Note: The following is based on notes taken during the talks and were not provided directly by the presenters.

Jennifer Krull - Mediation in Multilevel Models

Approximate standard error of the mediated effect: 

Second order Taylor Series: 

Common prevention design:
- Intervention administered to intact groups
- randomized at the group level
- individual level mediators and outcomes

Subjects = 1226 football players on 31 teams

Solution 1: ignore levels - treat as single data set
if on individual - ignores grouping
- ignores unit of randomization
- produces downwardly biased standard errors
if on group - number of observations significantly decreased
- limited to group means

Solution 2: better to look at multilevels
multilevel models: - take nesting into account
- can simultaneously examine effects of group level and individual level variables on individual level outcome

The single level mediation equations can be reexpressed as multilevel equations, for example:

OLS - Equation 2 - 

Multilevel Model - 

Fixed Parameters: 

Random Parameters:

Can obtain using multilevel models , however,
 

	OLS	Multilevel
Fixed Effects	(XX)-1XY	(XV-1X)-1XV-1Y
Cov	(XX)-1	(XV-1X)-1

The differences between OLS and multilevel estimators involve only a weighting matrix V, the elements of which are functions of the random parameters. Since different V matrices will be used in estimating the 3 mediation equations, theequivalence found in single level analyses will not hold in the multilevel case.

How different are estimates of the mediated effect? Is one better than the other?

How much do multilevel analyses improve the standard error relative to OLS?

How well do they work in small samples?

Simulation study:
- pretty similar, as sample size or group size increases, they become more similar, therefore they are asymptotically equivalent in multilevel analyses.

Should we prefer one over the other? Is there bias wtih either? Both are very small, smaller as group size and sample size increase. is very slightly more efficient than

Standard errors:
OLS vs. HLM - OLS smaller than they need to be, HLM is better
Relative bias - OLS underestimates, larger at larger sample and group sizes
HLM - don't see as much underestimation, second order are overestimating at small sample size, therefore first order Taylor series standard error should be used

ATLAS data - OLS standard error is smaller than HLM standard error - on average HLM is 42% higher