5. Ordinal regression: cumulative categories proportional odds. 6. Ordinal regression: comparison to single reference generalized logits

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1 Lecture Logistic regression with binary response 2. Proc Logistic and its surprises 3. quadratic model 4. Hosmer-Lemeshow test for lack of fit 5. Ordinal regression: cumulative categories proportional odds 6. Ordinal regression: comparison to single reference generalized logits Logistic regression using the SAS system: theory and application by Paul Allison, 2001, Wiley-SAS Categorical Data Analysis Using the SAS System, 2nd ed. by Stokes, Davis, and Koch, 2009, SAS Institute 1 Proc Logistic: Traps for the Unwary, by P.L. Flom (on course website) 2

2 Example: Obesity in NHANES 2004 NHANES 2004 data for children and adults people under age 50 (n = 6116) Event = obesity, defined as BMI 30, or 95th percentile for children Association between age and rate of obesity? P[obese age] = º(age) 3 Proc Logistic Proc Logistic data=under50; model obese = age; model format like Proc GLM Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age exp( ) = , Correct value, but does this odds ratio make sense? 4

3 First part of the output: The LOGISTIC Procedure Data Set WORK.ADULT Response Variable obese Number of Response Levels 2 Model binary logit Optimization Technique Fisher s scoring Number of Observations Read 6116 Number of Observations Used 6116 Response Profile Ordered Total Value obese Frequency Probability modeled is obese=0. 5 There is also a warning in the log NOTE: PROC LOGISTIC is modeling the probability that obese=0. One way to change this to model the probability that obese=1 is to specify the response variable option EVENT= 1. NOTE: Convergence criterion (GCONV=1E-8) satisfied. NOTE: There were 6116 observations read from the data set WORK.UNDER50. Proc Logistic models probability of no event by default. Opposite of what everyone expects. 6

4 Two fixes: 1. Use the descending option to make SAS fit the probability that y = 1. Works in both Logistic and Proc Genmod (which also fits logistic regression). Proc Logistic descending model obese = age; data=under50; 2. Define the event in the MODEL statement. Only works in Logistic. Proc Logistic data=under50; model obese (event = 1 ) = age; Clearer code, but unfortunately doesn t work in Proc Genmod. 7 Proc Logistic data=under50; model obese (event = 1 ) = age; NOTE: PROC LOGISTIC is modeling the probability that obese=1. NOTE: Convergence criterion (GCONV=1E-8) satisfied. NOTE: There were 6116 observations read from the data set WORK.UNDER50 8

5 Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age Odds of obesity increase by a factor of 1.03 per year for people under 50 (95% confidence interval ). Odds of obesity increase by 3% per year for people under 50 (95% CI %). 9 Two graphs of fitted (linear) logistic model: µ º(age) log odds of obesity (as function of age) = log = age 1 º(age) 0.4 Log odds of obesity Age (years) 10

6 exp( age) probability of obesity as function of age = º(age) = 1 + exp( age) 1.0 Fitted probability of obesity Age (years) 11 Fitted logistic curve extrapolated to show full S-shaped curve: 1 Rate of Obesity Range of data Age (years) 12

7 Changing units for odds ratios To get odds ratios for a 10-year change in age: Proc Logistic descending data=under50; model obese = age / rsquare CLodds = PL; units age = 10.0 ; Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age Profile Likelihood Confidence Interval for Odds Ratios Effect Unit Estimate 95% Confidence Limits age Two useful options to the model statement in Proc Logistic: Proc Logistic descending; model obese = age / CLodds = PL Rsquare ; Default (Wald) confidence intervals for OR are exp ˆØ ± SE( ˆØ) CLodds = PL profile likelihood confidence intervals" are more accurate than Wald for small samples. Gives same answer for large samples. Rsquare gives a version of R-square from linear regression Maximum possible value of generalized R 2 is not 1.0 as for linear regression. Max-rescaled R-Square divides by this maximum value to fix this. 14

