FIXED AND RANDOM EFFECTS IN HLM

Fixed effects produce constant impact on DV.

Random effects produce variable impact on DV.

FIXED VS RANDOM EFFECTS

OLS IS A FIXED EFFECT MODEL

0 1ij j j ij ijY X r

Here, only rij is random.

What if the β’s are random (variable)?

1 10 11 1j j jW

0 1ij j j ij ijY X r L1

L20 00 01 0j j jW

HLM

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

0ij j ijY r

0 00 0j ju

Level 1:

Level 2:

EXAMPLE, UNCONDITIONAL MODEL

Fixed effect: intercept

Random effect: intercept – significant variability between groups?

level-1 – significant variability within groups?

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

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.

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

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.

AS COMPARISON, FIXED VS RANDOM EFFECTS IN ANOVA

MATHEMATICAL EXPRESSIONS – FIXED EFFECT MODEL

μ is grand mean (constant). j is group j effect (constant). Єij is the residual or error (random).

Єij ~ N(0, σ2) and independent.

MATHEMATICAL EXPRESSIONS- RANDOM EFFECT MODEL

μ – grand mean (constant). Tj is group j effect (random).

Tj ~N(0, τ2) and independent.

Єij is the w/in groups residual or error (random).

Єij ~N(0, σ2) and independent.

Tj and Єij are independent, ie., cov(Uj ,Rij) = 0.

0ij j ijY r

0 00 0j ju

Level 1:

Level 2:

GO BACK TO THE EXAMPLE, UNCONDITIONAL MODEL

Fixed effect: intercept

Random effect: intercept

level-1

0ij j ijY r

0 00 0j ju

Level 1:

Level 2:

GO BACK TO THE EXAMPLE, UNCONDITIONAL MODEL

Fixed effect: intercept

Random effect: intercept

level-1

EXAMPLE, MEANS AS OUTCOME REGRESSION MODEL

Random effect: intercept

level-1

0ij j ijY r Level 1:

Level 2: 0 00 01 0j j jW u

Fixed effect: intercept

slope