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The Stata Journal (2009)
9, Number 2, pp. 175–196

Methods for estimating adjusted risk ratios

Peter Cummings
Department of Epidemiology
School of Public Health and
Harborview Injury Prevention and Research Center
University of Washington
Seattle, WA
peterc@u.washington.edu

Abstract. The risk ratio can be a useful statistic for summarizing the results of
cross-sectional, cohort, and randomized trial studies. I discuss several methods for
estimating adjusted risk ratios and show how they can be executed in Stata, in-
cluding 1) Mantel–Haenszel and inverse-variance stratified methods; 2) generalized
linear regression with a log link and binomial distribution; 3) generalized linear
regression with a log link, normal distribution, and robust variance estimator; 4)
Poisson regression with a robust variance estimator; 5) Cox proportional hazards
regression with a robust variance estimator; 6) standardized risk ratios from logis-
tic, probit, complementary log-log, and log-log regression; and 7) a substitution
method. Advantages and drawbacks are noted for some methods.
Keywords: st0162, risk ratio, odds ratio

1 Introduction
The case–control study design is typically (but not always) used when outcomes are
rare in the population from which study subjects are sampled. In 1951, Cornfield noted
that when outcomes are sufficiently rare, the odds ratio from a case–control study will
approximate the population risk ratio for the association of an exposure with a disease
outcome. It was later realized that if controls are sampled as each case arises in time,
the odds ratio will estimate the incidence-rate ratio even when outcomes are common
(Greenland and Thomas 1982; Rodrigues and Kirkwood 1990; Rothman, Greenland,
and Lash 2008, 113–114). In Stata, case–control data can be analyzed using Mantel–
Haenszel stratified methods (cc, tabodds, mhodds), logistic regression (logistic), or
conditional logistic regression (clogit) to estimate adjusted odds ratios that usually
can be interpreted either as risk ratios (when outcomes are rare) or incidence-rate ratios
(when incidence density sampling is used).
Cross-sectional, cohort, and randomized controlled trial designs with binary out-
comes can often be summarized by estimating odds ratios or risk ratios. If the study
outcome is sufficiently rare among exposed and unexposed study subjects, the odds
ratio for the exposure–outcome association will closely approximate the risk ratio. But
if the outcome is common and the risk ratio is not close to 1, the odds ratio will be
further from 1 compared with the risk ratio. Even if the outcome is rare in the entire
sample, if an adjustment is made for other variables, then the adjusted odds ratio will


c 2009 StataCorp LP st0162
176 Methods for estimating adjusted risk ratios

be further from 1 than the adjusted risk ratio if the outcome is common in adjustment
variable subgroups that contribute a noteworthy portion of the outcomes (Greenland
1987).
When summary odds and risk ratios differ, there is debate regarding which is prefer-
able. Some have argued that odds ratios are preferred because they are symmetric with
regard to the outcome definition (Walter 1998; Olkin 1998; Senn 1999; Newman 2001,
35–40; Cook 2002). Furthermore, when outcomes are common, a constant (homoge-
neous) adjusted odds ratio for all subjects may be more plausible than a constant risk
ratio (Levin 1991; Senn 1998; Cook 2002).
Some who favor risk ratios feel they are more easily understood by physicians
(Sackett, Deeks, and Altman 1996). Others have noted that risk ratios have a desir-
able feature called collapsibility; in the absence of confounding, a weighted average of
stratum-specific risk ratios will equal the ratio from one 2 × 2 table of the pooled (col-
lapsed) counts from the stratum-specific tables (Miettinen and Cook 1981; Greenland
1987, 1991b; Greenland, Robins, and Pearl 1999; Newman 2001, 52–55; Rothman,
Greenland, and Lash 2008, 62). This means that a crude (unadjusted) risk ratio will not
change if we adjust for a variable that is not a confounder. In the absence of confound-
ing, the risk ratio estimates the change in risk, on a ratio scale, for the entire exposed
group due to exposure. Because of collapsibility, this risk ratio has a useful interpre-
tation as the ratio change in the average risk in the exposed group due to exposure.
It is not the average ratio change in risk (i.e., the average risk ratio) among exposed
individuals, except in the unlikely event that the risk ratios for all individuals are the
same (Greenland 1987).
Odds ratios lack the property of collapsibility and therefore the interpretation of an
odds ratio is more limited; in the absence of confounding, it estimates the change in odds,
on a ratio scale, in the exposed group due to exposure. But it does not estimate either
the change in the average odds of the exposed due to exposure or the average change in
odds (i.e., the average odds ratio) among exposed individuals, not even if all individuals
had the same change in odds when exposed (Greenland 1987). The odds ratio will
estimate the average change in odds for exposed individuals only if all individual odds
ratios are the same and all individual risks without exposure are the same. Except in this
unlikely situation, the crude odds ratio will be closer to 1 than the average of stratum-
specific or individual odds ratios. Even in the absence of confounding, the adjusted
(conditional) odds ratio will be further from 1 than the crude (unadjusted or marginal)
odds ratio (Gail et al. 1984; Greenland 1987; Hauck et al. 1998; Steyerberg et al. 2000;
Newman 2001, 52–55; Rothman, Greenland, and Lash 2008, 62; Cummings 2009).
For analysts who wish to estimate odds ratios for the association of exposure with
disease in a cross-sectional study, cohort study, or randomized trial, the statistical meth-
ods in Stata’s cc, tabodds, mhodds, and logistic commands can be used. If the goal is
to estimate risk ratios, these same methods can be used if outcomes are sufficiently rare
that odds ratios will closely approximate risk ratios. But if risk ratios are desired when
outcomes are common, odds ratio estimates will not suffice. In this article, I describe
methods for estimating adjusted risk ratios with confidence intervals (CIs) in Stata.
P. Cummings 177

