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Logistic Logistic Regression Regression

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  • Slide 1
  • Logistic Regression
  • Slide 2
  • Outline Review of simple and multiple regressionReview of simple and multiple regression Simple Logistic RegressionSimple Logistic Regression The logistic functionThe logistic function Interpretation of coefficientsInterpretation of coefficients continuous predictor (X)continuous predictor (X) dichotomous categorical predictor (X)dichotomous categorical predictor (X) categorical predictor with three or more levels (X)categorical predictor with three or more levels (X) Multiple Logistic RegressionMultiple Logistic Regression ExamplesExamples
  • Slide 3
  • Simple Linear Regression Model the mean of a numeric response Y as a function of a single predictor X, i.e. E(Y|X) = o + 1 f(x) Here f(x) is any function of X, e.g. f(x) = X E(Y|X) = o + 1 X (line) f(x) = X E(Y|X) = o + 1 X (line) f(x) = ln(X) E(Y|X) = o + 1 ln(X) (curved) f(x) = ln(X) E(Y|X) = o + 1 ln(X) (curved) The key is that E(Y|X) is a linear in the parameters o and 1 but not necessarily in X.
  • Slide 4
  • 0 = Estimated Intercept 1 = Estimated Slope w units 00 1 w units Simple Linear Regression Interpretable only if x = 0 is a value of particular interest. Always interpretable! = -value at x = 0 = Change in for every unit increase in x x 0 ^ ^ ^ ^ = estimated change in the mean of Y for a unit change in X.
  • Slide 5
  • Multiple Linear Regression We model the mean of a numeric response as linear combination of the predictors themselves or some functions based on the predictors, i.e. E(Y|X) = o + 1 X 1 + 2 X 2 ++ p X p E(Y|X) = o + 1 X 1 + 2 X 2 ++ p X p Here the terms in the model are the predictors E(Y|X) = o + 1 f 1 (X)+ 2 f 2 (X)++ k f k (X) Here the terms in the model are k different functions of the p predictors
  • Slide 6
  • Multiple Linear Regression For the classic multiple regression model E(Y|X) = o + 1 X 1 + 2 X 2 ++ p X p the regression coefficients ( i ) represent the estimated change in the mean of the response Y associated with a unit change in X i while the other predictors are held constant. They measure the association between Y and X i adjusted for the other predictors in the model.
  • Slide 7
  • General Linear Models Family of regression modelsFamily of regression models Response Model Type ContinuousLinear regression CountsPoisson regression Survival timesCox model BinomialLogistic regression UsesUses Control for potentially confounding factorsControl for potentially confounding factors Model building, risk predictionModel building, risk prediction
  • Slide 8
  • Logistic Regression Models relationship between set of variables X iModels relationship between set of variables X i dichotomous (yes/no, smoker/nonsmoker,)dichotomous (yes/no, smoker/nonsmoker,) categorical (social class, race,... )categorical (social class, race,... ) continuous (age, weight, gestational age,...)continuous (age, weight, gestational age,...) and dichotomous categorical response variable Ydichotomous categorical response variable Y e.g. Success/Failure, Remission/No Remission Survived/Died, CHD/No CHD, Low Birth Weight/Normal Birth Weight, etc e.g. Success/Failure, Remission/No Remission Survived/Died, CHD/No CHD, Low Birth Weight/Normal Birth Weight, etc
  • Slide 9
  • Logistic Regression Example: Coronary Heart Disease (CD) and Age In this study sampled individuals were examined for signs of CD (present = 1 / absent = 0) and the potential relationship between this outcome and their age (yrs.) was considered. This is a portion of the raw data for the 100 subjects who participated in the study.
  • Slide 10
  • Logistic Regression How can we analyze these data?How can we analyze these data? The mean age of the individuals with some signs of coronary heart disease is 51.28 years vs. 39.18 years for individuals without signs (t = 5.95, p