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chapter14a - Stat 704 Data Analysis I Fall 2011 Generalized...

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Stat 704: Data Analysis I, Fall 2011 Generalized linear models Generalize regular regression to non-normal data { ( Y i , x i ) } n i =1 , most often Bernoulli or Poisson Y i . The general theory of GLMs has been developed to outcomes in the exponential family (normal, gamma, Poisson, binomial, negative binomial, ordinal/nominal multinomial). The i th mean is μ i = E ( Y i ) The i th linear predictor is η i = β 0 + β 1 x i 1 + · · · + β k x ik = x i β . A GLM relates the mean to the linear predictor through a link function g ( · ): g ( μ i ) = η i = x i β . 1
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Chapter 14: Logistic Regression 14.1, 14.2 Binary response regression Let Y i Bern( π i ). Y i might indicate the presence/absence of a disease, whether someone has obtained their drivers license or not, etc. (pp. 555-556) We wish to relate the probability of “success” π i to explanatory covariates x i = (1 , x i 1 , . . . , x ik ). Y i Bern( π i ) , and E ( Y i ) = π i and var( Y i ) = π i (1 π i ). 2
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The identity link gives π i = β x i . When x i = (1 , x i ) , this reduces to Y i Bern( β 0 + β 1 x i ) . When x i large or small, π i can be less than zero or greater than one. Appropriate for a restricted range of x i values. Can of course be extended to π i = β x i where x i = (1 , x i 1 , . . . , x ik ). Can be fit in SAS proc genmod . 3
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Individual Bernoulli vs. aggregated binomial Data can be stored in one of two ways: If each subject has their own individual binary outcome Y i , we can write model y=x1 x2 in proc genmod or proc logistic . If data are grouped, so that there are Y · j successes out of n j with covariate x j , j = 1 , . . . , c , then write model y/n=x1 x2 . This method is sometimes used to reduce a very large number of individuals n to a small number of distinct covariates c . 4
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Example : Association between snoring (as measured by a snoring score) and heart disease. Let s be someone’s snoring score, s ∈ { 0 , 2 , 4 , 5 } . Heart disease Proportion Snoring s yes no yes Never 0 24 1355 0.017 Occasionally 2 35 603 0.055 Nearly every night 4 21 192 0.099 Every night 5 30 224 0.118 This is fit in proc genmod : data glm; input snoring disease total @@; datalines; 0 24 1379 2 35 638 4 21 213 5 30 254 ; proc genmod; model disease/total = snoring / dist=bin link=identity; run; 5
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The GENMOD Procedure Model Information Description Value Distribution BINOMIAL Link Function IDENTITY Dependent Variable DISEASE Dependent Variable TOTAL Observations Used 4 Number Of Events 110 Number Of Trials 2484 Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 2 0.0692 0.0346 Pearson Chi-Square 2 0.0688 0.0344 Log Likelihood . -417.4960 . Analysis Of Parameter Estimates Parameter DF Estimate Std Err ChiSquare Pr>Chi INTERCEPT 1 0.0172 0.0034 25.1805 0.0001 SNORING 1 0.0198 0.0028 49.9708 0.0001 SCALE 0 1.0000 0.0000 . . 6
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The fitted model is ˆ π ( s ) = 0 . 0172 + 0 . 0198 s. For every unit increase in snoring score s , the probability of heart disease increases by about 2%.
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