STATISTICS IN MEDICINE
Statist. Med. 2003; 22:2071–2083 (DOI: 10.1002/sim.1405)
Condence limits for the ratio of two rates based
on likelihood scores: non-iterative method
P. L. Graham1; ∗; † , K. Mengersen1 and A. P. Morton2
1 Statistics;
School of Mathematical and Physical Sciences; University of Newcastle;
Callaghan NSW 2308; Australia
2 Infectious Diseases Department; Princess Alexandra Hospital; Brisbane Qld 4102; Australia
SUMMARY
This paper describes a method for creating a condence interval for the ratio of rates using the score
statistic. This non-iterative and easy to apply procedure produces condence intervals that are suitable
for use with Poisson data and simulation results indicate that it is close to the nominal level for a wide
range of scenarios. Copyright ? 2003 John Wiley & Sons, Ltd.
KEY WORDS:
condence interval, coverage probability; rate ratio; Poisson rates
1. INTRODUCTION
Rates are commonly used to describe the frequency of an event and are widely used in health
and many other applications. For example, consider the description of independent hospital
acquired infections such as device related bacteremias. Here, it is important to monitor the
number of such infections over particular time periods, but the baseline may dier over weeks,
months, or years depending on the size and practice of the hospital and the incidence of the
infection. Hence the rate, rather than a simple count, provides a standardized measure that can
be conveniently monitored or compared. A second example, also borrowed from biostatistics,
is taken from Rothman and Greenland [1] and involves the comparison of incidence of breast
cancer. Here two groups of women were compared to discover whether those who had been
examined using x-ray uoroscopy during treatment for tuberculosis had a higher rate of breast
cancer than those who had not been examined in that way.
It is a common aim in such applications not only to compute rates but to compare them.
Typically, such a comparison takes the form of a dierence or a ratio. While the latter loses
the absolute size of the comparison, it has a strong advantage in providing a natural reference
∗
Correspondence to: P. L. Graham, Statistics, School of Mathematical and Physical Sciences, University of
Newcastle, Callaghan NSW 2308, Australia.
† E-mail: [email protected]
Contract=grant sponsor: Australian Research Council SPIRT; contract=grant number: C10024120.
Published online 18 March 2003
Copyright ? 2003 John Wiley & Sons, Ltd.
Received September 2001
Accepted August 2002
2072
P. L. GRAHAM, K. MENGERSEN AND A. P. MORTON
point of unity. An example of a widely employed ratio is the relative risk used in areas such
as evidence-based medicine. The statistical vehicle for comparison may be a hypothesis test
of equality of the rates (or, equivalently, that the ratio is equal to unity), but considerably
more information can be gained from a condence interval for the ratio.
To date, focus has been concentrated on developing condence intervals for rate ratios under
the assumption that the underlying rates have a binomial distribution. Hence the probability,
p, of an event is described through P(x) = ( nx )p x (1 − p)n−x where x = 0; 1; : : : ; n is the number
of events of interest observed and n is the total number of observed events. In a comparison
of two such proportions, p0 and p1 , say, a condence interval may be based on the ratio
p1 =p0 . A widely favoured maximum likelihood approach to the derivation of such condence
intervals uses score methods [2–5], however these methods require iterative calculation for
their solution. Nam [2] recently introduced a non-iterative approach which is further discussed,
along with the other score-based methods, in Section 2. Many other approaches are also
available such as large-sample Wald-type approximations using logs and other derivations of
score-based intervals [6, 7] but these will not be discussed further in this paper.
For rare events, or in the situation where denominator data may be unavailable or dicult
to collect, a Poisson distribution may better describe the behaviour of the observed
events
x
or be an adequate approximation. The Poisson distribution is dened as P(x) = x! e− where
x = 0; 1; 2; : : : is the number of events and is the rate of an event. Now interest is in the
ratio of two such rates, 1 = 0 , and its associated condence interval.
Condence intervals on the ratio of Poisson rates have been developed by Jaech [8] and
Rothman and Greenland [1], as described in Section 2. However, of the two alternatives
proposed in Jaech’s paper, one of these initially treats the data as binomial and the other uses
a transformation of the Poisson statistic. Rothman and Greenland on the other hand use a
large-sample Wald-based approximation.
