Chapter 10 Exercises
Students' goals
Solutions

In Example 10.3.4 (Students' goals),

a) extend the model to allow the gender effect b.boy to vary between schools, giving a random coefficient model. Use the Comparison Tool to compare the odds ratio, and the probabilities that a boy (or a girl) prefers sports, between schools
b) use the DIC to compare this model with the original fixed coefficient model.

c) use conflict p-values (see Section 10.7, Example 10.7.4) to assess whether there are any schools whose effects are inconsistent with the random effects distribution.


a) The code added in bold shows how the original model is changed to add random gender effects.


model {
for (i in 1:npupil) {
Goals[i] ~ dcat(p[i,])
Goals.fix[i] ~ dcat(p.fix[i,])
for (k in 1:3) {
p[i,k] <- q[i,k]/sum(q[i,])
log(q[i,k]) <- a[i,k]
p.fix[i,k] <- q.fix[i,k]/sum(q.fix[i,])
log(q.fix[i,k]) <- a.fix[i,k]
}
a[i,1] <- b[School[i], 1] +
b.boy[School[i]] *Gender[i]
a[i,2] <- b[School[i], 2]
a[i,3] <- 0

a.fix[i,1] <- b.fix[School[i], 1] + b.boy.fix[School[i]]*Gender[i]
a.fix[i,2] <- b.fix[School[i], 2]
a.fix[i,3] <- 0
}
for (j in 1:nschool) {
b[j,1] ~ dnorm(mu[1], tau[1])
mub2[j] <- mu[2] + s[2]/s[1]*cor*(b[j,1] - mu[1])
b[j,2] ~ dnorm(mub2[j], taub2)
b[j, 3] <- 0
   for (k in 1:3) {    egirl[j,k] <- exp(b[j,k]) }
   eboy[j,1] <- exp(b[j,1] +
b.boy[j] ); for (k in 2:3) { eboy[j,k] <- exp(b[j,k]) }
   for (k in 1:3) {
      p.girl[j,k] <- egirl[j,k] / sum(egirl[j,])
      p.boy[j,k] <- eboy[j,k] / sum(eboy[j,])
   }
   b.boy[j] ~ dnorm(mu.boy, tau.boy)
   or.boy[j]    <- exp(b.boy[j])

   b.pred[j,1] ~ dnorm(mu[1], tau[1])
b.pred[j,2] ~ dnorm(mub2[j], taub2)
   b.boy.pred[j] ~ dnorm(mu.boy, tau.boy)
   b.fix[j,1] ~ dnorm(0, 0.0001)
   b.fix[j,2] ~ dnorm(0, 0.0001)
   b.boy.fix[j] ~ dnorm(0, 0.0001)
   p.base.fix[j,1] <- step(b.fix[j,1] - b.pred[j,1])
   p.base.fix[j,2] <- step(b.fix[j,2] - b.pred[j,2])
   p.boy.fix[j] <- step(b.boy.fix[j] - b.boy.pred[j])
}
vb2 <- (1 - cor*cor)*v[2]
tau[1] <- 1/v[1]
taub2 <- 1/vb2
for (k in 1:2) {
mu[k] ~ dnorm(0, 0.0001)
v[k] <- s[k]*s[k]
s[k] ~ dunif(0, 100)
}
cor ~ dunif(0, 1)
mu.boy ~ dnorm(0, 0.0001)
sig.boy ~ dunif(0, 10); tau.boy <- 1 / pow(sig.boy, 2)
}

list(mu=c(0,0), s=c(1,1), cor=0.5, mu.boy=0, sig.boy=1)

The gender odds ratio is fairly homogeneous between schools, with posterior means of around 3.

[exercises-ch10-studentsgoals-solutions0]
The biggest odds ratio is for school 6 (Brown Middle) where over 50% of boys preferred sports, compared to about 10% of girls. Though given the extent of the credible intervals around each school-specific odds ratio, this seems consistent with sampling variability.

The probabilities of preferring sports for a boy in each of the nine schools are shown in the plot below, and below again for a girl. Note the schools are in upside-down order compared to Figure 10.5 in the book, which was plotted in R. The probabilities are substantively the same as in the hierarchical model without random coefficients.

[exercises-ch10-studentsgoals-solutions1]

(note: use right click on plot, Properties, Axis Bounds to make x axis the same as for boys)

[exercises-ch10-studentsgoals-solutions2]

b) The DIC of 952 suggests that adding random coefficients doesn't improve the predictive ability of the model, compared to the hierarchical model with random intercepts but not random coefficients (DIC 951)

Dbar = post.mean of -2logL; Dhat = -2LogL at post.mean of stochastic nodes
Dbar   Dhat   pD   DIC   
Goals   937.052   921.775   15.277   952.328   
total   937.052   921.775   15.277   952.328   


c) To test whether there are any schools (e.g. school 6) that are inconsistent with the normal distribution for the random coefficients model, see the
code added in red in the model above. The resulting conflict p-values ( p.base.fix and p.boy.fix ) are all around 0.5 (see the posterior summary statistics below) suggesting all schools are consistent with this model.

