11H3.

library(rethinking)
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## - Online documentation and vignettes at mc-stan.org/cmdstanr
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library(MASS)
data(eagles) 
d <- eagles

(salamanders) = records of salmon pirating attempts by Bald Eagles in Washington State. While one eagle feeds, sometimes another will swoop in and try to steal the salmon from it.

victim = feeding eagle pirate = the thief

  1. build a binomial GLM of successful pirating attempts. Consider the following model:

yi ∼ Binomial(n_i, p_i) log (pi / 1 − p_i) = α + βPP_i + βVV_i + βAA_i α ∼ Normal(0, 1.5) βP ∼ Normal(0, 0.5) βV ∼ Normal(0, 0.5) βA ∼ Normal(0, 0.5)

y = number of successful attempts n = total number of attempts P = body size of pirate eagle (large, small) V = body size of victim eagle (large, small) A = age of pirate eagle (immature, adult)

Fit the model above to the eagles data, using both quap and ulam. Is the quadratic approximation okay?

d$pirateSize <- as.integer(ifelse(d$P == "S", 2, 1)) #small = 2, large = 1
d$victimSize <- as.integer(ifelse(d$V == "S", 2, 1)) #small = 2, large = 1
d$pirateAge <-  as.integer(ifelse(d$P == "I", 2, 1)) #immature = 2, adult = 1

(dat_list <- list(
  pirateSize = d$pirateSize,
  victimSize = d$victimSize,
  pirateAge = d$pirateAge,
  successfulThefts = d$y,
  attempts = d$n
  ))
## $pirateSize
## [1] 1 1 1 1 2 2 2 2
## 
## $victimSize
## [1] 1 2 1 2 1 2 1 2
## 
## $pirateAge
## [1] 1 1 1 1 1 1 1 1
## 
## $successfulThefts
## [1] 17 29 17 20  1 15  0  1
## 
## $attempts
## [1] 24 29 27 20 12 16 28  4
#yi ∼ Binomial(n_i, p_i)
#log (pi / 1 − p_i) = α + βPP_i + βVV_i + βAA_i
#α ∼ Normal(0, 1.5)
#βP ∼ Normal(0, 0.5)
#βV ∼ Normal(0, 0.5)
#βA ∼ Normal(0, 0.5)

#I coundn't get the code form the slides to work for me in this problem so I had to troubleshoot using different R syntax. I don't know why it wouldn't work??

m.ulam <- ulam(
  alist(
    successfulThefts ~ dbinom(attempts, p),
    logit(p) <- alpha + beta1*pirateSize + beta2*victimSize + beta3*pirateAge,
    alpha ~ dnorm(0, 1.5), 
    beta1 ~ dnorm(0, .5),
    beta2 ~ dnorm(0, .5),
    beta3 ~ dnorm(0, .5)
  ), data = dat_list, chains = 4, log_lik = TRUE, iter = 1e4)
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m.quap <- quap(
  alist(
    successfulThefts ~ dbinom(attempts, p),
    logit(p) <- alpha + beta1*pirateSize + beta2*victimSize + beta3*pirateAge,
    alpha ~ dnorm(0, 1.5),
    beta1 ~ dnorm(0, .5),
    beta2 ~ dnorm(0, .5),
    beta3 ~ dnorm(0, .5)
  ), data = dat_list)

(outcomes <- precis(m.ulam, depth = 2, prob = .95, hist = FALSE))
##              mean        sd       2.5%      97.5%    n_eff    Rhat4
## alpha  0.27938018 0.6986669 -1.0939612  1.6493540 10188.57 1.000441
## beta1 -1.60421510 0.3010733 -2.1943662 -1.0182148 11710.50 1.000042
## beta2  1.81265236 0.3101798  1.2185790  2.4280532 12270.26 1.000399
## beta3  0.02354015 0.4785971 -0.9150909  0.9688061 12483.65 1.000146
precis(m.quap, depth = 2, prob = .95, hist = FALSE)
##              mean        sd       2.5%      97.5%
## alpha  0.27537663 0.6968729 -1.0904692  1.6412225
## beta1 -1.59346371 0.2957625 -2.1731475 -1.0137799
## beta2  1.79456184 0.3104537  1.1860838  2.4030399
## beta3  0.03106679 0.4777213 -0.9052498  0.9673834

ANSWER the ulam and quap models give outputs very closely in line with one another. I’ll still use ulam to interpret.

