Chapter 12 - Monsters & 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.

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

12-1. 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)
p <- n/sum(n)
log <-log(cumsum(p)/(1-cumsum(p)))
log
## [1] -1.9459101  0.0000000  0.2937611        Inf

12-2. Make a version of Figure 12.5 for the employee ratings data given just above.

plot(1:4, cumsum(p), type="b")
for (i in 1:4) lines(c(i, i), c(0, cumsum(p)[i]), col = "gray", lwd = 4)
for (i in 1:4) lines(c(i, i) + 0.1, c(cumsum(p)[i] - p[i], cumsum(p)[i]), col = "red", lwd = 4)

12-3. 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.

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?

data(Hurricanes)
df <- Hurricanes
df$femininity <-scale(df$femininity)

m1 <- ulam(
  alist(
    deaths ~ dpois(lambda),
    log(lambda) <- a + b*femininity,
    a ~ dnorm(1, 1),
    b ~ dnorm(0, 1)
  ), data=df, chains=4, log_lik=TRUE)
## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
## 
## Chain 1 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 1 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 1 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 1 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 1 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 1 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 1 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 1 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 1 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 1 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 1 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 1 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 1 finished in 0.3 seconds.
## Chain 2 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 2 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 2 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 2 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 2 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 2 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 2 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 2 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 2 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 2 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 2 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 2 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 2 finished in 0.3 seconds.
## Chain 3 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 3 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 3 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 3 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 3 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 3 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 3 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 3 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 3 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 3 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 3 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 3 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 3 finished in 0.3 seconds.
## Chain 4 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 4 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 4 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 4 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 4 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 4 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 4 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 4 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 4 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 4 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 4 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 4 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 4 finished in 0.3 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.3 seconds.
## Total execution time: 1.6 seconds.
#intercept-only model
m2 <- ulam(
  alist(
    deaths ~ dpois(lambda),
    log(lambda) <- a,
    a ~ dnorm(1, 1)
  ), data=df, chains=4, log_lik=TRUE)
## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
## 
## Chain 1 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 1 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 1 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 1 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 1 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 1 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 1 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 1 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 1 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 1 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 1 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 1 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 1 finished in 0.1 seconds.
## Chain 2 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 2 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 2 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 2 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 2 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 2 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 2 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 2 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 2 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 2 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 2 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 2 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 2 finished in 0.1 seconds.
## Chain 3 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 3 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 3 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 3 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 3 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 3 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 3 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 3 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 3 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 3 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 3 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 3 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 3 finished in 0.1 seconds.
## Chain 4 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 4 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 4 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 4 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 4 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 4 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 4 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 4 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 4 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 4 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 4 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 4 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 4 finished in 0.1 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.1 seconds.
## Total execution time: 0.9 seconds.
compare(m1, m2, func=PSIS)
##        PSIS        SE    dPSIS      dSE     pPSIS       weight
## m1 4385.513  986.7924  0.00000       NA 115.08769 1.000000e+00
## m2 4423.513 1065.7257 38.00018 145.4873  64.52271 5.602285e-09
precis(m1)
##        mean         sd      5.5%     94.5%    n_eff    Rhat4
## a 2.9989212 0.02368678 2.9611689 3.0359106 1097.592 1.001739
## b 0.2395167 0.02601430 0.1976223 0.2810013 1239.055 1.000818
plot(df$femininity, df$deaths)
pred <- list( femininity = seq(from = -2, to = 1.5, length.out = 30) )
lambda <- link(m1, data=pred)
mu <- apply(lambda, 2, mean)
lines(pred$femininity, mu)

#The second model with intercept only does not fit as well with a large PSIS. There is a positive relationship between deaths and femininity. From the plot it seems that the positive relationship between deaths and femininity is not quite strong.

12-4. Counts are nearly always over-dispersed relative to Poisson. So fit a gamma-Poisson (aka negative-binomial) model to predict deaths using femininity. Show that the over-dispersed model no longer shows as precise a positive association between femininity and deaths, with an 89% interval that overlaps zero. Can you explain why the association diminished in strength?

m3 <- ulam(
  alist(
    deaths ~ dgampois(lambda, scale),
    log(lambda) <- a + b*femininity,
    a ~ dnorm(1, 1),
    b ~ dnorm(0, 1),
    scale ~ dexp(1)
  ), data=df, chains=4, log_lik=TRUE)
## Running MCMC with 4 sequential chains, with 1 thread(s) per chain...
## 
## Chain 1 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 1 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 1 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 1 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 1 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 1 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 1 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 1 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 1 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 1 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 1 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 1 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 1 finished in 0.4 seconds.
## Chain 2 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 2 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 2 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 2 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 2 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 2 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 2 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 2 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 2 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 2 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 2 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 2 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 2 finished in 0.5 seconds.
## Chain 3 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 3 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 3 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 3 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 3 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 3 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 3 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 3 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 3 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 3 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 3 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 3 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 3 finished in 0.4 seconds.
## Chain 4 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 4 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 4 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 4 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 4 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 4 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 4 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 4 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 4 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 4 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 4 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 4 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 4 finished in 0.5 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.4 seconds.
## Total execution time: 2.1 seconds.
precis(m3)
##            mean         sd        5.5%     94.5%    n_eff     Rhat4
## a     2.9664115 0.15140697  2.72734810 3.2121975 1982.744 0.9991328
## b     0.2144620 0.14752419 -0.02556021 0.4437835 1858.173 1.0001899
## scale 0.4552657 0.06365695  0.35847742 0.5675499 1865.463 1.0000276
#When we fit a gamma-Poisson model we have one more parameter.

