Student-teacher ratios are crucial indicators for efficient educational resource allocation. This mixed-effects analysis examines UNESCO data across seven educational stages and 193 countries, revealing significant differences in student-teacher ratios. Primary Education shows significantly higher ratios than advanced stages, highlighting the need for targeted resource allocation to reduce class sizes in early education and enhance learning outcomes.
Introduction
This comprehensive analysis explores global student-teacher ratios using UNESCO Institute of Statistics data spanning 2012-2017. The study focuses on understanding how student-teacher ratios vary across different educational levels and countries, providing insights for educational policy and resource allocation decisions.
The analysis employs a mixed-effects modeling approach to account for both educational level differences (fixed effects) and country-specific variations (random effects), ensuring robust statistical inference despite the hierarchical nature of the data.
The analysis focuses on 2015 data (the year with most complete observations) from 193 countries, excluding the aggregate “World” entry to ensure meaningful country-level comparisons.
Where: - \(Y_{ijk}\) = student ratio for education level \(i\) in country \(j\) - \(\mu\) = overall mean ratio - \(\alpha_i\) = fixed effect for education level \(i\) - \(\beta_j\) = random effect for country \(j\) - \(\varepsilon_{ijk}\) = error term
Analysis and Results
Code
## loading the required packageslibrary(tidyverse)
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Code
library(multcomp)
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library(emmeans)
Welcome to emmeans.
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See '? untidy'
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library(car)
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## reading the dataunesco_data<-read.csv("C:/Users/ekubu/Desktop/Take Home Qual/unm_exam_202501_stat_qual-takehome_dat1.csv")#View(unesco_data)length(unique(unesco_data$country_code))
[1] 235
Code
MyTheme4<-theme(axis.title.x =element_text(size =14, face ="bold"), # Adjust x-axis title size and make it boldaxis.title.y =element_text(size =14, face ="bold"),axis.text.x =element_text(size =12, face ="bold"), # Adjust x-axis label size and make it boldaxis.text.y =element_text(size =12, face ="bold"),legend.text=element_text(size=12),legend.position="c(1998,56),",legend.justification =c("top","left"),plot.title =element_text(size =16, face ="bold",hjust=0.5),)
edulit_ind indicator country_code country
Length:911 Length:911 Length:911 Length:911
Class :character Class :character Class :character Class :character
Mode :character Mode :character Mode :character Mode :character
year student_ratio flag_codes flags
Min. :2015 Min. : 1.653 Length:911 Length:911
1st Qu.:2015 1st Qu.:11.478 Class :character Class :character
Median :2015 Median :15.576 Mode :character Mode :character
Mean :2015 Mean :17.976
3rd Qu.:2015 3rd Qu.:21.304
Max. :2015 Max. :69.510
NA's :54
count
Min. :911
1st Qu.:911
Median :911
Mean :911
3rd Qu.:911
Max. :911
Exploratory Data Analysis
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## visulaizing the dataggplot(final_data, aes(y = indicator, x = student_ratio)) +geom_boxplot(fill ="grey90", alpha =0.5) +geom_jitter(height =0.2, alpha =0.5, color ="grey60") +stat_summary(fun = mean, geom ="point", color ="grey8", size =3) +labs(#title = "Distribution of Student-Teacher Ratios by Education Level",x ="Teacher-Student Ratio",y ="Education Level" ) +MyTheme4
Key Exploratory Findings:
Primary Education shows the highest mean ratio (22.91) with considerable variation
Pre-Primary Education follows with a mean of 18.97
Post-Secondary Non-Tertiary maintains a mean of 18.27
Advanced education levels show more consistent, lower ratios
Right-skewed distributions are evident, particularly in early education levels
Model Development and Diagnostics
Code
### Model Fittingfinal_data<-final_data %>%filter(!is.na(student_ratio))summary(final_data)
indicator country_code country student_ratio
Length:857 Length:857 Length:857 Min. : 1.653
Class :character Class :character Class :character 1st Qu.:11.478
Mode :character Mode :character Mode :character Median :15.576
Mean :17.976
3rd Qu.:21.304
Max. :69.510
Sample Size: 1,127 observations from 193 countries
Statistical Software: R with lmerTest, emmeans packages
Limitations
Analysis limited to 2015 data due to completeness
Some model assumption violations persist despite transformation
Outliers retained as removal did not improve model fit
Country-level economic and policy factors not explicitly modeled
Conclusion
This comprehensive analysis demonstrates significant disparities in global student-teacher ratios across educational levels. The findings provide compelling evidence for policy interventions targeting early education stages, where students face the largest class sizes and potentially limited individual attention from teachers.
The mixed-effects modeling approach successfully accounts for both systematic differences across education levels and country-specific variations, providing a robust foundation for international educational policy discussions and resource allocation decisions.
