LogitandProbitModels.ppt

1、1,個體數據之類別分析基本模型Logit and Probit Models: An Introduction,黃紀 政治大學 講座教授,2,Outline,I. Review of Basic Concepts: Cross-tables and Measures of Association Odds, Log of the odds=Logit, and Odds Ratio II. Binary Regression Models Logit Model (or Logisitic Regression) Probit Model III. Ordered Regression Mod

2、els Cumulative Probability (or Proportional Odds) Model Continuation Ratio (or Sequential Logit) Model IV. Multinomial Regression Models Multinomial Logit (MNL) Model Multinomial Probit (MNP) Model,Review of Basic Concepts,4,I. Basic Statistical Methods,常見之統計模型,5,II. Bivariate Discrete Variables: Me

3、asures of Association for 2x2 Tables（有差？冇差？） Sample data: nij and proportion,6,Estimated Joint Probabilities,Y,7,Estimated Conditional Probabilities,8,i. Scale of Measures of Association: 1.Unit scale: 2 Nominal Variables: between 0 and 1 0, 1 2 Ordinal Variables: between -1 and +1 -1, +1 2.0, ) mul

4、tiplicative scale ii. the Unit Scale: 1.Difference of Proportions: 2.Chi-Squared-Based Measures of Association 3.PRE Statistics: Proportional Reduction in Prediction Errors,9,iii. the Multiplicative Scale 1.The Concept of “Odds”勝算: The expected number of success for each failure odds=1 means equal c

5、hance of success and failure probability of success,10,2.Log of the odds = ln(odds) = logit (due to Joseph Berkson, 1944) 勝算之對數 , symmetric around 0 logit=0 means equal chance of success and failure exp(logit)=expln(odds)=odds,11,Probability, Odds, and Logit,13,3.Odds Ratio (Cross-Product Ratio) 勝算比

6、 When X and Y are independent, =1 The odds ratio treats the variables X and Y symmetrically,14,Sample odds ratio (cross-product ratio): 就22表而言，odds ratio的樣本估算式又稱為交叉相乘比（cross-product ratio），因為：,15,2008年立委與總統選舉投票模式之交叉分析,16,4.ln(odds ratio) = ln(odds1)-ln(odds2)=logit difference 5.Statistical Inference

7、 for odds ratio: Since sampling distribution of odds ratio is highly skewed, use 6.Relative Risk (RR)= 7.Odds Ratio= If the event of interest occurs infrequently, the odds ratio can be used as an estimate of RR.,17,Binary Regression Models,I. Historical Origins: Problems with linear probability mode

8、ls (LPM),18,19,20,II. Alternative Views of Binary Regression: 殊途同歸 Probability Model: where F is a cumulative distribution function (CDF),21,22,i. Latent Variable Regression: Threshold Model Identifying Assumptions of the Latent Variable Models and Implications the threshold is 0 the conditional mea

9、n of the error is 0,23,the conditional variance of the error is a constant: 1 in the probit and in the logit model, the magnitude of the slope depends on the scale of the dependent variable and cannot be interpreted directly. But identifying assumptions do not affect , which is an estimable function

10、. We can interpret changes in probabilities and odds.,24,ii. Random Utility Model (RUM) The Choice Set Ci and Random Utility: U=V+e Derivation of Choice Probability based on Utility Maximization Principle,25,The IID Type I Extreme Value Assumption of the Error Terms Implications of the RUM: Only Dif

11、ferences in Utility Matter: Difference between 2 Type I Extreme Value Distribution is Logistic Distribution The Overall Scale of Utility Is Irrelevant and thus Normalized to,26,iii. Generalized Linear Model (GLM) GLM for Binary Data Random (Stochastic) Component: Bernoulli Distribution,27,where,28,S

12、ystematic Component: Link Function: the logit link Alternative Link Function: Probit link Probit = , or,29,Logit Link: Logistic Regression based on Logistic Response Function Origin of the Logistic Growth Curve: Belgian mathematician Verhulst and the Effects of the Parameters and replace t with X,30

13、,Berksons (1944) “Logit” (log of the odds),31,Statistical Inference for Binary Regression Models: Estimation,Probability Model:,32,Log-Likelihood Function,33,Maximum Likelihood Estimation (MLE) For logit model: or,34,For simple logistic regression, the likelihood equations are,35,I. Hypothesis Testi

14、ng t-test and Likelihood Ratio Test of a Slope Coefficient CI: LR Test of a group of coefficients The Concept of “Deviance” between Two Nested Models LR Test of Model Comparison: lrtest command under identical sample size,36,II. Interpretation i. The Logit-Difference Interpretation: the Dependent V.

15、 is “Logit” Simple: the same as a linear regression Meaningless: change in the “logit” (log of the odds),37,ii. Predicted Probabilities Interpretation of Logit and Probit Models: the effect of a unit change in X on the predicted probability 1.The Effects of the Coefficients on the Logistic Growth Cu

16、rve (Long 1997, 62-64) the general (mathematical) logistic curve:,38,is the asymptotic upper limit (here we set =1.0); determines horizontal position or “takes-off point”; and controls steepness. (Hamilton 1992, 166) The Effects of the Intercept : When =0, the S-shaped curve passes through the =0.5

17、As gets larger, the curve shifts to the left; as gets smaller, the curve shifts to the right,39,The Effects of changing the Slope : When =0, the curve is simply a horizontal line When 0, the S-shaped curve stretches from lower left to the upper right: The smaller the , the more stretched out the cur

18、ve. The larger the , the curve increases more rapidly as x approaches 0.,40,When 0, the S-shaped curve is rotated 180o around x=0, i.e., stretches from upper left to the lower right,41,2.Marginal Effect (Partial Change): mfx/margins command Marginal Effect (Huang Skrondal and Rabe-Hesketh 2004, 32).

19、,79,Accordingly, the conditional probabilities of going beyond stage yj given being in stage yj or higher are: . Continuation ratio models are particularly useful for studying sequential processes in which research interest focuses on the conditional probabilities of continuing beyond a stage given

20、that the stage has been reached.,80,By taking the log of odds, I construct an upward continuation-ratio logit (or sequential logit) model (see Agresti 2002, 289-291; Amemiya 1985, 310-311; Hosmer and Lemeshow 2000, 290; McCullagh and Nelder 1989, 160-164) to estimate the effects of a vector of indep

21、endent variables xi of the ith individual voter on his or her sequential choice at the jth stage :,81,where the explanatory variables xi include voters presidential voting choice, interaction between such choice and the support for democracy, party preference, as well as four demographic variables (sex, age, education, and ethnicity).,82,83,Table 8.1 Continuation-Ratio Model of Voting on Referendum Issue #1 (Huang 2008, 130),Table 8.1 Continuation-Ratio Model of Voting on Referendum Issue #1 (continued),84,85,Table 8.1 Continuation-Ratio Model of Voting on