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
log(p/(1-p))
## [1] -0.6190392

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?

exp(1.7)
## [1] 5.473947
#It means the log will be increased exp(1.7) times

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

#Poisson regressions require the events to be in a comparable scale and offset can convert different scales into the same. An example could be using offset to convert daily activities to monthly.

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?

#This is because the aggregated form has an extra log factor and thus, the likelihood of data are different between the 2 forms.

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

#This implies that the lambda parameter of the Poisson regression will increase by exp(1.7)=5.473947 times when changing the predictor by 1 unit.

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

#This is because logit link maps a parameter that is defined as a probability mass and lies between 0 and 1, which is consistent with the binomial generalized linear model.

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

#log link returns positive values, and this is appropriate for the Poisson generalized linear model.

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?

# We can imply that the mean of a Poisson generalized linear model will always be positive, and the nature of the outcomes are positive values, such as counting the number of people.

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?

#In Binomial distribution the events are discrete and the expect value is constant. Poisson distributions have an expected value that is constant, but poisson distribution have higher constraints than binomial. 
#The constraints are not all different for Binomial and Poisson distributions because they have equal constraints at maximum entropy.

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 )

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(m11.1,m11.2,m11.3,m11.4)
##           WAIC        SE    dWAIC      dSE     pWAIC       weight
## m11.4 556.7123 18.343104   0.0000       NA 18.888965 1.000000e+00
## m11.2 680.5268  9.225503 123.8144 17.89693  2.013661 1.300275e-27
## m11.3 682.2240  9.299584 125.5117 17.83371  2.939049 5.565238e-28
## m11.1 687.9628  7.305875 131.2504 18.78808  1.007319 3.157387e-29