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Logistic Regression I HRP 261 2/09/04 Related reading: chapters 4.1-4.2 and 5.1-5.5 of Agresti

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Outline Introduction to Generalized Linear Models The simplest logistic regression (from a 2x2 table)—illustrates how the math works… Step-by-step examples Dummy variables – Confounding and interaction Introduction to model-building strategies

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Generalized Linear Models (chapter 4 of Agresti) Twice the generality! The generalized linear model is a generalization of the general linear model

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Why generalize? General linear models require normally distributed response variables and homogeneity of variances. Generalized linear models do not. The response variables can be binomial, Poisson, or exponential, among others. Allows use of linear regression and ANOVA methods on non-normal data

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Why not just transform? A traditional way of analyzing non-normal data is to transform the response variable so it is approximately normal, with constant variance. And then apply least squares regression. E.g., derivative[ (lnY i -(mx+b)) 2] =0 But then g(Y i ) has to be normal, with constant variance. “Maximum likelihood” is more general than least squares

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Example : The Bernouilli (binomial) distribution Smoking (cigarettes/day) Lung cancer; yes/no y n

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Could model probability of lung cancer…. = + 1 *X Smoking (cigarettes/day) The probability of lung cancer ( ) 1 0 But why might this not be best modeled as linear? [ ]

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Alternatively… log( /1- ) = + 1 *X Logit function

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Generalized Model G( )= + 1 *X + 2 *W + 3 *Z…. The link function =G( )= + 1 *X + 2 *W + 3 *Z…. Traditional linear regression, the identity link

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The link function The relationship between a linear combination of the predictors and the response is specified by a non-linear link function (example=log function, or the inverse of the exponential) For traditional linear models in which the response variable follows a normal distribution, the link function is the identity link. For Bernouilli/binomial, link function is: logit (or log odds)

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The Logit Model Logit function (log odds) Baseline odds Linear function of risk factors and covariates for individual i: 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 …

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Relating odds to probabilities oddsalgebraprobability

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Probabilities associated with each individual’s outcome: Individual Probability Functions Example:

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The Likelihood Function The likelihood function is an equation for the joint probability of the observed events as a function of

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Maximum Likelihood Estimates of Take the log of the likelihood function to linearize it Maximize the function (just basic calculus): Take the derivative of the log likelihood function Set the derivative equal to 0 Solve for

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“Adjusted” Odds Ratio Interpretation

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Adjusted odds ratio, continuous predictor

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Practical Interpretation The odds of disease increase multiplicatively by e ß for for every one-unit increase in the exposure, controlling for other variables in the model.

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Simple Logistic Regression

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2x2 Table (courtesy Hosmer and Lemeshow) Exposure=1Exposure=0 Disease = 1 Disease = 0

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(courtesy Hosmer and Lemeshow) Odds Ratio for simple 2x2 Table

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Example 1: CHD and Age (2x2) (from Hosmer and Lemeshow) =>55 yrs<55 years CHD Present CHD Absent 2122 651

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The Likelihood

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The Log Likelihood

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The Log Likelihood, cont.

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Derivative of the log likelihood

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Maximize

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Maximize

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Hypothesis Testing H 0 : =0 2. The Likelihood Ratio test: 1. The Wald test: Reduced=reduced model with k parameters; Full=full model with k+p parameters 3. The Score Test (deferred for later discussion)

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Hypothesis Testing H 0 : =0 2. What is the Likelihood Ratio test here? – Full model = includes age variable – Reduced model = includes only intercept Maximum likelihood ought to be (.43) 43 x(.57) 57 …does MLE yield this?… 1. What is the Wald Test here?

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Likelihood value for reduced model = marginal odds of CHD!

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Likelihood value of full model

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Finally the LR…

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Example 2: >2 exposure levels *(dummy coding) CHD status WhiteBlackHispanicOther Present5201510 Absent2010 (From Hosmer and Lemeshow)

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SAS CODE data race; input chd race_2 race_3 race_4 number; datalines; 0 0 0 0 20 1 0 0 0 5 0 1 0 0 10 1 1 0 0 20 0 0 1 0 10 1 0 1 0 15 0 0 0 1 10 1 0 0 1 10 end; run; proc logistic data=race descending; weight number; model chd = race_2 race_3 race_4; run; Note the use of “dummy variables.” “Baseline” category is white here.

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What’s the likelihood here?

