Biost 536 / Epi 536 Categorical Data Analysis in Epidemiology

Transcription

1 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 iost 536 / Epi 536 ategorical ata nalysis in Epidemiology Scott S. Emerson, M.., Ph.. Professor of iostatistics University of Washington Lecture 9: Inference with omplex Modeling of POI November 4, ategorical ata nalysis, UT

2 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Lecture Outline Modeling scientific question omplex Modeling of POI Effect Modification 2 ategorical ata nalysis, UT

3 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Modeling Predictor of Interest 3 ategorical ata nalysis, UT

4 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Uses of Regression Modeling questions about associations efining contrasts for associations between response and POI efining contrasts for detecting effect modification Enabling comparisons across POI groups that are otherwise similar with respect to other variables djusting for confounding djusting to gain precision 4 ategorical ata nalysis, UT

5 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 orrowing Information Use other groups to make estimates in groups with sparse data Intuitively: 67 and 69 year olds would provide some relevant information about 68 year olds ssuming straight line relationship in modeled covariates tells us how to adjust data from other (even more distant) age groups If we do not know about the exact functional relationship, we might want to borrow information only close to each group 5 ategorical ata nalysis, UT

6 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 efining ontrasts efine a comparison across groups to use when answering scientific question If straight line relationship in parameter, slope for POI compares parameter between groups differing by 1 unit in X when all other covariates in model are equal If nonlinear relationship in parameter, slope is average comparison of parameter between groups differing by 1 unit in X holding covariates constant Statistical jargon: a contrast across the groups If multiple regression predictors model the POI, interpetation of the contrast is more difficult 6 ategorical ata nalysis, UT

7 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Statistical Questions: lassification 1. lustering of observations Perhaps into groups that might be different diseases 2. lustering of variables Perhaps into groups representing biochemical pathways 3. Quantification of distributions Perhaps reporting mean life expectancy after diagnosis 4. omparing distributions Perhaps investigating associations between variables 5. Prediction of individual observations Perhaps diagnosing disease or estimating kidney function 7 ategorical ata nalysis, UT

8 Lecture 9: Inference with omplex Modeling of POI November 4, Investigating ssociations Our scientific questions can be at many different levels of detail 1. Is there an association? 2. What is the general (first order) trend in Y with higher X? 3. Is their a nonlinear trend in the association? 4. Is the general trend a particular shape? Increasing exponentially? Increasing to a threshold? onstant then decreasing? U-shaped? S-shaped? 5. What is the association at particular levels of X? E.g., What is the difference in odds of mortality between subjects with LL of 160 and 161 mg/dl? ny questions can be about associations independent of other mechanisms (i.e., adjusted for potential confounding) 8 ategorical ata nalysis, UT

9 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Stratification vs Regression Generally, any stratified analysis could be performed as a regression model Stratification adjusts for covariates and all interactions among those covariates E.g, sex, race, and the sex-race interaction ny covariates modeled in each stratum s analysis would have to be modeled as interactions E.g., sex stratified analyses of response adjusted for age could be modeled in an unstratified analysis with sex-age interaction Our habit in regression is to just adjust for the covariates (the main effect ), and consider interactions less often 9 ategorical ata nalysis, UT

10 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 omplex Modeling Predictor of Interest Univariate Transformations 10 ategorical ata nalysis, UT

11 Lecture 9: Inference with omplex Modeling of POI November 4, Investigating ssociations Our scientific questions can be at many different levels of detail 1. Is there an association? 2. What is the general (first order) trend in Y with higher X? 3. Is their a nonlinear trend in the association? 4. Is the general trend a particular shape? Increasing exponentially? Increasing to a threshold? onstant then decreasing? U-shaped? S-shaped? 5. What is the association at particular levels of X? E.g., What is the difference in odds of mortality between subjects with LL of 160 and 161 mg/dl? ny questions can be about associations independent of other mechanisms (i.e., adjusted for potential confounding) 11 ategorical ata nalysis, UT

12 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Transformations of Predictors We transform predictors to provide more flexible description of complex associations between the response and some scientific measure Threshold effects Exponentially increasing effects U-shaped functions S-shaped functions etc. Sometimes we transform to gain precision about specific questions ummy variables epartures from linearity 12 ategorical ata nalysis, UT

13 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 ommon hoices Nominal variables ichotomized ummy variables Ordered categorical (possibly categorized continuous) ichotomized ummy variables Grouped linear (possibly with nonlinear scores) ontinuous Linear Univariate transformations (e.g., log, square, ) Multivariate transformations Polynomial Linear splines (or cubic splines, ) 13 ategorical ata nalysis, UT

