updates.

pages/instructor-schedule.md

@@ -103,22 +103,43 @@ _Thurs, week 9_

 - simulation study presentations.

_Tues, week 10_
 _Tues, week 10_

 - Missing data

_Thurs, week 10_

 - Logistic regression lecture

_Tues, week 11_

 - power analysis lecture 
 
_Thurs, week 10_
_Thurs, week 11_

 - likelihood lecture

_Tues, week 11_
_Tues, week 12_

 - LDA lecture
 
_Thurs, week 11_
_Thurs, week 12_

 - cancelled
 
_Tues, week 13_

 - LDA lecture?
 - Logistic regression quiz
 - course evals
 - project prep time
 
_Thurs, week 13_

 - presentations
 
_Tues, week 14_

 - presentations
 

 

pages/quiz-logistic.Rmd

@@ -0,0 +1,91 @@
---
title: 'Quiz: Logistic Regression'
author: "Nicholas Reich, adapted from Project MOSAIC"
date: "April, 2016"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Overview

The NASA space shuttle Challenger had a catastrophic accident during launch on January 28, 1986. Photographic evidence from the launch showed that the accident resulted from a plume of hot flame from the side of one of the booster rockets which cut into the main fuel tank. US President Reagan appointed a commission to investigate the accident. The commission concluded that the jet was due to the failure of an O-ring gasket between segments of the booster rocket.

An important issue for the commission was whether the accident was avoidable. Attention focused on the fact that the ground temperature at the time of launch was 31 degrees Fahrenheit, much lower than for any previous launch. Commission member and Nobel laureate physicist Richard Feynman famously demonstrated, using a glass of ice water and a C-clamp, that the O-rings were very inflexible when cold. But did the data available to NASA before the launch indicate a high risk of an O-ring failure?

Here are some initial commands to load and look at the dataset, which has 23 observations, shown below sorted by temperature: 
```{r, message=FALSE, echo=FALSE}
library(pander)
oring = fetch::fetchData("oring-damage.csv")
oring_srted <- oring[order(oring$temp),]
row.names(oring_srted) <- NULL #seq_along(1:nrow(oring))
pander(oring_srted)
```

We fit a model to predict booster rocket damage as a function of temperature at launch. Before fitting the model, we centered our temperature variable by subtracting 60 from each observation so the model we fit is:

$$ logit[Pr(damage==1|temp)] = \beta_0 + \beta_1 \cdot (temp - 60). $$

This is the fitted model output from R.
```{r, echo=FALSE}
oring$temp_ctr <- oring$temp - 60
mod = glm(damage ~ temp_ctr, data = oring, family = "binomial")
pander(round(summary(mod)$coef,2))
```



```{r, echo=FALSE, include=FALSE}
cm <- coef(mod)
round(exp(cm),2)
```
We also note that $e^{`r round(cm[1], 2)`}$=`r round(exp(cm[1]), 2)`, and $e^{`r round(cm[2], 2)`}$=`r round(exp(cm[2]), 2)`.

And here are a few graphs illustrating possible fitted models to the data.

```{r, echo=FALSE}
library(ggplot2)
dat <- dplyr::data_frame(x = 30:90,
                  y1 = boot::inv.logit(cm[1] + (x-60)*cm[2]),
                  y2 = boot::inv.logit(cm[1] + (x-60)*-cm[2]),
                  y3 = boot::inv.logit(cm[1] + (x-45)*cm[2]))
p1 <- ggplot(dat, aes(x)) +
    geom_line(aes(y=y1)) + ylab(NULL) + xlab("temp.") + #geom_rug(aes(jitter(temp)), data=oring) +
    ggtitle("Figure B")
    #stat_smooth(aes(y=y1), se=FALSE, method='glm', method.args=list(family='binomial')) + ylab("prob. of damage") + xlab("temp.")
p2 <- ggplot(dat, aes(x)) +
    geom_line(aes(y=y2)) + ylab("prob. of damage") + xlab("temp.") +
    ggtitle("Figure A")
    #stat_smooth(aes(y=y2), se=FALSE, method='glm', method.args=list(family='binomial')) + ylab("prob. of damage") + xlab("temp.")
p3 <- ggplot(dat, aes(x)) +
    geom_line(aes(y=y3)) + ylab(NULL) + xlab("temp.") +
    ggtitle("Figure C")
    #stat_smooth(aes(y=y3), se=FALSE, method='glm', method.args=list(family='binomial')) + ylab("prob. of damage") + xlab("temp.")
gridExtra::grid.arrange(p2, p1, p3, ncol=3)
```


<!--
Question 1: what is the interpretation of $\beta_0$ in this model?

Question: What is the model estimated probability of booster rocket damage associated with a temperature of 70 degrees. Hint: inv.logit() is the inverse of the logit function or logit^{-1}()

 - 1.11 - 0.23 * 70
 - 1.11 - 0.23 * 10
 - inv.logit(1.11 - 0.23 * 10)
 - inv.logit(1.11 - 0.23 * 70)
  - - 0.23 + 1.11 * 10
 - inv.logit(- 0.23 + 1.11 * 70)


Question 2: Which graph shows the predicted probabilities of damage from the fitted model shown above?

Question: What are reasons for using the logit link function in logistic regression? 

 - prevents fitted probabilities from being outside the interval (0,1)
 - it gives you coefficients that have the (reasonably) intuitive interpretation of odds ratios
 - it is the only reasonable link function for binary data

-->