7/28/15. Review Homework. Overview. Lecture 6: Logistic Regression Analysis

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1 Lecture 6: Logistic Regression Analysis Christopher S. Hollenbeak, PhD Jane R. Schubart, PhD The Outcomes Research Toolbox Review Homework 2 Overview Logistic regression model conceptually Logistic regression model graphically Lots of equations Performing logistic regression in Stata Performing logistic regression in R Predicting SSI in liver transplantation 3 1

2 Logistic Regression Model Recall we use logistic regression when we have a binary dependent variable Independent variables could be continuous, binary, or categorical The goals becomes to estimate the effect of covariates on the probability that the dependent variable equals 1 Pr(y i = 1) = x 1i + + k x ki 4 When to Use Logistic Regression When you want to estimate or predict the probability that an event occurs Examples: What is the probability of developing a surgical site infection? What is the effect of age on risk of dying in the hospital? What do the data look like? 5 Pr(Died) Age 2

3 Logistic Regression We could fit a linear regression model to these data Called a linear probability model Often done in economics Almost never done in biomedical research! What would it look like? 7 Pr(Died) Age 8 What s Wrong with This? Does not fit the assumptions of linear regression model Not normally distributed Always heteroskedastic Predicted probabilities can be <0 or >1!! Need to fit a function that fits the data 9 3

4 Pr(Died) Age 10 Logistic Regression This functional form restricts the probability between 0 and 1 What functions look like this? Cumulative distributions functions! 11 Cumulative Distribution Functions CDF, Density x 4

5 Cumulative Distribution Function We just need to pick a probability distribution and use its CDF Which probability distribution? The Logistic Distribution Which is why we call this logistic regression 13 The Model The logistic CDF looks like this: e x 1i + + k x ki Pr(y i = 1) = 1+e x 1i + + k x ki If we instead model the ratio we get something simpler: Pr(y i = 1) Pr(y i = 0) = e x 1i + + k x ki The Model If we now take the natural log of both sides, we get something even simpler Pr(yi = 1) ln = x 1i + + k x ki Pr(y i = 0) This model is just a linear model of the log odds of risk 15 5

6 Quantifying Risk: One Event The best measure of risk that one event occurs is the probability of the event (p) Risk is not the only measure Could also use the odds of the event Odds is the ratio of the probability that it occurs to the probability that it doesn t occur p 1 p An odds of 2 (or 2:1) means that the event is twice as likely to occur as not to occur 16 Quantifying Risk: Two Events But what if I have two events (p 1 and p 2 )? How can I quantify relative risk? Risk Ratio p 1 p 2 If risk ratio < 1, event 1 is less likely to occur If risk ratio > 1, event 1 is more likely to occur If risk ratio = 1, the events are equally likely 17 Quantifying Risk: Two Events Another common alternative is the Odds Ratio p 1 1 p 1 p 2 1 p 2 If odds ratio < 1, event 1 is less likely to occur If odds ratio > 1, event 1 is more likely to occur If odds ratio = 1, the events are equally likely 18 6

7 Logistic Regression Coefficients Pr(yi = 1) ln = x 1i + + k x ki Pr(y i = 0) Raw coefficients tell you the effect of a one-unit change in the covariate on the log odds of risk But we cannot interpret this! But a simple transformation of the coefficient gives us the odds ratio of risk Odds Ratio = e k In logistic regression we always report and use odds ratios 19 Interpreting the Coefficients Be careful when using and reporting odds ratios Scale matters! = =2 Always report baseline probability so the reader knows what the odds apply to This will be the Intercept in the logistic regression output 20 Stata Code The Stata command for a logistic regression is logit depvar ind 1 ind 2, or The or options reports odds ratios instead of coefficients Important, otherwise you will have trouble with the interpretation 21 7

8 Example: Risk of SSI in Liver Tx Consider an example from our liver transplant data What factors are associated with risk of surgical site infection? logit ssi age4049 age5059 age60 female /// black ab0 ab1 ab2 ab3, or Example: Risk of SSI Without or option Logistic regression Number of obs = 777 LR chi2(9) = Prob > chi2 = Log likelihood = Pseudo R2 = ssi Coef. Std. Err. z P> z [95% Conf. Interval] age age age female black ab ab ab ab _cons Example: Risk of SSI With or option Logistic regression Number of obs = 777 LR chi2(9) = Prob > chi2 = Log likelihood = Pseudo R2 = ssi Odds Ratio Std. Err. z P> z [95% Conf. Interval] age age age female black ab ab ab ab _cons

9 Goodness of Fit How can you tell when you have a good model? Usual statistic is the C statistic Comparable to the R 2 in linear regression Lower bound of.5 Upper bound of 1 Computed as the area under the Receiver Operating Characteristic (ROC) curve Stata command: lroc Run on line right after logit statement Goodness of Fit Logistic model for ssi number of observations = 777 area under ROC curve = R Code The R function that performs logistic regression is glm() glm = Generalized Linear Model Must specify that the data come from the binomial family of distributions Create a glm object, then summarize lr1 <- glm(dat$depvar ~ dat$indvar dat$indvar k, family= binomial ) summary(lr1) confint(lr1) 9

10 R Results Call: glm(formula = data1$ssi ~ data1$age data1$age data1$age60 + data1$female + data1$black + data1$abmm, family = "binomial") Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) data1$age data1$age data1$age data1$female * data1$black data1$abmm Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 776 degrees of freedom Residual deviance: on 770 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 R Code R will not give you odds ratios You have to force them exp(cbind(or = coef(lr1), confint(lr1))) R Results > exp(cbind(or = coef(lr1), confint(lr1))) Waiting for profiling to be done... OR 2.5 % 97.5 % (Intercept) data1$age data1$age data1$age data1$female data1$black data1$abmm

11 R Code For area under the curve, you need an add-on package install.packages("deducer") library(deducer) This gives you a new function called rocplot() Apply the function to your glm object rocplot(lr1) R Results logit (data1$ssi ~ data1$age data1$age data1$age60 + data1$female + ) Sensitivity AUC= Specificity Reporting Logistic Regression Include the odds ratio, its 95% confidence interval, and p-value Do not include the Intercept!! Do mention the baseline probability in your text For p-values less than , indicate with < rather than the actual p-value Indicate reference group Include C-statistics in the table caption Usually don t include a plot of the ROC curve 33 11

12 34 The Narrative Women had 30% lower odds of developing SSI relative to men (OR = 0.70, p = 0.02), and African Americans had 4% greater odds of eveloping SSI, although this effect was not statistically significant (p = 0.94). 35 Summary Regression analysis for binary dependent variable Fits the data to a cumulative logistic distribution Raw parameters tell you the impact of a one-unit change in the covariates on the log odds of the event of interest Raising e to the power of your coefficients give you their odds ratios, or the impact of a one-unit change in your covariate on the odds of the event C-statistic measures goodness of fit 12

13 Homework Using the Liver Transplant data: What factors determine the probability of dying? Primary interest is in effect of SSI on risk of death Use the other covariates you used in your cost analysis Find at least one other *new* variable in the data set that you did not include in your cost analysis Is it better to use age as a continuous covariate or categories? Is it better to use HLA mismatches as a continuous covariate or categories? 37 13