1 Introduction to Modeling Beyond the Basics (Chapter 7)

2 Content Simple and multiple linear regression Simple logistic regression –The logistic function –Estimation of parameters –Interpretation of coefficients Multiple logistic regression –Interpretation of coefficients –Coding of variables

3 How can we analyse these data? Table 1 Age and systolic blood pressure (SBP) among 33 adult women

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

5 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

6 Multiple linear regression Relation between a continuous variable and a set of i continuous or categorical 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

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

8 Multivariate analysis Model Outcome Linear regression continous Poisson regression counts Cox model survival Logistic regression binomial Choice of the tool according to study, objectives, and the variables –Control of confounding –Model building, prediction

9 Logistic regression Models the relationship between a set of variables x i –dichotomous (eat : yes/no) –categorical (social class,... ) –continuous (age,...) and –dichotomous variable Y Dichotomous (binary) outcome most common situation in biology and epidemiology

10 How can we analyse these data? Table 2 Age and signes of Coronary Heart Disease (CHD), 33 women CHD

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

12 Dot-plot: Data from Table 2

13 NO YES Y = x AGE

14 Table 3 - Prevalence (%) of signs of CHD according to age group

Dot-plot: Data from Table 3

16 Dot-plot: Data from Table 3 Diseased % Age (years) P 1-P

17 Dot-plot: Data from Table 3 Diseased % Age (years)

18 The logistic function (2) logit of P(y|x) {

19 The logistic function (1) Probability of disease x

20 The logistic function (2) logit of P(y|x) {

21 The logistic function (3) Advantages of the logit –Simple transformation of P(y|x) –Linear relationship with x –Can be continuous (Logit between -  to +  ) –Known binomial distribution (P between 0 and 1) –Directly related to the notion of odds of disease

22 Interpretation of  (1)

23 Practice 1. MI and Hyperhomocysteinemia? Hyper Homocysteinemia noyesTotal control case Total

24 Practice 1 Normal Homocysteine High Homocysteine MI (%) Odds Ln(Odds)

25 Normal HC  X = 0  ln(Odds)=  +  x 0   = ln(Odds) …….  = High HC  X=1  ln(Odds)=  +  x 1   = ln(Odds)-  …….  = (-0.39) = 1.06 OR ? = e  = 2.88 SE  = 0.33 How can you interpret  /OR?

26 Interpretation of  (2)  = increase in log-odds for a one unit increase in x Test of the hypothesis that  =0 (Wald test) Interval testing OR

27 If you run Linear Regression … Y = x High HC % MI in High HC = % MI in Normal HC = Diff = 25.7 %  1 What is your interpretation about  1 ?

28 Example Age (<55 and 55+ years) and risk of developing coronary heart disease (CD)

29 Results of fitting Logistic Regression Model

30 Interpretation of  (1)

31 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

32 Multiple logistic regression Effect modification –Can be modelled by including interaction terms

33 Reference Hosmer DW, Lemeshow S. Applied logistic regression.Wiley & Sons, New York, 1989