14 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Examples: Patterns of Predicted Values It is useful to consider how the different choices model the response We consider the predicted values that might be obtained from regression models of 4 year V mortality on age or systolic P It is most interpretable to consider what the models say about the probability of an event ut RR and OR have nonlinear link functions This makes the form of some predicted relationships differ from what a naïve user might consider E.g., linear models on the log odds are not linear on the probability 14 ategorical ata nalysis, UT

15 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Linear ge ontinuous E.g.: logistic deadin4 age Predicted Values from R (black), RR (green), OR(blue) regression age Fitted values Pr(deadin4) Predicted number of events 15 ategorical ata nalysis, UT

16 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Linear SP ontinuous E.g.: logistic deadin4 sbp Predicted Values from R (black), RR (green), OR(blue) regression sbp Fitted values Pr(deadin4) Predicted number of events 16 ategorical ata nalysis, UT

17 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Squared ge ontinuous E.g.: logistic deadin4 agesqr Predicted Values from R (black), RR (green), OR(blue) regression age Fitted values Pr(deadin4) Predicted number of events 17 ategorical ata nalysis, UT

18 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Squared SP ontinuous E.g.: logistic deadin4 sbpsqr Predicted Values from R (black), RR (green), OR(blue) regression sbp Fitted values Pr(deadin4) Predicted number of events 18 ategorical ata nalysis, UT

19 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Log ge ontinuous E.g.: logistic deadin4 logage Predicted Values from R (black), RR (green), OR(blue) regression age Fitted values Pr(deadin4) Predicted number of events 19 ategorical ata nalysis, UT

20 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Log SP ontinuous E.g.: logistic deadin4 logsbp Predicted Values from R (black), RR (green), OR(blue) regression sbp Fitted values Pr(deadin4) Predicted number of events 20 ategorical ata nalysis, UT

21 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 omments: Univariate Transformations straight line can approximate many curvilinear functions over a small interval of their domains converse of sorts is also true in statistics: If the truth is a straight line, a statistical procedure will estimate the parameters in a way that makes it cover a small interval of the domain 21 ategorical ata nalysis, UT

22 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Table) From Homework #3, we found suggestion of a nonlinear effect 22 ategorical ata nalysis, UT

23 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Plots) From Homework #3, we found suggestion of a nonlinear effect I can use linear splines to mimic a smooth to the data (Linear splines can handle the link functions in RR and OR) Fitted Values: Step Function (heptiles) aai R (linear) RR (Poisson) OR (logistic) 23 ategorical ata nalysis, UT

24 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Linear) Linear continuous models might have seemed unusual fits Need to consider preponderance of the data Need to consider impact of link function on possible curves Need to consider influential points relative to link function Fitted Values: Linear Splines, Linear aai Rspl (linear) RRspl (Poisson) ORlin (logistic) ORspl (logistic) Rlin (linear) RRlin (Poisson) 24 ategorical ata nalysis, UT

25 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Log) Linear continuous models might have seemed unusual fits Need to consider preponderance of the data Need to consider impact of link function on possible curves Need to consider influential points relative to link function Fitted Values: Linear Splines, Linear aai Rspl (linear) RRspl (Poisson) ORlog (logistic) ORspl (logistic) Rlog (linear) RRlog (Poisson) 25 ategorical ata nalysis, UT

26 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 omments: Univariate Transformations Many univariate transformations are in some sense defined relative to 0 or 1 If you center the covariates you might get a different picture (nd because a quadratic function includes the linear term, that in some sense centers the curve) 26 ategorical ata nalysis, UT

27 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Log (SP-75) ontinuous E.g.: logistic deadin4 logsbp75 Predicted Values from R (black), RR (green), OR(blue) regression sbp Fitted values Pr(deadin4) Predicted number of events 27 ategorical ata nalysis, UT

28 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Quadratic ge ontinuous E.g.: logistic deadin4 age agesqr Predicted Values from R (black), RR (green), OR(blue) regression age Fitted values Pr(deadin4) Predicted number of events 28 ategorical ata nalysis, UT

29 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Quadratic SP ontinuous E.g.: logistic deadin4 sbp sbpsqr Predicted Values from R (black), RR (green), OR(blue) regression sbp Fitted values Pr(deadin4) Predicted number of events 29 ategorical ata nalysis, UT