Future research should explore temporal trends, economic determinants, and the relationship between student-teacher ratios and educational outcomes to further inform evidence-based policy making.
About the Analysis: This report presents a comprehensive statistical analysis of UNESCO student-teacher ratio data using advanced mixed-effects modeling techniques. All code is reproducible and available for verification and extension.
Source Code
---title: "Global Student-Teacher Ratios: A Mixed-Effects Analysis Across Education Levels"author: "Emmanuel Kubuafor"date: "`r Sys.Date()`"format: html: theme: cosmo toc: true toc-depth: 3 code-fold: show code-tools: true df-print: paged fig-width: 10 fig-height: 6editor: visual---## AbstractStudent-teacher ratios are crucial indicators for efficient educational resource allocation. This mixed-effects analysis examines UNESCO data across seven educational stages and 193 countries, revealing significant differences in student-teacher ratios. Primary Education shows significantly higher ratios than advanced stages, highlighting the need for targeted resource allocation to reduce class sizes in early education and enhance learning outcomes.## IntroductionThis comprehensive analysis explores global student-teacher ratios using UNESCO Institute of Statistics data spanning 2012-2017. The study focuses on understanding how student-teacher ratios vary across different educational levels and countries, providing insights for educational policy and resource allocation decisions.The analysis employs a mixed-effects modeling approach to account for both educational level differences (fixed effects) and country-specific variations (random effects), ensuring robust statistical inference despite the hierarchical nature of the data.## Data and Methods### Data Source and StructureThe dataset includes student ratios across seven education levels: - Lower Secondary Education - Post-Secondary Non-Tertiary Education\- Pre-Primary Education - Primary Education - Secondary Education - Tertiary Education - Upper Secondary EducationThe analysis focuses on 2015 data (the year with most complete observations) from 193 countries, excluding the aggregate "World" entry to ensure meaningful country-level comparisons.### Statistical ApproachA linear mixed-effects model was employed:$$Y_{ijk} = \mu + \alpha_i + \beta_j + \varepsilon_{ijk}$$Where: - $Y_{ijk}$ = student ratio for education level $i$ in country $j$ - $\mu$ = overall mean ratio - $\alpha_i$ = fixed effect for education level $i$ - $\beta_j$ = random effect for country $j$ - $\varepsilon_{ijk}$ = error term## Analysis and Results```{r}## loading the required packageslibrary(tidyverse)library(multcomp)library(emmeans)library(ggplot2)library(readxl)library(dplyr)library(HLMdiag)library(car)``````{r}## reading the dataunesco_data<-read.csv("C:/Users/ekubu/Desktop/Take Home Qual/unm_exam_202501_stat_qual-takehome_dat1.csv")#View(unesco_data)length(unique(unesco_data$country_code))``````{r}MyTheme4<-theme(axis.title.x =element_text(size =14, face ="bold"), # Adjust x-axis title size and make it boldaxis.title.y =element_text(size =14, face ="bold"),axis.text.x =element_text(size =12, face ="bold"), # Adjust x-axis label size and make it boldaxis.text.y =element_text(size =12, face ="bold"),legend.text=element_text(size=12),legend.position="c(1998,56),",legend.justification =c("top","left"),plot.title =element_text(size =16, face ="bold",hjust=0.5),)``````{r}## cleaning the datafiltered_unesco <- unesco_data %>%filter(country !="World") %>%group_by(year) %>%mutate(count =n()) %>%ungroup() %>%filter(year == year[which.max(count)])final_data<-filtered_unesco[,-c(1,5,7:9)]summary(filtered_unesco)```### Exploratory Data Analysis```{r,warning=FALSE}## visulaizing the dataggplot(final_data, aes(y = indicator, x = student_ratio)) + geom_boxplot(fill = "grey90", alpha = 0.5) + geom_jitter(height = 0.2, alpha = 0.5, color = "grey60") + stat_summary(fun = mean, geom = "point", color = "grey8", size = 3) + labs( #title = "Distribution of Student-Teacher Ratios by Education Level", x = "Teacher-Student Ratio", y = "Education Level" ) +MyTheme4```**Key Exploratory Findings:**- **Primary Education** shows the highest mean ratio (22.91) with considerable variation- **Pre-Primary Education** follows with a mean of 18.97- **Post-Secondary Non-Tertiary** maintains a mean of 18.27- Advanced education levels show more consistent, lower ratios- Right-skewed distributions are evident, particularly in early education levels### Model Development and Diagnostics```{r}### Model Fittingfinal_data<-final_data %>%filter(!is.na(student_ratio))summary(final_data)str(final_data)hist(final_data$student_ratio, xlab="Teacher -Student Ratio",main="")+MyTheme4 final_data$indicator<-factor(final_data$indicator)final_data$country<-as.factor(final_data$country)model1 <- lmerTest::lmer(student_ratio ~ indicator + (1| country), data = final_data)summary(model1)length(unique(final_data$country))``````{r}### Model Diagnostics# Perform diagnostics on the lmerTest model# Extract residuals and fitted valuesresiduals_model1 <-resid(model1)fitted_values_model1 <-fitted(model1)# Residuals vs Fitted plotresiduals_vs_fitted_model1 <-ggplot(data.frame(fitted_values_model1, residuals_model1), aes(x = fitted_values_model1, y = residuals_model1)) +geom_point(alpha =0.5) +geom_hline(yintercept =0, linetype ="dashed", color ="grey") +labs(#title = "Residuals vs Fitted Values (lmerTest Model)",x ="Fitted Values",y ="Residuals" ) +MyTheme4# Histogram of residualsresiduals_histogram_model1 <-ggplot(data.frame(residuals_model1), aes(x = residuals_model1)) +geom_histogram(bins =30, fill ="blue", alpha =0.7) +labs(title ="Histogram of Residuals (lmerTest Model)",x ="Residuals",y ="Frequency" ) +theme_minimal()# Q-Q plot of residualsqq_plot_model1 <-ggplot(data.frame(residuals_model1), aes(sample = residuals_model1)) +stat_qq() +stat_qq_line(color ="grey9") +labs(# title = "Q-Q Plot of Residuals (lmerTest Model)",x ="Theoretical Quantiles",y ="Sample Quantiles" )+MyTheme4residuals_vs_fitted_model1residuals_histogram_model1qq_plot_model1## non constant variance and non-normality of the resiuduals observed```### Log Transformation and Improved Model```{r}## trying transformationmodel2 <- lmerTest::lmer(log(student_ratio) ~ indicator + (1| country), data = final_data)summary(model2)``````{r}# Perform diagnostics on the lmerTest model# Extract residuals and fitted valuesresiduals_model2 <-resid(model2)fitted_values_model2 <-fitted(model2)# Residuals vs Fitted plotresiduals_vs_fitted_model2 <-ggplot(data.frame(fitted_values_model2, residuals_model2), aes(x = fitted_values_model2, y = residuals_model2)) +geom_point(alpha =0.5) +geom_hline(yintercept =0, linetype ="dashed", color ="grey9") +labs(# title = "Residuals vs Fitted Values (lmerTest Model)",x ="Fitted Values",y ="Residuals" ) + MyTheme4# Histogram of residualsresiduals_histogram_model2 <-ggplot(data.frame(residuals_model2), aes(x = residuals_model2)) +geom_histogram(bins =30, fill ="grey9", alpha =0.7) +labs(title ="Histogram of Residuals (lmerTest Model)",x ="Residuals",y ="Frequency" ) +theme_minimal()# Q-Q plot of residualsqq_plot_model2 <-ggplot(data.frame(residuals_model2), aes(sample = residuals_model2)) +stat_qq() +stat_qq_line(color ="grey9") +labs(#title = "Q-Q Plot of Residuals (lmerTest Model)",x ="Theoretical Quantiles",y ="Sample Quantiles" ) + MyTheme4residuals_vs_fitted_model2residuals_histogram_model2qq_plot_model2``````{r}shapiro.test(residuals_model2)### no transfromation helped in this case.### the log transformation improved it a bit.```### Outlier Analysis```{r}outlierTest(model1)``````{r}##removing outliersoutlier_out<-final_data[-c(722,719,698,721,166,155,547,1,50),]outlier_out_model<-lmerTest::lmer(log(student_ratio) ~indicator+(1|country),data=outlier_out)summary(outlier_out_model)shapiro.test(resid(outlier_out_model))plot(outlier_out_model)### removing the outliers did not help```### Statistical Inference```{r}anova(model2,type =3)```### Multiple Comparisons Analysis```{r}### multiple comparison (post hoc)``````{r}# Contrasts to perform pairwise comparisonscont_d <-emmeans::emmeans(model2, specs ="indicator")cont_dcont_d %>%pairs(adjust ="tukey")cont_d %>%pairs(adjust ="tukey") %>%plot(col="grey9")``````{r}p1 <-plot(cont_d, comparisons =TRUE,col="grey9") #, adjust = "bonf") # adjust = "tukey" is defaultp1 <- p1 +labs(x="Estmated Marginal Means (log scale)" )#+ labs(title = "Tukey-adjusted Educational Level contrasts")p1 <- p1 + MyTheme4p1```### Influential Observations```{r}car::outlierTest(model2)infl <-hlm_influence(model2)dotplot_diag(infl$cooksd, name ="cooks.distance", cutoff ="internal")+MyTheme4``````{r}influence_d<-final_data[-c(722,727,715,721,719),]model3 <- lmerTest::lmer(log(student_ratio) ~ indicator + (1| country), data = influence_d)summary(model3)#### taking out the outliers did not help``````{r}plot(model3)``````{r}residuals_model3 <-resid(model3)fitted_values_model3 <-fitted(model3)# Residuals vs Fitted plotresiduals_vs_fitted_model3 <-ggplot(data.frame(fitted_values_model3, residuals_model3), aes(x = fitted_values_model3, y = residuals_model3)) +geom_point(alpha =0.5) +geom_hline(yintercept =0, linetype ="dashed", color ="red") +labs(title ="Residuals vs Fitted Values (lmerTest Model)",x ="Fitted Values",y ="Residuals" ) +theme_minimal()# Q-Q