1.1.1 Goals of this section

• Multiple measures of the same thing or related things as an outcome
  • Possibly over time
• Want the variables separate: Not PCA / FA
• In this section:
  • Last time: MANOVA and repeated measures ANOVA
  • Mixed models (this week)

1.1.2 Goals of this lecture

• Mixed models as an approach to repeated measures
  • Focus on individual change
  • Fewer problems with missing data
  • Continuous and unevenly spaced time
  • Flexible with predictors (continuous and categorical)
• Way more complex & interesting than we have time to talk about!
  • Take another class: Longitudinal, Multilevel, Categorical, SEM

2.1.1 Linear mixed model

• Also known as random coefficient models, multilevel models, nested models, hierarchical linear models, random effects models

• Developed in different disciplines

  • Random coefficient models from statistics and biostatistics
  • Multilevel models from education

2.1.2 Linear mixed model

• Model for non-independent observations
  • Cross-sectional
    • Multiple schoolchildren with the same teacher
    • Employees who work in teams or workgroups
  • Longitudinal
    • Multiple observations from the same individual over time
• Observations from same class/team/person are more similar to one another than observations from different classes/teams/persons

2.1.3 How not to do it

https://xkcd.com/2533

2.1.4 Non-independence

• Non-independence means that there is some redundancy (or correlation) between observations
  • Effective sample size is smaller that the actual sample size
    • Collect 100 observations but we only have (for example) 72 obs’ worth of information, due to correlations between obs
• Smaller effective sample size means standard error is underestimated if you ignore non-independence
  • How much the standard errors are underestimated depends on how much the observations are related to one another

2.1.5 Linear mixed model: Motivation

• Linear mixed model (LMM): Extension of general linear model (GLM)
  • Partitions variation, just like ANOVA and regression
  • But more ways to partition and more control over the form

2.1.6 Linear mixed model: Motivation

• How are observations related to one another?
  • Linear regression: They’re not (Independence)
  • Between-subjects ANOVA: They’re not (Independence)
  • Repeated-measures ANOVA: According to “compound symmetry”
  • LMM has two ways to do this
    • Random effects (this class)
    • Correlated residuals (not this class)

2.1.7 Random effects

• Relationship between time and outcome
  • X axis = time
  • Y axis = outcome
• Random effects: Relationship can be different for each person
  • Differences in intercept = random intercepts
  • Differences in change over time = random slopes
  • Can have one or other or both

2.1.8 Data: Executive functioning dataset

2.1.9 Individual trajectories for first 4 people

2.1.10 All individual trajectories: Spaghetti plot

2.1.11 Assumptions

• Linear regression assumes independence of observations
  • Definitely not true here
  • What should we do?
• We can still use the regression lines for each person
  • Non-independence only a problem for estimating standard errors
  • Can use the estimates of individual intercepts and slopes

2.1.12 So I just report 100 intercepts and slopes?

• What do we do with all those regression lines?
  • Fixed effects: Average of intercepts and slopes
  • Random effects: Variance of intercepts and slopes

2.1.13 Mixed model: Output in R

Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
  method [lmerModLmerTest]
Formula: dlpfc ~ 1 + age12 + tx + age12 * tx + (1 + age12 | id)
   Data: ef_uni

     AIC      BIC   logLik deviance df.resid 
  3502.5   3543.9  -1743.3   3486.5     1296 

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.89823 -0.50229 -0.02245  0.51797  2.80891 

Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 id       (Intercept) 0.67110  0.8192        
          age12       0.05248  0.2291   -0.37
 Residual             0.48857  0.6990        
Number of obs: 1304, groups:  id, 342

Fixed effects:
             Estimate Std. Error        df t value Pr(>|t|)    
(Intercept)   0.58230    0.07792 341.09983   7.473 6.63e-13 ***
age12         0.11158    0.03059 335.26580   3.648 0.000307 ***
tx           -0.06801    0.10933 342.28563  -0.622 0.534346    
age12:tx      0.01185    0.04298 338.26053   0.276 0.782972    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
         (Intr) age12  tx    
age12    -0.550              
tx       -0.713  0.392       
age12:tx  0.391 -0.712 -0.552

