Date Assigned: Tuesdau, Movember 21, 2023

Date Due: Wednesday, December 13, 2023 at one minute before midnight

Instructions

Do using Rmarkdown. Upload two files to Canvas before the time Canvas says the homework is due

Rmarkdown file

and the corresponding output

cover the basics of Rmarkdown. For the basics of markdown text formatting, see

For further info on Rmarkdown, the source for all of the course notes is always available for seeing how anything in any of the course notes was done.

Assignment

Solve each problem. Explain your reasoning. No credit for answers with no explanation.

5-1

The data are the data in Table 10.9 in Agresti. These data can be read into R as follows

u <- "http://www.stat.umn.edu/geyer/5421/data/table-10.9.txt"
foo <- read.table(u, header = TRUE)
sapply(foo, class)
##    survival   gestation     smoking         age      counts 
## "character" "character" "character" "character"   "integer"

If you had trouble doing the above, then download the file in a separate step and then read it, as discussed in an announcement on the course home page.

  1. Which single hierarchical model has the lowest AIC?

  2. Which group of hierarchical models has the lowest AIC and has Akaike weights adding up to 0.95?

  3. Which models listed in part (b) are graphical?

  4. Which single graphical model has the lowest AIC?

  5. Which group of graphical models has the lowest AIC and has Akaike weights adding up to 0.95?

  6. For the five graphical models with the lowest AIC, draw the graphs and interpret them by giving the implied conditional independence relationships.

In all parts it may be helpful to know that the result of R generic function summary applied to the result of R function glmbb is a list, the results component of which is the printed data frame. For example

gout <- glmbb(...)
sout <- summary(gout)

then sout$results$criterion is the vector of criteria (AIC, BIC, or AICc, as the case may be), sout$results$weight is the vector of weights, and sout$results$formula is the vector of formulas expressed as character strings. The R function as.formula converts one character string to a formula. The R function isGraphical in the R package glmbb tells whether a formula corresponds to a graphical model.

5-2

For each of the top five models (according to AIC) in part (a) of problem 1 on this homework, what do the “observed equals expected” and “maximum entropy” principles say about the maximum likelihood estimates?

5-3

For the horseshoe crab data

library(CatDataAnalysis)
data(table_4.3)
names(table_4.3)
## [1] "color"  "spine"  "width"  "satell" "weight" "y"
sapply(table_4.3, class)
##     color     spine     width    satell    weight         y 
## "integer" "integer" "numeric" "integer" "integer" "integer"
table_4.3 <- transform(table_4.3, color = as.factor(color))

in problem 3.2 we did a Bayesian analysis of the model having formula

satell ~ 0 + color + weight

If instead we do a frequentist analysis

gout <- glm(satell ~ 0 + color + weight, family = poisson,
    data = table_4.3)
summary(gout)
## 
## Call:
## glm(formula = satell ~ 0 + color + weight, family = poisson, 
##     data = table_4.3)
## 
## Coefficients:
##          Estimate Std. Error z value Pr(>|z|)    
## color2 -4.978e-02  2.331e-01  -0.214   0.8309    
## color3 -2.549e-01  1.974e-01  -1.291   0.1967    
## color4 -4.996e-01  1.959e-01  -2.551   0.0108 *  
## color5 -5.018e-01  2.156e-01  -2.328   0.0199 *  
## weight  5.462e-04  6.811e-05   8.019 1.07e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 1050.8  on 173  degrees of freedom
## Residual deviance:  551.8  on 168  degrees of freedom
## AIC: 917.1
## 
## Number of Fisher Scoring iterations: 6

What does the “observed equals expected” principle say about this model? What are the submodel canonical sufficient statistics? Calculate their expected values. Interpret these submodel canonical sufficient statistics. What is their meaning?