1. Multi-level model
Level 1: \[ Y_{ij} = \beta_{0} + \epsilon_{ij} \]
Level 2: \[ \beta_{0} = \beta_{00} + b_{0i} \]
2. Composite Model \[ Y_{ij} = \beta_{00} + b_{0i} + \epsilon_{ij} \]
3. Model fit in R
lmer1 <- lmer(popularity ~ 1 + (1 | class), data = pop)
summary(lmer1)
## Linear mixed model fit by REML ['merModLmerTest']
## Formula: popularity ~ 1 + (1 | class)
## Data: pop
##
## REML criterion at convergence: 6330
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.566 -0.698 0.002 0.676 3.318
##
## Random effects:
## Groups Name Variance Std.Dev.
## class (Intercept) 0.702 0.838
## Residual 1.222 1.105
## Number of obs: 2000, groups: class, 100
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 5.0779 0.0874 98.9000 58.1 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
4. Intraclass Correlation
var.class <- 0.7021
var.resid <- 1.2218
rho = var.class/(var.class + var.resid)
rho
## [1] 0.3649
An estimated 36.5% of the total variation in popularity is attributable to differences between class
5. Multi-level model
Level 1: \[ Y_{ij} = \beta_{0i} + \beta_{1i}(female_{i}) + \beta_{2i}(extra_{i}) + \epsilon_{ij} \]
Level 2: \[ \beta_{0i} = \beta_{00} + b_{0j} \]
\[ \beta_{1i} = \beta_{10} + b_{1j} \]
\[ \beta_{2i} = \beta_{20} + b_{2j} \]
6. Composite Model \[ Y_{ij} = \beta_{00} + \beta_{10}(female_{i}) + \beta_{20}(extra_{i}) + [b_{0j} + b_{1j}(female_{i}) + b_{2j}(extra_{i}) + \epsilon_{ij}] \]
7. Model fit in R
lmer2 <- lmer(popularity ~ 1 + female + extra + (1 + female + extra | class),
data = pop)
summary(lmer2)
## Linear mixed model fit by REML ['merModLmerTest']
## Formula: popularity ~ 1 + female + extra + (1 + female + extra | class)
## Data: pop
##
## REML criterion at convergence: 4870
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0190 -0.6496 -0.0106 0.6710 3.1176
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## class (Intercept) 2.61044 1.6157
## female 0.00537 0.0733 -0.33
## extra 0.02984 0.1727 -0.93 -0.04
## Residual 0.55289 0.7436
## Number of obs: 2000, groups: class, 100
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 2.0879 0.1828 5.8000 6.91 0.00053 ***
## female 1.2448 0.0373 18.8000 28.23 < 2e-16 ***
## extra 0.4430 0.0234 14.8400 15.49 1.4e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) female
## female -0.117
## extra -0.907 -0.065
VarCorr(lmer2)$class
## (Intercept) female extra
## (Intercept) 2.61044 -0.0395683 -0.2596114
## female -0.03957 0.0053666 -0.0004443
## extra -0.25961 -0.0004443 0.0298421
## attr(,"stddev")
## (Intercept) female extra
## 1.61569 0.07326 0.17275
## attr(,"correlation")
## (Intercept) female extra
## (Intercept) 1.0000 -0.33430 -0.93015
## female -0.3343 1.00000 -0.03511
## extra -0.9301 -0.03511 1.00000
8. Random effects: variance explanations:
female coefficient is 0.005.extra (extraversion) coefficient is 0.03.9. Random effects: Corr values
The Corr column represents the correlations between the residuals of each factor included in the model; in this instance, it can be inferred that (1) classes with higher intercepts have lower slopes for the female and extraversion factors, and (2) female-dominate classes have lower extraversion slopes (this variance correlation is not nearly as strong as the intercept-factor variance correlations).
10. Why does the random effect for female not belong?
The variance is so small; it is likely that, on average, the ratio of females to males is roughly equal across classes, and thus wouldn't account for much of the between-class variability in popularity.
11. Interpret the fixed effect estimate in the extra row.
On average, each 1-unit increase in extraversion is associated with a 0.443 unit increase in popularity.
12. Interpret the fixed effect estimate in the female row.
On average, being female is associated with a 1.25 unit increase in popularity.