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SAS OUTPUT – model fit Intercept Intercept and Criterion Only Covariates AIC 140.629 132.587 SC 140.709 132.905 -2 Log L 138.629 124.587 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 14.0420 3 0.0028 Score 13.3333 3 0.0040 Wald 11.7715 3 0.0082

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SAS OUTPUT – regression coefficients Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.3863 0.5000 7.6871 0.0056 race_2 1 2.0794 0.6325 10.8100 0.0010 race_3 1 1.7917 0.6455 7.7048 0.0055 race_4 1 1.3863 0.6708 4.2706 0.0388

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SAS output – OR estimates The LOGISTIC Procedure Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits race_2 8.000 2.316 27.633 race_3 6.000 1.693 21.261 race_4 4.000 1.074 14.895 Interpretation: 8x increase in odds of CHD for black vs. white 6x increase in odds of CHD for hispanic vs. white 4x increase in odds of CHD for other vs. white

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Example 3: Prostrate Cancer Study Question: Does PSA level predict tumor penetration into the prostatic capsule (yes/no)? Is this association confounded by race? Does race modify this association (interaction)?

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1.What’s the relationship between PSA (continuous variable) and capsule penetration (binary)?

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Capsule (yes/no) vs. PSA (mg/ml) psa vs. capsule capsule 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 psa 0102030405060708090100110120130140

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Mean PSA per quintile vs. proportion capsule=yes S-shaped? proportion with capsule=yes 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 PSA (mg/ml) 01020304050

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logit plot of psa predicting capsule, by quintiles linear in the logit? Est. logit 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 psa 01020304050

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psa vs. proportion, by decile… 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 010203040506070 proportion with capsule=yes PSA (mg/ml)

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logit vs. psa, by decile Est. logit 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 psa 010203040506070

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model: capsule = psa model: capsule = psa Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 49.1277 1 <.0001 Score 41.7430 1 <.0001 Wald 29.4230 1 <.0001 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.1137 0.1616 47.5168 <.0001 psa 1 0.0502 0.00925 29.4230 <.0001

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Model: capsule = psa race Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -0.4992 0.4581 1.1878 0.2758 psa 1 0.0512 0.00949 29.0371 <.0001 race 1 -0.5788 0.4187 1.9111 0.1668 No indication of confounding by race since the regression coefficient is not changed in magnitude.

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Model: capsule = psa race psa*race Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2858 0.6247 4.2360 0.0396 psa 1 0.0608 0.0280 11.6952 0.0006 race 1 0.0954 0.5421 0.0310 0.8603 psa*race 1 -0.0349 0.0193 3.2822 0.0700 Evidence of effect modification by race (p=.07).

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---------------------------- race=0 ---------------------------- Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.1904 0.1793 44.0820 <.0001 psa 1 0.0608 0.0117 26.9250 <.0001 ---------------------------- race=1 ---------------------------- Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.0950 0.5116 4.5812 0.0323 psa 1 0.0259 0.0153 2.8570 0.0910 STRATIFIED BY RACE:

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How to calculate OR’s from model with interaction term Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.2858 0.6247 4.2360 0.0396 psa 1 0.0608 0.0280 11.6952 0.0006 race 1 0.0954 0.5421 0.0310 0.8603 psa*race 1 -0.0349 0.0193 3.2822 0.0700 Increased odds for every 5 mg/ml increase in PSA: If white (race=0): If black (race=1):

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Example 4: Model building and the prostate cancer study

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What’s the study goal? Model building precedes very differently depending on whether… (1) We are testing a primary hypothesis, and we are only interested in covariates so far as they may confound or modify the primary association of interest (e.g. psa predicts capsule penetration). OR (2) We are trying to find the best predictors of capsule penetration from a set of possible predictors (more like data dredging).

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Does PSA predict capsule penetration and in what setting?

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Univariate analysis Other variables in the dataset that we are going to consider today (besides race, psa): Age Tumor Volume (from ultrasound) Total Gleason Score, 0 - 10

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Univariate: Age 4245485154576063666972757881 age 0 0.02 0.04 0.06 D e n s i t y

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Univariate: Tumor Volume -420446892 vol 0 0.02 0.04 D e n s i t y **Note the huge 0 group. Make new variable: HasVol=1/0 Vol=g/cm 2

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Gleason score -0.42.04.46.89.2 gleason 0 0.2 0.4 D e n s i t y We might consider grouping Gleason score rather than treating as continuous

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Quadratic in the logit? Est. logit 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 gleason 123456789 Could use median (=6) cutoff or log of gleason?

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Race -0.050.250.550.85 race 0 2 4 6 8 D e n s i t y Note: small size of black group may effect estimates…keep in mind…

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PSA 03672108144 psa 0 0.02 0.04 D e n s i t y

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Bivariate analysis Race vs. capsule race capsule Frequency‚ 0‚ 1‚ Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 0 ‚ 204 ‚ 137 ‚ 341 ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 1 ‚ 22 ‚ 14 ‚ 36 ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total 226 151 377 Statistics for Table of race by capsule Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1 0.0225 0.8809

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Bivariate analysis Age and capsule Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -0.1491 1.0623 0.0197 0.8884 age 1 0.00824 0.0160 0.2642 0.6073

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Tumor volume and capsule Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 0.1206 0.1555 0.6017 0.4379 vol 1 0.00772 0.00949 0.6629 0.4155 HasVol 1 0.2736 0.3366 0.6605 0.4164

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Gleason score and capsule Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 8.4196 1.0024 70.5458 <.0001 gleason 1 -1.2388 0.1526 65.9389 <.0001

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Summary The following variables besides PSA appear to be associated with capsule: – Gleason Score – But age, race, tumor volume could still be confounders or effect modifiers…

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Bivariate analysis with PSA Prob > |r| under H0: Rho=0 Number of Observations psa age vol gleason psa 1.00000 0.00596 0.01221 0.38848 0.9078 0.8128 <.0001 380 380 379 380 age 0.00596 1.00000 0.10742 0.03378 0.9078 0.0366 0.5115 380 380 379 380 vol 0.01221 0.10742 1.00000 -0.06041 0.8128 0.0366 0.2407 379 379 379 379 gleason 0.38848 0.03378 -0.06041 1.00000 <.0001 0.5115 0.2407 380 380 379 380

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HasVol vs. PSA Lower CL Upper CL Variable HasVol N Mean Mean Mean psa Diff (1-2) -2.952 1.1272 5.2062 T-Tests Variable Method Variances DF t Value Pr > |t| psa Pooled Equal 377 0.54 0.5872 psa Satterthwaite Unequal 357 0.54 0.5865

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Race (white/black) vs. PSA PSA vs. race Lower CL Upper CL Variable race N Mean Mean Mean psa Diff (1-2) -17.15 -10.38 -3.603 Variable Method Variances DF t Value Pr > |t| psa Pooled Equal 375 -3.01 0.0028 psa Satterthwaite Unequal 38.6 -2.23 0.0313

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Summary The following variables besides appear to be associated with psa: – Gleason Score – Race Age and tumor volume do not appear to be related to psa or capsule penetration. These are unlikely to be confounders or effect modifiers, but still may be considered for biological reasons.

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One strategy (recommended by Kleinbaum and Klein): “ Hierarchial backward elimination procedure.” Pick all possible confounders and effect modifiers and higher order terms that make biological sense = biggest model. Eliminate backwards from there, assessing interaction before confounding. – Note: it is inappropriate to remove a main effect term if the model contains higher-order interactions involving that term.

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An epidemiologist’s perspective “Validity takes precedence over precision.” – Kleinbaum and Klein “The assessment of confounding is carried out without using statistical testing.”—Kleinbaum and Klein

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Possible “Full Model” prostate cancer proc logistic; Model capsule = age psa vol gleason HasVol race psa*age psa*race psa*vol vol*vol psa*psa age*age psa*age*race; run;

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Backward Elimination Procedure Intercept Intercept and Criterion Only Covariates -2 Log L 506.587 391.067 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 4.5565 10.9673 0.1726 0.6778 age 1 0.0857 0.3394 0.0638 0.8006 psa 1 -0.1424 0.1366 1.0864 0.2973 vol 1 -0.0561 0.0498 1.2690 0.2600 gleason 1 -1.0551 0.1682 39.3315 <.0001 HasVol 1 1.0823 0.7984 1.8377 0.1752 race 1 -0.4494 0.6553 0.4702 0.4929 age*psa 1 0.00169 0.00203 0.6908 0.4059 psa*race 1 0.0588 0.1789 0.1080 0.7425 psa*vol 1 -0.00021 0.000548 0.1468 0.7016 vol*vol 1 0.000995 0.000768 1.6757 0.1955 psa*psa 1 -0.00009 0.000332 0.0686 0.7935 age*age 1 -0.00063 0.00263 0.0583 0.8093 age*psa*race 1 -0.00019 0.00274 0.0046 0.9462

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subtracting away all the interaction and higher order terms...the model fit is... Intercept Intercept and Criterion Only Covariates -2 Log L 506.587 396.894 LRtest= 396.8 (reduced model) – 391(full model) = 5.8; chi-square of 7 df. NS. What Kleinbaum and Klein call the “Chunk Test.” Many, many other strategies are possible… eliminate one at a time as Agresti talks about (highest p-value first) Hosmer and Lemeshow put all main effects in the model and try interaction terms one at a time, keeping only those that are significant I.e., many possible strategies!! No one right answer…I’m choosing the one that saves class time! Less desirable automated computer selection Model capsule = age psa vol gleason HasVol race

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proc logistic data=kristin.psa descending; Model capsule = age psa vol gleason HasVol race; units age=5 psa=10 vol=10 gleason=1 HasVol=1 race=1; run; To get “meaningful” OR’s, adjust units…

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Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age 0.979 0.943 1.017 psa 1.028 1.009 1.047 vol 0.991 0.967 1.016 gleason 2.871 2.087 3.950 HasVol 0.855 0.380 1.921 race 0.652 0.270 1.573 Adjusted Odds Ratios Effect Unit Estimate age 5.0000 0.900 psa 10.0000 1.316 vol 10.0000 0.913 gleason 1.0000 2.871 HasVol 1.0000 0.855 race 1.0000 0.652

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PSA point estimate not affected by removal of vol or HasVol (=not confounders!) Effect Unit Estimate age 5.0000 0.881 psa 10.0000 1.314 gleason 1.0000 2.929 race 1.0000 0.601 Not surprising, given bivariate analysis…

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Model Diagnostics Partition for the Hosmer and Lemeshow Test (see Agresti 113-114) capsule = 1 capsule = 0 Group Total Observed Expected Observed Expected 1 38 3 3.02 35 34.98 2 38 4 5.07 34 32.93 3 38 9 8.08 29 29.92 4 38 11 9.64 27 28.36 5 38 7 11.03 31 26.97 6 38 19 15.64 19 22.36 7 38 18 18.98 20 19.02 8 38 26 21.30 12 16.70 9 38 22 26.50 16 11.50 10 35 32 31.75 3 3.25 Hosmer and Lemeshow Goodness-of-Fit Test Chi-Square DF Pr > ChiSq 8.9522 8 0.3463 No evidence of lack of fit

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Capsule vs. predicted 0.00.51.0 capsule -2 0 2 r e s i d u a l s Example of an observation that deviates a bit from the model a 68 yr old man with high psa and gleason=9 but no capsule

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Discussion What are some of the merits of the preceding model selection procedure? What are some problems that you see? – e.g. What happened to race*psa interaction? Model building is an art!

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Results of other strategies with the same variables AUTOMATE BACKEARD ELIMINATION WITH CUT-OFF p=.10 proc logistic descending data=kristin.psa; model capsule = psa age race gleason vol HasVol psa*age psa*race psa*vol vol*vol psa*psa age*age psa*age*race /selection=backward slstay=.10; run; Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -7.2839 1.0221 50.7898 <.0001 psa 1 0.0264 0.00904 8.5312 0.0035 gleason 1 1.0380 0.1594 42.4010 <.0001 vol 1 -0.0145 0.00750 3.7321 0.0534 Takes in volume but not “Has volume.” Gets rid of age and race. Note PSA and gleason OR’s don’t change much.

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Adding/removing interaction terms one by one (p cut=off.10) The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -7.6648 1.0206 56.4048 <.0001 psa 1 0.0354 0.0112 10.0270 0.0015 race 1 0.1935 0.5760 0.1129 0.7369 gleason 1 1.0530 0.1593 43.6758 <.0001 psa*race 1 -0.0313 0.0187 2.7865 0.0951 This actually gets the psa*race interaction, just barely...

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Helpful SAS code/options *Useful SAS options, first line: 1. descending - SAS quirk is to reverse 0/1 --use descending option to remedy this. Useful model options: 1. lackfit - gives Hosmer and Lemeshow goodness-of-fit statistic (see Agresti 113-114)--groups into 10 approximately even groups and compares expected vs. observed proportions with approximate chi-square statistic 2. risklimits or clodds - gives confidence limits for ORs 3. selection=backward slstay=p-value = use for automated backward selection with p-value criteria for remaining in model set by slstay Useful output options: after "output out=dataset"... 1. dfbetas = var list -- for all variables in your list, creates a new variable containing the change in the regression coefficient (divided by its s.e.) when each observation is deleted,(measure of observation influence). 2. difchisq= var name -- creates a new variable that is the difference in the chi-square goodness-of-fit statistic when each observation is deleted (influence) 3. reschi= var name -- creates a new variable that is the Pearson residual for each observation (observed - expected over standard error) 4. predprobs=i --gives individual predicted probabilities; EXAMPLE” proc logistic descending data=kristin.lbw; *smoke only; model low = smoke / lackfit ; output out=kristin.diag dfbetas=smoke reschi=residuals difchisq=deletions predprobs=i; run;

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