8 R-Square Max-rescaled R-Square Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age Profile Likelihood Confidence Interval for Odds Ratios Effect Unit Estimate 95% Confidence Limits age Profile Likelihood CI is identical here because sample size is large (n = 6116) 15 CLASS variables in Proc Logistic proc logistic descending data=under50; class gender; model obese = age gender; Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 gender F Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age gender F vs M For age, exp(0.0302) = 1.031, but for gender, exp(0.0779) =

9 The LOGISTIC Procedure Model Information Data Set WORK.ADULT Response Variable obese Number of Response Levels 2 Model binary logit Optimization Technique Fisher s scoring Class Level Information Design Class Value Variables gender F 1 M -1 Default coding for CLASS variables is not the same as Proc GLM (0/1) 17 If you want to work with regression coefficients, then request 0 ± 1 indicator variables. proc logistic descending data=under50; class gender / param = GLM ; model obese = age gender Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 gender F gender M Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age gender F vs M exp(0.1558) =

10 Logistic regression fits linear model on log(odds) scale. is parallel lines model. Distance between lines = log(odds ratio). obese = age gender Females 0.5 Log(odds of obesity) 1.0 Males Age (years) 19 Parallel lines model on probability scale Females Males Probability of Obesity Age (years) 20

11 Check for interaction (separate lines model): Proc Logistic descending data=under50; class gender; model obese = age gender age*gender ; Surprise: get regression coefficients but no odds ratios. 21 Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq age <.0001 gender age*gender Analysis of Maximum Likelihood Estimates reg coef Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 gender female age*gender female Proc Logistic does not calculate odds ratios for effects included in interaction terms: no odds ratios for age or gender 22

12 Why no odds ratio for terms in interaction? Proc Logistic descending data=young; class gender; model obese = age gender age*gender / rsquare CLodds = PL; Analysis of Maximum Likelihood Estimates reg coef Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age <.0001 gender female age*gender female Model fits separate line for each gender: 2 intercepts, 2 slopes 23 Separate line for each gender on log(odds) scale. Distance between gender lines is different at every age: no single gender effect 0.0 Females Log(odds of obesity) Males Age (years) 24

13 Separate line for each gender on probability scale Probability of Obesity Females Males Age (years) 25 ODDSRATIO statement This is like LSmeans in Proc GLM: gives odds ratios for combinations of factor-levels in the interaction. proc logistic descending data=under50; class gender; model obese = age gender age*gender / rsquare CLodds = PL; oddsratio age ; oddsratio gender / at (age= ); 26

14 Wald Confidence Interval for Odds Ratios Label Estimate 95% Confidence Limits age at gender=f age at gender=m separate age odds ratios for each gender) gender F vs M at age= gender F vs M at age= gender F vs M at age= gender F vs M at age= comparisons of genders adjusted for specific ages 27 Summary: surprises in Proc Logistic 1. Fits probability of no event by default. Opposite of what everyone expects. 2. Codes class variables using +1 ± 1 rather than 0 ± 1. Makes regression coefficients difficult to interpret. 3. Odds ratios are main effects, so no odds ratios for terms involved in interactions. Use oddsratio statement. 4. More: see Proc Logistic: Traps for the Unwary, by P.L. Flom (on course website) 28

15 Fitting a quadratic model Linear logistic regression model: obese = age 1.0 Fitted probability of obesity Age (years) 29 cannot capture the initial dip we see in the smooth: 30

16 To test for this initial decrease, fit a quadratic term in age: age2 = (age-15.)*(age-15.)/100.; add variable in data step proc logistic descending data=under50; model obese = age age2 / rsquare CLodds = PL ; Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 age age Conclusion? 31 Lack of fit test: Hosmer-Lemeshow Hosmer and Lemeshow (2000) proposed a test for lack of fit: 1. Divide the fitted probabilities into deciles (rank and divide into tenths). 2. Find the mean probability p i in each decile. 3. For each decile, calculate expected events as p i N i, for i = 1,..., Calculate chi-square test using observed and expected events: X 2 = 10X i=1 observed count expected count 2 expected count Null hypothesis: no lack of fit expected count = observed count. 32

