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

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: x 1 , … , x k model: SAS textbook Sec 8.5.

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

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  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(p) • k quantitative/ordinal factors: x1, … , xk • 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(b)— Not easy for investigators to understand

  5. 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

  6. 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

  7. Example: Insecticide vs Beetles See handout for SAS code and output

  8. 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

  9. 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

  10. 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

  11. Significance Tests • H0: b1=0 vs. H1: b1is 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

  13. 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

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