The aim of this paper is thus to develop condence intervals for the ratio of two independent
rates with underlying Poisson distributions using a non-iterative likelihood score method. The
approach is a direct but useful generalization of Nam’s [2] development under the binomial
assumption. We show by example a comparison of the new interval in contexts of interest in
Section 4.
The paper proceeds as follows. Section 2 provides a background on the relevant methods
developed to date for condence intervals of ratios of rates under binomial and Poisson
assumptions. Section 3 presents the new non-iterative condence intervals assuming Poisson
distributions for the rates. In Section 4 an illustration is given and the result is compared with
those obtained from alternative methods. Finally, a simulation study in Section 5 reveals the
true level of the condence interval for a range of values and allows comparison to be made
between the proposed condence interval and alternative condence intervals.
2. PREVIOUS WORK ON CONFIDENCE INTERVALS FOR
THE RATIO OF RATES
Much of the earliest work on condence intervals using likelihood score methods was by
Bartlett [3]. Bartlett’s theoretical development was subsequently used by Gart [4] in his development of a condence interval for the ratio of two independent binomial proportions. Gart
and Nam [5] further developed this work by implementing a skewness correction. However
the solution of both condence interval formulae requires an iterative calculation that could
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
CONFIDENCE LIMITS FOR THE RATIO OF TWO RATES
2073
be computationally laborious, and can be dicult with certain data sets (see Nam [2] for
details). Thus Nam [2] derived a non-iterative method to calculate the condence interval for
the ratio of two binomial proportions. All of the above score methods for proportions use the
same procedures initially:
Take two binomial variables x 0 and x1 with corresponding sample sizes n0 and n1 and
parameters p0 and p1 . Dene to be p1 =p0 and let n· = n0 + n1 and qi = 1 − pi for i = 0 or
1. They reparameterize the log-likelihood by p1 = p0 giving
ln L = (x 0 + x1 ) ln p0 + (n0 − x 0 ) ln q0 + x1 ln + (n1 − x1 ) ln(1 − p0 )
and nd that the score for is
S (; p0 ) =
x1 − n1p1
q1 with variance var{S (; p0 )} = [2 {q0 =(n0 p0 ) + q1 =(n1p1 )}]−1 . The approximate 1 − condence limits are the two solutions to
(x 0 − n0 p̃0 )2
n0 (1 − p̃0 )
2
2
1+
= z=2
(1)
X () =
n0 p̃0 q̃0
n1 q̃0
where p̃0 is the maximum likelihood estimate of p0 in
Sp0 (; p0 ) =
x 0 − n0p0 x1 − n1p1
+
q0
q1
(2)
At this point Gart [4] and Gart and Nam [5] nd the approximate condence limits satisfying
(1) and (2) using an iterative procedure whereas Nam [2] uses a series of substitutions to
nd the condence limits analytically.
Various methods for calculating the condence interval for the ratio of two independent
Poisson rates have also been developed. Now taking x 0 and x1 to be independent Poisson
variables, with associated rates 0 and 1 and denominators n0 and n1 , Jaech [8] describes
two dierent methods for nding a condence interval of the ratio = 1 = 0 . The rst (we
call this the conditional binomial method) treats the data as binomial by initially ignoring
the Poisson denominators and instead using x1 =(x 0 + x1 ). Tables or software are used to nd
condence limits (denoted L and U ) on the new numerator and denominator and the resultant
interval, [B lower = n2 L=n1 (1 − L); B upper = n2 U=n1 (1 − U )], is found to be conservative (that is,
the true ratio, , is rejected less often than expected). The second method (we call this
method Jaech’s transformation), which may be used as an alternative to the score method,
uses a transformation of the Poisson statistic to approximate
√ a standard√normal distribution
when is suciently large. The transformation used is 2[ (x + 0:5) − ] and after some
manipulation the limits, k lower and k upper are found through
√[(x + 0:5)(x + 0:5)] ± 0:5z √[x + x + 1 − (z 2 =4)]
√
1
0
1
0
=2
n0
=2
k=
2 =4)
n1
x 0 + 0:5 − (z=2
At the nominal = 0:05 level Jaech nds this interval to be less conservative than the conditional binomial method and sometimes a little liberal (that is, the true ratio, , is rejected
more often than expected).
