Multi-Level Models CSP-770 Fall 2016

• Regression by Ordinarly Least Squares (OLS) assumes observations are indendent
• When observations are not independent:
  • E.G. When participants are clustered by some grouping factor
  • Like a study with multiple therapists, clients are grouped into therapist clusters
  • E.G. When participants are measured at more than one time point (repeated measures)
  • The Type-I error rate gets inflated (we call it significant when it is not)
• Random Coefficients Regression, (RC), can handle dependency, (correlation), between observations

• When observations are clustered we may have predictor variables at different levels

  • E.G. - Multiple therapists treating clients in a trial
    • Level 1 factors: client charactersitics (baseline depression, age, gender, etc.)
    • Level 2 factors: therapist characteristics (degree, years of experience, etc.)

• Multi Level Models (also heirachical linear models, mixed-effects models)

  • take into account IVs at different levels and use RC Regression to model the data

• The question is, are members of a cluster more like themselves than they are like the members of other groups?
  • If so, ignoring this factor inflates alpha.
  • The more group members are alike one another, the greater the inflation.
• Clustering is categorized using the Intraclass correlation or ICC
• The ICC measures the proporton of total variance of a variable that is accounted by the group membership
• ICC ranges from 0 to 1

• 40 Groups of Women Enrolled in Weight Loss Program
• Question - does pounds lost depend on motivation?

aovLm <- aov(pounds~GR, data=c14e01Df)

\[ {ICC} = \frac{{MS}_{tx} - {MS}_{er}}{{MS}_{tx} + (\tilde{n}-1){MS}_{er}} \]

\[ {ICC} = \frac{60.03 - 16.02}{60.03 + (9.63*16.02)}=0.22 \]

The typical regression with one predictor

• \[ \hat{Y} = B_0 + B_TX_1 \]

Table 14.2.1.A

Table 14.2.1.B

Table 14.2.1.C

Just using group membership

• R² = 0.297, F(39, 346) = 3.75
  Adding motivation
  
• R² = 0.704, F(40, 345) = 20.48

• When data are clustered, RC:
  • Provides accurate estimates of individual level predictors to a dependent variable
  • Provides accurate estimates of the standard errors
• Supports multilevel modeling (aka hierachical linear modeling)
  • Distinct predictors at different levels of analysis
  • students in classrooms: achievement tests and class size

• \( \underline{Y_{i}} = \beta_{1}^*X_{1}+\beta_{0}^*+\underline{\epsilon_{i}} \)

• Random Variables are selected at random from a probability distribution

• Fixed Variables are assumed to take on fixed values

• In RC regression we still have a population regression equation
• However, we add complexity by adding hierarchical structure (clustering)
• The lowest level of aggregation is referred to as level 1 (micro level)
  • This is what we've been used to, level 1 refers to the individual cases (rows)
• The cluster (group) level is referred to as level 2 (macro level)
• There can be may levels
• Clusters (level 2) assumed to be random variables

• A. Coefficients in micro-level equationfor group j

  • \( \underline{\beta_{0j}} \) = level 1 regression intercept for group j
    
  • \( \underline{\beta_{1j}} \) = level 1 regression coefficient (slope) for group j
    

• B. Fixed population regression coefficients: the fixed part of the model

  • \( \gamma_{00} \) = the population intercept (what we are used to thinking about)
    
  • \( \gamma_{10} \) = the population coefficient for the level 1 IV
    
  • \( \gamma_{01} \) = the population coefficient for the level 2 IV
    
  • \( \gamma_{11} \) = the population coefficient for the interaction between level 1 & 2 IVs
    

• C. Residuals and variance components: the random part of the model

  1. Residuals.
     • \( \underline{r_{ij}} \) = level 1 error for subject i in group j (residuals for level 1 equation)
       
     • \( \underline{u_{0j}} \) = random deviation of the intercept of the individual group j from the population intercept (level 2)
       
     • \( \underline{u_{1j}} \) = random deviation of the regression coefficient of an individual group j from the population coefficient (level 2)
       
  2. Variance components
     
     • \( \sigma^2 \) = variance due to random error at level 1 (variance of residuals)
     • \( \tau_{00} \) = variance of the random intercepts
     • \( \tau_{11} \) = variance of the random regression coefficients
     • \( \tau_{01} \) = covariance between errors of the random regression coefficients and the random regression intercepts

# Calcualte the ICC from the HLM
(lmeICC <- tidyLmerMod$estimate[2]^2/sum(tidyLmerMod$estimate[2:3]^2))

[1] 0.233909

• Longitudinal data is conceptually a matter of nesting time points within individuals
• Hence the interest is in differences in outcome measured at time1, time2, … time(k)