   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   cor   0.7268   0.2447   0.001997   0.115   0.8042   0.9935   5000   390002
   mu[1]   -1.535   0.3223   0.003277   -2.183   -1.535   -0.9015   5000   390002
   mu[2]   -0.4928   0.2135   0.00118   -0.9037   -0.4995   -0.04512   5000   390002
   mu.boy   1.04   0.3053   0.004377   0.4312   1.041   1.633   5000   390002
   or.boy[1]   2.774   1.335   0.01407   0.7749   2.613   5.826   5000   390002
   or.boy[2]   3.086   1.301   0.01528   1.342   2.851   6.232   5000   390002
   or.boy[3]   2.757   1.044   0.01338   1.184   2.614   5.188   5000   390002
   or.boy[4]   2.97   1.178   0.01396   1.234   2.792   5.796   5000   390002
   or.boy[5]   3.326   1.641   0.01614   1.33   2.994   7.439   5000   390002
   or.boy[6]   4.301   2.295   0.02671   1.846   3.673   10.43   5000   390002
   or.boy[7]   3.345   1.498   0.01693   1.487   3.047   7.039   5000   390002
   or.boy[8]   2.773   1.069   0.01317   1.113   2.636   5.272   5000   390002
   or.boy[9]   2.718   1.193   0.0137   0.8665   2.578   5.499   5000   390002
   p.base.fix[1,1]   0.5058   0.5   8.116E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[1,2]   0.501   0.5   7.996E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[2,1]   0.506   0.5   7.98E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[2,2]   0.5021   0.5   8.005E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[3,1]   0.5053   0.5   7.898E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[3,2]   0.5017   0.5   8.219E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[4,1]   0.5054   0.5   7.856E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[4,2]   0.5002   0.5   8.043E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[5,1]   0.5056   0.5   8.074E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[5,2]   0.5012   0.5   8.152E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[6,1]   0.5062   0.5   7.96E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[6,2]   0.5026   0.5   8.0E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[7,1]   0.5064   0.5   7.887E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[7,2]   0.5048   0.5   8.318E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[8,1]   0.5072   0.4999   8.015E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[8,2]   0.5006   0.5   7.719E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[9,1]   0.5049   0.5   8.337E-4   0.0   1.0   1.0   5000   390002
   p.base.fix[9,2]   0.5032   0.5   8.087E-4   0.0   1.0   1.0   5000   390002
   p.boy[1,1]   0.3613   0.122   0.001122   0.1393   0.3548   0.6173   5000   390002
   p.boy[1,2]   0.3358   0.08841   7.601E-4   0.1894   0.3252   0.5355   5000   390002
   p.boy[1,3]   0.3028   0.09673   0.001207   0.1273   0.2992   0.4956   5000   390002
   p.boy[2,1]   0.2146   0.05296   3.093E-4   0.1191   0.2118   0.3254   5000   390002
   p.boy[2,2]   0.2491   0.04212   2.991E-4   0.1685   0.2486   0.3334   5000   390002
   p.boy[2,3]   0.5363   0.05763   4.418E-4   0.4251   0.5356   0.6513   5000   390002
   p.boy[3,1]   0.211   0.05007   3.05E-4   0.1199   0.2089   0.3152   5000   390002
   p.boy[3,2]   0.2745   0.03979   2.487E-4   0.2005   0.2731   0.3572   5000   390002
   p.boy[3,3]   0.5144   0.05165   3.41E-4   0.4141   0.5144   0.6152   5000   390002
   p.boy[4,1]   0.3588   0.08028   6.105E-4   0.2109   0.3553   0.5245   5000   390002
   p.boy[4,2]   0.3021   0.05744   4.563E-4   0.2008   0.298   0.4253   5000   390002
   p.boy[4,3]   0.3392   0.06757   7.284E-4   0.2119   0.3374   0.4735   5000   390002
   p.boy[5,1]   0.296   0.08778   4.733E-4   0.1453   0.2882   0.4899   5000   390002
   p.boy[5,2]   0.2516   0.05324   3.177E-4   0.1495   0.2511   0.3608   5000   390002
   p.boy[5,3]   0.4524   0.08033   5.352E-4   0.2943   0.4531   0.6104   5000   390002
   p.boy[6,1]   0.4011   0.08507   6.939E-4   0.2516   0.3952   0.5823   5000   390002
   p.boy[6,2]   0.2383   0.04864   3.718E-4   0.1463   0.2373   0.3376   5000   390002
   p.boy[6,3]   0.3607   0.06537   5.221E-4   0.2337   0.3605   0.4888   5000   390002
