LOGISTIC REGRESSION A statistical procedure to relate the probability of an event to explanatory variables Used in epidemiology to describe and evaluate the effect of a risk on the occurrence of a disease event. Example: Framingham Heart Study Coronary heart disease and blood pressure

LOGISTIC REGRESSION: AN EXAMPLE Event: Coronary Heart Disease Occurrence is the dependent variable, which takes 2 values: Yes or No. Risk factor: Blood pressure Systolic blood pressure is the independent variable X, a continuous measurement. The probability of getting coronary heart disease depends on blood pressure.

DATA

SCATTER PLOT

LINEAR REGRESSION FOR Prob.(CHD): NOT A GOOD IDEA!

PROPORTION WITH CHD BY SBP GROUP Systolic BP Range Proportion mmHg 0/ mmHg 2/ mmHg 3/3 1.00

LOGISTIC REGRESSION PROBABILITY MODEL 1 p(X) = exp (-  0 -   X) The probability of the event varies as an S-shaped function of the risk factor X: the logistic curve.

LOGISTIC CURVE MODEL: OCCURRENCE OF CHD AS A FUNCTION OF SBP

LOGISTIC MODEL: LOG ODDS p (X) log =  0 +  1 X 1 - p (X) The log of the odds of the event is a linear function of X. Log(odds of CHD) = (SBP)

ODDS The odds of an event is the chance that the event occurs divided by the chance of its not occurring: Odds = p/(1 - p) = p/q

  : KEY PARAMETER OF THE LOGISTIC MODEL p (X) log =  0 +  1 X 1 - p (X) The parameter   is like the slope of a linear regression model.   = 0 indicates that X has no effect on the probability, e.g., a man’s chance of CHD does not depend on his SBP.

 1 : KEY PARAMETER p (X) log =  0 +  1 X 1 - p (X) The coefficient  1 measures the amount of change in the log of the odds per unit change in X.

 1 : KEY PARAMETER log odds(X+1) =  0 +  1 (X+1) =  0 +  1 X+  1 log odds(X) =  0 +  1 X Difference in log odds =  1 E.g., the log of the odds of getting CHD increases by for an increase of 1 mmHg of systolic blood pressure. (Hard to explain to a patient!)

THE COEFFICIENT  1 AND THE ODDS RATIO Difference in log odds given by  1 translates into the odds ratio (OR). exp(  1 ) = OR = ratio of odds at risk level of X+1 to the odds when risk level is X  1 = 0  OR = 1.

THE COEFFICIENT $ 1 AND THE ODDS RATIO For example, the odds of CHD are multiplied by the factor exp(0.0243) = for every increase of 1 mmHg in SBP. A difference of 10 mmHg multiplies the odds of CHD by (1.025) 10, or

ESTIMATION OF THE PARAMETERS Technique: Maximum likelihood estimation For large sample sizes, the normal distribution is used to put a confidence interval around the estimate of the coefficient  .

HYPOTHESIS TESTING Ho:  1 = 0 No difference in risk at different levels of the risk factor X. No association between risk factor X and probability of occurrence.

HYPOTHESIS TESTING Ha:  1 =/= 0 or  1 > 0 (risk increases with X) or  1 < 0 (risk goes down as X increases)

HYPOTHESIS TESTING Ho: OR = 1 Ha: OR =/= 1 or OR > 1 (risk increases with X) or OR < 1 (X is protective)

RESULTS OF LOGISTIC REGRESSION OR with confidence interval and p value indicate whether there is a significant association between level of the risk factor and chance of occurrence OR = (1.015, 1.034), p < 0.001

RESULTS OF LOGISTIC REGRESSION Can be used to predict an individual’s risk: prob. of CHD when SBP = 180: p/q = exp{ (180)} Solve for p: prob. of CHD = 0.125

MULTIVARIATE LOGISTIC REGRESSION Model with additional risk factors: p (X) log =  0 +  1 X +  2 X 1 - p (X) Log(odds of CHD) =   0 +  1 (SBP) +  2 (CHOL) +  3 (smoker)