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Area of Science:

  • Statistics
  • Multilevel Modeling
  • Quantitative Psychology

Background:

  • Hierarchical linear modeling (HLM) often assumes infinite populations at higher levels, neglecting finite population effects.
  • Overestimation of standard errors occurs when Level-2 finite populations are treated as infinite, reducing statistical power and widening confidence intervals.

Purpose of the Study:

  • To propose a method for calculating finite-population-adjusted standard errors for Level-1 and Level-2 fixed effects in 2-level HLM.
  • To demonstrate the impact of ignoring finite population correction using real data and simulations.

Main Methods:

  • Developed a method for finite-population adjustment of standard errors in 2-level HLM.
  • Utilized a real data example and simulation studies (random intercept and random slope models) to evaluate the proposed method.

Main Results:

  • Bias in unadjusted standard errors was substantial when Level-2 sample size exceeded 10% of the Level-2 population.
  • Bias increased with higher intraclass correlation, more clusters, and larger average cluster size.
  • The proposed adjustment yielded unbiased standard errors, especially with ≥30 clusters and ≥10 average cluster size.

Conclusions:

  • Finite population correction is crucial in HLM when the Level-2 sample constitutes a significant portion of the population.
  • Researchers should consider target population characteristics and apply finite population adjustments for accurate standard errors and improved statistical inference.