{smcl}
{* 02 Feb 2008}{...}
{hline}
help for {hi:oddsrisk}
{hline}
{title:Conversion from Logistic Odds Ratios to Risk Ratios}
{p 8 12}{cmd:oddsrisk}
{it:y(1/0)} {it:riskfactor(1/0)} {it:varlist} {it:[fw=countvariable]} {it: }
{title:Description}
{p}
{cmd:oddsrisk} converts logistic regression odds ratios to relative risk ratios
by the formula described below. Source: Zhang and K. Yu, 1998. Frequency weights are
allowed in order to calculate odds and risk ratios from 2 x 2 tables. The response
must be binary, as does the first predictor, which is considered to be the risk factor
or exposure.
When the incidence of an outcome is common in the study population; i.e. greater than
10%, the logistic regression odds ratio no longer approximates the risk ratio. As the
incidence rate becomes more frequent, the more the odds ratio overestiamtes the risk
ratio when its value is greater than 1, and the more it underestiamtes the risk ratio
when under one. J. Zhang and K. Yu proposed a method of adjusting the logistic
regression odds ratio in a cohort study or clinical trial so that it approximates
the risk ratio. This is particularly important when the odds ratio is greater than
2.5 or under 0.5. The method has also been shown to be applicable for retrospective
and observational studies as well.
The method is based on the response having a binary risk factor, or primary predictor
of interest. Other confounding predictors may also be included in the model. The
risk ratio is defined as
Rho = pe / pu
and represents how much more or less likely it is for {it:y} to be the case; i.e.
(y==1) given a risk factor of 1, in comparison to {y==1} given a risk factor of 0.
For example, let us denote the risk of death within 48 hours of admission to a
hospital (y) occurring for those sustaining an anterior infarct as pe and the risk
of death (y) among those having an other-site infarct as pu. The subscripts e and
u signify exposed (1) and unexposed (0); i.e. e is having had an anterior infarct
and u is having an other-site infarct. The risk ratio, Rho, represents how much
more or less likely it is for an individual to die within 48 hours of admission
among those sustaining an anterior infarct in comparison to those who died having
an infarct at another site.
When pe is small; i.e. the incidence rate for the exposed group is under 0.1,
then the values of Rho, the risk ratio, and OR, the odds ratio, are nearly the
same. As the incidence rate grows, the odds ratio will overestimate the risk ratio
when the latter is greater than 1.0 and underestimate the risk ratio when less
than 1.0. Simulation studies have shown that values of Pe less than 0.1.
Under this formulation, the odds ratio of the risk factor, or predictor of
interest, is defined as:
pe/(1- pe)
OR = ------------
pu/(1- pu)
where {it:pe} is the incidence rate for exposed patients (RF==1),
and {it:pu} is the incidence rate for unexposed patients (RF==0),
where RF is the risk factor, a binary predictor of interest.
The formula for risk ratio in terms of the odds ratio is:
OR
RR ~ ---------------------
(1- pu) + (pu *OR)
One must first calculate the incidence rate of the unexposed group; ie when
the risk factor equals 0. This is the value of pu.
The fomula may be used to convert all predictors in the model to extimated
risk ratios. The same formula may also be used for the confidence intervals
of the odds ratio.
{title:Example 1}
The {cmd:anterior} data set was created from a tabulation of death and anterior
taken from the {it:heart01} data set. A listing of the data, together with a
schemata of the underlying table, is shown first, followed by the use of
{cmd:oddsrisk} to estimate the odds and risk ratios, as well as 95% confidence
intervals. These examples taken from
Hilbe J.M. (2008). {it:Logistic Regression Models}, Chapman & Hall/CRC Press
. use anterior
. list
+--------------------------+
| count death anterior |
|--------------------------|
1. | 120 1 1 |
2. | 67 1 0 |
3. | 2005 0 1 |
4. | 2504 0 0 |
+--------------------------+
Table represented by data
Response (death)
1 0
---------------
Risk 1 | 120 2005 | 2125
(Anterior MI) | |
0 | 67 2504 | 2571
---------------
187 4509 4696
Use of {cmd:oddsrisk}, using {it:count} as a frequency weight
. oddsrisk death anterior [fw=count]
---------------------------------------------------------------------
Incidence for unexposed risk group = 0.0261
---------------------------------------------------------------------
Predictor Odds Ratio Risk Ratio [95% Conf. Interval]
---------------------------------------------------------------------
anterior 2.2368 2.1670 1.6220 2.8807
---------------------------------------------------------------------
Use of the {cmd:csi} command. Risk ratio CIs calculated using Woolfe's method.