17 proc logistic descending data=under50; model obese = age / rsquare CLodds = PL lackfit ; requests Hosmer-Lemeshow test Partition for the Hosmer and Lemeshow Test obese = 1 obese = 0 Group Total Observed Expected Observed Expected Hosmer and Lemeshow Goodness-of-Fit Test Chi-Square DF Pr > ChiSq Hosmer-Lemeshow results when the quadratic age term is included: Partition for the Hosmer and Lemeshow Test obese = 1 obese = 0 Group Total Observed Expected Observed Expected Hosmer and Lemeshow Goodness-of-Fit Test Chi-Square DF Pr > ChiSq

18 Ordinal logistic regression In the logistic regression example, we looked at how rate of obesity related to age and gender in NHANES Obesity is a binary response, defined by BMI 30. However, there is an intermediate category, defined in adults as: Obese: BMI 30. Overweight: 25 BMI < 30 Normal weight: 18 BMI < 25 Examine how rates of obesity and overweight relate to age and gender, then three ordered categories: Normal weight < Overweight < Obese 35 Ordinal response Response variable has three or more ordered categories. Ordinal response categories may be defined by a continuous measurement scale, as obesity and overweight are defined with reference to the BMI scale. Or they may just be ordered: Worse < No Change < Recovered where it does not make sense to ask about the distance between categories. Ordinal models use only the ranks of the categories. 36

19 Proc SGplot data=ph6470.age_bmi; loess y = obese x = age /LINEATTRS= (pattern=1 color="red"); loess y = overweight x = age /LINEATTRS= (pattern=1 color="blue"); 37 Divide age into two categories: young aged years old, and old aged Frequency Row Pct 1_normal 2_overwt 3_obese Total old young Total Percent in each weight category for old: ô 11, ô 12, ô 13. Percent in each weight category for young: ô 21, ô 22, ô

20 Multinomial distribution (generalizes binomial distribution): parameters are the probabilities (º ij ) of being in each category. Logistic regression estimates the difference of odds (on the log scale). Ordinal regression will fit two logistic regression simultaneously: Odds of being in the top category vs the rest: obese vs (overweight + normal) Odds of being in the top two categories vs the rest: (overweight + obese) vs normal Difference in odds between young and old assumed to be same for both. 39 Proportional odds model for ordinal responses Proportional odds model forces > 2 ordinal categories into binary comparisons by combining categories in sequence from the top. Gives cumulative odds: 1. Odds of top category vs the rest: obese vs normal + overweight 2. Odds of top two categories vs the rest: obese + overweight vs normal 3. Odds of being in the top three categories vs the rest, etc. To define these odds we define cumulative probabilities: µ 3 = chance of obesity = º 3 µ 2 = chance of obesity or overweight = º 2 + º 3, 40

21 Frequency Row Pct 1_normal 2_overwt 3_obese Total old young Total Find odds ratios for old to young of: obesity overweight + obesity 41 Normal + Obese Overweight odds odds ratio Age Age Obese + Overweight Normal odds odds ratio Age Age

22 Proportional odds model combines two logistic regression models: logit(µ h3 ) logit(µ h2 ) = log odds of being in the top category vs the rest, for group h = log odds of being in the top two category vs the rest, for group h 8 >< >: µh3 logit(µ h3 ) = log 1 µ h3 µh2 logit(µ h2 ) = log 1 µ h2 = Æ 3 + x h Ø = Æ 2 + x h Ø Ø estimates the same covariate effect in both models: an average effect (odds ratio) of age for both BMI cut-points, ratio of the odds for someone 40+ of being in a heavier category to the odds for someone Proc Logistic tests this assumption. 43 Fitting the proportional odds model in Proc Logistic Proc Logistic descending data=mayod327.age_bmi_sample; class age_category /param=glm; model bmi_cat = age_category ; bmi_cat has 3 levels. Default when the response has > 2 levels is the proportional odds model. 44