2.1.14 Mixed model: Output in SPSS

2.1.15 Fixed effects: Means or averages

• R: \(t\)-tests with no df or \(p\)-values with just lme4 package
  • df and \(p\)-values with lmerTest package
• SPSS: \(t\)-tests

2.1.16 Fixed effects: Interpretation

• \(Y = 0.582 + 0.112 (age12) + -0.068 (tx) + 0.012 (age12*tx)\)

  • tx = 0 (in \(\color{blue}{blue}\)): \(Y = 0.582 + 0.112 (age12)\)
    • Intercept: Expected dlpfc when age12 = 0, for group tx = 0
    • Slope: Change in dlpfc for 1 unit change in age12, for group tx = 0
  • tx = 1 (in \(\color{red}{red}\)): \(Y = 0.514 + 0.123 (age12)\)
    • Intercept: Expected dlpfc when age12 = 0, for group tx = 1
    • Slope: Change in dlpfc for 1 unit change in age12, for group tx = 1

2.1.17 Fixed effects: Figure

2.1.18 Random effects: Variances

• R: No test statistics
• SPSS: \(z\)-tests (\(p\)-value/2 for variances)

2.1.19 Random effects: Interpretation

• Intercept variance: Variance of individual intercepts
• Slope variance: Variance of individual slopes
• Correlation between intercept and slope: Correlation between individual intercepts and slopes
  • How is a person’s intercept related to their slope?
• Residual: Error
  • How well we do at predicting the individual trajectory

2.1.20 Prediction interval: Fixed + random

• Average effects with individual variation
  • What do typical individual effects look like?
  • Assume normally distributed variance: \(estimate \pm 1.96\times SD\)
• Prediction intervals
  • Interval for likely values of individual intercepts and slopes
• Not confidence intervals
  • Sampling distribution of test statistic
  • You get those for fixed effects

2.1.21 Prediction interval: Fixed + random

• Average intercept (for tx = 0) = \(0.582\)
  • \(1.96 \times SD = 1.96 \times 0.819 = 1.606\)
  • 95% of individual intercepts are in [\(-1.023\), \(2.188\)]
• Average slope (for tx = 0) = \(0.112\)
  • \(1.96 \times SD = 1.96 \times 0.229 = 0.449\)
  • 95% of individual slopes are in [\(-0.337\), \(0.561\)]

2.1.22 Means + individual variability

3.1.1 Linear mixed model: Equations

\(Y = \underbrace{\textbf{X} \beta}_\text{fixed effects} + \underbrace{\textbf{Z} \gamma}_\text{random effects} + \underbrace{\epsilon}_\text{residual}\)

3.2.1 Adding predictors to the model

• The example model has two predictors
  • Treatment (tx): Categorical (dummy code)
    • Time-invariant predictor: Same value at all times
  • Age (age12): Continuous
    • Time-varying predictor: Different value at each time
• Two types of predictors are entered into model in different ways
  • Peugh, J. L. (2010). A practical guide to multilevel modeling. Journal of school psychology, 48(1), 85-112.

3.2.2 Data: Executive functioning dataset

3.2.3 LMM = Multilevel model

• LMM is also a multilevel model where
  • Level 1: Occasions
  • Level 2: Person
  • Multiple occasions are nested within each person (L1 w/in L2)
• LMM can be re-written in terms of L1 and L2
  • Time-varying predictors go in L1 (occasions) part
  • Time-invariant predictors go in L2 (person) part

3.2.4 Multilevel models: Adding predictors

• Level 1: Trajectories
  • \(Y_{ij} = \pi_{0i} + \pi_{1i} (age12_{ij}) + e_{ij}\)

3.3.1 Why center predictors?

3.3.2 Why does this matter so much for mixed models?

• Level 1 (occasion) observations have two kinds of information
  • Occasion (L1)
  • Person (L2)
• If you ask me one day if I’m depressed, that gives you information about
  • How depressed I am that day (occasion, L1)
  • How depressed I generally am (person, L2)