30 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Grouped Linear ge E.g.: logistic deadin4 age5yr Predicted Values from R (black), RR (green), OR(blue) regression age Fitted values Pr(deadin4) Predicted number of events 30 ategorical ata nalysis, UT

31 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Interpretation: Identity Link With categorical data, we use an identity link with R regression Linear regression on a binary response variable When modeling a scientific factor (POI) with a single predictor, interpretation of parameters is straightforward when predictor is inary: ifference in means between two groups Untransformed continuous: (verage) difference in means between two groups that differ in the predictor by one unit Log transformed continuous (verage) difference in means between two groups that differ by a e-fold difference in their predictor values Interpretation of parameters with other transformations is difficult 31 ategorical ata nalysis, UT

32 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Identity Link: Log Transformed Predictors Suppose we use the natural log transformation e= fold difference in ldl associated with.1477 lower probability of death (ould look at log(2) times the regression parameter for doubling). regress deadin5 logldl, robust Linear regression Number of obs = 725 F( 1, 723) = 8.14 Prob > F = R-squared = Root MSE = Robust deadin5 oef. StdErr t P> t [95% onf. Intvl] logldl _cons ategorical ata nalysis, UT

33 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Interpretation: log Link With categorical data, we use a logarithmic link (natural log) with RR (Poisson) and OR (logistic) regression When modeling a scientific factor (POI) with a single predictor, interpretation of exponentiated parameters is straightforward when predictor is inary: RR or OR between two groups Untransformed continuous: (verage) RR or OR two groups that differ in the predictor by one unit Log transformed continuous (verage) RR or OR between two groups that differ by a e-fold difference in their predictor values Interpretation of parameters with other transformations is difficult 33 ategorical ata nalysis, UT

34 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Log Link: Log Transformed Predictors Suppose we use the natural log transformation e= fold difference in ldl associated with OR only.3705 times as high (ould look at regression parameter raised to log(2) times for doubling). logistic deadin5 logldl Logistic regression Number of obs = 725 LR chi2(1) = 9.26 Prob > chi2 = Log likelihood = Pseudo R2 = deadin5 Odds Ratio StdErr. z P> z [95% onf. Intvl] logldl ategorical ata nalysis, UT

35 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Interpretable Estimates Sometimes 1 unit is outside the variability of our data E.g., age measured in centuries E.g., log height using natural log (a fold difference) Other times 1 unit is so small as to seem inconsequential E.g., adult ages measured in days The easiest way to deal with this is to transform your predictor to units that are more interesting Use log base 2 or log base 1.1 E.g., loght = log(height) / log(1.1) (see Supplementary Materials) Rescale to better units E.g. replace aai = aai / 0.1 E.g. replace age = age / 10 (decades) In any case, an OR or RR of should usually be regarded as 1 significant digit 35 ategorical ata nalysis, UT

36 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 ummy Variables Indicator variables for all but one group This is the only appropriate way to model nominal (unordered) variables E.g., for marital status create indicator variables for married (married = 1, everything else = 0) widowed (widowed = 1, everything else = 0) divorced (divorced = 1, everything else = 0) (single would then be the intercept) Often used for other settings as well escriptive statistics as in Homework #3 ut dummy variables ignore order of categories Equivalent to nalysis of Variance (NOV) 36 ategorical ata nalysis, UT

37 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Ex: Interpretation of Slopes ased on coding used Intercept corresponds to reference group Slope for other terms compare each group to the reference Tests for association must test all predictors together Need to be very careful in overinterpreting statistical significance among groups 37 ategorical ata nalysis, UT

38 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Stata: ummy Variables Stata has a facility to automatically create dummy variables Old way: Prefix regression commands with xi: Now just prefix variables to be modeled as dummy variables with i.varname Example: 4 year mortality rate by ankle: arm ratio of SP (I) xtiles aai5= aai, nq(5) logistic deadin4 aai5 Stata will drop the lowest category by default You can choose the category dropped by specifying, for instance, the third interval for logistic deadin4 ib3.aai5 This does not change the predicted values, only the interpretation of individual regression parameters 38 ategorical ata nalysis, UT

39 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Lowest Interval as aseline. logit deadin4 i.aaiq5 Logistic regression Number of obs = 4879 LR chi2(4) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 oef. Std. Err. z P> z [95% onf. Interval] aaiq _cons ategorical ata nalysis, UT

