The model’s below, used in Professor Song’s lecture, estimate cross-level interaction of student popularity level by gender and teachers experience using two different methods. The models are indentical models despite having a numerical difference.
library(nlme)
library(dplyr)
library(magrittr)
library(tidyr)
library(haven)
library(lmerTest)
library(ggplot2)
library(texreg)popula <- read_dta("https://stats.idre.ucla.edu/stat/stata/examples/mlm_ma_hox/popular.dta")m4_lme <- lme(popular ~ sex*texp, data = popula, random = ~ sex|school, method = "ML")
summary(m4_lme)Linear mixed-effects model fit by maximum likelihood
Data: popula
AIC BIC logLik
4261.85 4306.657 -2122.925
Random effects:
Formula: ~sex | school
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 0.6347377 (Intr)
sex 0.4692521 0.08
Residual 0.6264320
Fixed effects: popular ~ sex * texp
Value Std.Error DF t-value p-value
(Intercept) 3.313651 0.15954654 1898 20.769180 0e+00
sex 1.329479 0.13183479 1898 10.084432 0e+00
texp 0.110229 0.01013882 98 10.872007 0e+00
sex:texp -0.034025 0.00837995 1898 -4.060303 1e-04
Correlation:
(Intr) sex texp
sex -0.046
texp -0.909 0.042
sex:texp 0.042 -0.908 -0.046
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-2.93387434 -0.64733763 0.02294381 0.53272602 3.49150352
Number of Observations: 2000
Number of Groups: 100 This cross level interaction model evaluates student’s perception of populatiry level(between 0-10) by gender and teachers expereience while finding association of school and sex from the data. By adding the interaction, we were able to now measure on different levels. There is a clear interaction between gender and texp (teachers exprience). On average the popularity index for boys with a teacher of 0 expreience is 3.31. For every 1 year increase in experience of the teacher, boys average perception of popularility increases by .11 Whereas the other counterfactual, girls on average perceieve their popularity around 4.643 when the teacher has 0 years of experience. With a increase in teachers experience, females popularity perception decreases by .034. The random effect in this model suggests that on average the popularity perception for a random school is .6347 higher than the fixed effect. For females it is ,4692 higher.
popula %<>% mutate(ctexp = texp - mean(texp))m4a_lme <- lme(popular ~ sex*ctexp, data = popula, random = ~ sex|school, method = "ML")
summary(m4a_lme)Linear mixed-effects model fit by maximum likelihood
Data: popula
AIC BIC logLik
4261.85 4306.657 -2122.925
Random effects:
Formula: ~sex | school
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 0.6347377 (Intr)
sex 0.4692521 0.08
Residual 0.6264320
Fixed effects: popular ~ sex * ctexp
Value Std.Error DF t-value p-value
(Intercept) 4.885851 0.06660875 1898 73.35149 0e+00
sex 0.844178 0.05510824 1898 15.31855 0e+00
ctexp 0.110229 0.01013882 98 10.87201 0e+00
sex:ctexp -0.034025 0.00837995 1898 -4.06030 1e-04
Correlation:
(Intr) sex ctexp
sex -0.044
ctexp -0.006 0.000
sex:ctexp 0.000 -0.004 -0.046
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-2.93387434 -0.64733763 0.02294381 0.53272602 3.49150352
Number of Observations: 2000
Number of Groups: 100 The second model analyses the same interaction with a different apprach. We see here that by centering the covaiates we created a new center for the variable making the mean the starting point 0. In other words, sometimes you need to center your covariates. The major difference between ctexp and texp is assinging a better poisiton for techer exp. After centering, there is a ease in our interpretation and we reestimate the model. Anything below the mean will fall below 0 and those above will be higher than 0. On average, males perception of popularity is 4.88 when teachers experience is averaged whereas females percepton of populatiry is 5.724 when teachers experience is averaged. As teacher’s experience increases by 1 year, boys popularity increases by .11 while females perception decreases by .03.
htmlreg(list(m4_lme, m4a_lme))| Model 1 | Model 2 | ||
|---|---|---|---|
| (Intercept) | 3.31*** | 4.89*** | |
| (0.16) | (0.07) | ||
| sex | 1.33*** | 0.84*** | |
| (0.13) | (0.06) | ||
| texp | 0.11*** | ||
| (0.01) | |||
| sex:texp | -0.03*** | ||
| (0.01) | |||
| ctexp | 0.11*** | ||
| (0.01) | |||
| sex:ctexp | -0.03*** | ||
| (0.01) | |||
| AIC | 4261.85 | 4261.85 | |
| BIC | 4306.66 | 4306.66 | |
| Log Likelihood | -2122.92 | -2122.92 | |
| Num. obs. | 2000 | 2000 | |
| Num. groups | 100 | 100 | |
| p < 0.001, p < 0.01, p < 0.05 | |||
The two models tell similar stories while examining the interaction of teachers experience with gender to see the influence both variables have on average perception of popularity between males and females. The main difference between both models are the fixed effect calculations. This difference is seen in model 2 because we cnetered the variable teachers experience. We do this to control for outliers and to see how gender falls when the mean of teachers experience is to be a starting point.
Despite some numerical differences in the intercepts and slopes, the story line of the interaction of teachers experience and gender is the same which demostrates that a random school’s gender difference has teh same difference in both models. Both models are a good fit as they both have the same BIC and AIC.