Evaluation of Joint Model Imputation of Multilevel Models

Missing data is a pervasive challenge in research, particularly in studies involving multilevel models. When left unaddressed or handled poorly, missing data can reduce statistical power and introduce bias, undermining the validity of findings. Although multiple imputation has emerged as a preferred method for addressing missing data due to its ability to provide unbiased and efficient estimates, its application in multilevel modeling remains underexplored. This gap is particularly significant given the hierarchical nature of multilevel data, which requires specialized approaches to imputation. The purpose of the study is to evaluate the effectiveness of joint model imputation in handling missing data in multilevel models under various data conditions. This includes comparing three types of multilevel models—random intercept, random coefficient, and random intercept-and-slope models—by manipulating factors such as sample size, missing data percentage, and intraclass correlation coefficient (ICC). Specifically, the study addresses three research questions: (a) In what ways does varying level 1 and level 2 sample sizes influence the accuracy and reliability of outcomes obtained through joint model imputation across three different hierarchical linear models (HLM), as evaluated using bias, and root mean square error (RMSE) and confidence interval coverage (CI)? (b) To what extent does changing the proportion of missing data affect the outcomes of joint model imputation in terms of bias, root mean square error (RMSE), and confidence interval coverage? (c) How does the variation in intraclass correlation (ICC) impact the effectiveness of joint model imputation for achieving high-quality results, specifically looking at bias, root mean square error (RMSE), and confidence interval coverage? A simulation study was conducted to systematically manipulate sample sizes, proportions of missing data, and ICC values. Performance was evaluated using metrics such as relative bias, RMSE, and CI coverage rates. Additionally, the Trends in International Mathematics and Science Study (TIMSS) 2019 dataset for fourth graders in the United States was analyzed to demonstrate the practical application of joint model imputation under real-world conditions. Findings from the simulation study indicate that larger sample sizes at both levels improved parameter estimates, reducing bias and RMSE, particularly in the random-intercept-and-slope model. However, smaller sample sizes posed challenge to the imputation. Proportion of missing data significantly influenced imputation outcomes; while the method performed well at 10% and 20% missingness, it showed diminished effectiveness at 30%. ICC variations presented additional challenges, with higher ICC values exacerbating bias and RMSE, particularly in the random-coefficient model. The random-intercept model consistently demonstrated robust performance under small sample sizes and high ICC, while the random-intercept-and-slope model excelled in scenarios with moderate ICC and larger sample sizes. The empirical validation using the TIMSS dataset corroborated these findings, emphasizing the utility of joint model imputation in real-world multilevel contexts.