2 Data used to illustrate the methods

I will show how to reproduce the risk-ratio estimates and CIs that Greenland (2004a)
gave in a review of risk-ratio estimation. The data (table 1) are from table 5.3 in
Newman’s (2001, 98 and 126) textbook. Newman described 192 women who were
diagnosed with breast cancer in Canada and followed for 5 years; 28% (54/192) of the
women died, so the outcome was not rare. Greenland estimated the risk ratio for death
at 5 years among women with low estrogen-receptor levels in their breast cancer tissue
compared with women who had high receptor levels; these risk ratios were adjusted for
cancer stage (I, II, or III) so that women with the same cancer stage were compared.

Table 1. Deaths, total subjects, and risk of death for 192 women with breast cancer
followed for 5 years, by stage at diagnosis (I, II, III) and estrogen-receptor–level category
(low, high). Also, risk ratios within each cancer stage for death among women with low
versus high receptor levels.

Receptor levels Stage Died Total Risk Risk ratio for death
comparing women with low
versus high receptor levels
Low I 2 12 0.17 1.8
High I 5 55 0.09 1.0 (reference group)
Low II 9 22 0.41 1.8
High II 17 74 0.23 1.0 (reference group)
Low III 12 14 0.86 1.4
High III 9 15 0.60 1.0 (reference group)

3 Method 1: Mantel–Haenszel and inverse-variance

stratified methods
Mantel–Haenszel methods for odds ratios were described in 1959 (Mantel and Haenszel
1959) and extended to risk ratios in 1981 (Nurminen 1981; Tarone 1981; Kleinbaum,
Kupper, and Morgenstern 1982; Newman 2001, 148–149; Rothman, Greenland, and
Lash 2008, 274–275). We can estimate the adjusted risk ratio for death associated with
low estrogen-receptor levels (the low variable) compared with high estrogen-receptor
levels by using the cs command:

(Continued on next page)

178 Methods for estimating adjusted risk ratios

. use brcadat
(Breast cancer data)
. cs died low, by(stage) pool
Cancer stage RR [95% Conf. Interval] M-H Weight

1 1.833333 .4024728 8.35115 .8955224

2 1.780749 .9269437 3.420991 3.895833
3 1.428571 .8971062 2.274888 4.344828

Crude 2.225806 1.449035 3.418974

Pooled (direct) 1.554553 1.076372 2.245169
M-H combined 1.618421 1.093775 2.394719

Test of homogeneity (direct) chi2(2) = 0.339 Pr>chi2 = 0.8443

Test of homogeneity (M-H) chi2(2) = 0.385 Pr>chi2 = 0.8251

I invoked the pool option so that the output shows both the Mantel–Haenszel combined
risk ratio and the pooled risk ratio obtained using inverse-variance weights. These
methods require that variables be treated as categorical, not continuous.

4 Method 2: Generalized linear regression with a log link

and binomial distribution
Estimation of risk ratios using a generalized linear model with a log link and bino-
mial distribution was proposed in 1986 (Wacholder 1986). This approach has been
described in several articles (Robbins, Chao, and Fonseca 2002; McNutt et al. 2003;
Barros and Hirakata 2003); it has been called log-binomial (Blizzard and Hosmer 2006)
or binomial log-linear regression (Greenland 2004a). This approach can be implemented
in Stata:
P. Cummings 179

. glm died low stage2 stage3, family(binomial) link(log) eform difficult

Iteration 0: log likelihood = -154.81266 (not concave)
Iteration 1: log likelihood = -99.23223
Iteration 2: log likelihood = -94.764125
Iteration 3: log likelihood = -93.93508
Iteration 4: log likelihood = -93.050613
Iteration 5: log likelihood = -92.930394
Iteration 6: log likelihood = -92.927072
Iteration 7: log likelihood = -92.927069
Generalized linear models No. of obs = 192
Optimization : ML Residual df = 188
Scale parameter = 1
Deviance = 185.8541388 (1/df) Deviance = .9885858
Pearson = 190.2212968 (1/df) Pearson = 1.011815
Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = ln(u) [Log]
AIC = 1.009657
Log likelihood = -92.92706941 BIC = -802.555

OIM
died Risk Ratio Std. Err. z P>|z| [95% Conf. Interval]

low 1.558321 .3148624 2.20 0.028 1.048745 2.315497

stage2 2.538159 .9991488 2.37 0.018 1.17339 5.490288
stage3 5.868042 2.273768 4.57 0.000 2.745787 12.54064

