Handout18 - Lecture 18 1. Odds ratio and relative risk 2....

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Unformatted text preview: Lecture 18 1. Odds ratio and relative risk 2. Log-binomial regression 3. Ordinal regression: cumulative categories—proportional odds 4. Ordinal regression: comparison to single reference—generalized logits Review of log-binomial regression: Am J Epidemiol 2005; 162:199–200 Stokes, etal. (2000) Categorical Data Analysis Using the SAS System, 2nd Edition McCullagh, Nelder (1989) Generalized Linear Models, Second Edition Harrell (2001) Regression Modeling Strategies (Springer) 1 Odds and risk for rare events Logistic regression gives odds ratios from regression coefficients: ˆ exp(Ø X ) = odds ratio for 1-unit increase in X Odds ratio sometimes used as an estimate of risk ratio = relative risk Suppose number of events is A , number of non-events is B , odds = risk = number with event A = number without event B number with event A = number with event + number without event B+A 2 Relationship between odds ratio and relative risk odds = ± number with event = A B, number without event risk = ± number with event = A (A + B ) total number =) odds > risk. Let º1 and º2 be risk in groups 1 and 2, respectively. º1 relative risk = , º2 odds ratio = º1 1°º1 º2 1°º2 ≥ º ¥≥ 1 ° º ¥ 1 2 = 1 ° º1 º2 ≥ º ¥≥ 1 ° º ¥≥ º ¥ odds ratio 1 ° º2 1 2 2 = = relative risk 1 ° º1 º2 º1 1 ° º1 3 odds ratio 1 ° º2 = relative risk 1 ° º1 Fix relative risk as constant µ (theta): relative risk = µ = Substituting gives º1 º2 =) º1 = µº2 odds ratio 1 ° º2 1 ° º2 = = relative risk 1 ° º1 1 ° µº2 For relative risk = µ = 1, this ratio is one =) OR = RR For relative risk = µ > 1, this ratio is greater than one =) OR > RR For relative risk = µ < 1, this ratio is less than one =) OR < RR Odds ratio is farther than relative risk from null value 1. 4 ° ¢±° ¢ Graph of Odds Ratio/Relative Risk = 1 ° º2 1 ° µº2 for range of relative risk. 5 RR = 1.8 5 Ratio: Odds Ratio/Relative Risk 4 4 3 3 RR = 1.6 2 2 RR = 1.4 RR = 1.2 1 1 RR = 0.5 0.1 0.2 0.3 0.4 0.5 Reference risk (denominator), π2 rare event common event 5 Rare events: the number of events A is very small relative to the number of non-events B , B º B + A, so A A º B B+A that is, odds º risk (rate). When event is rare in both groups, odds ratio º relative risk. But when events are not rare, odds ratios are not good estimates of relative risk. Alternative: estimate relative risk directly. 6 Log-binomial regression: estimating relative risks directly Logistic regression: binomial response y , mean chance of event is º(x ) and µ ∂ º( x ) log = Ø0 + Ø1 x 1 ° º( x ) Log-binomial regression: binomial response y , mean chance of event is º(x ) and ° ¢ log º(x ) = Ø0 + Ø1 x Change the function that links the mean º to the linear regression Ø0 + Ø1 x 7 With log-binomial model, we estimate relative risk, not odds ratio. Regression coefficient for predictor x is ° ¢° ¢ Ø1 = Ø0 + Ø1(x + 1) ° Ø0 + Ø1(x ) ° ¢ ° ¢ = log p (x + 1) ° log p (x ) µ p (x + 1) = log p (x ) ∂ ° ¢ = log relative risk for unit increase in x =) exp(Ø1) = relative risk for 1-unit increase in X 8 Non-rare event: OR vs RR Example from Greenland (Am J Epidemiol 2004; 160:301–305) Model-based Estimation of Epidemiologic Measures 3 Cohort study followed 192 women with breast cancer, classified by breast-cancer TABLE II, III) relating receptor level (low, high) stage (I,1. Dataand receptor level (low, high). and stage (I, II, III) to 5-year breast cancer mortality (23), observed and model-based estimates of average risk (incidence Survival = alivereceptor level and stage, and distributions for standardizing receptorproportion) by 5 years after diagnosis level comparisons to total-cohort stage distribution Stage I Stage II Stage III Low High Low High Low High Deaths 2 5 9 17 12 9 Survivors 10 50 13 57 2 6 Total 12 55 22 74 14 15 409 230 857 600 422 226 816 639 376 242 870 558 386 237 905 555 Estimates of 5-year mortality not Event is death, risk rare in this example. (per 1,000) Observed* 167 91 Logistic 190 86 Binomial† 148 95 Poisson‡ 153 94 (Data source: Newman SC. Biostatistical methods in epidemiology. New York, NY: Wiley, 2001.) 