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Conversion from Logistic Odds Ratios to Risk Ratios

oddsrisky(1/0)riskfactor(1/0)varlist[fw=countvariable]<if> <in>

Description

oddsriskconverts 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. greate > r 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 approximat > es the risk ratio. This is particularly important when the odds ratio is greater t > han 2.5 or under 0.5. The method has also been shown to be applicable for retrospec > tive and observational studies as well.

The method is based on the response having a binary risk factor, or primary pre > dictor of interest. Other confounding predictors may also be included in the model. Th > e risk ratio is defined as Rho = pe / pu

and represents how much more or less likely it is for

yto 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 r > isk 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 infar > ct and u is having an other-site infarct. The risk ratio, Rho, represents how muc > h 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 havi > ng 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 ra > tio 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

peis the incidence rate for exposed patients (RF==1), andpuis 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.

Example 1The

anteriordata set was created from a tabulation of death and anterior taken from theheart01data set. A listing of the data, together with a schemata of the underlying table, is shown first, followed by the use ofoddsriskto estimate the odds and risk ratios, as well as 95% confidence intervals. These examples taken from Hilbe J.M. (2008).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

oddsrisk, usingcountas 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

csicommand. 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 (

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 ------------------------------------------------------------------------------

Example 2Use 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 ------------------------------------------------------------------------------

Joseph M. Hilbe, Arizona State University ---- Hilbe@asu.edu; jhilbe@aol.comAuthor

ReferencesJ. Zhang and K. Yu, 1998.

What's the Relative Risk, JAMA, Vol 280, No 19, pp 1690-1691.J.M. Hilbe, 2008

Logistic Regression Models, Chapman & Hall/CRC Press.