Above Stata reported that for iteration 0, the likelihood region was not concave.
When the command is run without the difficult option, Stata 10.0 will repeatedly
report a not-concave region and fail to converge. The difficult option changed Stata’s
convergence algorithm and solved the problem in this example, but that option may
not always work. The convergence problem arose because among women with Stage III
cancer and low estrogen-receptor levels, the risk of death was close to 1: 12/14 = 0.86.
When the risk is close to 1 in a stratum of the data, maximum-likelihood convergence
may fail. This problem has been discussed in several articles (Carter, Lipsitz, and Tilley
2005; Blizzard and Hosmer 2006; Lumley, Kronmal, and Ma 2006; Localio, Margolis,
and Berlin 2007).
Wacholder (1986; Lumley, Kronmal, and Ma 2006) described a method that modi-
fied the convergence by truncating estimated risks to values slightly greater than 0 and
less than 1. This is implemented in Stata’s binreg command:

(Continued on next page)

180 Methods for estimating adjusted risk ratios

. xi: binreg died low i.stage, rr

i.stage _Istage_1-3 (naturally coded; _Istage_1 omitted)
Iteration 1: deviance = 309.6253
Iteration 2: deviance = 190.79
Iteration 3: deviance = 185.9416
Iteration 4: deviance = 185.8543
Iteration 5: deviance = 185.8541
Iteration 6: deviance = 185.8541
Generalized linear models No. of obs = 192
Optimization : MQL Fisher scoring Residual df = 188
(IRLS EIM) Scale parameter = 1
Deviance = 185.8541388 (1/df) Deviance = .9885858
Pearson = 190.2164034 (1/df) Pearson = 1.011789
Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = ln(u) [Log]
BIC = -802.555

EIM
died Risk Ratio Std. Err. z P>|z| [95% Conf. Interval]

low 1.558326 .3067419 2.25 0.024 1.059515 2.291972

_Istage_2 2.538158 .9976133 2.37 0.018 1.174781 5.483782
_Istage_3 5.868047 2.259727 4.60 0.000 2.758699 12.48196

The risk ratios and standard errors estimated by glm, family(binomial)

link(log) and by binreg, rr are similar, but not identical, because the convergence
methods differ. The default for glm is maximum likelihood and the default for binreg
is iterated, reweighted least squares; by changing the default optimization, either com-
mand can produce the estimates obtained by the other, provided that both methods
achieve convergence. Because binreg constrains risk estimates to be greater than 0
and less than 1, it may converge when maximum likelihood will not. Sometimes both
methods will fail to converge.
Another approach to convergence difficulty is to make several copies of the original
data and append them into one data file (Deddens, Petersen, and Lei 2003; Petersen
and Deddens 2006; Deddens and Petersen 2008). Then make one more copy, but recode
all the 0 outcomes to 1 and all the 1 outcomes to 0 in that copy, and append this recoded
copy to all the other copies. Then analyze all these data together; including one set of
data with reversed outcomes may help the maximum-likelihood algorithm converge. If
the number of copies is sufficiently large, the risk-ratio estimates will approximate those
from maximum-likelihood methods. Because a set of records larger than the original
data is used, corrections must be made to the standard errors and CIs. This extra
step of correcting the standard errors can be avoided by using just two copies of the
data, one with recoded outcomes, with appropriate weights (Lumley, Kronmal, and Ma
2006). Below I used importance weights of 0.999 and 0.001, and produced the risk ratio
I would get by analyzing 999 copies of the original data with just 1 copy of recoded
data. The difficult option was still required.
P. Cummings 181

. generate iweight = .999

. append using brcadat
(label noyes already defined)
. recode died 0=1 1=0 if iweight==.
(died: 192 changes made)
. replace iweight = .001 if iweight==.
(192 real changes made)
. glm died low stage2 stage3 [iweight=iweight], family(binomial) link(log)
> eform difficult
Iteration 0: log likelihood = -155.22946 (not concave)
Iteration 1: log likelihood = -99.401459
Iteration 2: log likelihood = -94.953215
Iteration 3: log likelihood = -94.13882
Iteration 4: log likelihood = -93.23838
Iteration 5: log likelihood = -93.113209
Iteration 6: log likelihood = -93.109614
Iteration 7: log likelihood = -93.109611
Generalized linear models No. of obs = 384
Optimization : ML Residual df = 380
Scale parameter = 1
Deviance = 186.2192211 (1/df) Deviance = .4900506
Pearson = 190.253604 (1/df) Pearson = .5006674
Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = ln(u) [Log]
AIC = .5057792
Log likelihood = -93.10961053 BIC = -2075.025

OIM
died Risk Ratio Std. Err. z P>|z| [95% Conf. Interval]

low 1.556724 .3143539 2.19 0.028 1.047915 2.312583

stage2 2.523499 .9897355 2.36 0.018 1.169918 5.443157
stage3 5.823722 2.248418 4.56 0.000 2.732558 12.41172

