American Journal of Epidemiology
Copyright O 1998 by The Johns Hopkins University School of Hygiene and Public Health
All rights reserved
Vol. 147, No. 1
Printed In USA
Maximum (Max) and Mid-P Confidence Intervals and p Values for the
Standardized Mortality and Incidence Ratios
Pandurang M. Kulkami,1 Ram C. Tripathi,2 and Joel E. Michalek3
The authors present algorithms to compute the confidence interval for a Poisson parameter (A) and the p
value for testing the hypothesis that A is equal to a constant, which can be used to make inferences about the
standardized mortality ratio and the standardized incidence ratio. The p value and confidence interval always
agree, despite the discrete nature of the Poisson distribution. The authors also give simple equations to
compute numeric approximations of the confidence limits that do not require the use of any probability
distributions. An example based on data arising from a study of cancer incidence is given. Am J Epidemiol
1998:147:83-6.
confidence intervals; Poisson distribution; risk; significance tests
In epidemiologic studies, interest often lies in comparing the mortality rate of a certain cohort with that of
a standard population; a standard summary statistic in
this regard is the standardized mortality ratio (SMR).
In studies of disease incidence, the corresponding statistic is the standardized incidence ratio (SIR). Traditionally, the observed number of events is assumed to
have a Poisson distribution. Thus, inference procedures for the SMR (or SIR) can be formulated based
on those for the Poisson distribution. Many authors
have discussed methods for constructing confidence
intervals for the SMR. Some have provided tables
(1-3), and others give methods based on some approximations (3-5). An exact method of constructing confidence intervals was suggested by Garwood (6), and it
reappeared in an article by Ulm (7). Tests of hypothesis for the SMR have been considered by Monson (8),
Armitage (9), and Rothman and Boice (10). Many
tests are based on the normal approximation to the
Poisson distribution. These approximations work well
when the observed number of deaths is large, but they
require the use of normal and chi-square distribution
tables. Here, we provide simple equations to compute
numeric approximations of the confidence intervals
without requiring the use of tables. Further, exact tests
for the SMR and their relation to confidence intervals
are not well documented. Ideally, inferences based on
confidence intervals and hypothesis tests should lead
to the same decision. However, when using approximate procedures, this may not be the case. In this
paper, we also provide procedures to calculate p values that are inverted to obtain confidence intervals.
Therefore, the two procedures always lead to the same
decision.
Generally, with discrete distributions it is not possible to construct confidence intervals with specified
coverage (the probability that the interval contains the
parameter of interest). Therefore, one uses confidence
intervals with the nominal coverage as the lower
bound for the actual coverage. This results in conservative p values and conservative confidence intervals,
implying that the significance level of the test is less
and the coverage probability of the confidence interval
is greater than nominal. Procedures to give shorter
intervals are available (11, 12); however, those methods are computer intensive. Here, we propose a
method (known as the Mid-P method (13, 14)) to
obtain p values and confidence intervals that are less
conservative than the methods of Garwood (6) and
Ulm (7). For this case also, we provide simple equations to generate numerically approximate confidence
intervals without the use of tables. We apply the results
to cancer incidence data in Air Force air crews (15).
Received for publication February 13, 1997, and accepted for
publication June 26, 1997.
Abbreviations: Cl, confidence interval; Max, maximum; SIR, standardized incidence ratio; SMR, standardized mortality ratio.
1
Department of Mathematics and Statistics, University of South
Alabama, Mobile, A L
2
Department of Mathematics and Statistics, University of Texas
at San Antonio, San Antonio, TX
3
Armstrong Laboratory, Brooks Air Force Base, TX.
Reprint requests to Dr. Pandurang M. Kulkarni, Department of
Mathematics and Statistics, University of South Alabama, Mobile,
AL 36688-0002.
83
84
Kulkami et al.
Mid-P
Max
1-0.
