---
title: "Assignment 3"
subtitle: "The Anatomy of a Regression Model and Generalized Linear Models (Weeks 5–6)"
author: "YOUR NAME HERE"
date: "`r format(Sys.Date(), '%B %d, %Y')`"
urlcolor: blue
output:
  pdf_document:
    number_sections: true
    latex_engine: xelatex
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE)
# No package is required for this assignment. If you use any, load them here,
# for example: library(tidyverse)
```

# How to use this template {-}

1. Put your name in the `author:` field above and save this file as
   `assignment3_lastname.Rmd`.
2. Each part of each question is followed by an answer block that begins with
   **Answer.** Replace the italic placeholder with your answer. Leave the
   question text in place so that your peer reviewer can read your answer next
   to what was asked.
3. Write mathematics in LaTeX between dollar signs, for example
   `$\hat\beta_1 = \log(6/3)$` for inline math or `$$ ... $$` for a displayed
   equation. Where you use R, add a chunk (Ctrl/Cmd + Alt + I in RStudio) so
   that both your code and its output appear in the rendered file.
4. If you include an image, add it with `knitr::include_graphics("yourfile.png")` inside a
   chunk and upload the image file together with this `.Rmd`.
5. Knit to PDF before you submit (the **Knit** button, or
   `rmarkdown::render("assignment3_lastname.Rmd")` from the R console). The file 
   must render. If you do not have a LaTeX installation, run
   `tinytex::install_tinytex()` once (install the `tinytex` package first using `install.packages()`).
6. Submit the `.Rmd` file through Canvas.

# Question 1: Two Coding Standards in One Model

An analyst is trying to study whether neighborhood tree-canopy cover (continuous, in
percentage points of land area) affects a continuous depressive-symptom score,
using a linear model. The confounders are age (years), household income (six
ordered bands, coded 1 through 6 in the data file), and census region (four
regions, coded 1 through 4). The first regression model enters every variable as the
numeric variable in the data file (i.e., no dummy variables, no splines, no polynomials, etc):

$$E(Y \mid \text{canopy}, \text{age}, \text{income}, \text{region})
  = \beta_0 + \beta_1\,\text{canopy} + \beta_2\,\text{age}
  + \beta_3\,\text{income} + \beta_4\,\text{region}.$$

a) Split the right-hand side into its target and nuisance functions, and state
what each one is coded *for*.

**Answer.** *Write your answer here.*

b) Region is entered as a single numeric variable. What does that coding assert
about the outcome mean across the four regions? Is this a reasonable assertion?
What does the fix cost in parameters? Income is also a single numeric
variable; say in one sentence why the same objection applies less strongly for the income variable.

**Answer.** *Write your answer here.*

c) The canopy coefficient is 0.004 symptom-score points per percentage point of
canopy, with standard error 0.002, and a reviewer calls it "negligible."
Report the coefficient and standard error per 10 percentage points and say what
happens to the Wald test statistic (defined as the point estimate / standard error).
A colleague proposes a standardized effect, rescaling canopy to standard-deviation units
instead. The SD of canopy cover is 12 points in this cohort. In a separate 
replication cohort, the corresponding SD is 20. Why would the two cohorts' per-SD coefficients differ even if the
canopy effect was identical?

**Answer.** *Write your answer here.*

# Question 2: Coordinates or Filter?

A closed cohort is cross-classified by a binary exposure $X$, a binary
covariate $C$, and a binary outcome $Y$, with 100 people in each $(X, C)$ cell, with cell numerators representing the event tally:

| | $X = 0$ | $X = 1$ |
|:---|:---:|:---:|
| $C = 0$ | 10 / 100 | 20 / 100 |
| $C = 1$ | 30 / 100 | 40 / 100 |

a) Compute the risk difference, risk ratio, and odds ratio for $X$ within each
level of $C$, and say which contrast is the same in both strata. Then consider
the saturated binomial model
$g[P(Y = 1 \mid X, C)] = \beta_0 + \beta_1 X + \beta_2 C + \beta_3 XC$ under
the identity, log, and logit links. Without fitting anything, say what the
fitted risks in the four cells are in terms of model parameters, write the four
identity-link coefficients by hand. What does the link determine in a saturated model?

**Answer.** *Write your answer here.*

b) Now drop the product term, leaving three parameters for four cells. Name the
one pattern each link's model now keeps (one risk difference, one risk ratio,
or one odds ratio). Exactly one of the three reproduces all four observed
risks: say which, and explain how a three-parameter model can be lossless
here. For the other two, say what feature of the table ends up in the
residuals. (No fitting is needed; in R you would find fitted risks of
0.122 / 0.179 / 0.282 / 0.415 under the log link and 0.114 / 0.186 / 0.286 /
0.414 under the logit.)

**Answer.** *Write your answer here.*

# Question 3: Choosing the Family Is Choosing the Variance

An analyst models an indicator of above-median weight gain ($Y$) against
quitting smoking ($X$) with a log link, so that $\exp(\beta_1)$ is a risk
ratio, under three families with the same right-hand side.

- *Crude model* ($X$ only): risks are 0.467 among non-quitters and 0.596 among
  quitters, and the binomial, Poisson, and Gaussian families return the same
  risk ratio, 1.275517, to six decimals.
- *Adjusted model* ($X$ plus age, sex, race, exercise, marital status, and
  smoking intensity): the three families return 1.333, 1.355, and 1.348. The
  model-based standard error of $\hat\beta_1$ is 0.050 under the binomial and
  0.080 under the Poisson; a robust (sandwich) correction of the Poisson fit
  gives 0.052, with $\hat\beta_1$ unchanged.

a) Why are the three crude estimates identical, and the three adjusted
estimates close but not identical? Say what the family contributes to the
point estimate, and why that contribution disappears in a saturated model.

**Answer.** *Write your answer here.*

b) The Poisson model-based standard error is 60% larger than the binomial's.
Using the two variance functions, $\mu$ and $\mu(1 - \mu)$, explain the
direction of the error, and say which direction it would take instead for a
count outcome whose variance exceeds its mean.

**Answer.** *Write your answer here.*

# Question 4: Two Ways a GLM Fails to Converge

a) An analyst fits a log-binomial model, `family = binomial(link = "log")`,
to a binary outcome and receives the message *"no valid set of coefficients
has been found: please supply starting values."* Why can the binomial 
likelihood not be evaluated there? Why can a logistic model never produce this message?

**Answer.** *Write your answer here.*

b) With starting values supplied, the fit converges (risk ratio 1.33, largest
fitted probability 0.85). Give one reason to treat such a rescued fit
with caution. Name two routine alternatives, and detail how you would apply them to data.

**Answer.** *Write your answer here.*

c) In a 12-site cohort study that adjusts for site as a nuisance factor, one
site enrolled 9 patients, all of whom had the outcome. The logistic fit reports
a coefficient of 17.5 for that site's indicator, with standard error 789, and
fitted probabilities of 1.000 for those 9 patients. Name this phenomenon,
and give two remedies to the problem discussed (or to be discussed) in class. Say in one sentence how this failure
differs from the one in (a).

**Answer.** *Write your answer here.*