5 Method 3: Generalized linear regression with a log link,

normal distribution, and robust variance estimator
Convergence problems for generalized linear regression with a log link can also be re-
solved by using a Gaussian (normal) distribution (Lumley, Kronmal, and Ma 2006).
The resulting standard errors may be too big or small, but a robust variance estimator
will correct the standard errors. Below convergence was achieved without the difficult
option:

(Continued on next page)

182 Methods for estimating adjusted risk ratios

. use brcadat, clear

(Breast cancer data)
. glm died low stage2 stage3, family(gaussian) link(log) eform robust
Iteration 0: log pseudolikelihood = -169.42788
Iteration 1: log pseudolikelihood = -119.04978
Iteration 2: log pseudolikelihood = -98.365962
Iteration 3: log pseudolikelihood = -94.332246
Iteration 4: log pseudolikelihood = -94.242364
Iteration 5: log pseudolikelihood = -94.242364
Generalized linear models No. of obs = 192
Optimization : ML Residual df = 188
Scale parameter = .1595924
Deviance = 30.00336652 (1/df) Deviance = .1595924
Pearson = 30.00336652 (1/df) Pearson = .1595924
Variance function: V(u) = 1 [Gaussian]
Link function : g(u) = ln(u) [Log]
AIC = 1.023358
Log pseudolikelihood = -94.24236355 BIC = -958.4058

Robust
died exp(b) Std. Err. z P>|z| [95% Conf. Interval]

low 1.553274 .3193155 2.14 0.032 1.038154 2.323992

stage2 2.524622 1.005045 2.33 0.020 1.157006 5.508803
stage3 5.86819 2.312817 4.49 0.000 2.710329 12.70534

6 Method 4: Poisson regression with a robust variance

estimator
Poisson regression is a generalized linear model with a log link and a Poisson distribu-
tion. When the outcome is binary, the exponentiated coefficients are risk ratios instead
of incidence-rate ratios (Gourieroux, Monfort, and Trognon 1984a,b; Lloyd 1999, 85–
86; Wooldridge 2002, 648–649; Greenland 2004a; Zou 2004; Carter, Lipsitz, and Tilley
2005). Methods that rely on the Poisson distribution assume that the mean count and
its variance are equal. In the breast cancer data, the mean count of deaths per woman
was 54/192 = 0.28125. If the variance of the mean count is also 0.28125, then the stan-
dard error of the mean count is the square root of the variance divided by the square
root of the number of women (192) = 0.0382733. This is indeed the standard error
that Stata reports using the ci, poisson command or using lincom after the poisson
command:
. ci died, poisson
Poisson Exact
Variable Exposure Mean Std. Err. [95% Conf. Interval]

died 192 .28125 .0382733 .2112837 .3669702

P. Cummings 183

. poisson died, nolog

Poisson regression Number of obs = 192
LR chi2(0) = 0.00
Prob > chi2 = .
Log likelihood = -122.49961 Pseudo R2 = 0.0000

died Coef. Std. Err. z P>|z| [95% Conf. Interval]

_cons -1.268511 .1360828 -9.32 0.000 -1.535229 -1.001794

. lincom _cons, irr

( 1) [died]_cons = 0

died IRR Std. Err. z P>|z| [95% Conf. Interval]

(1) .28125 .0382733 -9.32 0.000 .2154064 .3672201

If the deaths were from a Poisson distribution, women would have nonnegative in-
teger counts of 0, 1, 2, 3, . . . , or more deaths. The data cannot be Poisson, because
no woman dies more than once. The data are from a binomial distribution, and the
binomial variance is assumed to be the proportion that died multiplied by 1 minus that
proportion. The standard error of the mean proportion is the square root of the variance
divided by the square root of the number of women, which is 0.0324477. This is the
standard error reported by ci, binomial:

. ci died, binomial
Binomial Exact
Variable Obs Mean Std. Err. [95% Conf. Interval]

died 192 .28125 .0324477 .2188833 .3505085

If we use Poisson methods for these binomial data, the standard error for the outcome
proportion (risk) is too large: 0.03827 instead of 0.03245. As the outcome becomes less
common, the Poisson standard error will converge toward the binomial standard error
(Armitage, Berry, and Matthews 2002, 71–76). But in the breast cancer data, use of
Poisson regression to estimate risk ratios will produce standard errors, p-values, and CIs
that are too large:

. poisson died low stage2 stage3, irr nolog

Poisson regression Number of obs = 192
LR chi2(3) = 26.71
Prob > chi2 = 0.0000
Log likelihood = -109.14601 Pseudo R2 = 0.1090

died IRR Std. Err. z P>|z| [95% Conf. Interval]

low 1.630775 .4688634 1.70 0.089 .9282513 2.864987

stage2 2.520742 1.074375 2.17 0.030 1.093288 5.811955
stage3 5.913372 2.645148 3.97 0.000 2.460814 14.20992
184 Methods for estimating adjusted risk ratios

Above, the 95% CI for the low variable is wide, 0.93 to 2.86, compared with the CIs
from other methods. We can obtain standard errors and CIs that are approximately
correct by using a robust variance estimator, which can relax the assumption that the
data are from a Poisson distribution (Wooldridge 2002, 650–651; Greenland 2004a; Zou
2004; Carter, Lipsitz, and Tilley 2005). The robust variance estimator is sometimes
called the Huber, White, Huber–White, sandwich, or survey estimator, as well as other
names (Hardin and Hilbe 2007, 35–36). In Stata, we can invoke this estimator with the
vce(robust) option and the CI for the low variable becomes narrower, 1.07 to 2.48:

