Topics

1. Hypothesis Testing
   • Wald t-tests for individual parameters
   • Drop in Deviance tests involving 1 or more parameters
2. Interpretation of estimates
3. Confidence Intervals
4. Residual Plots

Assignment

1. Conceptual Exercises 1-8 (Be prepared to discuss in lab)
2. Exercise 11 parts a-f, plus summary described below.

Commands for the Assignment

1. Download the O-Ring data, Ex2011.asc
2. Plot Failure versus Temperature. Scatter plots may not be very illuminating with binary outcomes. Add the ordinary least squares regression line to the plot. What problems are there with using OLS to model Failure probabilities here?
3. Fit a logistic regression with failure as the Response and temp as the explanatory varialbe:
   • Go to the Statistics menu and select Generalized Linear Models (under Splus 2000, go to the Regression Menu)
   • Select Ex2011 as the dataframe
   • Under Model Family, specify binomial. The default link is logit; leave that alone for now.
   • The model formula is specified the same as in linear regression; in this case
     failure ~ temp
   • click OK to run
   Using the output in the report window, you should be able to answer questions a-f. For some you will need to calculate p-values; either use the tables or the S-Plus. To get the area to the left of the point z [i.e. P(Z < z)] using a normal distribution, use pnorm(z); for a Chi-Squared distribution use pchisq(z, df) where df = degrees of freedom.
4. Create a new variable that is temp - 31. Fit the logistic regression model using this centered variable (i.e. review discussion on page 182-183). The intercept now corresponds to the logit of the failure probability when temp = 31. Use this to construct a 95% confidence interval for the probability of failure at temp = 31.

Using your results, write up on a separate sheet (one page max) a report that describes your statistical analysis model of O-ring failure. For this, imagine that you were assigned to write a report (prior to the Challenger disaster) regarding O-ring failure at low temperatures. Discuss what evidence there is to suggest that failure depends on temperature. Using this analysis what would your recommendation be for flights at 31 F? While predictions at 31 F are clearly extrapolations, collecting additional data is not always an option and decisions have to be made. While 31 F is historically important, make sure that you explain how the odds of failure change with increases in temperature. The report should be written so that an engineer or a person in a decision making capacity could understand the results.