Multi-Level Modelling Concepts

Multilevel Modeling (MLM)

• MLM is a statistical approach that can be used to predict Y (individual level) based on Z (group level) while holding constant X (individual level).
• Example 1: Maro predicts micro-level - individual performance, using higher resourced classrooms (Z) to predict higher individual performance.
• Example 2: Z predicts Y (holding constant score on X), using regression to predict Y.
• Example 3: X and Y are dependent on Z, using verbal IQ (X) to predict individual results (Y) at any school (Z).

Aggregation

• Aggregation in MLM refers to the procedure of taking all data points at the lowest level and aggregating them into a single data point at a higher level.
• It is okay to use aggregation if you are only interested in macro-level information.
• Issues with aggregation:
  • Shift of meaning: Variables aggregated to the macro level tell us about the macro level, not directly about the micro level.
  • Neglect of original data structure: Reduction of power, prevents examination of cross-level interactions.
  • Ecological fallacy: Mistaken attempts to interpret aggregated data at a lower level (e.g., micro level).

Disaggregation

• Disaggregation in MLM involves breaking down aggregate data into its individual components.
• Issues with disaggregation:
  • Measure of macro-level variable considered as micro-level.
  • Miraculous multiplication of the number of units.
  • Risks of type 1 errors.
  • Does not take into account that observations within a macro-unit could be correlated.

Random Effects ANOVA

• Random effects ANOVA is a type of analysis that assumes groups are samples from a population of possible macro units.
• The null hypothesis for a random effects ANOVA does not change.
• Equation for a random effects ANOVA: Yij = γ00 + u0j + εij.
• Equation for the total variance in a random effects ANOVA: Total variance = variance of intercept + variance due to residual.

Intraclass Correlation Coefficient (ICC)

• ICC is a measure of the proportion of variance in the outcome variable that is due to the group structure.
• ICC is also the correlation between two randomly drawn individuals in one randomly drawn group.
• ICC falls between 0 and 1.
• A high ICC indicates that the group structure is an important factor in explaining the variance in the outcome variable.

Random Intercept Model

• The random intercept model attempts to explain the variation in the outcome variable due to the group structure.
• The random intercept model is a null model that estimates the variance in the intercepts between groups.
• The random intercept model with one predictor estimates the relationship between the predictor and the outcome variable while taking into account the variance in the intercepts between groups.

Multilevel Modeling (MLM) Example

• A simple example of a random intercept model would be to investigate whether SES (social status) is associated with performance on a math test.
• The model would estimate the overall mean of the math test score, the variation in the intercepts between schools, and the relationship between SES and math test score while controlling for the interclass correlation.