SAS PROC MCMC example in R: Logistic Regression Random-Effects Model

January 18, 2015
By

(This article was first published on Wiekvoet, and kindly contributed to R-bloggers)

In this post I will run SAS example Logistic Regression Random-Effects Model in four R based solutions; Jags, STAN, MCMCpack and LaplacesDemon. To quote the SAS manual: ‘The data are taken from Crowder (1978). The Seeds data set is a 2 x 2 factorial layout, with two types of seeds, O. aegyptiaca 75 and O. aegyptiaca 73, and two root extracts, bean and cucumber. You observe r, which is the number of germinated seeds, and n, which is the total number of seeds. The independent variables are seed and extract.’
The point of this exercise is to demonstrate a random effect. Each observation has a random effect associated with it. In contrast the other parameters have non-informative priors. As such, the models are not complex. 
Previous post in the series PROC MCMC examples programmed in R were: example 1: sampling, example 2: Box Cox transformation, example 5: Poisson regression and example 6: Non-Linear Poisson Regression.

JAGS

To make things easy on myself I wondered if this data was already present as R data. That is when I discovered this post at Grimwisdom doing exactly what I wanted as JAGS program. Hence that code was the basis for this JAGS code.
One thing on the models and distributions. JAGS uses tau (1/variance) as hyperparameter for the delta parameters and tau has gamma distribution. In contrast, all other programs use standard deviation as hyperparameter for delta parameter and hence gets the inverse gamma distribution. This distribution is available in the MCMCpack package and also in STAN. Gelman has a publication on this kind of priors: Prior distributions for variance parameters in hierarchical models.
library(R2jags)
data(orob2,package=’aod’)
seedData <- list ( N  = 21,
    r  = orob2$y,
    n  = orob2$n,
    x1 = c(1,0)[orob2$seed],
    x2 = c(0,1)[orob2$root]
)

modelRandomEffect <- function() {
  for(i in 1:4) {alpha[i] ~ dnorm(0.0,1.0E-6)}
  for(i in 1:N) {delta[i] ~ dnorm(0.0,tau)}
 
  for (i in 1:N) {
    logit(p[i]) <- alpha[1] +
        alpha[2]*x1[i] +
        alpha[3]*x2[i] +
        alpha[4]*x1[i]*x2[i] +
        delta[i];
    r[i] ~ dbin(p[i],n[i]);
  }
  tau ~ dgamma(.01,0.01)
  s2 <- 1/tau
}
params <- c(‘alpha’,’s2′)
myj <-jags(model=modelRandomEffect,
    data = seedData,
    parameters=params)
myj
Inference for Bugs model at “/tmp/RtmpKURJBm/model70173774d0c.txt”, fit using jags,
 3 chains, each with 2000 iterations (first 1000 discarded)
 n.sims = 3000 iterations saved
         mu.vect sd.vect   2.5%    25%     50%     75%   97.5%  Rhat n.eff
alpha[1]  -0.538   0.191 -0.907 -0.664  -0.542  -0.419  -0.135 1.005   440
alpha[2]   0.078   0.300 -0.550 -0.111   0.083   0.269   0.646 1.005   440
alpha[3]   1.342   0.284  0.768  1.164   1.340   1.524   1.895 1.004  1400
alpha[4]  -0.807   0.441 -1.663 -1.098  -0.817  -0.521   0.046 1.009   300
s2         0.105   0.096  0.010  0.041   0.077   0.138   0.366 1.051    47
deviance 101.236   6.653 89.853 96.595 100.903 105.313 114.371 1.022    95

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 21.7 and DIC = 122.9
DIC is an estimate of expected predictive error (lower deviance is better).