. csi 120 67 2005 2504
| Exposed Unexposed | Total
-----------------+------------------------+------------
Cases | 120 67 | 187
Noncases | 2005 2504 | 4509
-----------------+------------------------+------------
Total | 2125 2571 | 4696
| |
Risk | .0564706 .0260599 | .0398211
| |
| Point estimate | [95% Conf. Interval]
|------------------------+------------------------
Risk difference | .0304107 | .0188244 .041997
Risk ratio | 2.166953 | 1.616054 2.905651
Attr. frac. ex. | .5385226 | .3812087 .655843
Attr. frac. pop | .345576 |
+-------------------------------------------------
chi2(1) = 28.14 Pr>chi2 = 0.0000
Simulation studies have demonstrated that a Poisson model with robust
SEs closely estimate the relative risk. Likewise, a log-binomial can be
used with the same result, but convergence problems many times occur
with multivariate models.
We use a Poisson with robust standard errors for comparison of results.
Note its close similarity to Woolfe's method, as well as to Zhang and Yu's
method ({cmd:oddsrisk}].
. poisson death anterior [fw=count], nolog irr robust
[ Header excluded from display ]
------------------------------------------------------------------------------
| Robust
death | IRR Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
anterior | 2.166953 .3243484 5.17 0.000 1.616003 2.905741
------------------------------------------------------------------------------
{title:Example 2}
Use of full data.
. use heart01
. logit death anterior hcabg kk2-kk4, nolog or
[ Header excluded from display ]
------------------------------------------------------------------------------
death | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
anterior | 2.05989 .3391907 4.39 0.000 1.4917 2.844505
hcabg | 2.051369 .7098963 2.08 0.038 1.041063 4.042131
kk2 | 2.573692 .4558429 5.34 0.000 1.818841 3.641818
kk3 | 3.244251 .8504555 4.49 0.000 1.94079 5.423135
kk4 | 18.55298 6.12317 8.85 0.000 9.715963 35.4276
------------------------------------------------------------------------------
. oddsrisk death anterior hcabg kk2-kk4
---------------------------------------------------------------------
Incidence for unexposed risk group = 0.0261
---------------------------------------------------------------------
Predictor Odds Ratio Risk Ratio [95% Conf. Interval]
---------------------------------------------------------------------
anterior 2.0599 2.0045 1.4728 2.7140
hcabg 2.0514 1.9967 1.0400 3.7452
kk2 2.5737 2.4723 1.7808 3.4072
kk3 3.2443 3.0650 1.8943 4.8626
kk4 18.5530 12.7299 7.9176 18.6738
---------------------------------------------------------------------
Compare with a Poisson model with robust standard errors, which estimates
incident rate ratios. Simulation studies have confirmed that this model,
as well as a log-binomial model estimate relative risk ratios.
. poisson death anterior hcabg kk2-kk4, nolog irr robust
[ Header excluded from display ]
------------------------------------------------------------------------------
| Robust
death | IRR Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
anterior | 1.963766 .3099786 4.28 0.000 1.441213 2.675784
hcabg | 1.937465 .5851856 2.19 0.029 1.071864 3.502096
kk2 | 2.464633 .4128878 5.38 0.000 1.774822 3.422551
kk3 | 3.044356 .7604601 4.46 0.000 1.865825 4.967295
kk4 | 12.33746 2.883896 10.75 0.000 7.80291 19.50721
------------------------------------------------------------------------------
{title:Author}
{p} Joseph M. Hilbe, Arizona State University ----
Hilbe@asu.edu; jhilbe@aol.com
{title:References}
{p}J. Zhang and K. Yu, 1998.
{it:What's the Relative Risk},
JAMA, Vol 280, No 19, pp 1690-1691.
{p}J.M. Hilbe, 2008
{it:Logistic Regression Models},
Chapman & Hall/CRC Press.