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. 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

11E1. If an event has probability 0.35, what are the log-odds of this event?

# if p=0.35
log( 0.35 / (1-0.35) )
## [1] -0.6190392

11E2. If an event has log-odds 3.2, what is the probability of this event?

logistic( 3.2 )
## [1] 0.9608343

11E3. Suppose that a coefficient in a logistic regression has value 1.7. What does this imply about the proportional change in odds of the outcome?

exp( 1.7 )
## [1] 5.473947
#it means predictor variable's every change , the odds of the event will change in  5.5 times

11E4. Why do Poisson regressions sometimes require the use of an offset? Provide an example.

# offset means the relative range that a count was collected.
# if all observed counts are from the same range, we don't need offset. but if some counts are from the longer periods, we can use offset to let counts from bigger periods range.

11M1. 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?

# aggregated binomial counts average over all the orders, it's consistent for the counts. 
# disaggregated binomial counts is 0 1 form. don't need to deal with the order

11M2. If a coefficient in a Poisson regression has value 1.7, what does this imply about the change in the outcome?

#if the coefficient is 1.7,  for one unit change, then the proportional change in the expected count is exp(1.7)=5.5 time

11M3. Explain why the logit link is appropriate for a binomial generalized linear model.

# we can use logit link for binomial GLM as we need to map continuous linear model value to a probability parameter that is in 0 to 1. the inverse logit function logistic, is one way to realize it.

11M4. Explain why the log link is appropriate for a Poisson generalized linear model.

# we can use log link for poission GLM, as we need to map continuous linear model value to a positive mean, that is bounded at 0. we can use log's inverse function exp for the linear model

11M5. What would it imply to use a logit link for the mean of a Poisson generalized linear model? Can you think of a real research problem for which this would make sense?

# can use a logit link in a poisson model.
#yi ∼ Poisson(μi)
#logit(μi) = α + βxi
#logit link limits the value 0-1. use this for the mean of a poisson generalized linear model, that the variable values will between 0-1

11M6. State the constraints for which the binomial and Poisson distributions have maximum entropy. Are the constraints different at all for binomial and Poisson? Why or why not?

#the constraints that make both binomial and poisson max entropy distibutions are: discrete binary outcomes.  constant probability of each event across trials .
#these distributions have same constraints.as  poisson is a simple form of binomial when the probability of event is low and trials is big.

11M7. Use quap to construct a quadratic approximate posterior distribution for the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330). Compare the quadratic approximation to the posterior distribution produced instead from MCMC. 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. Do the differences increase or decrease? Why?

library(rethinking) 
data(chimpanzees)
d <- chimpanzees
d$treatment <- 1 + d$prosoc_left + 2*d$condition
dat_list <- list(
pulled_left = d$pulled_left,
actor = d$actor,
treatment = as.integer(d$treatment) )


m11.4q <- quap(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + b[treatment] ,
a[actor] ~ dnorm( 0 , 1.5 ),
b[treatment] ~ dnorm( 0 , 0.5 )
) , data=dat_list )
# trimmed data list 11.10
dat_list <- list(
pulled_left = d$pulled_left,
actor = d$actor,
treatment = as.integer(d$treatment) )