13. Compute a 95% confidence interval for the fixed effect of extra.
For the fixed effect of extra, the estimated std. error is 0.0234; thus
se = 0.0234
two.se = 2 * 0.0234
extra.est = 0.443
CI.lwr = 0.443 - two.se
CI.upr = 0.443 + two.se
CI = c(CI.lwr, CI.upr)
CI
## [1] 0.3962 0.4898
the 95% CI is (0.3962, 0.4898)
14. Multi-level model:
Level 1: \[ Y_{ij} = \beta_{0i} + \beta_{1i}(female_{i}) + \beta_{2i}(extra_{i}) + \epsilon_{ij} \]
Level 2: \[ \beta_{0i} = \beta_{00} + \beta_{01}(G_{j}) + b_{0j} \]
\[ \beta_{1i} = \beta_{10} + \beta_{11}(G_{j}) + b_{1j} \]
\[ \beta_{2i} = \beta_{20} + \beta_{21}(G_{j}) + b_{2j} \]
15. Composite Model \[ Y_{ij} = \beta_{00} + \beta_{01}(teacherExp_{j}) + \beta_{10}(female_{i}) + \beta_{11}(teacherExp_{j})(female_{i}) + \beta_{20} + \beta_{21}(teacherExp_{j})(extra_{i}) + b_{0j} + b_{1j}(female_{i}) + b_{2j}(extra_{i}) + \epsilon_{ij} \]
16. Fit model in R
lmer3 <- lmer(popularity ~ 1 + female + extra + teacherExp + teacherExp:female +
teacherExp:extra + (1 + female + extra | class), data = pop)
summary(lmer3)
## Linear mixed model fit by REML ['merModLmerTest']
## Formula:
## popularity ~ 1 + female + extra + teacherExp + teacherExp:female +
## teacherExp:extra + (1 + female + extra | class)
## Data: pop
##
## REML criterion at convergence: 4787
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.1209 -0.6485 -0.0196 0.6870 3.0506
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## class (Intercept) 0.49710 0.7051
## female 0.00414 0.0643 -0.44
## extra 0.00558 0.0747 -0.60 -0.45
## Residual 0.55208 0.7430
## Number of obs: 2000, groups: class, 100
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) -1.21436 0.27420 0.02200 -1.29 0.94
## female 1.26564 0.09268 0.02300 4.28 0.91
## extra 0.80289 0.04042 0.35500 17.47 0.25
## teacherExp 0.22689 0.01699 0.02400 4.67 0.91
## female:teacherExp -0.00177 0.00592 0.02500 -0.11 0.98
## extra:teacherExp -0.02470 0.00257 0.49200 -8.60 0.22
##
## Correlation of Fixed Effects:
## (Intr) female extra tchrEx fml:tE
## female -0.102
## extra -0.840 -0.161
## teacherExp -0.916 0.107 0.770
## fml:tchrExp 0.103 -0.918 0.142 -0.136
## extr:tchrEx 0.748 0.142 -0.902 -0.826 -0.142
VarCorr(lmer3)$class
## (Intercept) female extra
## (Intercept) 0.49710 -0.020019 -0.031822
## female -0.02002 0.004139 -0.002156
## extra -0.03182 -0.002156 0.005583
## attr(,"stddev")
## (Intercept) female extra
## 0.70506 0.06434 0.07472
## attr(,"correlation")
## (Intercept) female extra
## (Intercept) 1.0000 -0.4413 -0.6041
## female -0.4413 1.0000 -0.4486
## extra -0.6041 -0.4486 1.0000
17. Which of the five fixed effects estimates should be interpreted? Explain.
Only the female:teacherExp and extra:teacherExp interaction terms should be interpreted. The addition of a Level-2 predictor yields these interactions, and the effects are interpreted directly from the cross-level interactions.
18. Based on deviance, which of the three models fitted thus far seems to fit the best? Explain.
Based on the deviance, LMER 3 seems to fit the best because the deviance criterion is lowest for this model.
19. Based on AIC, which of the three models fitted thus far seems to fit the best? Explain.
AIC(lmer1)
## [1] 6337
AIC(lmer2)
## [1] 4891
AIC(lmer3)
## [1] 4813
Based on the AIC for each model, LMER 3 has the lowest AIC value, and thus seems to fit the best.
20. Based on BIC, which of the three models fitted thus far seems to fit the best? Explain.
BIC(lmer1)
## [1] 6353
BIC(lmer2)
## [1] 4947
BIC(lmer3)
## [1] 4885
Based on the BIC for each model, LMER 3 has the lowest BIC value, and thus seems to fit the best.