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. 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.

#The difference between an ordered categorical variable and an unordered one is there is a clear ordering of the variable in the ordered categorical variable, but not in the unordered one.
#For example, economic status is an ordered categorical variable with three clear ordering of the categories low, medium and high, while gender is an unordered categorical variable having two categories (male and female) and there is no intrinsic ordering to the categories.

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

#Ordered logit
#It differs from an ordinary logit link as it is a log-cumulative-odds link probability model. The ordered logit model is a regression model for an ordinal response variable. The model is based on the cumulative probabilities of the response variable instead of a discrete probability of a single event.

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

#If using a model that ignores zero-inflation, the excess of zero cannot be accounted for, which may result in the underestimation of the rate of events, because a count distribution with extra zeros added to it will have a lower 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?

#example: The number of deaths, number of cigarettes smoked, and number of disease cases. For such count data the Poisson model is a commonly applied statistical model. A key feature of the Poisson model is that the mean and the variance are equal. However, this equal mean-variance relationship rarely happens with real-life data. In most cases, the observed variance is larger than the assumed variance, which is known as overdispersion

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.

n <- c( 12, 36 , 7 , 41 )
q <- n / sum(n)
q
## [1] 0.12500000 0.37500000 0.07291667 0.42708333
sum(q)
## [1] 1
p <- cumsum(q)
p
## [1] 0.1250000 0.5000000 0.5729167 1.0000000
log(p/(1-p))
## [1] -1.9459101  0.0000000  0.2937611        Inf

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

plot( 1:4 , p , xlab="rating" , ylab="cumulative proportion" ,
xlim=c(0.7,4.3) , ylim=c(0,1) , xaxt="n" )
axis( 1 , at=1:4 , labels=1:4 )
# plot gray cumulative probability lines
for ( x in 1:4 ) lines( c(x,x) , c(0,p[x]) , col="gray" , lwd=2 )
# plot blue discrete probability segments
for ( x in 1:4 )
lines( c(x,x)+0.1 , c(p[x]-q[x],p[x]) , col="slateblue" , lwd=2 )
# add number labels
text( 1:4+0.2 , p-q/2 , labels=1:4 , col="slateblue" )

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

#The probability of a zero, mixing together both processes, 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)

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. Compare the model to an intercept-only Poisson model of deaths. 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?