m11.4 <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + b[treatment] ,
a[actor] ~ dnorm( 0 , 1.5 ),
b[treatment] ~ dnorm( 0 , 0.5 )
) , data=dat_list , chains=4 , log_lik=TRUE )
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precis( m11.4 , depth=2 )
##             mean        sd        5.5%       94.5%     n_eff    Rhat4
## a[1] -0.46755119 0.3316707 -0.98098064  0.07840123  875.7382 1.000106
## a[2]  3.90297183 0.7494907  2.79324369  5.13520442 1135.2105 1.000860
## a[3] -0.76839905 0.3339479 -1.31717967 -0.24050664  809.8418 1.001714
## a[4] -0.76783087 0.3311050 -1.30461384 -0.22919156  778.8098 1.003631
## a[5] -0.46702468 0.3329995 -0.98031034  0.06449306  703.2498 1.001489
## a[6]  0.47147449 0.3318493 -0.04990034  1.00522556  732.1296 1.003058
## a[7]  1.94448970 0.4094211  1.29992969  2.59246098  856.9884 1.003695
## b[1] -0.02638938 0.2841064 -0.48210829  0.41426408  731.8928 1.002492
## b[2]  0.49519678 0.2893072  0.04220036  0.95239872  720.1162 1.003093
## b[3] -0.37076281 0.2873956 -0.83318140  0.07916147  701.3510 1.001808
## b[4]  0.38031517 0.2825230 -0.09095279  0.82958496  699.3794 1.004170
pr <- precis( m11.4 , 2 )[,1:4]
prq <- precis( m11.4q , 2 )
round( cbind( pr , prq ) , 2 )
##       mean   sd  5.5% 94.5%  mean   sd  5.5% 94.5%
## a[1] -0.47 0.33 -0.98  0.08 -0.44 0.33 -0.96  0.08
## a[2]  3.90 0.75  2.79  5.14  3.71 0.72  2.55  4.86
## a[3] -0.77 0.33 -1.32 -0.24 -0.73 0.33 -1.26 -0.20
## a[4] -0.77 0.33 -1.30 -0.23 -0.73 0.33 -1.26 -0.20
## a[5] -0.47 0.33 -0.98  0.06 -0.44 0.33 -0.96  0.08
## a[6]  0.47 0.33 -0.05  1.01  0.47 0.33 -0.06  1.00
## a[7]  1.94 0.41  1.30  2.59  1.91 0.41  1.24  2.57
## b[1] -0.03 0.28 -0.48  0.41 -0.04 0.28 -0.49  0.41
## b[2]  0.50 0.29  0.04  0.95  0.47 0.28  0.02  0.93
## b[3] -0.37 0.29 -0.83  0.08 -0.38 0.29 -0.83  0.08
## b[4]  0.38 0.28 -0.09  0.83  0.36 0.28 -0.09  0.82
post <- extract.samples( m11.4 )
postq <- extract.samples( m11.4q )
dens( post$a[,2] , lwd=2 )
dens( postq$a[,2] , add=TRUE , lwd=2 , col=rangi2 )

m11M7q <- quap(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + b[treatment] ,
a[actor] ~ dnorm( 0 , 10 ),
b[treatment] ~ dnorm( 0 , 0.5 )
) , data=dat_list )
m11M7u <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + b[treatment] ,
a[actor] ~ dnorm( 0 , 10 ),
b[treatment] ~ dnorm( 0 , 0.5 )
) , data=dat_list , cores=4 , chains=4 )
## Warning: There were 11 divergent transitions after warmup. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.
## Warning: Examine the pairs() plot to diagnose sampling problems
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
post <- extract.samples( m11M7u ) 
postq <- extract.samples( m11M7q )
dens( post$a[,2] , lwd=2 , ylim=c(0,0.12) )
dens( postq$a[,2] , add=TRUE , lwd=2 , col=rangi2 )

precis( glm( pulled_left ~ as.factor(actor) + as.factor(treatment) , 
data=dat_list , family=binomial ) )
##                                mean          sd          5.5%         94.5%
## (Intercept)           -5.235674e-01   0.3054725 -1.011771e+00   -0.03536339
## as.factor(actor)2      1.896261e+01 752.7210266 -1.184031e+03 1221.95618973
## as.factor(actor)3     -3.061147e-01   0.3506842 -8.665758e-01    0.25434639
## as.factor(actor)4     -3.061147e-01   0.3506842 -8.665758e-01    0.25434639
## as.factor(actor)5     -3.228618e-16   0.3447667 -5.510038e-01    0.55100379
## as.factor(actor)6      9.447990e-01   0.3499405  3.855265e-01    1.50407139
## as.factor(actor)7      2.498070e+00   0.4522240  1.775329e+00    3.22081170
## as.factor(treatment)2  6.182384e-01   0.2995263  1.395374e-01    1.09693931
## as.factor(treatment)3 -4.186088e-01   0.3063093 -9.081502e-01    0.07093265
## as.factor(treatment)4  4.857116e-01   0.2986337  8.437290e-03    0.96298589