40 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Second Interval as aseline. logit deadin4 ib2.aaiq5 Logistic regression Number of obs = 4879 LR chi2(4) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 oef. Std. Err. z P> z [95% onf. Interval] aaiq _cons ategorical ata nalysis, UT

41 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Third Interval as aseline. logit deadin4 ib3.aaiq5 Logistic regression Number of obs = 4879 LR chi2(4) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 oef. Std. Err. z P> z [95% onf. Interval] aaiq _cons ategorical ata nalysis, UT

42 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Highest Interval as aseline. logit deadin4 ib5.aaiq5 Logistic regression Number of obs = 4879 LR chi2(4) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 oef. Std. Err. z P> z [95% onf. Interval] aaiq _cons ategorical ata nalysis, UT

43 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 aveats The parameterization does not affect the fitted values The parameterization does affect interpretation of regression coefficients E.g., coefficient for second level compares to baseline group Group 2 vs group 1 Group 2 vs group 3 Group 2 vs group 5 Interpretation of the p values are very problematic Not adjusted for multiple comparisons nd nonsignificant p values to not prove equivalence Were we to believe the validity of the p values We cannot discriminate group 2 from groups 3 or 5 (but 4 is different) We cannot discriminate group 3 from groups 2, 4, or 5 43 ategorical ata nalysis, UT

44 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Multivariate Modeling of POI: Inference If multiple modeled covariates involve our POI, we must test for an association by simultaneously testing all such parameters If there are no other covariates in the model, the overall F test or chi square test provide this inference If there are other covariates, we use Stata s test post-estimation command 44 ategorical ata nalysis, UT

45 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Question When is it more powerful to model the POI multivariately rather than univariately? 45 ategorical ata nalysis, UT

46 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 ompare Linear vs ummy Variables Suppose the truth is a straight line relationship: We can consider an example in logistic regression Linear ontinuous Power ummy Variables Power The major loss of power is from the dummy variables ignoring the order of the groups Had we used grouped linear, the power was.172,.576, ategorical ata nalysis, UT

47 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Linear ontinuous Models orrows information across groups ccurate, efficient if model is correct If model incorrect, mixes random and systematic error an gain power from ordering of groups in order to detect a trend ut, no matter how low the standard error is, if there is no trend in the mean, there is no statistical significance 47 ategorical ata nalysis, UT

48 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Hypothetical Settings Linear: Highest Power; NOV: High Power Linear: Moderate Power; NOV: Low Power Response Response ose ose Linear: No Power; NOV: High Power Linear: No Power; NOV: Low Power Response Response ose ose ategorical ata nalysis, UT

49 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Perils of ata-riven Model Selection Question: an we use the data to help us decide which form is used to model the association between the POI and response? 49 ategorical ata nalysis, UT

50 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Perils of ata-riven Model Selection Question: an we use the data to help us decide which form is used to model the association between the POI and response? nswer: onsider again lecture 8 50 ategorical ata nalysis, UT

51 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 pplication to Evaluation of Risk We consider a population of candidate hypotheses We use scientific studies to diagnose truly beneficial hypotheses Use both frequentist and ayesian optimality criteria Innovator (sponsor): High probability of adopting a true hypothesis (frequentist power) Regulatory: Low probability of adopting false hypotheses (freq type 1 error) High probability that adopted hypotheses are true (posterior probability) Public Health (frequentist sample space, ayes criteria) Maximize the number of true hypotheses adopted Minimize the number of false hypotheses adopted 51 ategorical ata nalysis, UT

52 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Frequentist vs ayesian Frequentist and ayesian inference truly complementary Frequentist: esign so the same data not likely from null / alt ayesian: Explore updated beliefs based on a range of priors ayes rule tells us that we can parameterize the positive predictive value by the type I error and prevalence Maximize new information by maximizing ayes factor With simple hypotheses: power prevalence PPV power prevalence type I err 1 prevalence PPV 1 PPV power prevalence type I err 1 prevalence posterior odds ayes Factor prior odds 52 ategorical ata nalysis, UT

53 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 onsider lternative Models We can test each with a type I error of.05 Linear continuous Quadratic ummy variables What if we take the lowest of p values from multiple models Linear, quadratic 0.05 for each goes to Linear, dummy 0.05 for each goes to ll three 0.05 for each goes to Example simulations: With a true linear effect, power increases by less than or or ategorical ata nalysis, UT