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
2074
P. L. GRAHAM, K. MENGERSEN AND A. P. MORTON
Rothman and Greenland [1] describe a large-sample Wald-based approximation. Here again
x 0 and x1 are taken to be independent Poisson variables with respective denominators n0 and
n1 and associated rates 0 and 1 . The Wald condence limits are then found through


1 
x1 =n1
1

± z=2
+
explog

x 0 =n0
x1 x 0
Rothman and Greenland state that under certain conditions based on sample sizes that this
condence interval should be ‘adequate’, although they do not describe what they mean by
this term.
For many of the above intervals situations arise for which the intervals cannot be sensibly
calculated. Naturally, for all of these intervals nothing meaningful may be calculated when
both x 0 and x1 are 0. In other circumstances there are a variety of ways for intervals to have
aberrations. When these situations are encountered we advise following the directions of Gart
and Nam [5] or Koopman [7] in which either the oending x 0 or x1 is redened as 0.5 or
the lower and upper condence bounds are set at 0 and ∞, respectively.
Note that Newcombe [9] provides a useful discussion on these problems with respect to a
variety of condence intervals for single proportions.
3. CONFIDENCE LIMITS FOR THE RATIO OF RATES UNDER THE ASSUMPTION
OF POISSON DISTRIBUTIONS
Consider two independent Poisson variables x 0 and x1 with corresponding parameters 0 and
1 . The log-likelihood for this is
ln L( 0 ; 1 ) = x 0 ln 0 − ln(x 0 !) − 0 + x1 ln 1 − ln(x1 !) − 1
Since in practice rates may be computed using dierent denominators we allow for this by
n0 1
1
dening = 10=n
=n0 = n1 0 as the ratio of the standardized rates and let x· = x 0 + x1 . Because n0
and n1 are strictly greater than zero in any practical situation, we assume n0 ; n1 ¿0 for the
remainder of this paper. If the rates are already standardized this reduces to (the more usual)
= 10 .
The log-likelihood is reparameterized by 1 = 0 nn10 so that it becomes a function of and
0 where is now the parameter of interest and 0 is now a nuisance parameter
n1
n1
− ln(x 0 !x1 !)
ln L(; 0 ) = x· ln 0 − 0
+ x1 ln − 0 + x1 ln
n0
n0
Now the score for is
S (; 0 ) =
=
Copyright ? 2003 John Wiley & Sons, Ltd.
@ ln L(; 0 )
@
x1 n1
− 0
n0
(3)
Statist. Med. 2003; 22:2071–2083
CONFIDENCE LIMITS FOR THE RATIO OF TWO RATES
2075
and the score for 0 is
@ ln L(; 0 )
@ 0
x 0 + x1
n1
=
−
−1
0
n0
S 0 (; 0 ) =
(4)
To apply Bartlett’s method [3] we need the maximum likelihood estimator of 0 given ,
which is the solution to [S 0 (; 0 )] 0 = ˜ 0 = 0, that is
x 0 + x1
˜ 0 = n1
n0 + 1
(5)
and (using notation by Gart [4]) we nd
var{S (; ˜ 0 )} = I (; ˜ 0 ) −
=
1
2 ( nn10 + 1)
(I 0 (; ˜ 0 ))2
I 0 0 (; ˜ 0 )
where I (; ˜ 0 ) = 1 =2 is the variance of (3), I 0 0 (; ˜ 0 ) = ( nn10 + 1)= ˜ 0 is the variance of
(4) and I 0 (; ˜ 0 ) = nn10 is the covariance of (3) and (4). It is interesting to note here, that of
the three elements of the I matrix, it is only I 0 0 that involves the optimization of 0 .
Now, approximate 1 − condence limits are the two solutions to the equation
X 2 () =
{S (; ˜ 0 )}2
var{S (; ˜ 0 )}
2
= z=2
Substituting, we nd, for x 0 ¿0
n0
L =
n1
n0
U =
n1
2
x· −
2x 0 x1 + z=2
√
2
2
{z=2
x· (4x 0 x1 + z=2
x· )}
(6)
2x20
2
2x 0 x1 + z=2
x· +
√
2
2
{z=2
x· (4x 0 x1 + z=2
x· )}
2x20
(7)
Note that by earlier discussion n0 and n1 are greater than zero and hence is greater than
zero. In the special case x 0 = 0 one might follow the directions discussed in the previous
section in which we either redene x 0 as x 0 = 0:5 or dene the upper and lower condence
limits as 0 and ∞, respectively. Note also that although the previous situation had special
cases, setting @ ln L=@ 0 = 0 leads to the maximum likelihood estimator of 0 given as
shown in (5), always, with no special cases unlike related situations in Newcombe [10, 11]
and Tango [12].
