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. 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?

exp(1.7)
## [1] 5.473947

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

# Offset can help bring all observations on the same scale. 
# An example could be if the number of events is measured on the daily or weekly basis, the offset parameter can be used to convert all measurements to the 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?

#The likelihood are different in the aggregated and disaggregated forms. The c(n,m) multiplier will be converted to to a constant at the log scale when converting from aggregated format to disaggregated format.

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

# This indicates 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.

# Because the logit link maps a parameter that is defined as a probability mass, its value lies between 0 & 1. A binomial generalized model always generates binary outcome variables i.e. either 0 or 1. Thus making logit link most appropriate for the model.

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

# log link returns positive values, which 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?

#the likelihood will always be non-negative, and the outcome are counts and always positive values.

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?

# Binomial distribution constraints are that the events are discrete and the expected value is constant. Poisson distribution has more constraints than binomial because it is a special case of binomial distribution and it has its own contraints. For poisson, the variance is equal to 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). Plot and 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. Plot and compare the posterior distributions. 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
## m11.4 555.9121 18.510508   0.0000       NA 18.747188 1.000000e+00
## m11.2 680.4768  9.299058 124.5648 18.01408  1.990311 8.935183e-28
## m11.3 682.4094  9.450597 126.4974 17.95586  3.034637 3.399694e-28
## m11.1 687.9533  7.126435 132.0413 18.89620  1.006633 2.126172e-29
#  WAIC in model 11.4 gets much smaller than other models, which weight at 1.