Biostatistics Case Studies 2005

Peter D. Christenson

Biostatistician

http://gcrc.humc.edu/Biostat

Session 4:

Taking Risks and Playing the Odds: OR vs. RR

Case Study

What's the Relative Risk? A Method of Correcting the Odds Ratio in Cohort Studies of Common OutcomesJun Zhang, MB, PhD; Kai F. Yu, PhD

JAMA. 1998;280:1690-1691. ABSTRACT

Logistic regression is used frequently in cohort studies and clinical trials. When the incidence of an outcome of interest is common in the study population (>10%), the adjusted odds ratio derived from the logistic regression can no longer approximate the risk ratio. The more frequent the outcome, the more the odds ratio overestimates the risk ratio when it is more than 1 or underestimates it when it is less than 1. We propose a simple method to approximate a risk ratio from the adjusted odds ratio and derive an estimate of an association or treatment effect that better represents the true relative risk

Further OR to RR References

1. McNutt LA, Wu C, Xue X, et. al. Estimating the relative risk in cohort studies and clinical trials of common outcomes. American Journal of Epidemiology 2003 ; 157: 940-3.

2. Greenland S. Model-based estimation of relative risks and other epidemiologic measures in studies of common outcomes and in case-control studies. American Journal of Epidemiology 2004 ; 160: 301-5.

Goals

• Why is RR vs. OR an issue?

• Examine the solution given in our case study.

• Point out difficulties with this solution.

• Suggest other solutions.

Why Use Odds Ratio at All?

• In case-control studies, cannot measure RR.• This is due to selection of controls. Example:

Risk Factor Cases Controls1 Controls2 + 90 60 600 - 10 40 400 100 100 1000Ratio of (90/150) (90/690)Percents /(10/50) /(10/410) = 3.0 = 5.3

Odds [(90/150)/(60/150)] [(90/690)/(600/690)]Ratio /[(10/50)/(40/50)] /[(10/410)/(400/410)] = 6.0 = 6.0

Cohort Study Example

Risk Factor Diseased

+ 50/100 = 50%- 30/100 = 30%

RR = 50%/30% = 1.67.

OR = (50%/50%)/(30%/70%) = 2.33

Cohort Study Example – Adjust for Gender

Strata Risk Factor Diseased

Male + 50/100 = 50%- 30/100 = 30%

Female + 80/100 = 80%- 40/120 = 33%

RR = (Male Weight)(50%/30%) + (Female Weight)(80%/33%) = (200/420)1.67 + (220/420)2.40 = 2.06

Other weights may be used.

Need for Additional Adjustment

Limitations of stratification:• Only a few covariates can be adjusted.• Covariates need to be categorical, or made

so.

Regression adjustment:• Allows more covariates (up to ~ 10 subjects

per parameter).• Allows continuous covariates.• Logistic, log-linear, poisson regression.

Logistic RegressionModel:

Log[Odds of disease] = Log[ Prob(disease) / Prob(non-disease) ] =

function( exposure, covariates, interactions)

Thus, only functions of the odds can be estimated, e.g., antilog of Log[Odds of disease] for exposed minus Log[Odds of disease] for unexposed, i.e., odds ratio (OR).

E.g., if log(odds) = 10.3 + 0.8(exposed) + 0.2(covariate),

then OR = exp(0.8) = 2.23, if exposed =1 or 0 for Yes or No

RR is Preferred

To obtain RR rather than OR, we can:

1. Convert OR from logistic regression to RR, or2. Use a model other than logistic that also fits the

data.

We consider (2) later (down 6 slides).

The solution proposed in our case study is to apply (1). Their solution is displayed on the next slide.

Case Study, Page 1691, 1st column

In a cohort study, P0 indicates the incidence of the outcome of interest in the nonexposed group and P1 in the exposed group;

OR, odds ratio; and RR, risk ratio:

OR=(P1/(1-P1))/(P0/(1-P0));

thus, (P1/P0)=OR/[(1-P0)+(P0xOR)].

Since RR=P1/P0, the corrected                  

Table from Case Study

The authors perform a simulation, setting the RR to be constant over all strata of the covariates, and show that their conversion does correct the logistic OR to a RR close to the stratified (M-H) RR which is correct here, with categorical strata:

Difficulties with the OR to RR Conversion

• The P0 is assumed to be fixed and known, so confidence intervals for the RR are too narrow.

• The formula is only valid if RR is constant over all covariate patterns, or is used for one particular set of covariates. Thus, it does not do the intended job of adjustment, i.e., account for confounding.

