1.1 Project Proposal

Analyzing the Interplay Between Ethnicity,Literacy, and Numeracy Skills in Educational Outcomes.

1.2 Project Question/Estimand

This project aims to unravel the complex interactions between students' ethnicity, their literacy (reading score), and numeracy (math score) skills, to understand the causal pathways influencing educational outcomes. The central question guiding our investigation is: How does ethnicity and reading skills jointly impact math performance among students?

1.3 Data Description

Our analysis will leverage a dataset encompassing student demographics (with a focus on ethnicity), math scores, and reading scores across multiple schools over one academic year. It includes detailed performance metrics alongside background demographic information, allowing for a nuanced exploration of the proposed causal relationships.

# Model Description

Reading Score Model (E -> R)

The model for reading scores is formulated as follows: Ethnicity's Impact on Reading Scores: We model the baseline reading scores for different ethnic groups using a normal distribution, where the mean reading score for each ethnic group is determined by the parameter , with a common standard deviation.

The prior for $\alpha_r$ is set as a normal distribution Normal(70,10), indicating

our belief that the average reading score across ethnicities is around 70 with some

variability. \

$\alpha_r$ ∼ Normal(70,10) \

$\mu_r$ ~ $\alpha_r$[E] \

Math Score Model (R -> M, E -> M)

e assume the math score for each ethnicity is influenced directly by its reading score. The model uses a normal distribution for math scores, with the mean score μ depending on both the ethnicity-specific intercept a and a slope b which scales the influence of the reading scores (adjusted by their mean )

$\alpha$ ~ Normal(75, 10) \

$\beta$ ~ Lognormal(0, 2) \

$\mu$ ~ $\alpha$[E] + $\beta$[E] * ($R$ - $R_{bar}$) \

$\sigma$ ~ Uniform(0, 10) \

Math Score ~ Normal($\mu$, $\sigma$)

Project Overview

This project investigates the causal relationship between ethnicity and math scores, with reading scores acting as a mediator. The data comprises ethnicity, reading scores, and math scores of students. Using a Directed Acyclic Graph (DAG) to outline causal pathways, the statistical model incorporated normal priors for intercepts and log-normal priors for slopes, fitting the observed data well. Detailed mathematical notation and appropriate handling of confounding variables ensured clarity and precision in the model. Posterior sampling diagnostics, including trace and density plots, indicated good mixing and convergence, supporting the reliability of parameter estimates.

Posterior predictive checks confirmed that the model's predictions closely matched the observed data, demonstrating effective sampling and well-behaved posterior distributions. The analysis revealed significant differences in baseline math scores among ethnic groups, with Group A having the lowest intercept and Group E the highest. Positive slopes for all groups indicated that increases in reading scores are

associated with higher math scores, with Group E showing the strongest relationship. The study highlights the importance of reading scores as a mediator and underscores the need to consider individual group characteristics and variability in educational outcomes across ethnicities. The comprehensive model and its findings provide valuable insights into the complex relationship between ethnicity and math scores, emphasizing the impact of mediating factors like reading scores in educational research.

Statistical model analysis:-

Based on the provided parameter estimates:

1. Alpha (Intercept) Mean Values:

- The alpha values represent the baseline math score for each ethnicity group when the reading score is at its mean value.

- These values range from approximately 65.1 to 70.4, indicating the baseline math scores for different ethnicity groups.

2. Beta (Slope) Mean Values:

- The beta values represent the change in math score for each unit increase in reading score, accounting for ethnicity.

- These values range from approximately 0.760 to 0.894, indicating the strength of the relationship between reading score and math score for different ethnicity groups.

3. Sigma (Standard Deviation):

- Sigma represents the standard deviation of the model's error term.

- The value of sigma is approximately 8.524, indicating the average deviation of observed math scores from the predicted values.

4. Interpretation:

- The model suggests that there is a positive relationship between reading score and math score for all ethnicity groups.

- Higher beta values indicate a stronger positive association between reading and math scores, suggesting that improvements in reading score lead to larger increases in math score for those groups.

- The variability in baseline math scores (alpha) among ethnicity groups reflects the differences in average math performance before considering the effect of reading score.