11M8. Revisit the data(Kline) islands example. This time drop Hawaii from the sample and refit the models. What changes do you observe?

library(rethinking)
data(Kline)
d <- Kline
d$P <- standardize( log(d$population) )
d$contact_id <- ifelse( d$contact=="high" , 2 , 1 )
d2 <- d[ d$culture!="Hawaii" , ]
dat2 <- list(
T = d2$total_tools ,
P = d2$P ,
cid = d2$contact_id )
m11.10b <- ulam(
alist(
T ~ dpois( lambda ),
log(lambda) <- a[cid] + b[cid]*P,
a[cid] ~ dnorm( 3 , 0.5 ),
b[cid] ~ dnorm( 0 , 0.2 )
), data=dat2 , chains=4 )
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##           mean         sd        5.5%     94.5%    n_eff    Rhat4
## a[1] 3.1751184 0.13185265  2.95950062 3.3862078 1391.673 1.001997
## a[2] 3.6129991 0.07358988  3.49369702 3.7280077 1633.301 1.000302
## b[1] 0.1897783 0.13165723 -0.02031797 0.4008250 1447.543 1.002236
## b[2] 0.1869402 0.16079978 -0.06608848 0.4390262 1915.840 1.000317

compared to the model fit to the complete sample. we can’t find difference between the two slopes

11H1. 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.

library(rethinking)
data(chimpanzees)
d <- chimpanzees
d$treatment <- 1 + d$prosoc_left + 2*d$condition
dat_list <- list(
pulled_left = d$pulled_left,
actor = d$actor,

treatment = as.integer(d$treatment) )
m11.1 <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a ,
a ~ dnorm( 0 , 10 )
) , data=dat_list , chains=4 , log_lik=TRUE )
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## 
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m11.2 <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a + b[treatment] ,
a ~ dnorm( 0 , 1.5 ),
b[treatment] ~ dnorm( 0 , 10 )
) , data=dat_list , chains=4 , log_lik=TRUE )
## 
## SAMPLING FOR MODEL '0f70e9cccc329738a5df98436ac74236' NOW (CHAIN 1).
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## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## http://mc-stan.org/misc/warnings.html#tail-ess
m11.3 <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a + b[treatment] ,
a ~ dnorm( 0 , 1.5 ),
b[treatment] ~ dnorm( 0 , 0.5 )
) , data=dat_list , chains=4 , log_lik=TRUE )
## 
## SAMPLING FOR MODEL 'cf5b2fa0ef5b7e8dcc719a7be2632c49' NOW (CHAIN 1).
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## 
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m11.4 <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + b[treatment] ,
a[actor] ~ dnorm( 0 , 1.5 ),
b[treatment] ~ dnorm( 0 , 0.5 )
) , data=dat_list , chains=4 , log_lik=TRUE )
## recompiling to avoid crashing R session
## 
## SAMPLING FOR MODEL '80e2b6267e3dc4ff0c2916d0cf0879e8' NOW (CHAIN 1).
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compare( m11.1 , m11.2 , m11.3 , m11.4 , func=PSIS )
##           PSIS        SE    dPSIS      dSE     pPSIS       weight
## m11.4 532.4488 19.034784   0.0000       NA 8.6393934 1.000000e+00
## m11.3 682.1601  9.221281 149.7113 18.53238 3.4967178 3.094524e-33
## m11.2 683.0561  9.694330 150.6073 18.60050 4.0010950 1.977120e-33
## m11.1 687.8694  7.127876 155.4206 19.06758 0.9646651 1.781701e-34