1 Say good things, think good thoughts, and do good deeds

2 Categorical Data Analysis Chapter 5 (I): Logistic Regression for Quantitative Factors

3 Logistic Regression Binary response variable: Y ~ Bernoulli( ) k quantitative/ordinal factors: x 1, …, x k model: SAS textbook Sec 8.5

4 Interpretation (for Only One Factor) (Multiplicative effect on the odds) Increasing x by one unit is estimated to give the odds of response a increase by a factor of exp()— Not easy for investigators to understand

Interpretation (for Only One Factor) Interpretation of the effect of x on Y in terms of risk (or called response rate): –The bigger  (the effect of X on Y) is, the bigger the slope of a tangent line of the fitting curve (with respect to X) is: how the risk changes instantly at x 5

LD 50 LD50 (LD = lethal dose)= the dose level at which toxicity rate (dose) is 50% In the logistic regression with dose being the only x, LD50= - The instant change rate of risk () at LD50 is  6

7 Example: Insecticide vs Beetles dosage# of beetles exposed # of dead beetles ……… See handout for SAS code and output

8 Only One Factor The estimate of ,34.27, can be interpreted as: Increasing dose by one unit is estimated to give the odds of death a increase by a factor of exp(34.27) Interpret the effect on the risk of death at dose 1.70

9 More than 1 Factors The estimate of is can be interpreted as: Increasing x by one unit, keeping other factors fixed, is estimated to give the odds of death a increase by a factor of exp(34.27)  1 is called the logistic regression coefficient for x 1 adjusted for other factors

10 Confidence Intervals Based on Fisher information matrix and asymptotical results of mle Wald C.I. for  i : found by SAS Wald C. I. for (x): found by SAS PROC GENMOD with the OBSTATS option

11 Significance Tests H 0 :    vs. H 1 :   is not zero Wald test LR test

12 Examining the Fit of the Logit Model Plot fitted and observed rates on the same plot Residuals for logit models

Grouped Data Grouping data makes overall goodness of fit test sensible and possible Example: Crab data grouped by the width –Ungrouped: deviance=191.7, df=165, p-value=.076 –Grouped: deviance=6.25, df=6, p-value=.40 13