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

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

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**Say good things, think good thoughts,**and do good deeds**Categorical Data Analysis**Chapter 5 (I): Logistic Regression for Quantitative Factors**Logistic Regression**• Binary response variable: Y ~ Bernoulli(p) • k quantitative/ordinal factors: x1, … , xk • model: SAS textbook Sec 8.5**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(b)— 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 b (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**LD 50**• LD50 (LD = lethal dose)= the dose level at which toxicity rate p(dose) is 50% • In the logistic regression with dose being the only x, LD50= -a/b • The instant change rate of risk (p) at LD50 is b/4**Example: Insecticide vs Beetles**See handout for SAS code and output**Only One Factor**• The estimate of b,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**More than 1 Factors**• The estimate of b is 34.27 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) • b1 is called the logistic regression coefficient for x1 adjusted for other factors**Confidence Intervals**Based on Fisher information matrix and asymptotical results of mle • Wald C.I. for bi : found by SAS • Wald C. I. for p(x): found by SAS PROC GENMOD with the OBSTATS option**Significance Tests**• H0: b1=0 vs. H1: b1is not zero • Wald test • LR test**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