Multilevel Linear Models

Field, Chapter 19

Why use multilevel models?

• Meeting the assumptions of the linear model– Homogeneity of regression coefficients– Independence of observations–Missing data

• More power (sort of)

Random intercepts

• Yi=(b0 + u0i) + b1X1i + εi

• What are the coefficients?– b0: overall intercept

– u0: individual deviation

– b1: slope coefficient

• What does this look like?– To the board!

Random intercepts, continued

• What does u0 represent, substantively?– Individual differences in signal rate change

Changes in -2LogLikelihood

• Why do this?– Test if random coefficients “improve” the model

• Generally applicable technique for a range of models

• Caveats– Need to use ML, not REML– New model needs to be nested within old

Writing the population model

• What’s the population model for the data we used last week?

• Fixed effects:– Yi=b0 + b1Coili + b2Accelerationx2 +

b3Accelerationx3 + b4Resolution3+ εi

• Random intercept:Yi=(b0 + u0i) + b1X1i + b2Accelerationx2 +

b3Accelerationx3 + b3Resolution3+ εi

Setting up the dataset

• Currently in wide format – multiple outcome variables per subject

• Need to go to long – multiple cases per subject

• (What’s the difference?)• Before we start, take a snapshot of your data:

you’ll need it later!

In SPSS

• Same dataset as last time• Step 1: Data -> Restructure–Which option do we want?

In SPSS 2

• Decide how many measurements are taken under each condition.–Which option do we want, and when might we

choose differently?

In SPSS 3

• Which variables are: ID, transposed, and fixed?

In SPSS 4

• What are the conditions under which the outcome was measured in each subject?– coil and acceleration– choose two indices

In SPSS 5

• Which variable is the major grouping variable?– Coil, because levels of acceleration are nested

within levels of coil (only in the data set-up!)

In SPSS 6

• Let’s no drop any information• Paste the syntax, then run it• Make sense of the syntax• VARSTOCASES• /MAKE trans1 FROM a0_12 a2_12 a3_12 a0_32 a2_32 a3_32• /INDEX=Index1 "coil"(2) Index2 "acceleration"(3) • /KEEP=subj resolution • /NULL=KEEP.

Starting the analysis

• Analyze -> Mixed Models -> Linear• Select your variables

Identifying variables

• Select dependent variable, factors, and covariates

Fixed variables

• For now, we won't allow random intercepts• Which variables have fixed effects?

Log Likelihood

• Which method should we use for estimation?–Maximum likelihood

Output

• Fit the model• Interpret the estimated parameters: what’s

surprising, and how do we explain it?– Take note of the -2LL

Refitting with random intercepts

• Fit the model again, this time with random intercepts

• Be sure to “include intercept”

Testing random intercepts

• Determine the change in -2LL, and the change in degrees of freedom

• Test to see if this is statistically significant– Change in -2LL near 17.5, change in df of 1 is

highly significant (use Chi-Squared calculator or table)

• If there's time, try splitting the sample on coil and repeating the analysis