or Excel)
Assumption checks, result validation and data cleaning
Tables and figures formatted for your manuscript
Interpretation notes for the methods and discussion sections, for you to write up in your own words

Choose Package 2

Package 3

Full Guidance

US$194

excl. 11% VAT

Duration: 18-21 working days

Output: Code + analysis + guidance until ready to present

Includes all of Packages 1 & 2, plus

Comparison against one alternative method with the reasoning for the choice
Two 60-minute online coaching sessions so you fully understand the code and results
Revisions following your supervisor feedback (up to 3 rounds)
Practice Q&A to prepare for your seminar and defence
Full documentation (README, R package list (sessionInfo)) so the results can be reproduced

Choose Package 3

Payment: 50% down payment before work starts, balance after the deliverables are handed over
Tax: Prices exclude 11% VAT
Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions

Do I get the real project file?

Yes. The original project file is a ZIP containing the complete analisis.R script, a data_contoh.csv sample dataset and a BACA-DULU.txt guide. It can be downloaded after payment, and we send it by WhatsApp or email. For your own data (Package 2 and 3) the script is adapted to your dataset.

Is this suitable for a Doctoral dissertation (S3)?

Yes. Multilevel Regression (lme4) is commonly used at this level in Education and Learning. We adapt the analysis to your research question, your own data and your supervisor's requirements.

Is the result sheet real data?

No. The sheet is computed from illustrative synthetic data so you can see the kind of output R Studio gives. For your order we run the same method on your own data.

Will I understand the code?

That is the point of the packages. Every script is commented, and Package 3 includes two coaching sessions plus practice questions so you can explain the method and the results at your seminar and defence.

How is the price decided?

Each project has three packages: Package 1 (code and explanation) US$31, Package 2 (your data analysis and interpretation) US$88 and Package 3 (full guidance) US$194, excluding 11% VAT. Larger datasets, extra methods or tight deadlines adjust the quote after a short consultation.

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