54 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Linear Splines raw straight lines between pre-specified knots Model intercept and m+1 variables when using m knots Suppose knots are k 1,, k m, for variable X efine variables Spline0 SplineM Spline0 equals X for X < k 1 k 1 for k 1 < X Then, for J = 1.. m, SplineJ equals (define k 0 =0, k m+1 = ) 0 for X < k J X k J for k J < X < k J+1 k J+1 k J for k J+1 < X 54 ategorical ata nalysis, UT

55 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Linear splines in 7 intervals 4 Year Mortality vs I Logistic regression Number of obs = 4879 LR chi2(7) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR Std. Err. z P> z [95% onf. Intvl] saai saai saai saai saai saai saai ategorical ata nalysis, UT

56 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Plots) From Homework #3, we found suggestion of a nonlinear effect I can use linear splines to mimic a smooth to the data (Linear splines can handle the link functions in RR and OR) Fitted Values: Step Function (heptiles) aai R (linear) RR (Poisson) OR (logistic) 56 ategorical ata nalysis, UT

57 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Linear Splines: Parameter Interpretation With identity link Intercept β 0 : θ Y X when X = 0 Slope parameters β j : Estimated difference in θ Y X between two groups both between the same knots but differing by 1 unit in X (May want to make sure that interval contains 1 unit With log link Exponentiated intercept exp(β 0 ): θ Y X when X = 0 Exponentiated slope parameters exp(β j ) : Estimated ratio of θ Y X between two groups both between the same knots but differing by 1 unit in X Tests for association must test all predictors together 57 ategorical ata nalysis, UT

58 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Plots) From Homework #3, we found suggestion of a nonlinear effect I can use linear splines to mimic a smooth to the data (Linear splines can handle the link functions in RR and OR) Fitted Values: Step Function (heptiles) aai R (linear) RR (Poisson) OR (logistic) 58 ategorical ata nalysis, UT

59 Lecture 9: Inference with omplex Modeling of POI November 4, Investigating ssociations Our scientific questions can be at many different levels of detail 1. Is there an association? 2. What is the general (first order) trend in Y with higher X? 3. Is their a nonlinear trend in the association? 4. Is the general trend a particular shape? Increasing exponentially? Increasing to a threshold? onstant then decreasing? U-shaped? S-shaped? 5. What is the association at particular levels of X? E.g., What is the difference in odds of mortality between subjects with LL of 160 and 161 mg/dl? ny questions can be about associations independent of other mechanisms (i.e., adjusted for potential confounding) 59 ategorical ata nalysis, UT

60 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Testing Nonlinearity fter answering questions about an association, we may move on to more detailed questions In an independent sample, or Protected by gate-keeping procedures in order to avoid inflating the type 1 error Only perform the more detailed test if the primary question was statistically significant Generally we can test for nonlinearity by Fitting a model that includes a linear term and other term(s) Then test to see if the other terms are (jointly) not significant 60 ategorical ata nalysis, UT

61 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - R Using a quadratic model We test the squared term. regress deadin4 aai aaisqr, robust Linear regression Number of obs = 4879 F( 2, 4876) = Prob > F = R-squared = Root MSE = Robust deadin4 oef. StdErr t P> t [95% onf Interval] aai aaisqr _cons ategorical ata nalysis, UT

62 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using a quadratic model We test the squared term. logistic deadin4 aai aaisqr Logistic regression Number of obs = 4879 LR chi2(2) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR StdErr z P> z [95% onf. Intrvl] aai aaisqr ategorical ata nalysis, UT

63 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - RR Using a quadratic model We test the squared term. poisson deadin4 aai aaisqr, robust irr Poisson regression Number of obs = 4879 Wald chi2(2) = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = Robust deadin4 IRR StdErr z P> z [95% onf Intrvl] aai aaisqr ategorical ata nalysis, UT

64 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Fitted Values: Linear Splines vs Linear Fitted Values: Linear Splines, Linear aai Rspl (linear) RRspl (Poisson) ORlin (logistic) ORspl (logistic) Rlin (linear) RRlin (Poisson) 64 ategorical ata nalysis, UT

65 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Fitted Values: Linear Splines vs Quadratic Fitted Values: Linear Splines, Quadratic aai Rspl (linear) RRspl (Poisson) ORquad (logistic) ORspl (logistic) Rquad (linear) RRquad (Poisson) 65 ategorical ata nalysis, UT

66 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Fitted Values: Linear vs Quadratic Fitted Values: Linear, Quadratic aai Rlin (linear) RRlin (Poisson) ORquad (logistic) ORlin (logistic) Rquad (linear) RRquad (Poisson) 66 ategorical ata nalysis, UT

