Chapter 13 - Models with Memory

This chapter has been an introduction to the motivation, implementation, and interpretation of basic multilevel models. It focused on varying intercepts, which achieve better estimates of baseline differences among clusters in the data. They achieve better estimates, because they simultaneously model the population of clusters and use inferences about the population to pool information among parameters. From another perspective, varying intercepts are adaptively regularized parameters, relying upon a prior that is itself learned from the data.

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

13-1. Revisit the Reed frog survival data, data(reedfrogs), and add the predation and size treatment variables to the varying intercepts model. Consider models with either main effect alone, both main effects, as well as a model including both and their interaction. Instead of focusing on inferences about these two predictor variables, focus on the inferred variation across tanks. Explain why it changes as it does across models. Plot the sigma estimates.

data(reedfrogs)
df <- reedfrogs
df <- list(
  tank = 1:nrow(df),
  surv = df$surv,
  density = df$density,
  pred = ifelse(df$pred == "no", 0L, 1L),
  size_ = ifelse(df$size == "small", 1L, 2L)
)

m1 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a[tank] + b * pred,
    a[tank] ~ dnorm(a_bar, sigma),
    b ~ dnorm(-0.5, 1),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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m2 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a[tank] + s[size_],
    a[tank] ~ dnorm(a_bar, sigma),
    s[size_] ~ dnorm(0, 0.5),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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m3 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a[tank] + b * pred + s[size_],
    a[tank] ~ dnorm(a_bar, sigma),
    b ~ dnorm(-0.5, 1),
    s[size_] ~ dnorm(0, 0.5),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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m4 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a_bar + a[tank] * sigma + b[size_] * pred + s[size_], 
    a[tank] ~ dnorm(0, 1),
    b[size_] ~ dnorm(-0.5, 1),
    s[size_] ~ dnorm(0, 0.5),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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plot(coeftab(m1, m2, m3, m4),
  pars = "sigma",
  labels = c( "Predation alone", "Size alone", "Both main effects", "Main effect with interaction")
)

#The predation could explain a lot of variation across tanks, since when we add predation the variation decreases.

13-2. Compare the models you fit just above, using WAIC. Can you reconcile the differences in WAIC with the posterior distributions of the models? Show the WAIC table.

compare(m1, m2, m3, m4)
##        WAIC       SE    dWAIC      dSE    pWAIC    weight
## m1 199.0766 8.926028 0.000000       NA 19.23223 0.4262744
## m4 200.3006 9.066615 1.223975 3.038188 19.29041 0.2311567
## m2 200.6563 7.268269 1.579711 5.563477 21.16750 0.1934904
## m3 201.1778 8.817000 2.101222 2.036710 19.73410 0.1490784
precis(m1)
##             mean        sd       5.5%     94.5%     n_eff    Rhat4
## b     -2.4308258 0.3025079 -2.9199191 -1.940634  465.0746 1.002897
## a_bar  2.5337065 0.2374450  2.1684630  2.924411  511.0650 1.004665
## sigma  0.8259274 0.1438368  0.6159309  1.069210 1508.0483 1.000718
precis(m2)
##           mean        sd      5.5%    94.5%     n_eff    Rhat4
## a_bar 1.244783 0.4105667 0.6011881 1.907789  478.4155 1.010208
## sigma 1.616666 0.2138592 1.3072172 1.984984 1935.9997 1.000739
precis(m3)
##             mean        sd       5.5%     94.5%     n_eff    Rhat4
## b     -2.4401937 0.2989547 -2.9228628 -1.964179  896.9907 1.005314
## a_bar  2.4095941 0.4253376  1.7225315  3.069880  263.2166 1.012522
## sigma  0.7844143 0.1456207  0.5695766  1.035922 1113.3159 1.002511
precis(m4)
##            mean        sd      5.5%     94.5%    n_eff    Rhat4
## a_bar 2.3051013 0.4091339 1.6589378 2.9620976 1818.088 1.000990
## sigma 0.7458555 0.1455165 0.5327259 0.9905894 1572.189 1.002217
#The only model (m2) without predation has a large sigma value, other models with predation have small sigma values, indicating that models with predation would have small variation across tanks.

13-3. Re-estimate the basic Reed frog varying intercept model, but now using a Cauchy distribution in place of the Gaussian distribution for the varying intercepts. That is, fit this model: \[\begin{align} s_i ∼ Binomial(n_i, p_i) \\ logit(p_i) = α_{tank[i]} \\ α_{tank} ∼ Cauchy(α, σ) \\ α ∼ Normal(0, 1) \\ σ ∼ Exponential(1) \\ \end{align}\]

(You are likely to see many divergent transitions for this model. Can you figure out why? Can you fix them?) Plot and compare the posterior means of the intercepts, αtank, to the posterior means produced in the chapter, using the customary Gaussian prior. Can you explain the pattern of differences? Take note of any change in the mean α as well.

m0 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a[tank],
    a[tank] ~ dnorm(a_bar, sigma),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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#Cauchy distribution
m5 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a[tank],
    a[tank] ~ dcauchy(a_bar, sigma),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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a1 <- apply(extract.samples(m0)$a, 2, mean)
a2 <- apply(extract.samples(m5)$a, 2, mean)
plot(a1, a2,pch = 10)
abline(a = 0, b = 1)

#Most of the posterior means using a Cauchy or Gaussian prior are the same, only a small amount of posterior means from a Cauchy prior are larger than those from a Gaussian prior.

