Unformatted text preview: Lecture 18 1. Odds ratio and relative risk
2. Logbinomial regression
3. Ordinal regression: cumulative categories—proportional odds
4. Ordinal regression: comparison to single reference—generalized logits Review of logbinomial 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 1unit increase in X
Odds ratio sometimes used as an estimate of risk ratio = relative risk
Suppose number of events is A , number of nonevents 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
nonevents 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 Logbinomial 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 )
Logbinomial 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 logbinomial 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 1unit increase in X 8 Nonrare event: OR vs RR Example from Greenland (Am J Epidemiol 2004; 160:301–305)
Modelbased Estimation of Epidemiologic Measures 3
Cohort study followed 192 women with breast cancer, classiﬁed by breastcancer
TABLE II, III) relating receptor level (low, high)
stage (I,1. Dataand receptor level (low, high). and stage (I, II, III) to 5year breast cancer mortality (23), observed and modelbased estimates of average risk (incidence
Survival = alivereceptor level and stage, and distributions for standardizing receptorproportion) by 5 years after diagnosis
level comparisons to totalcohort 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 5year 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.
† Loglinear risk model
‡ Loglinear 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 modelbased 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 loglinear 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 loglin
regression; and 1.07, 2.47 for RR10 and 0.021, 0.299 for RD
using Poisson regression.
CASECONTROL STUDIES 10 Cumulative casecontrol studies sample cases and contr
from cohort members who do and do not get disease by
end of followup (5, pp. 110–111). Given a valid estimate
the crude (overall) risk rc in the target population or of
ratio of casecontrol sampling fractions rf, one can estim
the covariatespecific 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 casecont cancer mortality (23), observed and modelbased estimates of average risk (incidence
proportion) by receptor level and stage, and distributions for standardizing receptorlevel comparisons to totalcohort 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 5year data greenland; * from Am J Epidemiol (2004) 160:301305;
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 † Loglinear risk model with receptor level and stage, fit by binomial maximum likelihood.
‡ Loglinear 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 loglin
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 modelbased 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
CASECONTROL 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 thecasecontrol 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 followup (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 casecontrol
class
are 1.09, 2.57 and for RD10 are 0.013, 0.303 (8, 9); and,
the covariatespecific risks in the target (and hence th
0
model is used, the limits for RR
/ CLodds=PL;
e binomial loglinear 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 casecont
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 MantelHaenszel limits for RR10 are
, 2.39 and for RD10 are 0.016, 0.316 (5, p. 271). The log 12 casecohort studies, in which controls are sampled from
ar model fit by binomial maximumlikelihood supplies
cohort members, not just noncases (5, pp. 417, 419; 33).
narrowest riskratio interval because it is the only one of
In density casecontrol studies, controls are sampled lon
e methods that is fully efficient under the model.
tudinally from those at risk, in proportion to persontime 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 ChiSquare 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
ChiSquare 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 logbinomial regression. 16 Proc Genmod: logbinomial 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
ChiSquare
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 backtransform. 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 logbinomial 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 nonrare events, odds ratio is farther than relative risk from null value 1.
21 Why doesn’t everyone use logbinomial instead of logistic regression?
1. Logistic is numerically more stable: logbinomial 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_normal2_overwt3_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_normal2_overwt3_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 cutpoints, 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 categorycomparison.
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
ChiSquare 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
ChiSquare 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 cutpoint?
Ordinal logistic regression also usually gives greater precision (more power): the
standard error for the regression coefﬁcient is smaller. Parameter Standard
Error Wald
ChiSquare 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
ChiSquare 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
ChiSquare
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
ChiSquare
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
ChiSquare 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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 Fall '11
 WilliamThomas
 Regression Analysis, Proc Logistic

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