  1. Interpret the estimates.

to do this I need to first build contrasts…

# i had to figure this out without the slides code--wouldn't work for my model
#size of pirate
(Sp_prob <- inv_logit(outcomes[2,1]))
## [1] 0.1673933
(Lp_prob <- 1-Sp_prob)
## [1] 0.8326067
(diff_prob <- Lp_prob - Sp_prob)
## [1] 0.6652134
#size of victim
(Sv_prob <- inv_logit(outcomes[3,1]))
## [1] 0.8596821
(Lv_prob <- 1 - Sv_prob)
## [1] 0.1403179
(diff_prob <- Lv_prob - Sv_prob)
## [1] -0.7193643
#age of pirate
(Ip_prob <- inv_logit(outcomes[4,1]))
## [1] 0.5058848
(Ap_prob <- 1 - Ip_prob)
## [1] 0.4941152
(diff_prob <- Ip_prob - Ap_prob)
## [1] 0.01176953
  1. Now interpret the estimates. If the quadratic approximation turned out okay, then it’s okay to use the quap estimates. Otherwise stick to ulam estimates. Then plot the posterior predictions. Compute and display both (1) the predicted probability of success and its 89% interval for each row (i) in the data, as well as (2) the predicted success count and its 89% interval. What different information does each type of posterior prediction provide?

ANSWER As body size of pirate eagles increases, they are more likely to be successful thieves, p(success given large) = .83. In contrast, smaller victim eagles are more likely to lose their food, p(success given small) = .86. Age of pirate eagle doesn’t seem to matter, p(success given adult) = .49.

postcheck(m.ulam)

Purple dots = raw data Circles = model predictions with 89% intervals Crosses = predicted probabilities

  1. Now try to improve the model. Consider an interaction between the pirate’s size and age (immature or adult). Compare this model to the previous one, using WAIC. Interpret.
m.ulam.int <- ulam(
  alist(
    successfulThefts ~ dbinom(attempts, p),
    logit(p) <- alpha + beta1*pirateSize + beta2*victimSize + beta3*pirateAge + beta4*pirateAge*pirateSize,
    alpha ~ dnorm(0, 1.5), 
    beta1 ~ dnorm(0, .5),
    beta2 ~ dnorm(0, .5),
    beta3 ~ dnorm(0, .5),
    beta4 ~ dnorm(0, .5)
  ), data = dat_list, chains = 4, log_lik = TRUE, iter = 1e4)
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## Total execution time: 7.5 seconds.
precis(m.ulam.int)
##              mean        sd       5.5%      94.5%    n_eff     Rhat4
## alpha  0.64336708 0.7100770 -0.4860052  1.7798987 13187.53 1.0000284
## beta1 -0.97564874 0.3909612 -1.6015411 -0.3496857 13439.08 0.9998985
## beta2  1.87634927 0.3138518  1.3742196  2.3814632 15000.44 1.0003276
## beta3  0.08336638 0.4775583 -0.6894643  0.8378146 14371.38 0.9999609
## beta4 -0.98685923 0.3925373 -1.6083417 -0.3596568 13776.34 1.0000080
compare(m.ulam, m.ulam.int)
##                WAIC       SE    dWAIC      dSE    pWAIC     weight
## m.ulam.int 57.67800 14.52790 0.000000       NA 7.639628 0.93195048
## m.ulam     62.91209 16.07169 5.234088 4.812959 8.163386 0.06804952

ANSWER Given the WAIC values are almost exactly the same, I don’t think the added interaction adds to the model.