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. Problems are labeled Easy (E), Medium (M), and Hard(H).

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

11E1. If an event has probability 0.35, what are the log-odds of this event?

p <- 0.35
p/(1-p)
## [1] 0.5384615

11E2. If an event has log-odds 3.2, what is the probability of this event?

lo <- 3.2
lo/(1+lo)
## [1] 0.7619048

11E3. Suppose that a coefficient in a logistic regression has value 1.7. What does this imply about the proportional change in odds of the outcome?

# The log odds will increase as exp(1.7) times
exp(1.7)
## [1] 5.473947

11E4. Why do Poisson regressions sometimes require the use of an offset? Provide an example.

# The offset can bring all observations on the same scale. 
# Example: if the number of events is either daily or weekly basis, the offset parameter can convert all measurements to daily basis.

11M1. 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?

#Because likelihood in these two formats are different. When converting likelihood in the aggregated form to the non-aggregated format, the c(n,m) multiplier is converted to an constant at the log-scale.

11M2. If a coefficient in a Poisson regression has value 1.7, what does this imply about the change in the outcome?

# If the predictor change by 1 unit, the outcome will change by exp(1.7) = 5.4739 times.

11M3. Explain why the logit link is appropriate for a binomial generalized linear model.

# Because logit link maps a parameter that is defined as a probability mass and therefore lies between 0 and 1. This is very useful when working with binormial GLMs.

11M4. Explain why the log link is appropriate for a Poisson generalized linear model.

#Because using a log link can constrain the parameter to be positive.

11M5. What would it imply to use a logit link for the mean of a Poisson generalized linear model? Can you think of a real research problem for which this would make sense?

# Using logit link implies that a lambda parameter of the Poisson likelihood always falls in [0, +inf) range.

11M6. State the constraints for which the binomial and Poisson distributions have maximum entropy. Are the constraints different at all for binomial and Poisson? Why or why not?

# Constrains for binomial distribution is that the events are discrete and the expected value is constant. Poisson distributions adds more constraints than binomial because it's a special case of binomial distribution, and it has it's own constrains. For Poisson, its variance is equal to the expected value and both are constant.

11M7. Use quap to construct a quadratic approximate posterior distribution for the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330). Compare the quadratic approximation to the posterior distribution produced instead from MCMC. 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. Do the differences increase or decrease? Why?

data("chimpanzees")
d <- chimpanzees
d$recipient <- NULL

# map
q2 <- map(alist(
  pulled_left ~ dbinom( 1 , p ) ,
  logit(p) <- a[actor] + (bp + bpC*condition)*prosoc_left ,
  a[actor] ~ dnorm(0,10),
  bp ~ dnorm(0,10),
  bpC ~ dnorm(0,10)
) ,
data=d)
pairs(q2)

11M8. Revisit the data(Kline) islands example. This time drop Hawaii from the sample and refit the models. What changes do you observe?

data(Kline)
d <- Kline
d$P <- scale( log(d$population) )
d$contact_id <- ifelse( d$contact=="high" , 2 , 1 )

11H1. 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.

data("chimpanzees")

d <- chimpanzees

m11.1 <- map(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a ,
    a ~ dnorm(0,10)
  ),
  data=d )

## 10.4
m11.2 <- map(
  alist(
    pulled_left ~ dbinom(1, p) ,
    logit(p) <- a + bp*prosoc_left ,
    a ~ dnorm(0,10) ,
    bp ~ dnorm(0,10)
  ),
  data=d )

m11.3 <- map(
  alist(
    pulled_left ~ dbinom(1, p) ,
    logit(p) <- a + (bp + bpC*condition)*prosoc_left ,
    a ~ dnorm(0,10) ,
    bp ~ dnorm(0,10) ,
    bpC ~ dnorm(0,10)
  ), data=d )

m11.4 <- map(
  alist(
    pulled_left ~ dbinom(1, p),
    logit(p) <- a[actor] + (bp + bpC*condition)*prosoc_left,
    a[actor] ~ dnorm(0, 10),
    bp ~ dnorm(0, 10),
    bpC ~ dnorm(0, 10)
  ),
  data = d)


# compare
compare(m11.1,m11.2,m11.3,m11.4)
##           WAIC        SE    dWAIC      dSE     pWAIC       weight
## m11.4 553.8187 18.449249   0.0000       NA 17.091460 1.000000e+00
## m11.2 680.5770  9.261371 126.7583 17.93842  2.039741 2.983940e-28
## m11.3 682.1368  9.340388 128.3180 17.88239  2.895896 1.368013e-28
## m11.1 687.9866  7.191298 134.1678 18.77941  1.022712 7.342134e-30