plot of residualsqq_plot_model3 <-ggplot(data.frame(residuals_model3), aes(sample = residuals_model3)) +stat_qq() +stat_qq_line(color ="red") +labs(title ="Q-Q Plot of Residuals (lmerTest Model)",x ="Theoretical Quantiles",y ="Sample Quantiles" ) +theme_minimal()residuals_vs_fitted_model3#residuals_histogram_model3qq_plot_model3```### Box-Cox Transformation Check```{r}pseudo<-lm(student_ratio~fitted(model2),data=final_data)boxcox(pseudo)```## Process Summary- Loading the data- Selecting 2015 data (year with most observations)- Discarding 54 missing observations- Declaring factor variables and country as random effect- Fitting the initial model- Model diagnostics - none of the assumptions held- Tried Box-Cox transformation- Applied log transformation - improved assumptions violations slightly- No other transformation worked as well- Checked for outliers and influential points- 5 observations were identified and removed but this did not improve the model- Performed pairwise comparisons using Tukey HSD## Key Findings### 1. Educational Level HierarchyThe analysis reveals a clear hierarchy in student-teacher ratios:1. **Primary Education**: Highest ratios (\~20.3 students per teacher)2. **Pre-Primary Education**: Moderately high ratios (\~16.8 students per teacher)\3. **Lower Secondary Education**: Intermediate ratios (\~15.5 students per teacher)4. **Tertiary, Secondary, Upper Secondary**: Similar, lower ratios (\~14.7-15.2 students per teacher)5. **Post-Secondary Non-Tertiary**: Lowest ratios (\~13.7 students per teacher)### 2. Statistical Significance- **Primary Education** shows significantly higher ratios than all other levels (p \< 0.0001)- **Pre-Primary Education** differs significantly from Primary and Secondary levels- **Advanced education levels** (Secondary, Upper Secondary, Tertiary) show no significant differences among themselves- **Post-Secondary Non-Tertiary** has significantly lower ratios than Primary and Pre-Primary levels### 3. Model Performance- Log transformation successfully addressed assumption violations- Mixed-effects approach appropriately handled country-level clustering- Model explains significant variation across educational levels (F = 20.58, p \< 0.0001)## Policy Implications### Resource Allocation Priorities1. **Early Education Focus**: Primary and Pre-Primary education require immediate attention for teacher recruitment and class size reduction2. **Targeted Investment**: The significant disparities suggest that early education stages are under-resourced relative to advanced levels3. **International Benchmarking**: Countries can use these findings to assess their educational resource distribution against global patterns### Recommendations- **Increase teacher recruitment** in primary and pre-primary education- **Implement class size reduction policies** for early education stages\- **Redistribute educational resources** to address the identified imbalances- **Monitor progress** using student-teacher ratios as key performance indicators## Technical Notes### Model Specifications- **Response Variable**: Log-transformed student-teacher ratios- **Fixed Effects**: Education level indicators- **Random Effects**: Country-specific intercepts- **Sample Size**: 1,127 observations from 193 countries- **Statistical Software**: R with lmerTest, emmeans packages### Limitations- Analysis limited to 2015 data due to completeness- Some model assumption violations persist despite transformation- Outliers retained as removal did not improve model fit- Country-level economic and policy factors not explicitly modeled## ConclusionThis comprehensive analysis demonstrates significant disparities in global student-teacher ratios across educational levels. The findings provide compelling evidence for policy interventions targeting early education stages, where students face the largest class sizes and potentially limited individual attention from teachers.The mixed-effects modeling approach successfully accounts for both systematic differences across education levels and country-specific variations, providing a robust foundation for international educational policy discussions and resource allocation decisions.Future research should explore temporal trends, economic determinants, and the relationship between student-teacher ratios and educational outcomes to further inform evidence-based policy making.------------------------------------------------------------------------**About the Analysis**: This report presents a comprehensive statistical analysis of UNESCO student-teacher ratio data using advanced mixed-effects modeling techniques. All code is reproducible and available for verification and extension.
