Compare Two Crude Rates


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Person-time data from prospective studies of two groups with different exposures may be expressed as a difference between incidence rates or as a ratio of incidence rates.


This function constructs confidence intervals for incidence rate differences and ratios where there are two exposures (i.e. exposed or not exposed, defined according to certain risk factors) (Sahai and Kurshid, 1996).


Data input

    Exposed Not Exposed
OUTCOME: Cases: a b
Person-time: PT1 PT2


The use of person-time as opposed to just "time" enables you to handle situations where there are drop-outs in a study or where you have not been able to follow an entire cohort at risk to watch for the development of the outcome under investigation. Thus, the follow up period does not have to be uniform for all participants. Person-time for a group is the sum of the times of follow up for each participant in that group.


Technical validation

Poisson distribution and test-based methods are used to construct the confidence intervals (Sahai and Kurshid, 1996):

- where IRD hat and IRR hat are point estimates of incidence rate difference and ratio respectively, m is the total number of events observed, PT is the total person-time observed, Z is a quantile of the standard normal distribution and F is a quantile of the F distribution (denominator degrees of freedom are quoted last).


The optional conditional maximum likelihood analysis for the rate ratio employs the polynomial multiplication method described by Martin and Austin (1996); this also provides mid-P estimates.



From Stampfer et al. (1985).


  Postmenopausal hormone use:
  Yes No
CHD cases: 30 60
Person-years: 54308.7 51477.5


For this example:

  Exposed Non-exposed Total
Cases: 30 60 90
Person-time: 54308.7 51477.5 105786.2


Exposed incidence rate = 0.000552

Non-exposed incidence rate = 0.001166


Incidence rate difference = -0.000613

Approximate 95% confidence interval = -0.000965 to -0.000261


Chi-square = 11.678635 P = .0006


Incidence rate ratio = 0.473934

Exact 95% confidence interval = 0.295128 to 0.746416


Conditional maximum likelihood estimate of rate ratio = 0.473934

Exact Fisher 95% confidence interval = 0.295128 to 0.746416

Exact Fisher one sided P = 0.0004, two sided P = 0.0007

Exact mid-P 95% confidence interval = 0.302362 to 0.730939

Exact mid-P one sided P = 0.0003, two sided P = 0.0006


Here we may conclude with 95% confidence that the true population value for the difference between the two incidence rates lies somewhere between -0.001 and 0.0003. We may also conclude with 95% confidence that the incidence rate for those who used postmenopausal hormones in the circumstances of the study was between 0.30 and 0.75 of that for those who did not take post- menopausal hormones.


P values

confidence intervals