The three options that are proposed in riskratio() refer to an asymptotic or large sample approach, an approximation for small sample, a resampling approach (asymptotic bootstrap, i.e. not based on percentile or bias-corrected). The former is described in Rothman's book (as referenced in the online help), chap. 14, pp. 241-244. The latter is relatively trivial so I will skip it. The small sample approach is just an adjustment on the calculation of the estimated relative risk.
If we consider the following table of counts for subjects cross-classififed according to their exposure and disease status,
Exposed Non-exposed Total
Cases a1 a0 m1
Non-case b1 b0 m0
Total n1 n0 N
the MLE of the risk ratio (RR), $\text{RR}=R_1/R_0$, is $\text{RR}=\frac{a_1/n_1}{a_0/n_0}$.
In the large sample approach, a score statistic (for testing $R_1=R_0$, or equivalently, $\text{RR}=1$) is used, $\chi_S=\frac{a_1-\tilde a_1}{V^{1/2}}$, where the numerator reflects the difference between the oberved and expected counts for exposed cases and $V=(m_1n_1m_0n_0)/(n^2(n-1))$ is the variance of $a_1$. Now, that's all for computing the $p$-value because we know that $\chi_S$ follow a chi-square distribution. In fact, the three $p$-values (mid-$p$, Fisher exact test, and $\chi^2$-test) that are returned by riskratio() are computed in the tab2by2.test() function. For more information on mid-$p$, you can refer to
Berry and Armitage (1995). Mid-P
confidence intervals: a brief
review. The Statistician, 44(4),
417-423.
Now, for computing the $100(1-\alpha)$ CIs, this asymptotic approach yields an approximate SD estimate for $\ln(\text{RR})$ of $(\frac{1}{a_1}-\frac{1}{n_1}+\frac{1}{a_0}-\frac{1}{n_0})^{1/2}$, and the Wald limits are found to be $\exp(\ln(\text{RR}))\pm Z_c \text{SD}(\ln(\text{RR}))$, where $Z_c$ is the corresponding quantile for the standard normal distribution.
The small sample approach makes use of an adjusted RR estimator: we just replace the denominator $a_0/n_0$ by $(a_0+1)/(n_0+1)$.
As to how to decide whether we should rely on the large or small sample approach, it is mainly by checking expected cell frequencies; for the $\chi_S$ to be valid, $\tilde a_1$, $m_1-\tilde a_1$, $n_1-\tilde a_1$ and $m_0-n_1+\tilde a_1$ should be $> 5$.
Working through the example of Rothman (p. 243),
sel <- matrix(c(2,9,12,7), 2, 2)
riskratio(sel, rev="row")
which yields
$data
Outcome
Predictor Disease1 Disease2 Total
Exposed2 9 7 16
Exposed1 2 12 14
Total 11 19 30
$measure
risk ratio with 95% C.I.
Predictor estimate lower upper
Exposed2 1.000000 NA NA
Exposed1 1.959184 1.080254 3.553240
$p.value
two-sided
Predictor midp.exact fisher.exact chi.square
Exposed2 NA NA NA
Exposed1 0.02332167 0.02588706 0.01733469
$correction
[1] FALSE
attr(,"method")
[1] "Unconditional MLE & normal approximation (Wald) CI"
By hand, we would get
$\text{RR} = (12/14)/(7/16)=1.96$, $\tilde a_1 = 19\times 14 / 30= 8.87$, $V = (8.87\times 11\times 16)/ \big(30\times (30-1)\big)= 1.79$, $\chi_S = (12-8.87)/\sqrt{1.79}= 2.34$, $\text{SD}(\ln(\text{RR})) = \left( 1/12-1/14+1/7-1/16 \right)^{1/2}=0.304$, $95\% \text{CIs} = \exp\big(\ln(1.96)\pm 1.645\times0.304\big)=[1.2;3.2]\quad \text{(rounded)}$.