67 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using a dummy variable model We add dummy variable terms to a linear model. logistic deadin4 i.aaictg aai Logistic regression Number of obs = 4879 LR chi2(7) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR StdErr z P> z [95% onf Intrvl] aaictg aai ategorical ata nalysis, UT

68 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using a dummy variable model We add dummy variable terms to a linear model Then we jointly test all dummy variables (Wald test in this case, though LR also possible with classical logistic regression). testparm i.aaictg ( 1) [deadin4]55.aaictg = 0 ( 2) [deadin4]75.aaictg = 0 ( 3) [deadin4]95.aaictg = 0 ( 4) [deadin4]115.aaictg = 0 ( 5) [deadin4]135.aaictg = 0 ( 6) [deadin4]155.aaictg = 0 chi2( 6) = Prob > chi2 = ategorical ata nalysis, UT

69 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using linear splines model We do not have to add a variable linear is special case. logistic deadin4 saai* Logistic regression Number of obs = 4879 LR chi2(7) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR StdErr z P> z [95% onf Intrvl] saai saai saai saai saai saai saai ategorical ata nalysis, UT

70 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using linear splines model We do not have to add a variable linear is special case We test for equality of slopes. test saai025 = saai055 = saai075 = saai095 = saai115 = saai135 = saai155 ( 1) [deadin4]saai025 - [deadin4]saai055 = 0 ( 2) [deadin4]saai025 - [deadin4]saai075 = 0 ( 3) [deadin4]saai025 - [deadin4]saai095 = 0 ( 4) [deadin4]saai025 - [deadin4]saai115 = 0 ( 5) [deadin4]saai025 - [deadin4]saai135 = 0 ( 6) [deadin4]saai025 - [deadin4]saai155 = 0 chi2( 6) = Prob > chi2 = ategorical ata nalysis, UT

71 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using linear splines model What if we do add a linear term?. logistic deadin4 aai saai* note: saai155 omitted because of collinearity Logistic regression Number of obs = 4879 LR chi2(7) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR StdErr z P> z [95% onf Intrvl] aai saai saai saai saai saai saai e+08 saai155 (omitted) 71 ategorical ata nalysis, UT

72 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR Using linear splines model If we add the linear term, then we test all the linear splines term are 0. testparm saai* ( 1) [deadin4]saai025 = 0 ( 2) [deadin4]saai055 = 0 ( 3) [deadin4]saai075 = 0 ( 4) [deadin4]saai095 = 0 ( 5) [deadin4]saai115 = 0 ( 6) [deadin4]saai135 = 0 chi2( 6) = Prob > chi2 = ategorical ata nalysis, UT

73 Lecture 9: Inference with omplex Modeling of POI November 4, Investigating ssociations Our scientific questions can be at many different levels of detail 1. Is there an association? 2. What is the general (first order) trend in Y with higher X? 3. Is their a nonlinear trend in the association? 4. Is the general trend a particular shape? Increasing exponentially? Increasing to a threshold? onstant then decreasing? U-shaped? S-shaped? 5. What is the association at particular levels of X? E.g., What is the difference in odds of mortality between subjects with LL of 160 and 161 mg/dl? ny questions can be about associations independent of other mechanisms (i.e., adjusted for potential confounding) 73 ategorical ata nalysis, UT

74 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 More etailed Questions Testing particular questions about a shape is more difficult We have to distinguish between testing in a model that presumes a particular shape and testing in a model that includes a particular shape E.g.: Quadratic model presumes U-shaped, but linear splines could include a U-shaped function as well as many others Have to isolate the properties of your question E.g.: U-shaped functions will have different slopes at the extremes of the predictor values We will rarely have good power nd we need to avoid using too flexible a design, because then we will have very little information We also have to worry about influential points 74 ategorical ata nalysis, UT

75 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR U-shaped functions using linear splines model We would need to test that simultaneously saai025 slope is negative (respectively, positive ) saai155 slope is positive (respectively, negative ). logistic deadin4 saai* Logistic regression Number of obs = 4879 LR chi2(7) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR StdErr z P> z [95% onf Intrvl] saai saai saai saai saai saai saai ategorical ata nalysis, UT

76 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Plots) From Homework #3, we found suggestion of a nonlinear effect I can use linear splines to mimic a smooth to the data (Linear splines can handle the link functions in RR and OR) Fitted Values: Step Function (heptiles) aai R (linear) RR (Poisson) OR (logistic) 76 ategorical ata nalysis, UT

77 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: From Homework #3 - OR U-shaped functions using linear splines model We would need to test that simultaneously s2aai025 slope is negative (respectively, positive ) s2aai110 slope is positive (respectively, negative ). logistic deadin4 s2aai* Logistic regression Number of obs = 4879 LR chi2(3) = Prob > chi2 = Log likelihood = Pseudo R2 = deadin4 OR StdErr z P> z [95% onf Intrvl] s2aai s2aai s2aai ategorical ata nalysis, UT

78 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: Mortality vs I (Plots) From Homework #3, we found suggestion of a nonlinear effect Linear splines with 2 knots Estimates suggest U-shaped trend, but not statistically significant Fitted Values: Linear Splines at 0.9 and 1.10 Pr(deadin4) aai 78 ategorical ata nalysis, UT

79 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Effect Modification 79 ategorical ata nalysis, UT

80 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Effect Modifier The association between Response and POI differs in strata defined by effect modifier Statistical term: Interaction epends on the measurement of effect Summary measure Mean, geometric mean, median, proportion, odds, hazard, etc. omparison across groups ifference, ratio 80 ategorical ata nalysis, UT

81 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 nalysis of Effect Modification When the scientific question involves effect modification, analyses must be within each stratum separately If we want to estimate degree of effect modification or test for its existence: regression model will typically include Predictor of interest Effect modifier covariate modeling the interaction (usually product) 81 ategorical ata nalysis, UT

82 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Model for Effect Modification Typical model for effect modification Include main effects (can be bad not to) X (or predictors that involve only X) W (or predictors that involve only W) Include interactions Predictor(s) derived from both X and W g X, W X W XW i i 0 0 X X X i i W W i W i XW XW X i i W i 82 ategorical ata nalysis, UT

83 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Interpretation of Parameters X i, Wi 0 X X i W Wi XW X i Wi Usual approach a bit more difficult We can try using the idea of comparison of across groups differing by 1 unit in corresponding predictor but agreeing in other modeled predictors However, terms involving two scientific variables makes this approach difficult 83 ategorical ata nalysis, UT

84 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Intercept X i, Wi 0 X X i W Wi XW X i Wi Interpretation of intercept straightforward 0 corresponds to X= 0, W= 0 May not be scientifically meaningful 84 ategorical ata nalysis, UT

85 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Slopes for Main Effects X i, Wi 0 X X i W Wi XW X i Wi Interpretation of main effects X corresponds to 1 unit difference in X holding W and (X W) constant So 1 unit difference in X when W= 0 May not be scientifically meaningful W corresponds to 1 unit difference in W holding X and (X W) constant So 1 unit difference in W when X= 0 May not be scientifically meaningful 85 ategorical ata nalysis, UT

86 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Slope for Interaction X i, Wi 0 X X i W Wi XW X i Wi Interpretation of interaction difficult XW corresponds to 1 unit difference in (X W) holding X and W constant Impossible, so we need another way to interpret this slope parameter 86 ategorical ata nalysis, UT

87 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 onsider Scientific Predictors g X i, w 0 X X w w w 0 W i W X XW XW X i w X i In stratum with Intercept : Slope : XW W w 0 X W XW w w by 1 unit in X correspond s to X compares groups differing is difference in X slope per 1 unit difference in W i 0 87 ategorical ata nalysis, UT

88 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 onsider Scientific Predictors g x, W i 0 X x W x x 0 X W W i XW XW x W W i i In stratum with X Intercept : 0 Slope : XW W x X XW x x by 1 unit in W correspond s to W compares groups differing is difference in W slope per 1 unit difference in X i 0 88 ategorical ata nalysis, UT

89 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Symmetry of Effect Modification Note that if X modifies the association between Y and W, then W modifies the association between Y and X side: onfounding need not be symmetric W can confound the association between Y and X, but X not confound the association between Y and W W and X associated in the sample Y and X not associated after adjusting for W Y and W associated after adjusting for X 89 ategorical ata nalysis, UT

90 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Inference for Effect Modification X i, Wi 0 X X i W Wi XW X i Wi No effect modification if XW = 0 Hence, inference about existence of effect modification tests that XW = 0 We can perform such inference using standard regression output for the corresponding slope parameter 90 ategorical ata nalysis, UT

91 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Inference for Main Effect Slope X i, Wi 0 X X i W Wi XW X i Wi Interpretation of X = 0 Same intercept in all strata defined by W Generally a very uninteresting question We rarely make inference on main effect slopes by themselves 91 ategorical ata nalysis, UT

92 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 g Inference bout Effect of X X i, Wi 0 X X i W Wi XW X i Wi Response parameter not associated with X if X = 0 N XW = 0 We will need to construct special tests that both parameters are simultaneously 0 The t tests given in regression output consider only one slope parameter at a time 92 ategorical ata nalysis, UT

93 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 Example: SEP Normal Ranges We want to find normal ranges for somatosensory evoked potential (SEP) Time that it takes a signal to reach your brain from your ankle Method of analysis Not recommended: Prediction intervals assuming same distribution (Gaussian) within each group Recommended: 2.5 th and 97.5 th quantile within groups s a first step, we want to consider important predictors of nerve conduction times If any variables such as sex, age, height, race, etc. are important predictors of nerve conduction times, then it would make most sense to obtain normal ranges within such groups 93 ategorical ata nalysis, UT

94 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Important Predictors Scientifically, we might expect that height, age, and sex are related to the nerve conduction time Nerve length should matter, and height is a surrogate for nerve length ge might affect nerve conduction times: People slow down with age Sex: Men are SO fragile 94 ategorical ata nalysis, UT

95 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Height ge Interaction? Prior to looking at the data, we can also consider the possibility that interactions between these variables might be important Height - age interaction? o we expect the difference in conduction times between 6 foot tall and 5 foot tall 20 year olds to be the same as the difference in conduction times between 6 foot tall and 5 foot tall 80 year olds? 95 ategorical ata nalysis, UT

96 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Height ge Interaction Rationale We might suspect such an interaction due to the fact that height may not be as good a surrogate for nerve length in older people With age, some people tend to shrink due to osteoporosis and compression of intervertebral discs It is not clear that nerve length would be altered in such a process Thus, in young people, differences in height probably are a better measure of nerve length than in old people Tall old people probably have been tall always Short old people will include some who were much taller when they were young 96 ategorical ata nalysis, UT

97 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Height ge - Sex Interaction? We can also consider the possibility of three way interactions between height, age, and sex Osteoporosis affects women far more than men Hence, we might expect the height - age interaction to be greatest in women and not so important in men two way interaction between height and age that is different between men and women defines a three way interaction between height, age, and sex 97 ategorical ata nalysis, UT

98 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 ategorical ata nalysis, UT Stratified Scatterplots Scatterplots of p60 by height for each sex, lowess in age strata verage Time to p60 Peak: Females Height (in.) Time to p60 Peak ,a = yo,b = yo,c = yo,d = yo a a a a a a a a a b b b b b b b b b b b b c c c c c c c c c d d d d d d d d d d verage Time to p60 Peak: Males Height (in.) Time to p60 Peak a a a a a a a a a a a a a b b b b b b b b b b b b b c c c c c c c c c d d d d d d d d d d d

99 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: efinition of Model efining a regression model with interactions We must create variables to model the three way interaction term Furthermore, it is a VERY GOO idea to include all main effects and lower order interactions in the model as well main effects : the individual variables which contribute to the interaction lower order terms : all interactions that involve some combination of the variables which contribute to the 3-way interaction 99 ategorical ata nalysis, UT

100 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Modeling Interactions Most often, we lack sufficient information to be able to guess what the true form of an interaction might be The most popular approach is thus to consider multiplicative interactions reate a new variable by merely multiplying the two (or more) interacting predictors 100 ategorical ata nalysis, UT

101 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: reating New Interaction Terms Thus for this problem we could create variables H = Height * ge HM = Height * Male M = ge * Male HM = Height * ge * Male 101 ategorical ata nalysis, UT

102 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Interpretation of Parameters In the presence of higher order terms (powers, interactions) interpretation of parameters is not easy We can no longer use the change associated with a 1 unit difference in predictor holding other variables constant It is generally impossible to hold other variables constant when changing a covariate involved in an interaction (If not impossible, it is certainly uninteresting) 102 ategorical ata nalysis, UT

103 Lecture 9: Inference with omplex Modeling of POI November 4, 2014 SEP: Interpretation Using Lines in Strata Interpretation of the model in terms of the SEP height relationship within age-sex strata E p60 Ht, H ge, Male H HM Ht ge Male 0 HM M H M HM HM M p60 - Height relationship for ge a : Sex Intercept F M 0 a H Ha a 0 M M Slope H HM H HM a 103 ategorical ata nalysis, UT