Chapter 11 - God Spiked the Integers

This chapter described some of the most common generalized linear models, those used to model counts. It is important to never convert counts to proportions before analysis, because doing so destroys information about sample size. A fundamental difficulty with these models is that parameters are on a different scale, typically log-odds (for binomial) or log-rate (for Poisson), than the outcome variable they describe. Therefore computing implied predictions is even more important than before.

Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them.

Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.

Questions

11-1. As explained in the chapter, binomial data can be organized in aggregated and disaggregated forms, without any impact on inference. But the likelihood of the data does change when the data are converted between the two formats. Can you explain why?

#In aggregated form the likelihood has a factor that contains data but no parameters. But in disaggregated term there is no such factor. So the two likelihoods would be different. And since the factor is not a function of parameter, the inferences would be the same.

11-2. Use quap to construct a quadratic approximate joint posterior distribution for the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330). Plot and compare the quadratic approximation to the joint posterior distribution produced instead from MCMC. Do not use the ‘pairs’ plot. Can you explain both the differences and the similarities between the approximate and the MCMC distributions? Relax the prior on the actor intercepts to Normal(0,10). Re-estimate the posterior using both ulam and quap. Plot and compare the posterior distributions. Do not use the ‘pairs’ plot. Do the differences increase or decrease? Why?

data("chimpanzees")
df=chimpanzees
df$treatment <- 1 + df$prosoc_left + 2 * df$condition
data <- list(
  pulled_left = df$pulled_left,
  actor = df$actor,
  treatment = as.integer(df$treatment)
)
#MCMC
m1<- ulam(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a[actor] + b[treatment],
    a[actor] ~ dnorm(0, 1.5),
    b[treatment] ~ dnorm(0, 0.5)
  ),
  data = data, chains = 4, log_lik = TRUE
)
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#quadratic approximate
m2 <- quap(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a[actor] + b[treatment],
    a[actor] ~ dnorm(0, 1.5),
    b[treatment] ~ dnorm(0, 0.5)
  ),
  data = data
)

post <- extract.samples(m1)
postq <- extract.samples(m2)
dens(post$a[, 2], lwd = 2)
dens(postq$a[, 2], add = TRUE, lwd = 2, col = 'red')

#The MCMC posterior in black is more right skewed 



#Relax the prior on the actor intercepts to Normal(0,10)
#MCMC
m3<- ulam(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a[actor] + b[treatment],
    a[actor] ~ dnorm(0, 10),
    b[treatment] ~ dnorm(0, 0.5)
  ),
  data = data, chains = 4, log_lik = TRUE
)
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#quadratic approximate
m4 <- quap(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a[actor] + b[treatment],
    a[actor] ~ dnorm(0, 10),
    b[treatment] ~ dnorm(0, 0.5)
  ),
  data = data
)

post <- extract.samples(m3)
postq <- extract.samples(m4)
dens(post$a[, 2], lwd = 2)
dens(postq$a[, 2], add = TRUE, lwd = 2, col = 'red')

#After change the prior on actor to Normal(0,10), the MCMC posterior becomes more right skewed

11-3. Revisit the data(Kline) islands example. This time drop Hawaii from the sample and refit the models. Plot the joint posterior. What changes do you observe?

data("Kline")
df2 <- Kline
df2$contact <- ifelse( df2$contact=="high" , 2 , 1)
df3<- subset(df2, df2$culture != "Hawaii")

m <- alist(
  total_tools ~ dpois(p),
  log(p) <- a + b * contact,
  a ~ dnorm(0, 1),
  b ~ dnorm(0, 1)
)
m_old<- ulam(m, data = df2, chains = 4)
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post <- extract.samples(m_old)
post2 <- extract.samples(m_new)
dens(post$a, lwd = 2)
dens(post2$a, add = TRUE, lwd = 2, col = 'red')

9-4. Use WAIC or PSIS to compare the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330), to the simpler models fit in the same section. Interpret the results.

m5 <- quap(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a + b[treatment],
    a ~ dnorm(0, 1.5),
    b[treatment] ~ dnorm(0, 0.5)
  ),
  data = df
)

compare(m1, m5)
##        WAIC        SE    dWAIC      dSE    pWAIC       weight
## m1 531.9954 18.899477   0.0000       NA 8.365104 1.000000e+00
## m5 682.5343  9.111197 150.5388 18.36138 3.661923 2.046003e-33
#m1 that includes a unique intercept for each actor performs better with a smaller WAIC.

11-5. Explain why the logit link is appropriate for a binomial generalized linear model?

#The logit is the log-odds for the probability of success. The logit link converts the linear combination of independent variables to the probability of success, which is between 0 and 1.