. poisson died low stage2 stage3, irr nolog vce(robust)

Poisson regression Number of obs = 192
Wald chi2(3) = 53.61
Prob > chi2 = 0.0000
Log pseudolikelihood = -109.14601 Pseudo R2 = 0.1090

Robust
died IRR Std. Err. z P>|z| [95% Conf. Interval]

low 1.630775 .3480542 2.29 0.022 1.073305 2.477792

stage2 2.520742 .9937819 2.35 0.019 1.16399 5.458932
stage3 5.913372 2.28568 4.60 0.000 2.772187 12.61386

7 Method 5: Cox proportional hazards regression with a

robust variance estimator
Cox proportional hazards regression can estimate risk ratios if we set the follow-up time
to 1, or any quantity that is the same for all subjects, and use the Breslow method
to break ties. The robust variance estimator should be used, because otherwise the
standard errors will be too large:

. generate byte time = 1

. stset time, failure(died) noshow
failure event: died != 0 & died < .
obs. time interval: (0, time]
exit on or before: failure

192 total obs.

0 exclusions

192 obs. remaining, representing

54 failures in single record/single failure data
192 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 1
P. Cummings 185

. stcox low stage2 stage3, hr breslow vce(robust) nolog

Cox regression -- Breslow method for ties
No. of subjects = 192 Number of obs = 192
No. of failures = 54
Time at risk = 192
Wald chi2(3) = 53.61
Log pseudolikelihood = -270.55115 Prob > chi2 = 0.0000

Robust
_t Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]

low 1.630775 .3480542 2.29 0.022 1.073305 2.477792

stage2 2.520742 .9937819 2.35 0.019 1.16399 5.458932
stage3 5.913372 2.28568 4.60 0.000 2.772187 12.61386

The results above reproduce exactly the results from Poisson regression with the
robust variance estimator; the Poisson and Cox methods are identical when implemented
in this way. Options other than the Breslow method for dealing with tied survival times
will produce risk-ratio estimates for exposure to a low estrogen-receptor–level tumor
that are too large: 1) the efron option produces a risk ratio of 1.91, 2) the exactm
option yields 2.04, and 3) the exactp option risk ratio is 2.49.

8 Method 6: Regression-based standardized risk ratios

Any regression model for binomial outcomes can estimate the probability (risk) of death
for women with high and low estrogen-receptor–level tumors within each cancer stage.
With this information, we can estimate the average risk of death that would be ex-
pected if all 192 women had low estrogen-receptor–level tumors and the distribution
of cancer stages observed in the data. This estimate is said to be standardized to
the distribution of the other variables, cancer stage in this example, in the regres-
sion model (Lane and Nelder 1982; Flanders and Rhodes 1987; Joffe and Greenland
1995; Greenland 1991a, 2004a; Localio, Margolis, and Berlin 2007; Rothman, Green-
land, and Lash 2008, 442–446). The average risk can also be estimated assuming that
all 192 women had a high estrogen-receptor–level tumor. The ratio of the low estrogen-
receptor–level average risk divided by the high estrogen-receptor–level average risk can
be calculated and the standard error for this risk ratio can be estimated using the delta
method (Casella and Berger 2002, 240–245). The word “standardized” is used just as
for standardized mortality rates or any statistic standardized to a given population
distribution. We can estimate this standardized risk ratio by using logistic regression:

(Continued on next page)

186 Methods for estimating adjusted risk ratios

. use brcadat, clear

(Breast cancer data)
. logistic died low stage2 stage3, nolog
Logistic regression Number of obs = 192
LR chi2(3) = 42.27
Prob > chi2 = 0.0000
Log likelihood = -92.939847 Pseudo R2 = 0.1853

died Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]

low 2.508065 .9916923 2.33 0.020 1.155507 5.443836

stage2 3.109772 1.44851 2.44 0.015 1.248087 7.748406
stage3 18.8389 11.03231 5.01 0.000 5.978344 59.36498

. #delimit ;
delimiter now ;
. predictnl lnrr =
> ln(
> sum(1/
> (1+exp(-(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3+_b[low]))))
> /
> sum(1/
> (1+exp(-(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3)))))
> , se(lnrr_se);
. #delimit cr
delimiter now cr
. scalar rr = exp(lnrr[_N])
. scalar upper = exp(lnrr[_N] + invnormal(1-.05/2)*lnrr_se[_N])
. scalar lower = exp(lnrr[_N] - invnormal(1-.05/2)*lnrr_se[_N])
. display "Risk ratio = " rr " 95% CI = " lower ", " upper
Risk ratio = 1.6755988 95% CI = 1.0935713, 2.5673969