9 Comparison distributions§ Model: p1(x) p0(x) death rate = receptor stage 0.349 0 0.500 0 0.151 0 0 0.349 0 0.500 0 0.151 * Deaths/total (nonparametric risk estimate). Compare results from with receptor level and stage, fit by binomial maximum likelihood. † Log-linear risk model ‡ Log-linear risk model fit by incorrect Poisson maximum likelihood. § p1(x) puts the total cohort at a low receptor level; p0(x) puts the total cohort at a high 1. Logistic regression odds ratios receptor level. 2. Log binomial relative risks is 1.89; both overestimate the risk ratio, as expected n that the outcome is not rare (over a quarter of the ents died). Another invalid model-based adjustment First: get data into SAS icts an expected number of exposed cases E from a el without exposure, then divides E into the observed ber of exposed cases to get a standardized mortality ratio sec. 4.3). This approach underestimates risk ratios (24); g a logistic model with only x2 and x3 yields E = 17.35 a standardized mortality ratio of 23/17.35 = 1.33. the observed proportions are used, 95 percent confie limits for RR10 are 1.06, 2.58 and for RD10 are 0.006, 4 (5, p. 263); if the logistic model is used, the limits for 0 are 1.09, 2.57 and for RD10 are 0.013, 0.303 (8, 9); and, e binomial log-linear model is used, the limits for RR10 xp(b1 ± 1.96v11/2) = 1.05, 2.30, where v1 is the estimated 0.023, 0.312 for RD10 using logistic regression; 1.05, 2.30 RR10 and 0.023, 0.312 for RD10 using binomial log-lin regression; and 1.07, 2.47 for RR10 and 0.021, 0.299 for RD using Poisson regression. CASE-CONTROL STUDIES 10 Cumulative case-control studies sample cases and contr from cohort members who do and do not get disease by end of follow-up (5, pp. 110–111). Given a valid estimate the crude (overall) risk rc in the target population or of ratio of case-control sampling fractions rf, one can estim the covariate-specific risks in the target (and hence th differences and ratios) even if the disease is common. If data are not sparse, one can use results from case-cont cancer mortality (23), observed and model-based estimates of average risk (incidence proportion) by receptor level and stage, and distributions for standardizing receptorlevel comparisons to total-cohort stage distribution Stage I Stage II Stage III Low High Low High Low High Deaths 2 5 9 17 12 9 Survivors 10 50 13 57 2 6 Total 12 55 22 74 14 15 Estimates of 5-year data greenland; * from Am J Epidemiol (2004) 160:301-305; mortality risk (per 1,000) input receptor$ stage deaths survivors; Observed* 167 409 230 857 total = deaths + survivors; 91 compute, don’t type Logistic 190 86 422 226 816 cards; 148 95 376 242 870 lowBinomial† 1 2 10 Poisson‡ 153 94 386 237 905 high 1 5 50 low 2 9 13 Comparison distributions§ high 2 17 57 0.349 0 0.500 0 0.151 lowp1(x)12 2 3 0 0.349 0 0.500 0 p3 high0(x) 9 6 ; * Deaths/total (nonparametric risk estimate). 600 639 558 555 0 0.151 † Log-linear risk model with receptor level and stage, fit by binomial maximum likelihood. ‡ Log-linear risk model fit by incorrect Poisson maximum likelihood. 11 § p1(x) puts the total cohort at a low receptor level; p0(x) puts the total cohort at a high receptor level. Grouped binary responses (binomial) 0.023, 0.312 for RD10 using logistic regression; 1.05, 2.30 is 1.89; both overestimate the risk ratio, as expected RR10 and 0.023, 0.312 for RD10 using binomial log-lin n that the outcome is not rare (over a quarter of the Proc Logistic ; regression; and 1.07, 2.47 for RR10 and 0.021, 0.299 for RD ents died). Another invalid model-based adjustment using Poisson regression. icts an expected number of number of events E/ from a of subjects = predictors ; exposed cases model number el without exposure, then divides E into the observed ber of exposed cases to get a standardized mortality ratio CASE-CONTROL STUDIES sec. 4.3). This approach underestimates risk ratios (24); NO descending x3 yields E = 17.35 g a logistic model with only x2 and option since events are specified in