Chapter 12 - Monsters and Mixtures

This chapter introduced several new types of regression, all of which are generalizations of generalized linear models (GLMs). Ordered logistic models are useful for categorical outcomes with a strict ordering. They are built by attaching a cumulative link function to a categorical outcome distribution. Zero-inflated models mix together two different outcome distributions, allowing us to model outcomes with an excess of zeros. Models for overdispersion, such as beta-binomial and gamma-Poisson, draw the expected value of each observation from a distribution that changes shape as a function of a linear model.

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

12E1. What is the difference between an ordered categorical variable and an unordered one? Define and then give an example of each.

#Ordered Categorial variable provides rank or order. E.g. is NPS score, 5 star customer review rating. 
#Unordered categorial variable is a varaible with no intrinsic ordering. E.g. gender, marital status

12E2. What kind of link function does an ordered logistic regression employ? How does it differ from an ordinary logit link?

# Ordered logistic regression uses cumulative log-adds. This provides the odds of the value or smaller values. 
# Ordinary logit link provides the odds of a particular value.

12E3. When count data are zero-inflated, using a model that ignores zero-inflation will tend to induce which kind of inferential error?

# Ignoring zero-inflation means excluding zero-inflation data. This underestimates rate of events, as zero-inflation data still impacts overall mean.

12E4. Over-dispersion is common in count data. Give an example of a natural process that might produce over-dispersed counts. Can you also give an example of a process that might produce underdispersed counts?

#Dispersion takes place when theoretical variance and observed variance are different. Over-dispersion takes place when observed vairance is higher than theoretical variance. Example is eating more calories for a long run than required. Underdispersed is working out more than calories can manage for muscle growth.

12M1. At a certain university, employees are annually rated from 1 to 4 on their productivity, with 1 being least productive and 4 most productive. In a certain department at this certain university in a certain year, the numbers of employees receiving each rating were (from 1 to 4): 12, 36, 7, 41. Compute the log cumulative odds of each rating.

r = c(12, 36, 7, 41)
for (i in c(1, 2, 3, 4)){
  print(log(sum(r[1:i])/sum(r[(i+1):4])))
}
## [1] -1.94591
## [1] 0
## [1] 0.2937611
## [1] NA

12M2. Make a version of Figure 12.5 for the employee ratings data given just above.

p= r/sum(r)

p=sapply(1:4,function(i){sum(p[1:i])})
plot(y=p,x=1:4,type="b", ylim=c(0,1))
segments(1:4,0,1:4,p)
for(i in 1:4){segments(i+0.05,c(0,p)[i],i+0.05,p[i], col = "red")}

12M3. Can you modify the derivation of the zero-inflated Poisson distribution (ZIPoisson) from the chapter to construct a zero-inflated binomial distribution?

# Probability of a zero is: 
# Pr(0|p0, q, n) = p0 + (1 − p0)(1 − q)^n

# The probability of any particular non-zero observation y is: 
# Pr(y|p0, q, n) = (1 − p0)(n!/(y!(n − y)!)(q^y)((1 − q)^(n−y))

12H1. In 2014, a paper was published that was entitled “Female hurricanes are deadlier than male hurricanes.”191 As the title suggests, the paper claimed that hurricanes with female names have caused greater loss of life, and the explanation given is that people unconsciously rate female hurricanes as less dangerous and so are less likely to evacuate. Statisticians severely criticized the paper after publication. Here, you’ll explore the complete data used in the paper and consider the hypothesis that hurricanes with female names are deadlier. Load the data with:

data(Hurricanes)

data1 <- Hurricanes
m1 <- map(
  alist(
    deaths ~ dpois( lambda ),
    log(lambda) <- a + bF*femininity,
    a ~ dnorm(0,10),
    bF ~ dnorm(0,5)
  ) ,
  data=data1)

m2 <- map(
  alist(
    deaths ~ dpois( lambda ),
    log(lambda) <- a ,
    a ~ dnorm(0,10)
  ) ,
  data=data1)

compare(m1,m2)
##        WAIC       SE    dWAIC      dSE     pWAIC       weight
## m1 4420.794 1002.387  0.00000       NA 133.43442 9.999159e-01
## m2 4439.562 1071.042 18.76788 137.4033  80.46761 8.405626e-05
y <- sim(m1)

y.mean <- colMeans(y)
y.PI <- apply(y, 2, PI)

plot(y=data1$deaths, x=data1$femininity, col=rangi2, ylab="deaths", xlab="femininity", pch=16)
points(y=y.mean, x=data1$femininity, pch=1)
segments(x0=data1$femininity, x1= data1$femininity, y0=y.PI[1,], y1=y.PI[2,])

lines(y= y.mean[order(data1$femininity)],  x=sort(data1$femininity))
lines( y.PI[1,order(data1$femininity)],  x=sort(data1$femininity), lty=2 )
lines( y.PI[2,order(data1$femininity)],  x=sort(data1$femininity), lty=2 )

# The relationship partially shows that death Hurricane name seems more feminine

Acquaint yourself with the columns by inspecting the help ?Hurricanes. In this problem, you’ll focus on predicting deaths using femininity of each hurricane’s name. Fit and interpret the simplest possible model, a Poisson model of deaths using femininity as a predictor. You can use quap or ulam. Plot your results of prior predictive simulation. Compare the model to an intercept-only Poisson model of deaths. Plot the results over the raw data. How strong is the association between femininity of name and deaths? Which storms does the model fit (retrodict) well? Which storms does it fit poorly?