The following papers also addresses the construction of the test statistic for the RR or the OR:
- Miettinen and Nurminen (1985). Comparative analysis of two rates. *Statistics in Medicine, 4: 213-226.
- Becker (1989). A comparison of maximum likelihood and Jewell's estimators of the odds ratio and relative risk in single 2 × 2 tables. Statistics in Medicine, 8(8): 987-996.
- Tian, Tang, Ng, and Chan (2008). Confidence intervals for the risk ratio under inverse sampling. Statistics in Medicine, 27(17), 3301-3324.
- Walter and Cook (1991). A comparison of several point estimators of the odds ratio in a single 2 x 2 contingency table. Biometrics, 47(3): 795-811.
Notes
- As far as I know, there's no reference to relative risk in Selvin's book (also referenced in the online help).
- Alan Agresti has also some code for relative risk.
You can use the Delta method to obtain an approximate distribution of your relative risk, as shown by that link. Then you can define a pivot and use this to obtain a CI.
I understand that there might be some confusion regarding the use of the Delta method, so here are a few simple steps that show how to construct an approximate CI for the relative risk.
- Estimate the RR from the data
- Find the natural log of RR: $\log(RR)$
- The confidence coefficient is from the standard
normal distribution: 1.96 for a 95% confidence
interval
Now you need the standard error. Using the Delta method for sample sizes $n$ and $m$ with probabilities $p$ and $q$ respectively, this is found to be
$$SE=\sqrt{\frac{1-p}{pn}+\frac{1-q}{qm}}$$
Of course you need to replace the unknown quantities with your estimates, let's denote them by $\widehat{p}$ and $\widehat{q}$. You might notice that this is the second approximation we are using.
Now that you have the formula, compute the standard error: $SE$
Calculate the lower and upper limits on the log
scale: $\log(RR) scale: \log(RR)
±
1.96 \times SE \log(RR)$
Exponentiate!
You can find plenty such information throughout the internet and the above steps are taken from here. We all have Fisher to thank for these approximations!
Best Answer
Check out the R Epi and epitools packages, which include many functions for computing exact and approximate CIs/p-values for various measures of association found in epidemiological studies, including relative risk (RR). I know there is also PropCIs, but I never tried it. Bootstraping is also an option, but generally these are exact or approximated CIs that are provided in epidemiological papers, although most of the explanatory studies rely on GLM, and thus make use of odds-ratio (OR) instead of RR (although, wrongly it is often the RR that is interpreted because it is easier to understand, but this is another story).
You can also check your results with online calculator, like on statpages.org, or Relative Risk and Risk Difference Confidence Intervals. The latter explains how computations are done.
By "exact" tests, we generally mean tests/CIs not relying on an asymptotic distribution, like the chi-square or standard normal; e.g. in the case of an RR, an 95% CI may be approximated as $\exp\left[ \log(\text{rr}) - 1.96\sqrt{\text{Var}\big(\log(\text{rr})\big)} \right], \exp\left[ \log(\text{rr}) + 1.96\sqrt{\text{Var}\big(\log(\text{rr})\big)} \right]$, where $\text{Var}\big(\log(\text{rr})\big)=1/a - 1/(a+b) + 1/c - 1/(c+d)$ (assuming a 2-way cross-classification table, with $a$, $b$, $c$, and $d$ denoting cell frequencies). The explanations given by @Keith are, however, very insightful.
For more details on the calculation of CIs in epidemiology, I would suggest to look at Rothman and Greenland's textbook, Modern Epidemiology (now in it's 3rd edition), Statistical Methods for Rates and Proportions, from Fleiss et al., or Statistical analyses of the relative risk, from J.J. Gart (1979).
You will generally get similar results with
fisher.test(), as pointed by @gd047, although in this case this function will provide you with a 95% CI for the odds-ratio (which in the case of a disease with low prevalence will be very close to the RR).Notes: