1. Introduction to Logistic Regression

2. Simple linear regression Table 1 Age and systolic blood pressure (SBP) among 33 adult women

3. SBP (mm Hg) Age (years) adapted from Colton T. Statistics in Medicine. Boston: Little Brown, 1974

4. Simple linear regression Relation between 2 continuous variables (SBP and age) Regression coefficient  1 –Measures association between y and x –Amount by which y changes on average when x changes by one unit –Least squares method y x Slope

5. Multiple linear regression Relation between a continuous variable and a set of i continuous variables Partial regression coefficients  i –Amount by which y changes on average when x i changes by one unit and all the other x i s remain constant –Measures association between x i and y adjusted for all other x i Example –SBP versus age, weight, height, etc

6. Multiple linear regression Predicted Predictor variables Response variable Explanatory variables Outcome variable Covariables Dependent Independent variables

7. General linear models Family of regression models Outcome variable determines choice of model Outcome Model ContinuousLinear regression CountsPoisson regression SurvivalCox model BinomialLogistic regression

8. Logistic regression Models relationship between set of variables x i –dichotomous (yes/no) –categorical (social class,... ) –continuous (age,...) and –dichotomous (binary) variable Y Dichotomous outcome very common situation in many fields

9. Logistic regression (1) Table 2 Age and signs of coronary heart disease (CD)

10. How can we analyse these data? Compare mean age of diseased and non- diseased –Non-diseased: 38.6 years –Diseased: 58.7 years (p<0.0001) Linear regression?

11. Dot-plot: Data from Table 2

12. Logistic regression (2) Table 3 Prevalence (%) of signs of CD according to age group

13. Dot-plot: Data from Table 3 Diseased % Age group

14. Logistic function (1) Probability of disease x

15. Transformation logit of P(y|x) {  = log odds of disease in unexposed  = log odds ratio associated with being exposed e  = odds ratio

16. Fitting equation to the data Linear regression: Least squares Logistic regression: Maximum likelihood Likelihood function –Estimates parameters  and  –Practically easier to work with log-likelihood

17. Maximum likelihood Iterative computing –Choice of an arbitrary value for the coefficients (usually 0) –Computing of log-likelihood –Variation of coefficients’ values –Reiteration until maximisation (plateau) Results –Maximum Likelihood Estimates (MLE) for  and  –Estimates of P(y) for a given value of x

18. Multiple logistic regression More than one independent variable –Dichotomous, ordinal, nominal, continuous … Interpretation of  i –Increase in log-odds for a one unit increase in x i with all the other x i s constant –Measures association between x i and log-odds adjusted for all other x i

19. Interpreting Odds Odds tells us that a single unit increase in x 1, holding x 2 ; : : : ; x 12 constant, is associated with an increase in the odds that a customer accepts the offer by a factor of exp(β 1 ) Odd interpretation applies to continuous and dummy variables