- The relatively low value of sigma indicates that the model's predictions are relatively close to the observed math scores on average.

In summary, the model provides insights into the relationships between ethnicity, reading score, and math score. It suggests that reading score is a significant predictor of math score, with differences observed among ethnicity groups. The model captures the associations between these variables while considering individual group characteristics and the variability in math scores.

Causal relationship between ethnicity and math score

To conclude about the causal relationship between ethnicity and math score based on the provided parameter estimates, we need to analyze the intercepts (a) and slopes (b) for each ethnicity group in the model.

1. Intercept Estimates (a):

- The intercept estimates represent the baseline math scores for each ethnicity group when the reading score is at its mean value.

- Differences in intercepts among ethnic groups provide insights into the average differences in math scores across ethnicities, independent of reading score.

2. Slope Estimates (b):

- The slope estimates represent how the math score changes for each unit Increase in the centered reading score (reading score minus its mean), accounting for ethnicity.

- Differences in slopes among ethnic groups indicate how the relationship between reading score and math score varies across ethnicities.

Based on the provided parameter estimates:

- Intercept Estimates (a):

- Ethnic groups have different baseline math scores when the reading score is at its mean value.

- For example, Group A has the lowest intercept (65.117), while Group E has the highest intercept (70.385), suggesting differences in baseline math scores across ethnicities.

- Slope Estimates (b):

- All ethnic groups show positive slopes, indicating a positive relationship between centered reading score and math score.

- The magnitudes of the slopes vary among ethnic groups, indicating differences in the strength of the relationship between reading score and math score across ethnicities.

- For instance, Group E has the highest mean slope (0.895), indicating a relatively stronger positive relationship between reading score and math score compared to other groups.

Overall, based on these parameter estimates:

- There is evidence to suggest that ethnicity has a causal relationship with math score, as indicated by the differences in intercepts among ethnic groups.

- Additionally, the positive slopes for all ethnic groups suggest that improvements in reading score are associated with increases in math score for each ethnicity.

- The varying magnitudes of intercepts and slopes among ethnic groups highlight the complexity of the relationship between ethnicity and math score, indicating potential differences in baseline proficiency and the strength of the relationship between reading and math scores across ethnicities.

Conclusion

Based on the provided estimates, we can interpret the direct and total effects of ethnicity on math score, considering reading score as a mediator.

1. Direct Effects:

- Alpha mean estimates represent the direct effect of ethnicity on math score, independent of reading score.

- Differences in alpha among ethnic groups indicate average differences in math scores when reading score is not considered.

2. Total Effect (Direct + Indirect):

- Beta mean estimates represent the total effect of ethnicity on math score, considering the direct effect of ethnicity and the indirect effect mediated through reading score.

- The total effect combines both the direct effect captured by alpha and the indirect effect mediated through reading score captured by beta.

3. Interpretation:

- The differences in alpha values suggest that there are average differences in math scores among ethnic groups, independent of reading score. For example, Group E tends to have higher baseline math scores compared to other groups.

- The beta values indicate the strength of the relationship between reading score and math score for each ethnicity group. Higher beta values suggest that improvements in reading score have a larger impact on math score for that ethnicity. For instance, Group E has the highest beta value, indicating that improvements in reading score have a stronger effect on math score for Group E compared to other groups.

- Considering both alpha and beta values together, it seems that ethnicity contributes to the average differences in math scores among groups, and improvements in reading score further amplify these differences. However, the extent to which reading score mediates the relationship between ethnicity and math score varies among ethnic groups.

4. Potential Confound Explanations:

- Despite accounting for ethnicity and reading score, there may still be unobserved confounds influencing math scores.

-The model's estimates provide insights into the direct and total effects of ethnicity on math score, but it's important to consider potential confounding Variables when interpreting the results.

- One potential confound could be socioeconomic status (SES), which may affect both ethnicity and academic performance. Students from higher SES backgrounds may have access to better resources and support, leading to higher math scores regardless of ethnicity.

- Additionally, other factors such as cultural differences in educational opportunities or teaching styles could also impact math scores and may not be fully captured in the model.

In summary, while ethnicity and reading score contribute to differences in math scores, there may be other unobserved factors such as socioeconomic status that influence the observed patterns.