Germán Rodríguez
Multilevel Models Princeton University

Modeling Intercepts and Slopes

We conclude our analysis of the Snijders and Bosker data by letting the intercept and slopes depend on a school-level predictor. Here are the data one last time.

. use http://data.princeton.edu/pop510/snijders, clear
(Scores in language test from Snijders and Bosker, 1999)

. quietly sum iq_verb

. gen iqvc = iq_verb - r(mean)
> library(foreign)

>   snijders <- read.dta("http://data.princeton.edu/pop510/snijders.dta") %>%
+   mutate(iqvc = iq_verb - mean(iq_verb))

A Level-2 Predictor: School SES

A useful exploratory step is to plot the estimated intercept and slope residuals obtained in the last model against potential predictors at the school level. Here an obvious candidate is school SES. As usual we will center this variable. We also store the school-level standard deviation for later use.

. bysort schoolnr: gen first = _n==1

. sum schoolses if first // just one      

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
   schoolses |        131    18.40458    4.433749         10         29

. gen sesc = schoolses - r(mean)

. scalar sesc_sd = r(sd)
> ses <- (group_by(snijders, schoolnr) %>% slice(1))$schoolses

> school <- data.frame(meanses = mean(ses), sdses = sd(ses))

> snijders <- mutate(snijders, sesc = schoolses - school$meanses)   

We are now ready to fit the model. We specify all the fixed effects first, including a cross-level interaction to let both the intercept and slope depend on school-level SES. These are followed by the random effects, which consists of the constant and the slope for verbal IQ:

. mixed langpost iqvc sesc c.iqvc#c.sesc || schoolnr: iqvc ///
>   , mle covariance(unstructured)

Performing EM optimization: 

Performing gradient-based optimization: 

Iteration 0:   log likelihood = -7608.5763  
Iteration 1:   log likelihood = -7607.8425  
Iteration 2:   log likelihood = -7607.8383  
Iteration 3:   log likelihood = -7607.8383  

Computing standard errors:

Mixed-effects ML regression                     Number of obs     =      2,287
Group variable: schoolnr                        Number of groups  =        131

                                                Obs per group:
                                                              min =          4
                                                              avg =       17.5
                                                              max =         35

                                                Wald chi2(3)      =    1059.58
Log likelihood = -7607.8383                     Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
     langpost |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
         iqvc |   2.515011   .0789096    31.87   0.000     2.360351    2.669671
         sesc |   .2410466   .0658997     3.66   0.000     .1118856    .3702077
              |
c.iqvc#c.sesc |  -.0470625   .0174818    -2.69   0.007    -.0813263   -.0127988
              |
        _cons |   40.71326   .2910762   139.87   0.000     40.14276    41.28376
-------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
schoolnr: Unstructured       |
                   var(iqvc) |   .1489215   .0932377      .0436548    .5080228
                  var(_cons) |    8.22376   1.367463      5.936506    11.39226
             cov(iqvc,_cons) |  -.8862652   .2848344      -1.44453   -.3280001
-----------------------------+------------------------------------------------
               var(Residual) |   41.51683   1.295675      39.05346    44.13558
------------------------------------------------------------------------------
LR test vs. linear model: chi2(3) = 215.75                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.
> library(lme4)

> cli <- lmer(langpost ~ iqvc*sesc + (1 + iqvc | schoolnr), 
+   data = snijders, REML = FALSE)

> cli
Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: langpost ~ iqvc * sesc + (1 + iqvc | schoolnr)
   Data: snijders
      AIC       BIC    logLik  deviance  df.resid 
15231.677 15277.557 -7607.838 15215.677      2279 
Random effects:
 Groups   Name        Std.Dev. Corr 
 schoolnr (Intercept) 2.8677        
          iqvc        0.3859   -0.80
 Residual             6.4434        
Number of obs: 2287, groups:  schoolnr, 131
Fixed Effects:
(Intercept)         iqvc         sesc    iqvc:sesc  
   40.71326      2.51501      0.24105     -0.04706  

We find that the relationship between language scores and verbal IQ varies substantially from school to school, depending on the school's SES and unobserved factors.

For the average school at mean SES, the mean language score for a child with average verbal IQ is 40.71 and increases an average of 2.52 points per point (or 5.20 points per standard deviation) of verbal IQ.

The expected score for an average kid increases 0.24 points per point (or about one point per standard deviation) of school SES, but this effect has a standard deviation across schools of 2.87 points, so there remain substantial unobserved school effects.

The average gain in language scores per point of verbal IQ decreases 0.05 points per point (or about 0.2 points per standard deviation) of school SES. This effect has a standard deviation of 0.39 across schools. Comparatively speaking, SES explains a bigger share of the variation in slopes than in intercepts.

Finally, we note that the random intercept and slope have a negative correlation of -0.80. This means that schools than tend to show higher language scores for average kids also tend to show smaller gains in language scores per point of IQ.

