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1 What you've always wanted to know about logistic regression analysis, but were afraid to ask... Februari, Gerrit Rooks Sociology of Innovation Innovation Sciences & Industrial Engineering Phone:

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This Lecture Why logistic regression analysis? The logistic regression model Estimation Goodness of fit An example 2

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3 What's the difference between 'normal' regression and logistic regression? Regression analysis: –Relate one or more independent (predictor) variables to a dependent (outcome) variable

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4 What's the difference between 'normal' regression and logistic regression? Often you will be confronted with outcome variables that are dichotomic: –success vs failure –employed vs unemployed –promoted or not –sick or healthy –pass or fail an exam

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5 Example Relationship between hours studied for exam and success Hours# Failed exam # Passed exam? Total # students Prob. pass exam

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6 Linear regression analysis Why is this wrong?

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7 Logistic Regression The better alternative

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9 The logistic regression equation predicting probabilities predicted probability (always between 0 and 1) similar to regression analysis

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10 The Logistic function Sometimes authors rearrange the model or also

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11 How do we estimate coefficients? Maximum-likelihood estimation Parameters are estimated by `fitting' models, based on the available predictors, to the observed data The chosen model fits the data best, i.e. is closest to the data Fit is determined by the so-called log likelihood statistic

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12 Maximum likelihood estimation The log-likelihood statistic Large values of LL indicate poor fit of the model HOWEVER, THIS STATISTIC CANNOT BE USED TO EVALUATE THE FIT OF A SINGLE MODEL

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13 Quantity of Study HoursOutcome An example to illustrate maximum likelihood and the log likelihood statistic Suppose we know hours spent studying and the outcome of an exam

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14 Quantity of Study HoursOutcome Predicted probability (b 0 =0; b 1 = 0.05) Predicted probability (b 0 =-6.44; b 1 = 0.39) In ML different values for the parameters are `tried' Lets look at two possibilities: 1; b 0 = 0 & b 1 = 0.05; 2, b 0 = 0 & b 1 = 0.05

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15 Quantity of Study HoursOutcome Predicted probability (b0=0; b1 = 0.05) LL (b0=0; b1 = 0.05) We are now able to calculate the log likelihood statistic

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16 Outcome Pr (b0=0; b1 = 0.05) LL (b0=0; b1 = 0.05) Pr (b0=-6.44; b1 = 0.39) LL (b0=-6.44; b1 = 0.39) Two models and their log likelihood statistic Based on a clever algorithm the model with the best fit (LL closest to 0) is chosen

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17 After estimation How do I determine significance? Obviously SPSS does all the work for you How to interpret output of SPSS Two major issues 1.Overall model fit –Between model comparisons –Pseudo R-square –Predictive accuracy / classification test 2.Coefficients –Wald test –Likelihood ratio test –Odds ratios

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18 Model fit: Between model comparison The log-likelihood ratio test statistic can be used to test the fit of a model The test statistic has a chi-square distribution Model fit reduced model Model fit full model

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19 Model fit The log-likelihood ratio test statistic can be used to test the fit of a model Model fit reduced modelModel fit full model

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Between model comparison Estimate a null model Baseline model Estimate an improved model This model contains more variables Assess the difference in - 2LL between the models This difference follows a chi-square distribution degrees of freedom = # estimated parameters in proposed model – # estimated parameters in null model 20 Model fit reduced model Model fit full model

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21 Overall model fit R and R 2 R2 in multiple regression is a measure of the variance explained by the model SS due to regression Total SS

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22 Overall model fit pseudo R 2 Just like in multiple regression, logit R 2 ranges 0.0 to 1.0 –Cox and Snell cannot theoretically reach 1 –Nagelkerke adjusted so that it can reach 1 log-likelihood of model before any predictors were entered log-likelihood of the model that you want to test NOTE: R2 in logistic regression tends to be (even) smaller than in multiple regression

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23 What is a small or large R and R 2 ? Strength of correlation Small0.10 to 0.29 Medium0.30 to 0.49 Large0.50 to 1.00

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24 Overall model fit Classification table How well does the model predict outcomes? This means that we assume that if our model predicts that a player will score with a probability of.51 (above.5) the prediction will be a score (lower than.50 is a miss). spss output

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25 Testing significance of coefficients The Wald statistic: not really good In linear regression analysis this statistic is used to test significance In logistic regression something similar exists however, when b is large, standard error tends to become inflated, hence underestimation (Type II errors are more likely) t-distribution standard error of estimate estimate

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26 Likelihood ratio test an alternative way to test significance of a coefficient To avoid type II errors for some variables you best use the Likelihood ratio test model with variablemodel without variable

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27 Before we go to the example A recap Logistic regression –dichotomous outcome –logistic function –log-likelihood / maximum likelihood Model fit –likelihood ratio test (compare LL of models) –Pseudo R-square –Classification table –Wald test

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28 Illustration with SPSS Penalty kicks data, variables: –Scored: outcome variable, 0 = penalty missed, and 1 = penalty scored –Pswq: degree to which a player worries –Previous: percentage of penalties scored by a particulare player in their career

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29 SPSS OUTPUT Logistic Regression Tells you something about the number of observations and missings

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30 Block 0: Beginning Block this table is based on the empty model, i.e. only the constant in the model these variables will be entered in the model later on

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31 Block 1: Method = Enter Block is useful to check significance of individual coefficients, see Field New model this is the test statistic after dividing by -2 Note: Nagelkerke is larger than Cox

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32 Block 1: Method = Enter (Continued) Predictive accuracy has improved (was 53%) estimates standard error estimates significance based on Wald statistic change in odds

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33 How is the classification table constructed? oops wrong prediction

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34 How is the classification table constructed? pswqpreviousscoredPredict. prob

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35 How is the classification table constructed? pswqprevio us scoredPredict. prob. predict ed

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