12-5. In the data, there are two measures of a hurricane’s potential to cause death: damage_norm and min_pressure. Consult ?Hurricanes for their meanings. It makes some sense to imagine that femininity of a name matters more when the hurricane is itself deadly. This implies an interaction between femininity and either or both of damage_norm and min_pressure. Fit a series of models evaluating these interactions. Interpret and compare the models. In interpreting the estimates, it may help to generate counterfactual predictions contrasting hurricanes with masculine and feminine names. Are the effect sizes plausible?

df$min_pressure <-scale(df$min_pressure)
df$damage_norm <-scale(df$damage_norm)

#model with femininity, min_pressure and interaction
m4 <- ulam(
  alist(
    deaths ~ dgampois(lambda, scale),
    log(lambda) <- a + b1 * femininity + b2 * min_pressure + b3 * femininity * min_pressure,
    a ~ dnorm(1, 1),
    b1 ~ dnorm(0, 1),
    b2 ~ dnorm(0, 1),
    b3 ~ dnorm(0, 1),
    scale ~ dexp(1)
  ),
  data = df, cores = 4, chains = 4, log_lik = TRUE
)
## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
## 
## Chain 1 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 1 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 2 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 2 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 3 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 3 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 4 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 4 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 1 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 1 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 1 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 2 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 2 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 2 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 3 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 3 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 3 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 4 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 4 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 4 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 1 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 1 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 1 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 2 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 2 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 2 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 3 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 3 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 3 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 4 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 4 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 4 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 1 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 1 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 2 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 2 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 3 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 3 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 4 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 4 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 1 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 1 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 2 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 2 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 3 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 3 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 4 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 4 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 1 finished in 0.7 seconds.
## Chain 2 finished in 0.6 seconds.
## Chain 3 finished in 0.6 seconds.
## Chain 4 finished in 0.6 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.6 seconds.
## Total execution time: 0.8 seconds.
#model with femininity, damage_norm and interaction
m5 <- ulam(
  alist(
    deaths ~ dgampois(lambda, scale),
    log(lambda) <- a + b1 * femininity + b2 * damage_norm + b3 * femininity * damage_norm,
    a ~ dnorm(1, 1),
    b1 ~ dnorm(0, 1),
    b2 ~ dnorm(0, 1),
    b3 ~ dnorm(0, 1),
    scale ~ dexp(1)
  ),
  data = df, cores = 4, chains = 4, log_lik = TRUE
)
## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
## 
## Chain 1 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 1 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 2 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 2 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 3 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 3 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 4 Iteration:   1 / 1000 [  0%]  (Warmup) 
## Chain 4 Iteration: 100 / 1000 [ 10%]  (Warmup) 
## Chain 1 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 1 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 1 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 1 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 1 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 2 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 2 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 2 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 3 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 3 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 3 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 3 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 3 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 4 Iteration: 200 / 1000 [ 20%]  (Warmup) 
## Chain 4 Iteration: 300 / 1000 [ 30%]  (Warmup) 
## Chain 4 Iteration: 400 / 1000 [ 40%]  (Warmup) 
## Chain 1 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 2 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 2 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 2 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 3 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 4 Iteration: 500 / 1000 [ 50%]  (Warmup) 
## Chain 4 Iteration: 501 / 1000 [ 50%]  (Sampling) 
## Chain 4 Iteration: 600 / 1000 [ 60%]  (Sampling) 
## Chain 1 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 1 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 2 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 2 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 3 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 3 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 4 Iteration: 700 / 1000 [ 70%]  (Sampling) 
## Chain 4 Iteration: 800 / 1000 [ 80%]  (Sampling) 
## Chain 1 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 1 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 2 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 2 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 3 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 3 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 4 Iteration: 900 / 1000 [ 90%]  (Sampling) 
## Chain 1 finished in 0.6 seconds.
## Chain 2 finished in 0.7 seconds.
## Chain 3 finished in 0.6 seconds.
## Chain 4 Iteration: 1000 / 1000 [100%]  (Sampling) 
## Chain 4 finished in 0.7 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.7 seconds.
## Total execution time: 0.9 seconds.
precis(m4)
##             mean         sd        5.5%      94.5%    n_eff     Rhat4
## a      2.7503232 0.14196841  2.53025975  2.9828053 2323.012 1.0004052
## b1     0.3001554 0.13993403  0.07557063  0.5228865 2753.851 0.9988032
## b2    -0.6744762 0.13638247 -0.89867834 -0.4627696 2250.724 0.9989078
## b3     0.3058106 0.14608376  0.07799079  0.5446330 2139.264 1.0006430
## scale  0.5539973 0.08084259  0.43607295  0.6921885 2842.044 0.9997469
precis(m5)
##             mean        sd        5.5%     94.5%    n_eff     Rhat4
## a     2.56816144 0.1271954  2.37451230 2.7758266 2032.649 1.0023484
## b1    0.08380725 0.1248234 -0.11423859 0.2748708 1749.786 1.0029359
## b2    1.25612912 0.2169873  0.91971658 1.6110315 1956.272 0.9988961
## b3    0.31152112 0.2012850 -0.02289794 0.6360890 2385.057 0.9990814
## scale 0.68591888 0.1030890  0.53203658 0.8593599 2603.928 0.9988626
compare(m4, m5, func=PSIS)
##        PSIS       SE    dPSIS      dSE    pPSIS       weight
## m5 670.4894 34.11528  0.00000       NA 6.748360 9.999956e-01
## m4 695.1622 39.22100 24.67282 19.58087 8.805596 4.388983e-06
#In the first model we fit with femininity, min_pressure and the interaction between femininity and min_pressure. The model shows a strong positive relationship between deaths and femininity. In the second model we fit with femininity, damage_norm and the interaction between femininity and damage_norm. The positive relationship between deaths and femininity becomes weak. The first model performs better with a smaller PSIS.