thecase-control studies sample cases and contr Cumulative model statement. a standardized mortality ratio of 23/17.35 = 1.33. from cohort members who do and do not get disease by the observed proportions are used, 95 percent confiend of follow-up (5, pp. 110–111). Given a valid estimate Proc Logistic data=greenland; e limits for RR10 are 1.06, 2.58 and for RD10 are 0.006, the crude (overall) risk rc in the target population or of 4 (5, p. 263); if the logistic receptor (ref="high") stage (ref="1") ; sampling fractions rf, one can estim model is used, the limits for ratio of case-control class are 1.09, 2.57 and for RD10 are 0.013, 0.303 (8, 9); and, the covariate-specific risks in the target (and hence th 0 model is used, the limits for RR / CLodds=PL; e binomial log-linear modeldeaths/total = receptor stage differences and ratios) even if the disease is common. If 10 xp(b1 ± 1.96v11/2) = 1.05, 2.30, where v1 is the estimated data are not sparse, one can use results from case-cont ance of b1, and for RD10 are 0.023, 0.312 (9). Standard modeling to estimate risks or rates and their contrasts (5, Set reference level in the class statement. son regression overestimates the variance of b1, yielding 418–419; 26–32). For models (such as the logistic) in wh ts for RR10 of 0.93, 2.87; nonetheless, GEE Poisson the baseline odds is a multiplicative factor, ln(rc) or ln ession with the robust variance estimate (available in becomes a simple adjustment term to the model intercept a proc xtgee and SAS proc genmod (25)) yields limits for pp. 417–419; 26, 27); other models can be used, howe (28, 31). Similar methods can be used to estimate risks fr 0 of 1.07, 2.48. The Mantel-Haenszel limits for RR10 are , 2.39 and for RD10 are 0.016, 0.316 (5, p. 271). The log- 12 case-cohort studies, in which controls are sampled from ar model fit by binomial maximum-likelihood supplies cohort members, not just noncases (5, pp. 417, 419; 33). narrowest risk-ratio interval because it is the only one of In density case-control studies, controls are sampled lon e methods that is fully efficient under the model. tudinally from those at risk, in proportion to person-time Number of Observations Read 6 Number of Observations Used 6 Sum of Frequencies Read 192 Sum of Frequencies Used 192 Response Profile Ordered Binary Value Total Outcome 1 Event 2 Frequency 54 Nonevent 138 13 Start by checking for interaction: proc logistic data=greenland; class receptor(ref="high") stage(ref="1"); model deaths/total = receptor stage receptor*stage Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq receptor 1 4.3873 0.0362 stage 2 22.8100 <.0001 receptor*stage 2 0.3381 0.8445 14 / CLodds=PL; Proc Logistic data=greenland; class receptor (ref="high") stage (ref="1") ; model deaths/total = receptor stage / CLodds=PL; Regression coefficients: Parameter Intercept receptor low stage 2 stage 3 DF Estimate Standard Error Wald Chi-Square Pr > ChiSq 1 1 1 1 -0.5503 0.4598 -0.2223 1.5791 0.2212 0.1977 0.2499 0.3231 6.1906 5.4080 0.7914 23.8796 0.0128 0.0200 0.3737 <.0001 15 Odds ratio estimates from logistic regression. Profile Likelihood Confidence Interval for Adjusted Odds Ratios Effect receptor low vs high stage 2 vs 1 stage 3 vs 1 Unit 1.0000 1.0000 1.0000 Estimate 2.508 3.110 18.839 95% Confidence Limits 1.148 5.454 1.306 8.303 6.299 63.727 Next: relative risk estimates from log-binomial regression. 16 Proc Genmod: log-binomial regression Use Proc Genmod, specify binomial distribution and log link: Proc Genmod data=greenland; class receptor stage ; model deaths/total = receptor stage / type3 dist=binomial link= log ; To fit logistic regression (same results as previous page), specify logit link: Proc Genmod data=greenland; class receptor stage ; model deaths/total = receptor stage / type3 dist=binomial link= logit ; 17 Reference level is supposed to work in Proc Genmod but doesn’t. Either recode levels, or use ESTIMATE statement. Parameter Intercept receptor receptor stage stage stage Scale high low 1 2 3 DF 1 1 0 1 1 0 0 Estimate -0.1390 -0.4436 0.0000 -1.7695 -0.8381 0.0000 1.0000 Parameter Intercept receptor receptor stage stage stage Scale Standard Error 0.0955 0.2021 0.0000 0.3875 0.2154 0.0000 0.0000 high low 1 2 3 18 Wald 95% Confidence Limits -0.3262 0.0482 -0.8396 -0.0476 0.0000 0.0000 -2.5290 -1.0101 -1.2604 -0.4158 0.0000 0.0000 1.0000 1.0000 Pr > ChiSq 0.1457 0.0281 . <.0001 0.0001 . Wald Chi-Square 2.12 4.82 . 