MCMCpack

MCMCpack had some difficulty with this particular prior for s2. In the end I chose inverse Gamma(0.1,0.1) that worked. Therefor parameters estimates turn out slightly different. For conciseness only the first parameters are displayed in the output.
library(MCMCpack)
xmat <- cbind(rep(1,21),seedData$x1,seedData$x2,seedData$x1*seedData$x2)
MCMCfun <- function(parm) {
    beta=parm[1:4]
    s2=parm[5]
    delta=parm[5+(1:21)]
    if(s2<0 ) return(-Inf)
    step1 <-  xmat %*% beta + delta
    p <- LaplacesDemon::invlogit(step1)
    LL <- sum(dbinom(seedData$r,seedData$n,p,log=TRUE))
    prior <- sum(dnorm(beta,0,1e3,log=TRUE))+
            sum(dnorm(delta,0,sqrt(s2),log=TRUE))+
            log(dinvgamma(s2,.1,.1))
    LP=LL+prior
    return(LP)
}
inits <- c(rnorm(4,0,1),runif(1,0,1),rnorm(21,0,1))
names(inits) <- c(paste(‘beta’,0:3,sep=”),
        ‘s2’,
        paste(‘delta’,1:21,sep=”))
mcmcout <- MCMCmetrop1R(MCMCfun,
        inits)
summary(mcmcout)

Iterations = 501:20500
Thinning interval = 1
Number of chains = 1
Sample size per chain = 20000

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

          Mean      SD  Naive SE Time-series SE
 [1,] -0.59046 0.30456 0.0021535        0.03613
 [2,]  0.01476 0.47526 0.0033606        0.04866
 [3,]  1.48129 0.38227 0.0027031        0.03883
 [4,] -0.93754 0.66414 0.0046962        0.07025
 [5,]  0.35146 0.08004 0.0005659        0.05020

STAN

Stan had no problems at all with this model.

library(rstan)
smodel <- ‘
data {
    int N;
    int n[N];
  int r[N];
    real x1[N];
  real x2[N];
}
parameters {
    real Beta[4];
    real s2;
  real Delta[N];
}
transformed parameters {
    vector[N] mu;
    for (i in 1:N) {
      mu[i] <- inv_logit(
        Beta[1] + Beta[2]*x1[i] +
        Beta[3]*x2[i]+Beta[4]*x1[i]*x2[i]+
        Delta[i]);
    }
}
model {
        r ~ binomial(n,mu);
        s2 ~ inv_gamma(.01,.01);
        Delta ~ normal(0,sqrt(s2));
    Beta ~ normal(0,1000);
}

fstan <- stan(model_code = smodel,
        data = seedData,
        pars=c(‘Beta’,’s2′))
fstan

Inference for Stan model: smodel.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.

           mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff Rhat
Beta[1]   -0.56    0.01 0.20   -0.97   -0.68   -0.55   -0.43   -0.14  1370 1.00
Beta[2]    0.08    0.01 0.33   -0.58   -0.12    0.08    0.29    0.72  1385 1.00
Beta[3]    1.36    0.01 0.29    0.83    1.18    1.36    1.54    1.97  1355 1.00
Beta[4]   -0.83    0.01 0.46   -1.76   -1.12   -0.82   -0.53    0.04  1434 1.00
s2         0.12    0.00 0.11    0.02    0.06    0.10    0.16    0.42   588 1.01
lp__    -523.70    0.34 6.86 -537.26 -528.19 -523.73 -519.21 -509.87   403 1.01

Samples were drawn using NUTS(diag_e) at Sat Jan 17 18:27:38 2015.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).

LaplacesDemon

While in the MCMCpack code I borrowed from LaplacesDemon the invlogit() function, in LaplacesDemon I borrow the InvGamma distribution. I guess it evens out. For a change no further tweaking in the code. Note that the core likelihood function is the same as MCMCpack. However, LaplacesDemon is able to use the correct prior. For brevity again part of the output has not been copied in the blog.
library(‘LaplacesDemon’)
mon.names <- “LP”
parm.names <- c(paste(‘beta’,0:3,sep=”),
        ‘s2’,
        paste(‘delta’,1:21,sep=”))
PGF <- function(Data) {
    x <-c(rnorm(21+4+1,0,1))
    x[5] <- runif(1,0,2)
    x
}
MyData <- list(mon.names=mon.names,
        parm.names=parm.names,
        PGF=PGF,
        xmat = xmat,
        r=seedData$r,
    n=seedData$n)
N<-1
Model <- function(parm, Data)
{
    beta=parm[1:4]
    s2=parm[5]
    delta=parm[5+(1:21)]
#    if(s2<0 ) return(-Inf)
     step1 <-  xmat %*% beta + delta
    p <- invlogit(step1)
    LL <- sum(dbinom(seedData$r,seedData$n,p,log=TRUE))
    tau <- 1/s2
    prior <- sum(dnorm(beta,0,1e3,log=TRUE))+
            sum(dnorm(delta,0,sqrt(s2),log=TRUE))+
            log(dinvgamma(s2,.01,.01))
    LP=LL+prior
    Modelout <- list(LP=LP, Dev=-2*LL, Monitor=LP,
            yhat=p,
            parm=parm)
    return(Modelout)
}
Initial.Values <- GIV(Model, MyData, PGF=TRUE)
Fit1 <- LaplacesDemon(Model,
        Data=MyData,
        Initial.Values = Initial.Values
)
Fit1