100
4 *
|
86
0
10 20 30 40 50
0
10 20 30 40 60
EOQ
0
10 20 30 40 GO
O
a
I
10 20 30 40 SO
E(X)
FIGURE 1. Coverage of Max and Mid-P (1 - a) x 100 percent
confidence intervals versus E(X), for a = 0.05 and a = 0.10. Max
refers to the Max method (equations 1 and 2), and Mid-P refers to
the Mid-P method (equations 3 and 4) for computing confidence
intervals. 5(X) is the expectation of a Poisson random variable,
denoted by A In the text. Coverage is the probability that the confidence interval contains A.
test When the underlying distributions are continuous,
one can obtain a confidence interval and a p value that
would result in the exact nominal significance level
and coverage probability, whereas with discrete distributions it is generally not possible to do so (13). For
discrete distributions, an exact p value can be computed using the formula p = 2{yPr(X = x) + min[Pr(X < x), Pr(X > x)]}, where 0 < y < 1 and the
probabilities are obtained under Ho (13, 14). If X is
continuously distributed, then Pr(X = x) = 0 and the
term associated with y is zero. One can use a randomization process to choose the value of y via a uniform
distribution (16, 17); however, this procedure can be
disturbing to a practitioner for one must add random
noise to the data to produce a p value (12). Instead, it
is a general practice (13) to choose 7 = 1 . This choice
leads to the most conservative p values and hence to
the most conservative confidence intervals obtained by
inverting the p value formula. Because these procedures result from y obtaining its maximum (Max)
value, we refer to the resultant procedures as Max
confidence intervals and p values. The confidence
intervals, with (1 — a) 100 percent confidence, with
7 = 1 are given by
1
"•Max, lower
CONFIDENCE INTERVALS AND p VALUES
Max method
2'
and
Let X denote the observed number of deaths in a
cohort study and assume that X follows a Poisson
distribution with expectation A. One is usually interested in knowing if A is the same as AQ, the corresponding parameter from a reference population.
Therefore, one is interested in testing the null hypothesis Ho: A = AQ. In a particular study, based on the
observed X = x, one wants to decide if Ho is to be
rejected. This decision can be made by constructing a
confidence interval for A/AQ, where we reject Ho if the
interval does not include 1. In some studies, it is usual
to report the p value associated with the hypothesis
TABLE 1.
(1)
^ AiZcl-a/2
' M n , upper
(2)
2(jr+l),a/2>
where x^fJ, denotes the upper b cutoff point of a
chi-square distribution with df degrees of freedom and
for df = 0, x^b = 0. This is derived using a wellknown relation between Poisson and chi-square distributions (7). Let PM^ denote the corresponding Max p
value. Then, j w = 2[p, - p2 + min(l - pu
withpj = PrO& < 2An) andp2 = Pr(xf(;c+1)
Further, because the confidence intervals are obtained
by simply inverting the p value formula, both the p
Coaffldenta for conawvattva (Max) approximate confidence intervals (equation* 5 and 6)
a
0.01
0.05
0.10
TABLE 2.
a
0.01
0.05
0.10
1.975642
1.060019
0.613212
0.742982
0.809256
0.841582
0.000600
0.0004360.000359
-3523624
-2.617304
-2552430
4.799837
2.464286
1.537202
1.247166
1.187111
1.156689
-0.000556
-0.000420
-0.000351
3.502510
2.718319
2.300006
Coefficients for the less conservative (Mid-P) appraxiinate confidence intervals (equations 7 and 8)
<••
S
0.216878
0.086808
0.034804
0.425475
0.436806
0.442301
0.485901
0.497343
0.689090
-1543794
-0.936323
-0.778045
2.085711
1.285344
0.940450
0.442950
0.438945
0.437537
0.642367
0.626021
0.616913
Am J Epidemiol
1.271724
0.991096
0.841053
Vol. 147, No. 1, 1998
Inference for the SMR and SIR
value and the confidence interval always lead to the
same decision for any chosen level of significance. In
figure 1 we have plotted the coverage of the conservative (Max) confidence intervals for values of A
between 0.1 and 50 with increments of 0.1 with a =
0.05. Figure 1 also shows the corresponding plot with
a — 0.10. It is clear that the coverage is much more
than the desired level; that is, the intervals are conservative.
