Assignment #8

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.

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

11-1. 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?

##Because the aggregated model contains an extra factor in its log probabilities. When the data is converted between the two formats c(n,m) multiplier is converted to constant.

11-2. Use quap to construct a quadratic approximate joint 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 joint posterior distribution produced instead from MCMC. Do not use the ‘pairs’ plot. 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 not use the ‘pairs’ plot. Do the differences increase or decrease? Why?

data("chimpanzees")
d<- chimpanzees
d$recipient <- NULL

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)

## There is a small increase.

11-3. Revisit the data(Kline) islands example. This time drop Hawaii from the sample and refit the models. Plot the joint posterior. What changes do you observe?

data(Kline)
d <- Kline
d$P <- scale( log(d$population) )
d$id <- ifelse( d$contact=="high" , 2 , 1 )
d

##       culture population contact total_tools mean_TU            P id
## 1    Malekula       1100     low          13     3.2 -1.291473310  1
## 2     Tikopia       1500     low          22     4.7 -1.088550750  1
## 3  Santa Cruz       3600     low          24     4.0 -0.515764892  1
## 4         Yap       4791    high          43     5.0 -0.328773359  2
## 5    Lau Fiji       7400    high          33     5.0 -0.044338980  2
## 6   Trobriand       8000    high          19     4.0  0.006668287  2
## 7       Chuuk       9200    high          40     3.8  0.098109204  2
## 8       Manus      13000     low          28     6.6  0.324317564  1
## 9       Tonga      17500    high          55     5.4  0.518797917  2
## 10     Hawaii     275000     low          71     6.6  2.321008320  1

d <- subset(d, d$culture != "Hawaii")
d$P <- scale( log(d$population) )
d$id <- ifelse( d$contact=="high" , 2 , 1 )
d

##      culture population contact total_tools mean_TU          P id
## 1   Malekula       1100     low          13     3.2 -1.6838108  1
## 2    Tikopia       1500     low          22     4.7 -1.3532297  1
## 3 Santa Cruz       3600     low          24     4.0 -0.4201043  1
## 4        Yap       4791    high          43     5.0 -0.1154764  2
## 5   Lau Fiji       7400    high          33     5.0  0.3478956  2
## 6  Trobriand       8000    high          19     4.0  0.4309916  2
## 7      Chuuk       9200    high          40     3.8  0.5799580  2
## 8      Manus      13000     low          28     6.6  0.9484740  1
## 9      Tonga      17500    high          55     5.4  1.2653019  2

##lau fiti becomes positive, every p value increase a little bit except for Tikopia

9-4. 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
## m11.4 566.5828 17.987052   0.0000       NA 23.2399524 1.000000e+00
## m11.2 680.5478  9.343863 113.9650 17.66099  2.0258187 1.789832e-25
## m11.3 682.3629  9.334910 115.7801 17.60590  3.0048651 7.222128e-26
## m11.1 687.9379  7.178520 121.3551 18.57200  0.9985719 4.446970e-27

Assignment #8

Chapter 11

Shiyu Bai

2023-01-15

Chapter 11 - God Spiked the Integers

Questions

m11.4 has the lowest WAIC, and it seems like m11.4 is more flexible fit to the data.