23 It is critical to check that SAS is combining categories in the right direction: use the descending option to reverse the order. Response Profile Ordered Total Value bmi_cat Frequency 1 3_obese _overwt _normal 308 Probabilities modeled are cumulated over the lower Ordered Values. From the log file: NOTE: PROC LOGISTIC is fitting the cumulative logit model. The probabilities modeled are summed over the responses having the lower Ordered Values in the Response Profile table. 45 Test of proportional odds assumption SAS tests H 0 : odds are proportional, against a larger model with separate effects of age for each category-comparison. In our example, this means two Ø values instead of one, so this larger model has 1 extra parameter and the test has 1 degree of freedom. Score Test for the Proportional Odds Assumption Chi-Square DF Pr > ChiSq

24 Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 3_obese <.0001 Intercept 2_overwt <.0001 age_category old <.0001 age_category young Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age_category old vs young Age effect: the old have 1.7 times the odds of being obese compared to the young, and 1.7 times the odds of being overweight or obese. Better: those have 1.7 times higher odds of being in a heavier category than those Intercepts are Æ 2 and Æ 3 : not informative. 47 Why not just do separate logistic regressions for each cut-point? Ordinal logistic regression also usually gives greater precision (more power): the standard error for the regression coefficient is smaller. Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq obese vs rest age_category old obese+overweight vs rest age_category old <.0001 Proportional Odds age_category old <.0001 Often a good model check. Regression coefficient from proportional odds is essentially an average of the regression coefficients in the 2 logistic regression models, but they are quite close. 48

25 Proportional odds model with age group and gender Proc Logistic descending data=mayod327.age_bmi_sample; class age_category gender /param=glm; model bmi_cat = age_category gender ; Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 3_obese <.0001 Intercept 2_overwt <.0001 age_category old <.0001 age_category young gender female gender male Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age_category old vs young gender female vs male With 2 age categories and 2 genders, we have 4 subgroups. We assume that the odds ratios between any two are the same for all cumulative comparisons of categories. Score Test for the Proportional Odds Assumption Chi-Square DF Pr > ChiSq It appears that this assumption fails for this data. What now? 50

26 Generalized logits model: unranked categories Proportional odds makes odds from adjoining ordered categories. If categories are not ordered, then proportional odds cannot be applied. Generalized logits makes odds between one reference category and all the other categories. Handles categories without order, eg. vanilla, strawberry, chocolate Proc Logistic descending data=mayod327.age_bmi_sample; class age_category gender / param=glm; model bmi_cat = age_category gender / link=glogit ; 51 Default is to use the highest category as reference: Ordered Total Value bmi_cat Frequency 1 3_obese _overwt _normal 308 Logits modeled use bmi_cat= 1_normal as the reference category. Notice that degrees of freedom are twice as large as they should be: Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq age_category <.0001 gender We are essentially fitting two separate models (normal vs overweight, normal vs obese). H 0 : reg coef for both models = 0 52

27 Here are the regression coefficients: note the doubling Standard Wald Parameter bmi_cat DF Estimate Error Chi-Square Pr > ChiSq Intercept 3_obese Intercept 2_overwt age_category old 3_obese <.0001 age_category old 2_overwt age_category young 3_obese age_category young 2_overwt gender female 3_obese gender female 2_overwt gender male 3_obese gender male 2_overwt If c response categories, then usual degrees of freedom are multiplied by (c 1). 53 Odds Ratio Estimates Point 95% Wald Effect bmi_cat Estimate Confidence Limits age_category old vs young 3_obese age_category old vs young 2_overwt gender female vs male 3_obese gender female vs male 2_overwt Averaging across genders, those over 40 have 2 times greater odds of being obese and 1.6 times greater odds of being overweight. Averaging across ages, women have about half the men s odds of being overweight, but about the same odds for obesity. 54