20.85 15.13 . Proc Genmod data=greenland; class receptor(ref=first) stage(ref=first); * ref doesn’t work; model deaths/total = receptor stage / type3 dist=bin link=log; estimate "receptor low vs. high" receptor -1 1 / exp ; estimate "stage2 vs stage1" stage -1 1 0 / exp; estimate "stage 3 vs stage1" stage -1 0 1/exp; ODS output ParameterEstimates = reg_coef; Genmod doesn’t calculate relative risks for you. Use ODS to save regression coefficients to data set, then back-transform. 19 Exponentiating the estimated difference gives the relative risk: Contrast Estimate Results Label Estimate Standard Error Alpha Confidence Limits receptor low vs. high Exp(receptor low vs. high) 0.4436 1.5583 0.2021 0.3149 0.05 0.05 0.0476 1.0487 0.8396 2.3155 stage2 vs stage1 Exp(stage2 vs stage1) 0.9314 2.5382 0.3937 0.9991 0.05 0.05 0.1599 1.1734 1.7030 5.4903 stage 3 vs stage1 Exp(stage 3 vs stage1) 1.7695 5.8680 0.3875 2.2738 0.05 0.05 1.0101 2.7458 2.5290 12.5406 20 Relative risks from log-binomial model Label Estimate Confidence Limits Exp(receptor low vs. high) 1.5583 1.0487 2.3155 Exp(stage2 vs stage1) 2.5382 1.1734 5.4903 Exp(stage 3 vs stage1) 5.8680 2.7458 12.5406 Odds ratios from logistic regression Effect Estimate receptor low vs high 95% Confidence Limits 2.508 1.148 5.454 stage 2 vs 1 3.110 1.306 8.303 stage 3 vs 1 18.839 6.299 63.727 For non-rare events, odds ratio is farther than relative risk from null value 1. 21 Why doesn’t everyone use log-binomial instead of logistic regression? 1. Logistic is numerically more stable: log-binomial does not always converge to produce an answer. 2. Logistic is conventional approach, software more developed. 22 Ordinal logistic regression In the logistic regression example, we looked at how rate of obesity related to age and gender in NHANES 2004. Obesity is a binary response, defined by BMI ∏ 30. However, there is an intermediate category: • 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 23 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. 24 Probability of being obese, as function of age. 25 Probability of being overweight or obese, as function of age. 26 Divide age into two categories: young aged 20–39 years old, and old aged 40–60. Frequency| Row Pct |1_normal|2_overwt|3_obese | Total ---------+--------+--------+--------+ old | 118 | 153 | 183 | 454 | 25.99 | 33.70 | 40.31 | ---------+--------+--------+--------+ young | 190 | 155 | 142 | 487 | 39.01 | 31.83 | 29.16 | ---------+--------+--------+--------+ Total 308 308 325 941 ˆ ˆ ˆ Percent in each weight category for old: º11, º12, º13. ˆ ˆ ˆ Percent in each weight category for young: º21, º22, º23. 27 Multinomial distribution (generalizes binomial distribution): parameters are the probabilities (ºi j ) 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. 28 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: µ3 = chance of obesity = º3 µ2 = chance of obesity or overweight = º2 + º3, 29 Frequency| Row Pct |1_normal|2_overwt|3_obese | ---------+--------+--------+--------+ old | 118 | 153 | 183 | ---------+--------+--------+--------+ young | 190 | 155 | 142 | ---------+--------+--------+--------+ Total 308 308 325 Find odds ratios for old to young of: obesity overweight + obesity 30 Total 454 487 941 Normal + Obese Overweight odds odds ratio Age 40+ 183 271 0.6753 Age 20–39 142 345 0.4116 1.64 Overweight Normal odds odds ratio Age 40+ 336 118 2.85 Age 20–39 297 190 1.56 Obese + 1.84 31 Proportional odds model combines two logistic regression models: logit(µh 3) = log odds of being in the top category vs the rest, for group h logit(µh 2) = log odds of being in the top two category vs the rest, for group h ∑ ∏ 8 µh 3 > > logit(µh 3) = log = Æ3 + x h Ø > > > 1 ° µh 3 < > ∑ ∏ > > > > logit(µh 2) = log µh 2 : = Æ2 + x h Ø 1 ° µh 2 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 20–39. Proc Logistic tests this assumption. 