Breast Cancer Example

• See table on next slide.• Outcome: 5-year mortality.• Predictor: receptor level, low vs. high; low is

suspected mortality risk factor.• Covariate: CA staging I, II, III.• Want: RR of death (low over high receptor),

adjusted for stage.• For this study, do not need to model; can use

observed death rates. Thus, we can compare the logistic OR to RR results with actual RRs.

• Since the RRs increase with stage, the OR to RR conversion does not perform well.

Greenland (2004) Comparison of RRs

Adjusted RR

1.651.89*1.561.63

* Using Zhang & Yu:

OR=2.51 and

p0=(5+17+9)/

(55+74+15)=0.215:

Converted RR= 2.51/[(1-.215) + (0.215*2.51)]=1.89

2/121.84 1.78 1.43RR by Strata:

Alternative #1 to Zhang & Yu Conversion

E.g., if log(p/(1-p)) = 10.3 + 0.8(exposed) + 0.2(covariate) = u,

then p=eu/(1+eu), from algebra, where p = prob (death).

• The OR to RR conversion over-estimates RR.• An alternative is to find Prob[death] for a low

receptor and for a high receptor population, using the distribution of staging (“Standardized RRs”), and take the ratio of these probabilities.

• These probabilities can be found from the logistic equation:

• In fact, these probabilities are given in the row for logistic in the previous table. We now find the standardized risk ratio using these probabilities:

Alternative #1 continued

• If all women were at the low receptor level, the standardized (to the staging distribution) risk of death is:0.190(0.349) + 0.422(0.500) + 0.816(0.151) = 0.401

• Similarly, at the high receptor level,0.086(0.349) + 0.226(0.500) + 0.639(0.151) = 0.239

• Thus, the standardized RR is 0.401/0.239 = 1.68, which is close to the adjusted RR from the observed death rates.

• Note that the weights (0.349, 0.500, 0.151) are the relative proportions of women at each stage, e.g., 0.349=(12+55)/(12+55+22+74+14+15).

Alternative #2 to Zhang & Yu Conversion

Logistic: log(p/(1-p)) = 10.3 + 0.8(exposed) + 0.2(covariate)

Log-linear: log(p) = 7.1 + 0.6(exposed) + 0.1(covariate)

(made-up numbers)

• Use a model other than the logistic, provided that it fits as well.

• The Greenland(2004) table used two other models, log-linear (binomial) and Poisson regression.

• The log-linear binomial fits log(prob(death), compared to the logistic that fits log(odds):

• From the log-linear model, adjusted RR is found directly as antilog(0.6) = 1.82. This avoids the unwanted OR entirely.

SAS Code for Log-Linear Model of Breast Cancer

data ca; input receptor stage death survive total;datalines;0 1 2 10 121 1 5 50 550 2 9 13 221 2 17 57 740 3 12 2 141 3 9 6 15;proc genmod; class receptor stage; model death/total = receptor stage/dist=binomial link=log; estimate 'r1' intercept 1 receptor 1 0 stage 1 0 0/exp; estimate 'r2' intercept 1 receptor 0 1 stage 1 0 0/exp; estimate 'r3' intercept 1 receptor 1 0 stage 0 1 0/exp; estimate 'r4' intercept 1 receptor 0 1 stage 0 1 0/exp; estimate 'r5' intercept 1 receptor 1 0 stage 0 0 1/exp; estimate 'r6' intercept 1 receptor 0 1 stage 0 0 1/exp; estimate 'rr' receptor 1 -1 /exp;run; *r1-r6 are in the Greenland(2004) table;

SAS Code for Logistic Model of Breast Cancer

data ca; input receptor stage death survive total;datalines;0 1 2 10 121 1 5 50 550 2 9 13 221 2 17 57 740 3 12 2 141 3 9 6 15;proc logistic data=ca; class receptor stage; model death/total = receptor stage/lackfit; output out=out1 pred=predicted l=lower u=upper;run;proc print data=out1;run; *values for predicted are in the Greenland(2004) table;

Conclusions

• The JAMA OR to RR conversion is faulty.• Can use logistic regression, and find standardized

risks, then take ratio to get RR.• Can use log-linear models that model risk directly,

rather than odds.• Should check whether any model adequately fits the

data. Often, several do.• The major advantage of the log-linear model is that

confidence intervals for the adjusted RR are much easier. I know of no software that gives CIs for the standardized RRs.