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
2076
P. L. GRAHAM, K. MENGERSEN AND A. P. MORTON
4. EXAMPLE
This illustration examines an example used in Rothman and Greenland [1]. Here two groups
of women were compared to discover whether those who had been examined using x-ray
uoroscopy during treatment for tuberculosis had a higher rate of breast cancer than those
who had not been examined using x-ray uoroscopy. In the given data set, the treatment
group consisted of women who had received x-ray uoroscopy and this group had 41 cases
of breast cancer in 28 010 person-years at risk. The control group consisted of women who
had not received x-ray uoroscopy and in this group there were 15 cases of breast cancer in
19 017 person-years at risk. In summary we have
x 0 = 15;
n0 = 19 017;
x1 = 41
and
n1 = 28 010:
Hence a 95 per cent condence interval on using (6) and (7) is [1:0358; 3:3249].
Because this data set has large denominators and small rates it is reasonable to consider
a comparison of the above interval with alternatives based on the binomial assumption [14].
Strictly speaking, methods such as that of Nam, which assume underlying binomial distributions, are inapplicable when denominators are person-years at risk. We employ it here simply
for comparison. Applying Nam’s method, which is the binomial equivalent of the above approach, we nd the condence interval to be [1.0361, 3.3241] which is almost the same as
that derived under the Poisson score method. Gart and Nams’ [5] score-based condence interval with skewness correction for binomial proportions gives a condence interval of [1.0440,
3.4364]. This interval has a wider upper limit than our original interval perhaps due to the
method or the skewness correction. Using Minitab [13] to nd binomial condence limits on
41=56 we then nd that Jaech’s [8] conditional binomial method gives an approximate interval
of [1.0057, 3.6093]. This is, as expected, wider than that obtained under the other methods.
Using Jaech’s transformation on the Poisson statistic we nd a 95 per cent condence interval
on to be [1.0436, 3.4339]. This interval appears to yield similar results to those found using
Gart and Nam’s iterative score method.
For further comparison, the Wald-based method described by Rothman and Greenland [1]
produces a 95 per cent condence interval of [1.0272, 3.3526]. This method appears to produce
a slightly wider interval than that obtained using the Poisson score method. Rothman and
Greenland describe the numbers used in this calculation as being suciently large enough to
produce an adequate approximation.
Note that for the conditional binomial method, Jaech [8] refers readers to the abridged tables
by Hald [15]. These tables are not recommended due to limited numerator and denominator
values. Newcombe (personal communication) recommended the use of more comprehensive
tables such as the Geigy Scientic Tables [16] or software programs such as SAS or Minitab
to nd the required limits. Note that the beta function in Excel may also be used to do this.
For the above comparison we have used moderate numerators and large denominators and it
is reasonable to expect that all of the methods will broadly agree. When small numerators are
used larger dierences between the methods are observed. Although this dierence varies a
great deal depending on the dierence between the numerators (the further apart the numerators
are, the more the methods agree), we found in general that Nam’s method and the Poisson
score-based method are very similar and are always the narrowest intervals and the conditional
binomial method is always widest (generally at least twice as wide as the score-based interval).
