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# Finite Population Corrections for Two-Level Hierarchical Linear Models with Binary Predictors

#### Abstract

Answering social science research questions about clustered data necessitates collecting data using sampling schemes, which may result in hierarchical data structures. Hierarchical liner modeling (HLM) techniques are required to account for the interdependency of observations due to clustering. However, traditional HLM assumes the target population is infinitely large or near enough to infinitely large for practical purposes (i.e., the sample consists of less than 5% of the target population). Often times, the assumption of an infinitely large target population may not hold. The current study was conducted in two separate phases using Monte Carlo simulation methods. First, the continuous predictors study evaluated a finite population correction (FPC) method for a few number of large clusters. The degree of relative bias in unadjusted standard error estimates exceeded .05 and was non-ignorable when the number of clusters sampled was greater than 20. The finite population correction adjusted standard error estimates exhibited acceptable levels of relative bias across most simulation conditions. However, finite population correction adjusted standard error estimates were negatively biased when the number of clusters sampled was few (i.e., 20 clusters). The continuous predictors study also examined standard error estimates from a finite population bootstrapping alternative. The finite population bootstrap estimates did not perform well and severely underestimated the empirical standard errors across all conditions. Second, the binary predictor study evaluated the efficiency of the finite population correction method for a level-2 binary predictor. Standard errors for a balanced binary predictor (i.e., binary predictors with a relatively constant 50:50 prevalence between groups) functioned similarly in terms of bias as continuous predictors. The relative bias in the finite population correction adjusted standard errors for a balanced predictor was smaller than the relative bias in unadjusted standard errors when at least 30 clusters were sampled. For a discrepant or unbalanced binary predictor (i.e., 20:80 prevalence), finite population correction adjusted standard errors were only acceptable when 60 clusters were sampled. The current study demonstrates the need for applied researchers to explicitly state their target populations, examine their sampling fraction, and consider the FPC adjustment. Doing so yields more accurate inferences for finite populations.

#### Subject Area

Educational psychology

#### Recommended Citation

Svoboda, Steven J, "Finite Population Corrections for Two-Level Hierarchical Linear Models with Binary Predictors" (2020). *ETD collection for University of Nebraska-Lincoln*. AAI27837465.

https://digitalcommons.unl.edu/dissertations/AAI27837465