APPROXIMATE CONFIDENCE INTERVALS
We derived simple equations to approximate the
percentile points of chi-square distributions, as in the
paper by Gilbert (20). These equations can be used to
obtain the confidence limits when the observed number of events is less than or equal to 50. Let v = 2x.
The Max lower and upper limits are approximated by
x, lower
= 0.5[oo + axv
Mid-P method
85
(5)
Max, upper
Now, we propose a method for obtaining less conservative p values and corresponding intervals. We
obtain these by choosing y = 0.5 instead of y = 1.0.
This approach has gained popularity in epidemiologic
studies and has been called the Mid-P procedure (13,
18). In this case, thep value i s p ^ d = 2[0.5(p1 — p2)
+ min(l - Pi,p2)]- The lower and upper Mid-P limits
for A are solutions of two equations; \Mid iower is the
solution of
= O.5[>o + bi(v + 2) + b2{v + 2)2
2)].
(6)
The Mid-P limits are approximated by
^Mid, lower = Co + CiA M a x ,i o w e r + C ^ + C 3 k>g(x)
(7)
Mid, upper
= <k +
, upper
+ <*2* +
(8)
The coefficients for equations 5 through 8 are listed in
tables 1 and 2 for a = 0.01, a = 0.05, and a = 0.10.
+ Pr|>L =£ 2AMi(Uower] = a
(3)
EXAMPLE
and
j
u p p e r
-
2
is the solution of
W upper]
L
= 2-a.
(4)
In figure 1, we have also plotted the coverage of the
Mid-P confidence intervals for values of A between
0.1 and 50 with increments of 0.1 with a = 0.10.
Figure 1 additionally shows the corresponding plot
with a = 0.05. It is clear from figure 1 that the Mid-P
procedure is less conservative than the Max procedure.
The average coverage, for the values of A considered,
for a = 0.10 were 92.4 percent for Max and 90.5
percent for Mid, and for a — 0.05 were 96.3 percent
for Max and ,95.3 percent for Mid.
The Mid-P method requires the solution of nonlinear equations; therefore, one must use a computer to
obtain the values of A
lower and A , ^ upperr. Because
many epidemiologic researchers use SAS software
(19), we have written a SAS code to compute confidence intervals and p values for both Max and Mid-P
methods. This code is available on request. Alternatively, in the next section we give simple equations
that can be used to calculate numerically approximate
intervals for both procedures. These do not require any
distributional tables and can be calculated with a hand
calculator.
Am J Epidemiol
Vol. 147, No. 1, 1998
We apply these methods to data arising from a study
of cancer in Air Force air crews (15). Grayson and
Lyons report 19 cases of cancer of the endocrine
system in air crews with an SIR of 1.61 when compared with nonflying Air Force officers. They report a
99 percent confidence interval (CI), based on a normal
approximation (3), of 0.81-2.83. The Max procedure
gives 99 percent CI 0.82-2.83 with p = 0.065. The
Mid-P procedure gives 99 percent CI 0.84-2.78 with
p = 0.05. The Max approximation (equations 5 and 6)
gives 99 percent CI 0.82-2.83, and the Mid-P approximation (equations 7 and 8) gives 99 percent CI 0.842.78. We obtained our confidence intervals by computing the 99 percent CI for A and dividing the
endpoints by the expected number of cases, computed
as the observed number of cases divided by the SIR.
Even though we have referred to the confidence intervals arising from equations 5 through 8 as approximate, we have found that both the Max and Mid-P
approximations are correct to within two decimals
when x is between 2 and 50 and the expected value of
X is greater than or equal to 5.
ACKNOWLEDGMENTS
Dr. Kulkarni's research was partially funded by the National Research Council.
86
Kulkarni et al.
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Am J Epidemiol
Vol. 147, No. 1, 1998