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
CONFIDENCE LIMITS FOR THE RATIO OF TWO RATES
2077
5. MONTE CARLO SIMULATION
A Monte Carlo simulation study was undertaken to examine the performance of the proposed
score-based interval, Jaech’s transformation interval and Rothman and Greenland’s Waldbased interval for all integer values of 0 and 1 between 1 and 40. The equivalence of
n1 = n0 was maintained for simplicity and without loss of generality. A methodology similar
to that described by Miettinen and Nurminen [6] was adopted. This involved generating a
random value from each of Poisson( 0 ) and Poisson(1 ) distributions. A nominal level = 0:05
condence interval was calculated based on these observed values and a test performed to see
whether the true ratio 10 was contained in the interval. This was repeated 1 000 000 times and
the number of times the ratio was not in the interval was recorded. As discussed in Sections
2 and 3, when x 0 is 0 there is a division by zero in the score-based interval and also for
Rothman and Greenland’s interval when either x 0 or x1 is 0. When this happened we followed
the directions of Gart and Nam [5] and Koopman [7] and tried two dierent approaches to
calculating the coverage probabilities for these intervals, rst dening x 0 = 0:5 for the scorebased interval and xi = 0:5 for i = 0; 1 in Rothman and Greenland’s Wald-based interval and
second dening the lower and upper bounds as 0 and ∞, respectively. The simulations under
the amendments produced very similar results so we will only discuss the case in which we
replaced xi = 0 with xi = 0:5. Jaech’s condence interval could only produce zero values with
certain non-integer values of x 0 depending on the value of z=2 . Since the chosen values for
0 ; 1 and z=2 for this interval could not produce zero values, amendments were not required.
Table I illustrates in detail the simulated coverage probabilities when 0 and 1 are small
(each between 1 and 8). The results of this simulation indicated that when 0 = 1 or 2 and
1 ¿4 (and vice versa) then the score-based interval is closest to nominal. When 0 = 1 then
Jaech’s method is closer to nominal, however the results for the score-based interval and
Jaech’s interval are very close when both rates are greater than 4. Elsewhere, Jaech’s interval
is usually closer in absolute value to the nominal 95 per cent level but this is not entirely
consistent and both are generally close to the nominal level. We also note that results are
very similar at both the 90 per cent and 99 per cent levels. The only major dierence occurs
at the 99 per cent level when 0 and 1 are small. Here, when one rate is equal to 1 and the
other rate is between 2 and 5 (see Table II), we nd that Jaech’s interval is very liberal and
the score-based interval is conservative.
Figure 1 shows a scatter plot comparing the coverage probabilities for the score method
and Jaech’s method. Here each point represents the coverage probability for a 0 − 1 pair.
This plot clearly presents the dierences in true levels and it is easy to see that many of the
intervals fall close to the 95 per cent level. Where this is not true the score-based intervals
are closer to the nominal level than Jaech’s intervals. A further comparison of the score-based
and Jaech’s intervals in a small simulation (100 iterations and for rates varying over 1 to 8
for each ) at the 95 per cent level showed that Jaech’s interval is usually wider than the
score-based interval.
Similarly, Figure 2 shows a comparison of the simulated coverage probabilities for the
score-based interval and Rothman and Greenland’s large-sample approximation method based
on Wald limits at the nominal 95 per cent level. Here we see that most of the score intervals
are around the 95 per cent level whereas the Wald-based intervals are almost all conservative
and on average at least half a per cent too large. Similar results are found when comparing
Jaech’s interval with the Wald interval. Lastly a box plot (Figure 3) shows that the distribution
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
2078
P. L. GRAHAM, K. MENGERSEN AND A. P. MORTON
Table I. True simulated coverage probability at the nominal 95 per cent level. For each rate: numbers in the rst row correspond to Jaech’s interval; numbers in the second row correspond to
the proposed score interval; numbers in the third row correspond to the Wald-based interval.