3.3.3 Centering is more complicated

• Grand mean centering (GMC)
  • Center all observations at the grand mean of all observations
  • Doesn’t change the relationships among variables
• Centering within cluster (CWC)
  • Center each person’s observations at the mean of that person
  • Does change the relationships among variables

3.3.4 Figure: Uncentered

3.3.5 Figure: GMC

3.3.6 Figure: CWC

3.3.7 GMC vs CWC

• Centering changes the context for the different clusters (L2: People)
  • GMC maintains mean differences between people on L1 predictor
    • What is a person like compared to other people?
  • CWC leliminates differences between people on L1 predictor
    • What are people like compared to their own mean?
• Different contexts means different interpretations for both level 1 and level 2 predictors

3.3.9 Centering time

• “Time” is a special predictor
  • Center time variable so that 0 is at a meaningful point
  • Intercept: Expected value of outcome when \(X\) = 0
    • Age = 0?
    • Baseline?
    • Centered age at specific time

3.4.1 Is it linear?

http://www.xkcd/605

3.4.2 If not linear, then what?

https://xkcd.com/2048/

3.4.3 Shape of change: L1 equation

• Linear change
  • \(Y_{ij} = \pi_{0i} + \pi_{1i} (age12_{ij}) + e_{ij}\)
• Quadratic change
  • \(Y_{ij} = \pi_{0i} + \pi_{1i} (age12_{ij}) + \pi_{2i} (age12_{ij})^2 + e_{ij}\)
• Logarithmic change
  • \(Y_{ij} = \pi_{0i} + \pi_{1i} (ln(age12_{ij})) + e_{ij}\)

3.4.4 Non-linear change

• Many phases of development or change are non-linear
  • Increase followed by plateau / maintanence
  • Decrease to a set point
  • Sometimes reflect floor or ceiling effects
• Non-linear change in inherently more complex
  • Straight lines are easy

3.5.1 Intraclass correlation (ICC)

• Quantifies non-independence in repeated outcome
• Use “random effects ANOVA” or “unconditional mixed model”
  • Like “no predictors” model from logistic regression
• Ratio of L1 and L2 variability:
  • \(ICC = \frac{\sigma_{r_{0i}}^2}{\sigma_{r_{0i}}^2 + \sigma_{e}^2}\)
  • Proportion of variance due to differences between people

3.5.2 Variance explained and variance reduction

• When you compare your model to the unconditional model
  • Variance explained
  • How much variance does my model explain?
  • Like \(R^2\)
• When you compare your model to some other (simpler) model
  • Variance reduction
  • How much is (error) variance reduced by adding whatever you added?
  • Like \(R_{change}^2\)

3.5.3 Variance explained and variance reduction

• Model 1 is simpler, Model 2 is more complex

  • The model you “care about” is Model 2

• Reduction in variance =

\[\frac{\sigma_{e}^2(Model1) - \sigma_{e}^2(Model2)}{\sigma_{e}^2(Model1)}\]

3.5.4 Missing data

• Missing on outcome: OK (assuming MAR)
  • Uses all observations for a person to create trajectories
• Missing on predictor: Case is dropped
  • Make sure no missing or use multiple imputation

3.5.5 Extensions of mixed models

• Change in multiple variables at once
  • Baldwin et al. (2014): Complicated but possible
• Nonnormal outcomes
  • More difficult in unexpected ways when outcomes are non-normal
• SEM framework (Latent growth models)
  • Growth as a predictor, simultaneous growth of multiple processes, and other more complex models

4.1.1 Summary of this week

• Mixed models as an approach to repeated measures
  • Focus on individual change
  • Fewer problems with missing data
  • Continuous and unevenly spaced time
  • Flexible with predictors (continuous and categorical)
• Way more complex & interesting than we have time to talk about!
  • Adding predictors, shape of change, multiple outcomes
  • Take Longitudinal or Multilevel models or Categorical or SEM

4.1.2 Summary: RM ANOVA vs mixed models

• RM ANOVA focuses on group level differences in means at each time point
  • Only uses complete cases on the outcome
  • Categorical predictors only
• LMM focuses on individual trajectories over time
  • Uses all observations available on the outcome
  • Continuous or categorical predictors
• In general, I would always use some version of a mixed model