32 Fitting the proportional odds model in Proc Logistic Proc Logistic class descending age_category data=mayod327.age_bmi_sample; /param=glm; model bmi_cat = age_category ; bmi_cat has 3 levels. Default when the response has > 2 levels is the proportional odds model. 33 It is critical to check that SAS is combining categories in the right direction: use the descending option to reverse the order. Ordered Value 1 2 3 Response Profile bmi_cat 3_obese 2_overwt 1_normal Total Frequency 325 308 308 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. 34 Test of proportional odds assumption SAS tests H0 : 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 0.5562 1 0.4558 35 DF Parameter Intercept Intercept age_category age_category 3_obese 2_overwt old young Estimate Standard Error Wald Chi-Square Pr > ChiSq 1 1 1 0 -0.9162 0.4680 0.5451 0 0.0926 0.0886 0.1210 . 97.7973 27.8689 20.2876 . <.0001 <.0001 <.0001 . Odds Ratio Estimates Effect age_category old vs young Point Estimate 1.725 95% Wald Confidence Limits 1.361 2.187 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 40–60 have 1.7 times higher odds of being in a heavier category than those 20–39. Intercepts are Æ2 and Æ3: not informative. 36 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. Parameter Standard Error Wald Chi-Square DF Estimate 1 0.4951 0.1382 12.8353 0.0003 obese+overweight vs rest age_category old 1 0.5996 0.1417 17.9064 <.0001 Proportional Odds age_category old 0.5451 0.1210 20.2876 <.0001 obese vs rest age_category old 1 Pr > ChiSq 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. 37 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 DF Parameter Intercept Intercept age_category age_category gender gender gender ; 3_obese 2_overwt old young female male Estimate Standard Error Wald Chi-Square Pr > ChiSq 1 1 1 0 1 0 -0.9528 0.4318 0.5440 0 0.0775 0 0.1097 0.1058 0.1211 . 0.1204 . 75.4200 16.6533 20.1875 . 0.4148 . <.0001 <.0001 <.0001 . 0.5195 . Odds Ratio Estimates Effect age_category old vs young gender female vs male Point Estimate 1.723 1.081 38 95% Wald Confidence Limits 1.359 2.184 0.854 1.368 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 13.2908 DF 2 Pr > ChiSq 0.0013 It appears that this assumption fails for this data. What now? 39 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 / 40 link=glogit ; Default is to use the highest category as reference: Ordered Value 1 2 3 bmi_cat 3_obese 2_overwt 1_normal Total Frequency 325 308 308 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 Effect age_category gender DF 2 2 Wald Chi-Square 20.6261 12.8702 Pr > ChiSq <.0001 0.0016 We are essentially fitting two separate models (normal vs overweight, normal vs obese). H0: reg coef for both models = 0 41 Here are the regression coefficients: note the doubling Parameter bmi_cat DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Intercept Intercept 3_obese 2_overwt 1 1 -0.3337 0.00496 0.1394 0.1314 5.7279 0.0014 0.0167 0.9699 age_category age_category age_category age_category old old young young 3_obese 2_overwt 3_obese 2_overwt 1 1 0 0 0.7278 0.4754 0 0 0.1621 0.1643 . . 20.1676 8.3743 . . <.0001 0.0038 . . gender gender gender gender female female male male 3_obese 2_overwt 3_obese 2_overwt 1 1 0 0 0.0814 -0.4576 0 0 0.1613 0.1633 . . 0.2545 7.8491 . . 0.6139 0.0051 . . If c response categories, then usual degrees of freedom are multiplied by (c ° 1). 42 Odds Ratio Estimates Effect bmi_cat age_category old age_category old gender gender Point Estimate 95% Wald Confidence Limits vs young vs young 3_obese 2_overwt 2.071 1.609 1.507 1.166 2.845 2.220 female vs male female vs male 3_obese 2_overwt 1.085 0.633 0.791 0.459 1.488 0.872 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. 43 ...
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This note was uploaded on 11/21/2011 for the course PUBH 6470 taught by Professor Williamthomas during the Fall '11 term at University of Florida.

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