1. FIXED AND RANDOM EFFECTS IN HLM

2. Fixed effects produce constant impact on DV. Random effects produce variable impact on DV. F IXED VS RANDOM EFFECTS

3. OLS IS A FIXED EFFECT MODEL Here, only r ij is random. What if the β’s are random (variable)?

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

5. Level 1: Level 2: EXAMPLE, UNCONDITIONAL MODEL Fixed effect: intercept Random effect: intercept – significant variability between groups? level-1 – significant variability within groups?

6. W HAT 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. 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. A S COMPARISON, FIXED VS RANDOM EFFECTS IN ANOVA

8. 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 A S COMPARISON, FIXED VS RANDOM EFFECTS IN ANOVA

9. 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. A S COMPARISON, FIXED VS RANDOM EFFECTS IN ANOVA

10. M ATHEMATICAL EXPRESSIONS – FIXED EFFECT MODEL μ is grand mean (constant). j is group j effect (constant). Є ij is the residual or error (random). Є ij ~ N(0, σ 2 ) and independent.

11. M ATHEMATICAL EXPRESSIONS - RANDOM EFFECT MODEL μ – grand mean (constant). T j is group j effect (random). T j ~N(0, τ 2 ) and independent. Є ij is the w/in groups residual or error (random). Є ij ~N(0, σ 2 ) and independent. T j and Є ij are independent, ie., cov(Uj,Rij) = 0.

12. Level 1: Level 2: G O BACK TO THE EXAMPLE, UNCONDITIONAL MODEL Fixed effect: intercept Random effect: intercept level-1

13. Level 1: Level 2: G O BACK TO THE EXAMPLE, UNCONDITIONAL MODEL Fixed effect: intercept Random effect: intercept level-1

14. E XAMPLE, MEANS AS OUTCOME REGRESSION MODEL Random effect: intercept level-1 Level 1: Level 2: Fixed effect: intercept slope