Fitted Lines

Next we will compute the predicted random effects and regression lines and plot the results:

. predict yhat3, fitted

. sort schoolnr iqvc

. line yhat3 iqvc, connect(ascending) ///
>    title(Random Coefficient Model with SES) ///
>    xtitle(Verbal IQ (centered)) ytitle(Language Score)

. graph export fig1lang3.png, width(500) replace
(file fig1lang3.png written in PNG format)

Contextual model

> snijders <- mutate(snijders, yhat3 = predict(cli))

> ggplot(group_by(snijders, schoolnr), aes(iqvc, yhat3, group=schoolnr)) + 
+   geom_line() + ggtitle("Random Coefficient Model with SES")

> ggsave("fig1lang3r.png", width = 500/72, height = 400/72, dpi = 72)

Contextual model The figure looks pretty much the same as in the previous analysis. We see that differences across schools in language scores are larger at low verbal IQs and smaller at high verbal IQs. This is all done at observed values of school SES. We could, of course, generate lines setting centered school SES to zero, to see what the model implies for schoools with average SES.

Plotting Scenarios

A better way to compare the effects of school SES and unobserved school characteristics is to plot the regression lines for 4 schools which represent different scenarios. We set two to have school SES one standard deviation above and one standard deviation below the mean. We then combine these with school effects on the intercept and slope also set one standard deviation above and below the mean.

Because of the high negative correlation between the slope and intercept, it makes more sense to combine a positive intercept residual with a negative slope residual, reflecting a "better" school where scores are higher and less dependent on verbal IQ. The contrast is provided by a negative intercept residual combined with a positive slope residual, reflecting a "worse" school where scores are lower and more dependent on verbal IQ.

Here are the four scenarios, using the standard deviations saved earlier and the parameter estimates from the last run.

. scalar a0 = _b[_cons]

. scalar b0 = _b[iqvc]

. scalar af = _b[sesc] * sesc_sd

. scalar bf =  _b[c.iqvc#c.sesc] * sesc_sd //_b[iqvcXsesc] * sesc_sd

. scalar ar =  exp(_b[lns1_1_2:_cons])

. scalar br = -exp(_b[lns1_1_1:_cons]) // negative sign to pair +ve with -ne

. capture drop arit* // to prevent ambiguity with ar

. local range range(-7.8 6.1)     

. twoway ///
>     function y=(a0+af+ar) + (b0+bf+br)*x, lpat(solid) lc(green) `range' ///
>  || function y=(a0+af-ar) + (b0+bf-br)*x, lpat(solid) lc(red)   `range' ///
>  || function y=(a0-af+ar) + (b0-bf+br)*x, lpat(dash)  lc(green) `range' ///
>  || function y=(a0-af-ar) + (b0-bf-br)*x, lpat(dash)  lc(red)   `range' ///     
>   , title(Predicted Regression Lines Given School SES and RE) ///
>    xtitle(verbal IQ (centered)) ytitle(Language score) ///
>    legend(order(1 "+ses +re" 2 "+ses -re" 3 "-ses +re" 4 "-ses -re") ///
>     ring(0) pos(5))

. graph export fig2lang3.png, width(500) replace
(file fig2lang3.png written in PNG format)

Four scenarios

> b <- fixef(cli)

> sr <- attr(VarCorr(cli)$schoolnr, "stddev")

> af <-  b[3] * school$sdses

> bf <-  b[4] * school$sdses

> x <- seq(-7.8, 6.1, 0.2)

> d <- data.frame(iqvc = x,
+   f1 = (b[1] + af + sr[1]) + (b[2] + bf - sr[2]) * x,
+       f2 = (b[1] + af - sr[1]) + (b[2] + bf + sr[2]) * x,
+           f3 = (b[1] - af + sr[1]) + (b[2] - bf - sr[2]) * x,
+   f4 = (b[1] - af - sr[1]) + (b[2] - bf + sr[2]) * x)

> x = d$iqvc[1] - 0.5

> ggplot(d, aes(iqvc, f1)) + xlim(-10, 6) + ylab("language score") +
+   geom_line() + geom_line(aes(iqvc,f2)) +
+   geom_line(aes(iqvc,f3), linetype=2) + geom_line(aes(iqvc,f4), linetype=2)  +
+   geom_text(aes(x, f1[1]),label="+ses +re", hjust=1) +  
+   ge .... [TRUNCATED] 

> ggsave(   "fig2lang3r.png", width = 500/72, height = 400/72, dpi = 72)

Four scenarios Compare first the top two lines, which are schools with SES one sd above (solid) and below (dashed) the mean, which happen to have favorable unobserved characteristics leading to higher scores and a shallower slope. We see differences in language scores by school SES at low verbal IQs, but these become smaller at higher verbal IQs and vanish at the top end.

Compare now the bottom two lines, which are schools with SES one sd above (solid) and below (dashed) the mean, which happen to have adverse unobserved characteristics leading to lower scores and a steeper slope. We see essentially the same story, with differences by school SES declining with child's verbal IQ.

Compare next the two solid lines, which represent schools with high SES with favorable (green) or adverse (red) conditions. The differences in language scores are larger than the differences by SES, and although they decline with verbal IQ, they never dissappear.

The same is true for the two dashed lines, which correspond to schools with low SES and favorable or adverse unobserved characteristics, which tell the same story but with generally lower language scores.

Graphs like these are essential to sort out the effects of observed and unobserved characteristics. They are a key component of my analysis of infant and child mortality in Kenya in the multilevel handbook.