1 Chapter 16 logistic Regression Analysis

2 Content Logistic regression Conditional logistic regression Application

3 Purpose: Work out the equations for logistic regression which are used to estimate the dependent variable (outcome factor) from the independent variables (risk factors). Logistic regression is a kind of nonlinear regression. Data: 1.The dependent variable is a binary categorical variable that has two values such as "yes" and "no“. 2.All of the independent variables, at least, most of which should be categories. Of course, some of them can be numerical variable. The categorical variable should be quantified.

4 Implication: Logistic regression can be used to study the quantitative relations between the happening of some diseases or phenomena and many risk factors. There are some demerits to use test (or u test ): 1. can only study one risk factor. 2. can only educe the qualitative conclusion.

5 Category: 1.Between-subjects (non-conditional) logistic regression equation 2. Paired (conditional) logistic regression equation

6 § 1 logistic regression （ non-conditional logistic regression ）

7 I Basic Conception The probability of positive outcome under the function of m independent variables can be marked like this:

8 If: Regression model Probability: P ： 0 ～ 1 ， logitP ：－ ∞ ～ ∞ 。 Scale:

9 Figure 16-1 the figure of logistic function

10 The meaning of model parameter By constant we mean the natural logarithm of likelihood ratio between happening and non-happening when exposure dose is zero. By regression coefficient we mean the change of logitP when the independent variable changes by one unit.

11 The statistical indicator--odds ratio which is used to measure the function of risk factor in the epidemiology,the formula of computation is: Odds ratio (OR)

12 The relationship with logistic P ：

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14 II the parametric estimation of logistic regression model 1. parametric estimation Theory ： the estimation of likelihood

15 2.Estimation of OR It can show the OR of two different levels （ c 1 ， c 0 ） of one factor.

16 e.g.: 16-1 Table 16-1 is a case-control data which is used to study the relations among smoking 、 drinking and esophagus cancer, please try running logistic regression analysis. Definite every variable’s code

17 Table16-1 the case-control data of the relation between smoking and esophagus cancer

18 Results: 95  confidence interval of The OR of smoking and nonsmoking ： The OR of drinking and no drinking

19 III the hypothesis test of logistic regression model 1. Likelihood test 2. Wald test comparing the estimations of parameters with zero, the control is its standard error, statistics are: Both of are more than 3.84, that is to say that esophagus cancer 、 smoking and drinking have relations with each other. The conclusion is same as above.

20 methods ： forward selection 、 backward elimination and stepwise regression. Test statistics ： it is not F statistic ， but one of likelihood 、 Wald test and score test statistics. IV variable selection e.g.: 16-2 In order to discuss the risk factors that relate to coronary heart disease, to take case-control study on 26 coronary heart disease patients and 28 controllers, table 16-2 and table 16-3 show the definition of all factors and the data. Please try using logistic stepwise regression to select the risk factors.

21 Table 16-2 eight probable risk factors of coronary heart disease and valuation

22 Table 16-3 the case-control data of heart disease’s risk factors

23 Table 16-4 e.g.16-2 the independent variables which are entering equation and estimations of related parameters Learn how to see the results ！

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25 Content Logistic regression Conditional logistic regression Application

26 I Principle §2 conditional logistic regression

27 Table 16-5 the data format of 1: M conditional logistic regression * t = 0 is the case and the others are the control.

28 The model of conditional logistic

29 II applied example

30 Table 16-7 the data table of 1:2 paired case-control study about larynx cancer P344:

31 Table16-8 e.g.16-3 The Estimation of independent variables and related parameters which have entered the equation Using stepwise Six risk factors variable selection four factors enter equation ， Table16-9 shows the results 。

32 Content Logistic regression Conditional logistic regression Application

33 I the application of logistic regression 1 ． The analysis of epidemiologic risk factors One feature of logistic regression is that the meaning of parameter is clear, so logistic regression is suitable for epidemiologic study. § 3 the application of logistic regression and the notice

34 2 ． Analysis of clinical experiment The goal of clinical experiment is to assess the effect of some drugs or cure methods, if there are some confounding factors, and they are not balance among teams, the final results will be wrong. So it is necessary to adjust these factors during the process of analysis. when dependent variable is binary, we can use logistic regression to analyze and get the adjusted results.

35 3 ． Analyze dose–response of drugs or poisons In the studies about dose–response of some drugs or poisons, if the date is the logarithm of dose,the Probability distribution close to normal. The distribution of normal function is very similar to logistic regression, then we can express their relation through the following model. (While P is the positive rate; X is dose.)

36 4 ． Forecast and discrimination logistic regression is a model of probability ， so we can use it to predict the probability of something. For example in clinical we can discriminate the probability of some diseases under some index. please refer to the chapter 18 about discrimination.

37 II the notice of application of logistic regression

38 summary ： Purpose: Work out the equations for logistic regression which are used to estimate the dependent variable (outcome factor) from the independent variable (risk factor). Logistic regression belong to probability type and nonlinear regression. Data: 1.The dependent variable is a binary categorical variable that has two values such as "yes" and "no“. 2.All of the independent variables, at least, most of which should be categories. Of course, some of them can be numerical variable. The categories variable should be measure by number.

39 Implication: Logistic regression can be used to study the quantitative relations between the happening of some disease or phenomena and many risk factors Category: 1.Between-subjects (non-conditional) logistic regression equation 2. Paired (conditional) logistic regression equation

40 Thinking ： In order to analysis the influent factors of the rescue of AMI patients, a hospital collected five years’ data of AMI patients (there are many related factors,this case only lists three ones for the limited space), which has 200 cases in total, the data has been shown in the following table, P=0 means successful rescue ， P=1 means death ； X 1 =1 means shock before rescue ， X 1 =0 means no shock before rescue ； X 2 =1 means heart failure before rescue ， X 2 =0 means no heart failure before rescue ； X 3 =1 means that it has been more than 12 hours from the beginning of AMI symptom to rescue ， X 3 =0 means the time has not passed 12 hours. which analysis method is the best one? why? which analysis method is the best one? why? which result can we got ？ which result can we got ？

41 The data of the rescue risk factor of the AMI patients The data of the rescue risk factor of the AMI patients P=0(successfully rescued) P=1(death) X1X2X3N X1X2X3N