0
1
1
2
3
4
5
6
7
8
1
98.59
99.27
100.00
98.12
95.77
99.82
95.10
96.56
99.03
97.46
94.86
97.24
97.86
95.86
97.08
97.41
95.58
96.48
97.49
95.91
96.52
97.51
95.67
96.22
2
98.13
98.26
99.82
95.83
97.56
99.93
96.25
97.04
99.54
95.65
96.54
98.84
96.70
96.07
98.34
96.46
96.12
97.53
96.85
96.50
96.93
97.04
96.50
96.63
3
95.10
96.57
99.04
96.21
96.93
99.53
95.29
96.64
99.54
95.48
95.89
99.36
95.11
96.42
98.18
94.72
96.23
97.82
95.41
96.10
97.87
95.20
96.52
97.56
4
97.43
95.51
97.20
95.60
96.42
98.84
95.46
95.82
99.34
95.54
96.18
98.81
95.00
95.85
98.43
95.40
95.97
97.98
95.79
96.18
97.62
94.91
95.71
97.38
5
97.86
96.12
97.09
96.69
95.88
98.35
95.13
95.72
98.18
94.99
95.39
98.44
95.64
95.88
97.99
95.34
95.55
97.61
94.99
95.33
97.57
95.58
94.76
97.83
6
97.41
95.61
96.44
96.47
95.88
97.54
94.76
95.46
97.85
95.35
95.43
97.95
95.31
95.35
97.63
95.55
95.62
97.26
95.21
95.24
97.25
95.15
95.30
96.72
7
97.51
95.93
96.53
96.84
96.32
96.93
95.43
95.05
97.88
95.78
95.67
97.62
94.99
95.15
97.56
95.19
95.21
97.26
95.40
95.41
96.76
95.68
95.14
96.70
8
97.48
95.65
96.20
97.09
96.44
96.66
95.23
95.80
97.58
94.86
95.19
97.36
95.57
94.55
97.82
95.13
95.25
96.72
95.66
95.13
96.66
95.18
95.16
96.32
Table II. True simulated coverage probability at the nominal 99 per cent level. For each
rate: numbers in the rst row correspond to Jaech’s interval; numbers in the second row
correspond to the proposed score interval.
0
1
1
2
3
4
5
6
7
8
1
99.96
100.00
89.77
99.64
94.36
99.22
96.95
98.49
98.41
98.46
97.57
98.34
98.58
98.32
99.15
98.34
2
89.76
99.85
99.54
99.94
99.51
99.65
98.42
99.25
98.98
99.37
99.44
99.16
99.42
98.79
99.42
98.79
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
CONFIDENCE LIMITS FOR THE RATIO OF TWO RATES
2079
1.00
0.99
JAECH
0.98
0.97
0.96
0.95
0.94
0.94
0.95
0.96
0.97
SCORE
0.98
0.99
1.00
Figure 1. Coverage probabilities of the score-based interval versus the
Jaech’s interval at the 95 per cent level.
1.00
0.99
ROTHMAN
0.98
0.97
0.96
0.95
0.94
0.94
0.95
0.96
0.97
SCORE
0.98
0.99
1.00
Figure 2. Coverage probabilities of the score-based interval versus the
Wald-based interval at the 95 per cent level.
of coverage probabilities for the simulated intervals discussed are all skewed right and all are
generally conservative, however, the Wald-based interval is more conservative than the others.
Table III shows the summary statistics for the coverage probabilities at the nominal 95 per
cent level. For example, ‘Max error-’ means the maximum amount the coverage was below
nominal. Here the minimum coverage probability for the score-based interval was 0.9455
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
2080
P. L. GRAHAM, K. MENGERSEN AND A. P. MORTON
1.00
0.99
0.98
0.97
0.96
0.95
0.94
JAECH
ROTHMAN
SCORE
Figure 3. Box plots of coverage probabilities at the nominal 95 per cent level.
Table III. Summary statistics for the coverage probabilities at the 95 per cent level.
Median
Mean
Max error −
Max error +
Mean error −
Mean error +
SD errors −
SD errors +
Proportion −
Proportion ¡94:9 per cent
Proportion +
Proportion ¿96 per cent
Score
Jaech
Rothman
0.9512
0.9525
0.0045
0.0427
0.0009
0.0033
0.0009
0.0041
0.1638
0.0046
0.8150
0.0838
0.9509
0.9532
0.0061
0.0359
0.0008
0.0041
0.0010
0.0077
0.1650
0.0034
0.8069
0.0944
0.9564
0.9593
0.0012
0.0500
0.0010
0.0093
0.0002
0.0072
0.0013
0.0006
0.9988
0.3581
+ indicates above nominal.
− indicates below nominal.
which is 0.0045 below nominal. Note that this table also shows the proportion of times
coverage probabilities are above or below nominal for each interval and also the proportion
of times the intervals are above 96 per cent or below 94.9 per cent.
