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FIXED AND RANDOM EFFECTS IN HLM

FIXED AND RANDOM EFFECTS IN HLM. Fixed vs random effects. Fixed effects produce constant impact on DV. Random effects produce variable impact on DV. OLS is a fixed effect model. Here, only r ij is random. What if the β ’s are random (variable)?. HLM. L1. L2.

margaret-emerson
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FIXED AND RANDOM EFFECTS IN HLM

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  1. FIXED AND RANDOM EFFECTS IN HLM

  2. Fixed vs random effects Fixed effects produce constant impact on DV. Random effects produce variable impact on DV.

  3. OLS is a fixed effect model Here, only rij is random. What if the β’s are random (variable)?

  4. HLM L1 L2 Predictors (W’s) at level 2 are used to model variation in intercepts and slopes between the j units

  5. example, unconditional model Level 1: Level 2: Fixed effect: intercept Random effect:intercept – significant variability between groups? level-1 – significant variability within groups?

  6. What about ? • It is random when estimation of variance components of is statistically significant. The model is one-way ANOVA with random effects. • It is fixed when estimation of variance components of is not statistically significant. We don’t need HLM, but simply a one-way ANOVA! • Therefore, the difference between fixed and random coefficients in level

  7. As comparison, fixed vs random effects in ANOVA Fixed effects: factor levels are assigned by researchers in an experiment. Example: we are interested in the effects of three HLM textbooks on students’ achievement Note: The study is to compare only three groups –not generalize to other textbooks that we didn't include although there are more than three textbooks for HLM.

  8. As comparison, fixed vs random effects in ANOVA Fixed effects: all levels of a variable in a non-experimental setting. Example: comparing students’ achievement between male & female (gender as a fixed effect). Note: The study includes all possible levels of the variable in the study

  9. As comparison, fixed vs random effects in ANOVA Random effects: the levels of a variable that we included in a study are treated as sample from a population of possible levels. Example: we select three HLM textbooks from many possible textbooks and want to draw a conclusion that different HLM textbooks have various contributions to students’ achievement. Note: The study is to compare all different HLM, but only three are selected to make inference.

  10. Mathematical expressions – fixed effect model • μ is grand mean (constant). • j is group j effect (constant). • Єijis the residual or error (random). • Єij ~ N(0, σ2) and independent.

  11. Mathematical expressions- random effect model • μ – grand mean (constant). • Tjis group j effect (random). • Tj ~N(0, τ2) and independent. • Єijis the w/in groups residual or error (random). • Єij ~N(0, σ2) and independent. • Tjand Єijare independent, ie., cov(Uj ,Rij) = 0.

  12. Go back to the example, unconditional model Level 1: Level 2: Fixed effect: intercept Random effect: intercept level-1

  13. Go back to the example, unconditional model Level 1: Level 2: Fixed effect: intercept Random effect: intercept level-1

  14. Example, means as outcome regression model Level 1: Level 2: Fixed effect: intercept slope Random effect: intercept level-1

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