# Calculation of the Confidence Interval for Incidence Rate Ratio using Exact Approach

I have incidence rate ratio equal to 0.04:

If I am not mistaken, normal computation of CI is:

Lower bound: e^[ln(0.04) - 2 * sqrt(1/866 + 1/877)] = 0.03634539
Upper bound: e^[ln(0.04) + 2 * sqrt(1/866 + 1/877)] = 0.04402209


But because the number of events in the treated group is small (1) we should use "exact approach" which should give us CI equal to (0.01, 0.27).

So, my question is: what are the formulas for the "exact approach" which will allow to arrive at (0.01, 0.27)?

P.S. Additional slide about the study if it will help:

• Is this a crude estimate for the IR ratio or do the authors report results from a stratified analysis?
– chl
Commented Nov 9, 2020 at 8:29
• I don't know. These are the slides from one online course. Lecturer mentioned in video that CI was obtained using "exact approach" (not usual) and that's all. I added one additional slide if it will help. Commented Nov 9, 2020 at 9:36
• Your normal computation of CI should be revised. The standard error of ln(IRR) is sqrt(1/E1 + 1/E2) where E1 is the event count in group 1 ; E2 is the event count in group 2 Commented yesterday
• In R, >exp(log(0.04) - 2 * sqrt(1/1 + 1/27)) [1] 0.005218339 > exp(log(0.04) + 2 * sqrt(1/1 + 1/27)) [1] 0.306611 with rounding the 95% CI is (0.01,0.31) (CI without using the exact approach) Commented yesterday

The authors of this study apparently carried out both univariate and multivariate stratified Cox proportional hazards models. They report hazard ratios with their 95% CI from the observed results (see Table 3). The hazard ratio, which is the ratio of "chance of an event occurring in the treatment arm" / "chance of an event occurring in the control arm", can be approximated by the incidence rate ratio (IRR, replace chance of an event with "risk of observing the outcome of interest"), provided the assumptions of the Cox model are met (constant and proportional hazard).

They report the following result (for univariate and multivariate analyses):(1)

Through viral genetic analysis, 28 transmissions were linked to the HIV-1–infected participant (incidence rate, 0.9 per 100 person-years; 95% CI, 0.6 to 1.3), with 1 transmission in the early-therapy group (incidence rate, 0.1 per 100 person-years; 95% CI, 0.0 to 0.4) and 27 transmissions in the delayed-therapy group (incidence rate, 1.7 per 100 person-years; 95% CI, 1.1 to 2.5), for a hazard ratio in the early-therapy group of 0.04 (95% CI, 0.01 to 0.27; P<0.001). (...) In the stratified multivariate analysis according to site, the adjusted hazard ratio for linked transmission in the early-therapy group was 0.04 (95% CI, 0.01 to 0.28; P<0.001).

An asymptotic confidence interval for the IRR based on the Normal approximation can be built using your formula, or we can rely on exact confidence intervals, based on the Poisson distribution, and a comparison of Cox and Poisson models is available in this related thread.

Using Stata, which uses the following formula from Rothman, Greenland & Lash, Modern Epidemiology (2008, 3rd ed), I get the 95% CI highlighted below as ***:

. iri 1 27 866 877

|   Exposed   Unexposed  |      Total
-----------------+------------------------+------------
Cases |         1          27  |         28
Person-time |       866         877  |       1743
-----------------+------------------------+------------
|                        |
Incidence rate |  .0011547    .0307868  |   .0160643
|                        |
|      Point estimate    |    [95% Conf. Interval]
|------------------------+------------------------
Inc. rate diff. |         -.029632       |   -.0414632   -.0178009
Inc. rate ratio |         .0375075       |    .0009161    .2275604 (exact)   ***
Prev. frac. ex. |         .9624925       |    .7724396    .9990839 (exact)
Prev. frac. pop |         .4782091       |
+-------------------------------------------------
(midp)   Pr(k<=1) =                     0.0000 (exact)
(midp) 2*Pr(k<=1) =                     0.0000 (exact)


The same can be done in R, using these alternative formulae:

e1 = 1; n1 = 866   ;; early therapy
e2 = 27; n2 = 877  ;; controls
irr = (e1/n1) / (e2/n2)
lb = n2/n1 * (e1/(e2+1)) * 1/qf(2*(e2+1), 2*e1, p = 0.05/2, lower.tail = FALSE)
ub = n2/n1 * ((e1+1)/e2) * qf(2*(e1+1), 2*e2, p = 0.05/2, lower.tail = FALSE)
cat(round(c(irr, lb, ub), 3), "\n")


(1) This is largely commented out on the Coursera course you probably refer to (0.04 is $$\exp(\beta_1)$$ in the Cox model $$\log\left(\lambda(t, x_1)\right) = \log\left(\hat\lambda_0(t)\right) + \hat\beta_1x_1$$).

• Thank you very much for your reply. So, 0.04 (0.01, 0.27) came from Cox regression. But why do you define irr as e1/e2? According to the link which you gave they use (e1/n1) / (e2/n2) for irr. Commented Nov 10, 2020 at 0:37
• Good catch! This should be fixed now.
– chl
Commented Nov 10, 2020 at 3:46