The adjusted odds ratio for death among women with a low estrogen-receptor level–
tumor, compared with women with a high estrogen-receptor–level tumor, was 2.5. Be-
cause the outcome of death was common, this odds ratio does not closely approximate
the risk ratio.
Above I used predictnl to estimate the ln of the risk ratio (lnrr variable); this
command can estimate nonlinear comparisons from regression coefficients. The se op-
tion estimated the standard error for the ln risk ratio using the delta method. To make
the output less cluttered, I used delimit to change how Stata recognizes the end of
a command line. To estimate the risk or probability of death, I used the expression
1/{1 + exp(−linear predictor)}. The first sum used by predictnl is for risk estimates
if all women had a low estrogen-receptor–level tumor, because the ln odds term for low
receptor status, b[low], is included in the sum, regardless of each woman’s actual recep-
tor status. In this first sum, the regression coefficients (which are ln odds estimates) for
Istage 2 and Istage 3 were both multiplied by each woman’s observed cancer stage,
thereby standardizing the estimate to the observed distribution of cancer stage. Stata’s
sum() function is the running sum from the first record to the last, so the sum in the
last record of the data is the sum of all the estimated risks if all 192 women had a low
estrogen-receptor–level tumor but each had her observed cancer stage. The second sum
P. Cummings 187

in the expression, after the line with only a division sign, is the sum of all estimated
risks if all women had a high estrogen-receptor–level tumor, again standardized to the
observed cancer-stage distribution. The first sum is divided by the second and the ln
taken of this ratio so that the ln of the risk ratio is estimated. I then estimated the risk
ratio, which is exp(lnrr), and the 95% upper and lower confidence limits for the risk
ratio, and used the display command to show these results for the last record by using
the subscript [ N]: risk ratio = 1.7, 95% CI is [1.1, 2.6].
We can use simpler commands to estimate the risk ratio, but they do not provide
a CI. Still, these commands show how the risks and risk ratio may be estimated and
are shown below to clarify how Stata is using the regression estimates. After fitting the
logistic model, the commands are

. replace low=0
(48 real changes made)
. predict risk0
(option pr assumed; Pr(died))
. summ risk0, meanonly
. local avrisk0 = r(mean)
. replace low=1
(192 real changes made)
. predict risk1
(option pr assumed; Pr(died))
. summ risk1, meanonly
. local avrisk1 = r(mean)
. local rr = `avrisk1´/`avrisk0´
. display "Risk1 = " `avrisk1´ " Risk0 = " `avrisk0´ " Risk ratio = " `rr´
Risk1 = .40087948 Risk0 = .23924549 Risk ratio = 1.6755989

Risks for binomial outcomes can also be estimated after probit regression. In the
probit model, the outcome risk estimate applies the cumulative standard normal distri-
bution function (normal()) to the linear predictor, instead of the ln odds function used
in logistic regression:

. use brcadat, clear

(Breast cancer data)
. probit died low stage2 stage3, nolog
Probit regression Number of obs = 192
LR chi2(3) = 42.21
Prob > chi2 = 0.0000
Log likelihood = -92.968357 Pseudo R2 = 0.1850

died Coef. Std. Err. z P>|z| [95% Conf. Interval]

low .5386148 .2343501 2.30 0.022 .079297 .9979327

stage2 .6290485 .2503224 2.51 0.012 .1384256 1.119671
stage3 1.739085 .3302966 5.27 0.000 1.091715 2.386454
_cons -1.376363 .2165793 -6.36 0.000 -1.800851 -.9518752
188 Methods for estimating adjusted risk ratios

. #delimit ;
delimiter now ;
. predictnl lnrr = ln(
> sum(normal(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3+_b[low]))
> /
> sum(normal(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3)))
> , se(lnrr_se);
. #delimit cr
delimiter now cr
. scalar rr = exp(lnrr[_N])
. scalar upper = exp(lnrr[_N] + invnormal(1-.05/2)*lnrr_se[_N])
. scalar lower = exp(lnrr[_N] - invnormal(1-.05/2)*lnrr_se[_N])
. display "Risk ratio = " rr " 95% CI = " lower ", " upper
Risk ratio = 1.6751332 95% CI = 1.0913484, 2.5711965

Nelder (2001) has suggested that when a dichotomous outcome is common, the
complementary log-log regression model may fit the data well:

. use brcadat, clear

(Breast cancer data)
. cloglog died low stage2 stage3, nolog
Complementary log-log regression Number of obs = 192
Zero outcomes = 138
Nonzero outcomes = 54
LR chi2(3) = 42.60
Log likelihood = -92.771237 Prob > chi2 = 0.0000

died Coef. Std. Err. z P>|z| [95% Conf. Interval]

low .7138795 .2936374 2.43 0.015 .1383607 1.289398

stage2 1.022028 .426798 2.39 0.017 .1855198 1.858537
stage3 2.302261 .4522776 5.09 0.000 1.415813 3.188709
_cons -2.369669 .3882655 -6.10 0.000 -3.130655 -1.608683

. #delimit ;
delimiter now ;
. predictnl lnrr = ln(
> sum(1-exp(-exp(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3+_b[low])))
> /
> sum(1-exp(-exp(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3))))
> , se(lnrr_se);
. #delimit cr
delimiter now cr
. scalar rr = exp(lnrr[_N])
. scalar upper = exp(lnrr[_N] + invnormal(1-.05/2)*lnrr_se[_N])
. scalar lower = exp(lnrr[_N] - invnormal(1-.05/2)*lnrr_se[_N])
. display "Risk ratio = " rr " 95% CI = " lower ", " upper
Risk ratio = 1.6652749 95% CI = 1.1009763, 2.5188013

Hardin and Hilbe (2007, 147) note that if most subjects either have or do not have
the outcome, the complementary log-log and log-log models may fit better than logistic
or probit models. We can fit the log-log model using the glm command:
P. Cummings 189

. use brcadat, clear

(Breast cancer data)
. glm died low stage2 stage3, nolog family(bin) link(loglog) eform
Generalized linear models No. of obs = 192
Optimization : ML Residual df = 188
Scale parameter = 1
Deviance = 186.5949538 (1/df) Deviance = .9925263
Pearson = 192.8460376 (1/df) Pearson = 1.025777
Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = -ln(-ln(u)) [Log-log]
AIC = 1.013515
Log likelihood = -93.2974769 BIC = -801.8142

OIM
died exp(b) Std. Err. z P>|z| [95% Conf. Interval]

low 1.642276 .3938706 2.07 0.039 1.026362 2.627796

stage2 1.695259 .3494023 2.56 0.010 1.131876 2.539063
stage3 6.133979 2.415435 4.61 0.000 2.835022 13.27174

. #delimit ;
delimiter now ;
. predictnl lnrr = ln(
> sum(exp(-exp(-(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3+_b[low]))))
> /
> sum(exp(-exp(-(_b[_cons]+_b[stage2]*stage2+_b[stage3]*stage3)))))
> , se(lnrr_se);
. #delimit cr
delimiter now cr
. scalar rr = exp(lnrr[_N])
. scalar upper = exp(lnrr[_N] + invnormal(1-.05/2)*lnrr_se[_N])
. scalar lower = exp(lnrr[_N] - invnormal(1-.05/2)*lnrr_se[_N])
. display "Risk ratio = " rr " 95% CI = " lower ", " upper
Risk ratio = 1.6312705 95% CI = 1.0545183, 2.5234682

9 Method 7: A substitution method

A crude (unadjusted) odds ratio can be converted to a risk ratio: crude risk ratio =
(crude odds ratio)/{(1−Po)+(Po×crude odds ratio)}, where Po is the proportion of all
unexposed subjects who had the outcome in the data. Zhang and Yu (1998) suggested
that, in a cohort study, one can use this same formula to convert an adjusted odds
ratio to an adjusted risk ratio. This substitution method was described by Holland
(1989), who used it to estimate an adjusted risk difference from a Mantel–Haenszel
summary odds ratio. Greenland and Holland (1991) reported that this will produce
ratio estimates biased away from 1 when outcomes are common and risk among those
not exposed varies substantially. The bias occurs because the summary odds ratio is
not a weighted average of stratum-specific odds ratios; odds ratios lack the property of
collapsibility (Greenland 1987). The method can be implemented in Stata:
190 Methods for estimating adjusted risk ratios

. use brcadat, clear

(Breast cancer data)
. summ died if low==0, meanonly
. local p0 = r(mean)
. display `p0´
.21527778
. logistic died low stage2 stage3, nolog
Logistic regression Number of obs = 192
LR chi2(3) = 42.27
Prob > chi2 = 0.0000
Log likelihood = -92.939847 Pseudo R2 = 0.1853

died Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]

low 2.508065 .9916923 2.33 0.020 1.155507 5.443836

stage2 3.109772 1.44851 2.44 0.015 1.248087 7.748406
stage3 18.8389 11.03231 5.01 0.000 5.978344 59.36498

. scalar rr = exp(_b[low])/[(1-`p0´)+(`p0´*exp(_b[low]))]
. scalar lower = exp(_b[low]-invnormal(1-.05/2)*_se[low])/[(1-`p0´)+
> (`p0´*exp(_b[low]-invnormal(1-.05/2)*_se[low]))]
. scalar upper = exp(_b[low]+invnormal(1-.05/2)*_se[low])/[(1-`p0´)+
> (`p0´*exp(_b[low]+invnormal(1-.05/2)*_se[low]))]
. display "Risk ratio = " rr " 95% CI = " lower ", " upper
Risk ratio = 1.8933751 95% CI = 1.1180767, 2.7822097

10 Bootstrap CIs
In some examples above, approximately correct CIs were obtained using robust or delta
methods. Bootstrap methods can also be used for CIs. Here are commands to estimate
bootstrap CIs for the risk ratio by using a logistic model:

. use brcadat, clear

(Breast cancer data)
. program stlogit, rclass
1. version 10
2. logistic died low stage2 stage3, nolog
3. preserve
4. replace low=0
5. predict risk0
6. summ risk0, meanonly
7. scalar avrisk0 = r(mean)
8. replace low=1
9. predict risk1
10. summ risk1, meanonly
11. scalar avrisk1 = r(mean)
12. return scalar lnrr = ln(avrisk1/avrisk0)
13. restore
14. end
. set seed 93514
P. Cummings 191

. bootstrap lnrr=r(lnrr), saving(bsanrr3a, replace) reps(400001) nowarn nodots:

> stlogit
(output omitted )
. estat bootstrap, all eform
Bootstrap results Number of obs = 192
Replications = 400001
command: stlogit
lnrr: r(lnrr)

Observed Bootstrap
exp(b) Bias Std. Err. [95% Conf. Interval]

lnrr 1.6755989 -.0049751 .37575458 1.079661 2.600476 (N)

1.068839 2.587837 (P)
1.071832 2.594932 (BC)

(N) normal confidence interval

(P) percentile confidence interval
(BC) bias-corrected confidence interval

Stata’s bootstrap command simplifies the task of estimating bootstrap CIs by using
four methods: 1) normal, 2) percentile, 3) bias corrected, and 4) bias corrected and ac-
celerated. Other methods are available (Carpenter and Bithell 2000). For the risk ratios
estimated in this article, the choice among Stata’s four methods makes little difference.
But for some epidemiologic data, the normal and percentile methods should be used
with caution because they may have substantial coverage error (Efron and Tibshirani
1993; Carpenter and Bithell 2000; Greenland 2004b).

11 Risk-ratio methods for matched data

Adjusted risk ratios for matched data can be estimated using conditional Poisson re-
gression, which Stata implements in the xtpoisson, fe command. I have previously
reviewed the analysis of matched cohort data in the Stata Journal (Cummings and McK-
night 2004) and elsewhere (Cummings, McKnight, and Weiss 2003; Cummings, McK-
night, and Greenland 2003).

12 Summary
When the risk-ratio estimates in this article are rounded to one decimal, nearly all the
methods produced estimates of 1.6 or 1.7 (table 2). They also differed little with regard
to estimated CIs: the 95% lower bound was 1.0 or 1.1 and the upper bound, 2.3 to 2.6.
The risk ratio that stands out as different came from the substitution method: risk
ratio = 1.9 and 95% CI is [1.1, 3.1]. The substitution method has nothing to recommend
it; it will usually produce estimates biased away from 1 when outcomes are common,
and in Stata it offers little advantage in terms of simplicity. Stata users who wish to
estimate an adjusted risk ratio have better methods that they can use, all of which are
fairly easy to implement.
192 Methods for estimating adjusted risk ratios

Table 2. Risk-ratio estimates for death within 5 years among 192 women with breast
cancer, comparing women with low estrogen-receptor–level tumors with women with
high estrogen-receptor–level tumors, adjusted for cancer stage at diagnosis. Results are
shown using the methods described in this article.

Method Risk ratio 95% CI Bootstrap Akaike

95% CI† information
criteria‡
1. Stratified methods
Mantel–Haenszel 1.62 1.09, 2.39 1.07, 2.48 ...
Inverse-variance weights 1.55 1.08, 2.25 ... ...
2. Generalized linear regression
with log link and
binomial distribution
Maximum likelihood 1.56 1.05, 2.32 ... 193.9
Wacholder’s truncated
method 1.56 1.06, 2.29 1.03, 2.44 ...
Copy method 1.56 1.05, 2.31 ... ...
3. Generalized linear regression
with log link, Gaussian
distribution, robust variance
estimator 1.55 1.04, 2.32 1.04, 2.43 196.5
4. Poisson regression with
robust variance estimator 1.63 1.07, 2.48 1.06, 2.55 226.3
5. Cox proportional hazards
with robust variance estimator 1.63 1.07, 2.48 1.06, 2.55 547.1
6. Regression-based
standardized risk ratios
Logistic 1.68 1.09, 2.57 1.07, 2.59 193.9
Probit 1.68 1.09, 2.57 1.07, 2.59 193.9
Complementary log-log 1.67 1.10, 2.52 1.08, 2.56 193.5
Log-log 1.63 1.05, 2.52 1.03, 2.54 194.6
7. Substitution method 1.89 1.12, 2.78 1.08, 3.11 ...

† Bias-corrected bootstrap CIs based upon 400001 replications. Not estimated for the inverse-variance
stratified method because the cs command does not return the pooled risk ratio from this method.
Not estimated for the maximum-likelihood version of the generalized linear model with a log link and
binomial distribution because convergence failed in many bootstrap samples. Convergence also failed in
100 bootstrap samples (0.025%) using Wacholder’s truncated method (binreg), 10 samples (0.0025%)
using the generalized linear model with a log link and Gaussian distribution, and 5 samples (0.00125%)
using the complementary log-log method.
‡ Akaike information criteria statistic for models fit using maximum likelihood. The statistic compares
the fitted model with a model that has only the outcome variable. In Stata, smaller Akaike information
criteria statistics indicate better fit.
P. Cummings 193

13 Acknowledgment
This work was supported by grant R49/CE000197-04 from the Centers for Disease
Control and Prevention, Atlanta, GA.

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About the author

Peter Cummings is a professor in the Department of Epidemiology in the School of Public
Health and a core faculty member at the Harborview Injury Research and Prevention Research
Center, University of Washington, Seattle, WA.