   p.boy[7,1]   0.2371   0.06103   3.736E-4   0.1304   0.2325   0.3692   5000   390002
   p.boy[7,2]   0.2094   0.04436   4.293E-4   0.1251   0.2086   0.2968   5000   390002
   p.boy[7,3]   0.5535   0.06438   5.733E-4   0.4286   0.5535   0.6793   5000   390002
   p.boy[8,1]   0.2625   0.06554   4.174E-4   0.1417   0.2599   0.3994   5000   390002
   p.boy[8,2]   0.2874   0.04573   2.989E-4   0.2048   0.2845   0.3854   5000   390002
   p.boy[8,3]   0.4501   0.05962   4.014E-4   0.3333   0.4501   0.567   5000   390002
   p.boy[9,1]   0.1979   0.06946   4.764E-4   0.07354   0.1946   0.3427   5000   390002
   p.boy[9,2]   0.2887   0.05009   3.398E-4   0.1992   0.2852   0.3979   5000   390002
   p.boy[9,3]   0.5134   0.06758   5.224E-4   0.3808   0.5134   0.6458   5000   390002
   p.boy.fix[1]   0.4956   0.5   7.733E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[2]   0.4957   0.5   7.765E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[3]   0.4972   0.5   7.998E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[4]   0.497   0.5   7.936E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[5]   0.4981   0.5   8.041E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[6]   0.4947   0.5   8.092E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[7]   0.496   0.5   7.89E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[8]   0.496   0.5   8.396E-4   0.0   0.0   1.0   5000   390002
   p.boy.fix[9]   0.4952   0.5   8.02E-4   0.0   0.0   1.0   5000   390002
   p.girl[1,1]   0.1884   0.06963   7.699E-4   0.08556   0.1774   0.3528   5000   390002
   p.girl[1,2]   0.4278   0.08772   8.996E-4   0.2833   0.419   0.6196   5000   390002
   p.girl[1,3]   0.3838   0.09995   0.001317   0.1879   0.3859   0.5661   5000   390002
   p.girl[2,1]   0.0894   0.02974   3.616E-4   0.03916   0.08695   0.154   5000   390002
   p.girl[2,2]   0.2889   0.04609   3.431E-4   0.1987   0.289   0.379   5000   390002
   p.girl[2,3]   0.6217   0.05505   5.204E-4   0.5154   0.6209   0.7301   5000   390002
   p.girl[3,1]   0.09568   0.02869   3.402E-4   0.04663   0.09373   0.1576   5000   390002
   p.girl[3,2]   0.3147   0.04216   2.676E-4   0.234   0.3141   0.4005   5000   390002
   p.girl[3,3]   0.5896   0.04935   4.073E-4   0.4922   0.5896   0.6855   5000   390002
   p.girl[4,1]   0.1718   0.05091   5.989E-4   0.09217   0.1654   0.2882   5000   390002
   p.girl[4,2]   0.3904   0.06147   5.735E-4   0.2813   0.3866   0.52   5000   390002
   p.girl[4,3]   0.4379   0.07153   8.679E-4   0.2958   0.4391   0.5722   5000   390002
   p.girl[5,1]   0.1238   0.04173   3.943E-4   0.05731   0.1183   0.2215   5000   390002
   p.girl[5,2]   0.3138   0.05826   3.901E-4   0.1978   0.3139   0.4314   5000   390002
   p.girl[5,3]   0.5625   0.07172   5.727E-4   0.4175   0.5632   0.7029   5000   390002
   p.girl[6,1]   0.1514   0.04518   5.236E-4   0.07934   0.1456   0.2546   5000   390002
   p.girl[6,2]   0.3375   0.0522   3.559E-4   0.2379   0.3362   0.4451   5000   390002
   p.girl[6,3]   0.5111   0.06362   5.959E-4   0.3826   0.5126   0.6327   5000   390002
   p.girl[7,1]   0.09323   0.03012   3.617E-4   0.04308   0.09025   0.1599   5000   390002
   p.girl[7,2]   0.249   0.05003   5.046E-4   0.1523   0.2486   0.3454   5000   390002
   p.girl[7,3]   0.6578   0.05829   6.612E-4   0.544   0.6582   0.7699   5000   390002
   p.girl[8,1]   0.1222   0.03414   3.599E-4   0.06507   0.1186   0.199   5000   390002
   p.girl[8,2]   0.3422   0.04754   3.148E-4   0.254   0.34   0.4425   5000   390002
   p.girl[8,3]   0.5356   0.05555   4.447E-4   0.4232   0.5371   0.6409   5000   390002
   p.girl[9,1]   0.09069   0.03121   3.359E-4   0.03669   0.08865   0.1575   5000   390002
   p.girl[9,2]   0.3275   0.0513   3.653E-4   0.2329   0.325   0.4377   5000   390002
   p.girl[9,3]   0.5818   0.05921   5.093E-4   0.462   0.5824   0.6959   5000   390002
   s[1]   0.6625   0.3286   0.004298   0.1014   0.6215   1.429   5000   390002
   s[2]   0.478   0.2447   0.002787   0.09566   0.4449   1.054   5000   390002
   sig.boy   0.3674   0.2939   0.004177   0.0161   0.3048   1.092   5000   390002