13-4. Now use a Student-t distribution with ν = 2 for the intercepts: \[\begin{align} α_{tank} ∼ Student(2, α, σ) \end{align}\]

Refer back to the Student-t example in Chapter 7 (page 234), if necessary. Plot and compare the resulting posterior to both the original model and the Cauchy model in 13-3. Can you explain the differences and similarities in shrinkage in terms of the properties of these distributions?

m6 <- ulam(
  alist(
    surv ~ dbinom(density, p),
    logit(p) <- a[tank],
    a[tank] ~ dstudent(2, a_bar, sigma),
    a_bar ~ dnorm(0, 1.5),
    sigma ~ dexp(1)
  ),
  data = df, chains = 4, cores = 4, log_lik = TRUE, iter = 2e3
)
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#compare original model with Student-t distribution
a1 <- apply(extract.samples(m0)$a, 2, mean)
a2 <- apply(extract.samples(m6)$a, 2, mean)
plot(a1, a2, pch = 10)
abline(a = 0, b = 1)

#compare Cauchy model with Student-t distribution
a1 <- apply(extract.samples(m5)$a, 2, mean)
a2 <- apply(extract.samples(m6)$a, 2, mean)
plot(a1, a2, pch = 10)
abline(a = 0, b = 1)

#When compare three models pairwise, most of the posterior means are the same. Comparing the new model with original model, only a small amount of posterior means from a student-t prior are larger than those from the original model; Comparing the new model with Cauchy model, only a small amount of posterior means from a student-t prior are smaller than those from the Cauchy model;

13-5. Sometimes the prior and the data (through the likelihood) are in conflict, because they concentrate around different regions of parameter space. What happens in these cases depends a lot upon the shape of the tails of the distributions. Likewise, the tails of distributions strongly influence can outliers are shrunk or not towards the mean. I want you to consider four different models to fit to one observation at y = 0, this is your data, do not use any other data set. The models differ only in the distributions assigned to the likelihood and prior. Here are the four models:

\[\begin{align} Model \;NN: y &∼ Normal(μ, 1) & Model \;TN: y &∼ Student(2, μ, 1) \\ μ &∼ Normal(10, 1) & μ &∼ Normal(10, 1) \\ Model \;NT: y &∼ Normal(μ, 1) & Model \;TT: y &∼ Student(2, μ, 1) \\ μ &∼ Student(2, 10, 1) & μ &∼ Student(2, 10, 1) \\ \end{align}\]

Estimate and plot the posterior distributions against the likelihoods for these models and compare them. Can you explain the results, using the properties of the distributions?

df2 <- as.data.frame(rep(0,50),ncol=1)
names(df2)[1] <- 'y'

m_nn <- ulam(
  alist(
    y ~ dnorm(mu, 1),
    mu ~ dnorm(10, 1)
  ),
  data = df2, chains = 1, cores = 1,  iter = 1e3
)
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m_tn <- ulam(
  alist(
    y ~ dstudent(2, mu, 1),
    mu ~ dnorm(10, 1)
  ),
  data = df2, chains = 1, cores = 1,  iter = 1e3
)
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m_nt <- ulam(
  alist(
    y ~ dnorm(mu, 1),
    mu ~ dstudent(2, 10, 1)
  ),
  data = df2, chains = 1, cores = 1,  iter = 1e3
)
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m_tt <- ulam(
  alist(
    y ~ dstudent(2, mu, 1),
    mu ~ dstudent(2, 10, 1)
  ),
  data = df2, chains = 1, cores = 1,  iter = 1e3
)
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precis(m_nn)
##             result
## mean    0.19003150
## sd      0.13636209
## 5.5%   -0.02117481
## 94.5%   0.40353429
## n_eff 142.77430793
## Rhat    1.00221239
precis(m_nt)
##              result
## mean    0.002812639
## sd      0.144815073
## 5.5%   -0.235232490
## 94.5%   0.255147755
## n_eff 157.197810660
## Rhat    1.004901380
precis(m_tn)
##             result
## mean    0.13777957
## sd      0.11505455
## 5.5%   -0.03928672
## 94.5%   0.31016200
## n_eff 208.21074065
## Rhat    1.00955618
precis(m_tt)
##              result
## mean    0.003445696
## sd      0.123202658
## 5.5%   -0.185919440
## 94.5%   0.191514570
## n_eff 181.759464656
## Rhat    1.015865164
dens(extract.samples(m_nn)$mu, col="black")
dens(extract.samples(m_nt)$mu, col="red", add = TRUE)
dens(extract.samples(m_tn)$mu, col="blue", add=TRUE)
dens(extract.samples(m_tt)$mu, col="green", add=TRUE)

#Model NN in black color has the largest posterior mean, while model TT in green color has the smallest posterior mean and more close to 0. The shape of densities from four models are almost the same. The student-t distribution has heavier tails than the normal distribution.