Call:
LaplacesDemon(Model = Model, Data = MyData, Initial.Values = Initial.Values)

Acceptance Rate: 0.67594
Algorithm: Metropolis-within-Gibbs
Covariance Matrix: (NOT SHOWN HERE; diagonal shown instead)
   beta0    beta1    beta2    beta3       s2   delta1   delta2   delta3
0.218082 0.218082 0.218082 0.218082 0.218082 0.218082 0.218082 0.218082
  delta4   delta5   delta6   delta7   delta8   delta9  delta10  delta11
0.218082 0.218082 0.218082 0.218082 0.218082 0.218082 0.218082 0.218082
 delta12  delta13  delta14  delta15  delta16  delta17  delta18  delta19
0.218082 0.218082 0.218082 0.218082 0.218082 0.218082 0.218082 0.218082
 delta20  delta21
0.218082 0.218082

Covariance (Diagonal) History: (NOT SHOWN HERE)
Deviance Information Criterion (DIC):
         All Stationary
Dbar 100.063    100.063
pD    26.630     26.630
DIC  126.692    126.692
Initial Values:
 [1]  1.385256609 -0.634946833  1.456635236 -0.041162276  1.883504417
 [6] -1.380783003 -0.688367493  0.210060822  0.127231904  0.710367572
[11] -0.865780359 -1.649760777 -0.005532662 -0.114739142  0.642440639
[16] -0.919494616 -0.829018195 -0.938486769  0.302152995 -1.877933490
[21]  1.170542660  0.131282852  0.210852443  0.808779058 -2.115209547
[26]  0.431205368

Iterations: 10000
Log(Marginal Likelihood): -27.92817
Minutes of run-time: 0.81
Model: (NOT SHOWN HERE)
Monitor: (NOT SHOWN HERE)
Parameters (Number of): 26
Posterior1: (NOT SHOWN HERE)
Posterior2: (NOT SHOWN HERE)
Recommended Burn-In of Thinned Samples: 0
Recommended Burn-In of Un-thinned Samples: 0
Recommended Thinning: 240
Specs: (NOT SHOWN HERE)
Status is displayed every 100 iterations
Summary1: (SHOWN BELOW)
Summary2: (SHOWN BELOW)
Thinned Samples: 1000
Thinning: 10

Summary of All Samples
                  Mean        SD       MCSE       ESS            LB
beta0     -0.544001376 0.2305040 0.03535316  93.03421   -0.95266664
beta1      0.060270601 0.3458235 0.04387534 106.92244   -0.67629607
beta2      1.360903746 0.3086684 0.04838620  60.40225    0.76275725
beta3     -0.828532715 0.4864563 0.06323965  99.93646   -1.74744708
s2         0.134846805 0.1055559 0.01599649  68.64912    0.02549966
(…)
Summary of Stationary Samples
                  Mean        SD       MCSE       ESS            LB
beta0     -0.544001376 0.2305040 0.03535316  93.03421   -0.95266664
beta1      0.060270601 0.3458235 0.04387534 106.92244   -0.67629607
beta2      1.360903746 0.3086684 0.04838620  60.40225    0.76275725
beta3     -0.828532715 0.4864563 0.06323965  99.93646   -1.74744708
s2         0.134846805 0.1055559 0.01599649  68.64912    0.02549966

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