Newcombe [9] points out that many studies (including this one) use integer values in the
simulations and the extreme discreteness of the coverage behaviour that this causes may invite
discussion about the conclusions. Thus, to provide a fuller picture, we performed a simulation
study similar to the previous one but now using values of 0 and 1 between 1 and 10 in
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
CONFIDENCE LIMITS FOR THE RATIO OF TWO RATES
2081
1.00
0.99
JAECH
0.98
0.97
0.96
0.95
0.94
0.94
0.95
0.96
0.97
SCORE
0.98
0.99
1.00
Figure 4. Coverage probabilities of the score-based interval
versus Jaech’s interval at the 95 per cent level.
1.00
0.99
ROTHMAN
0.98
0.97
0.96
0.95
0.94
0.94
0.95
0.96
0.97
SCORE
0.98
0.99
1.00
Figure 5. Coverage probabilities of the score-based interval versus the
Wald-based interval at the 95 per cent level.
steps of 0.1. Figure 4 shows that the majority of coverage probabilities produced by both the
score-based method and Jaech’s method are close to nominal. However, for those coverage
probabilities that are not close to nominal, Jaech’s interval has more points scattered between
97 per cent and 98 per cent and the score-based interval has some points that are a lot
more conservative than Jaech’s interval. On the other hand, Figure 5 shows that Rothman and
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
2082
P. L. GRAHAM, K. MENGERSEN AND A. P. MORTON
Greenland’s interval is always more conservative than the score-based interval. Note also for
these small rates that Rothman and Greenland’s interval is always at least 1 per cent above
the nominal 95 per cent level.
6. DISCUSSION
The score-based condence interval for the ratio of two Poisson rates developed here is
competitive with Jaech’s interval as evidenced by the example and simulations shown in previous sections. Furthermore, we may determine that the score-based method is preferable over
Jaech’s method, in many situations, because the coverage probabilities are less variable, are
more often closer to the nominal level and also because the intervals are generally narrower.
The summary statistics and plots shown in Section 5 helped to illustrate the similarities
between the score-based method and Jaech’s method and further conrm the conservative
nature of Rothman and Greenland’s interval. Table III also showed that although the scorebased method and Jaech’s method may be liberal at times, less than 0.5 per cent of simulated
coverage probabilities are less than 94.9 per cent. Thus we may conclude that provided it
is acceptable to have the occasional slightly liberal interval, then both Jaech’s transformation
and the score-based intervals are clearly preferable to the Wald-based interval of Rothman
and Greenland.
Finally, the score-based interval is easy to apply, requires no iterative calculations and is
maintained at close to the nominated coverage.
ACKNOWLEDGEMENTS
The authors are greatly indebted to the reviewers, including R. Newcombe, for their extremely useful
comments and suggestions. They would also like to thank Dr Nam for his helpful comments.
REFERENCES
1. Rothman KJ, Greenland S. Modern Epidemiology. 2nd edn. Lippincott-Raven: Philadelphia, 1998.
2. Nam J. Condence limits for the ratio of two binomial proportions based on likelihood scores: non-iterative
method. Biometrical Journal 1995; 37:375– 379.
3. Bartlett MS. Approximate condence intervals, II. More than one unknown parameter. Biometrika 1953; 40:
306– 317.
4. Gart JJ. Approximate tests and interval estimation of the common relative risk in the combination of 2×2 tables.
Biometrika 1985; 72(3):673 – 677.
5. Gart JJ, Nam J. Approximate interval estimation of the ratio of binomial parameters: a review and corrections
for skewness. Biometrics 1988; 44:323 – 338.
6. Miettinen O, Nurminen M. Comparative analysis of two rates. Statistics in Medicine 1985; 4:213 – 226.
7. Koopman PAR. Condence intervals for the ratio of two binomial proportions. Biometrics 1984; 40:513 – 517.
8. Jaech JL. Comparing two methods of obtaining a condence interval for the ratio of Poisson parameters.
Technometrics 1970; 12(2):383 – 387.
9. Newcombe RG. Two-sided condence intervals for the single proportion: comparison of seven methods.
Statistics in Medicine 1998; 17:857– 872.
10. Newcombe RG. Interval estimation for the dierence between independent proportions: comparison of eleven
methods. Statistics in Medicine 1998; 17:873 – 890.
11. Newcombe RG. Improved condence intervals for the dierence between binomial proportions based on paired
data. Statistics in Medicine 1998; 17:2635– 2650.
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083
CONFIDENCE LIMITS FOR THE RATIO OF TWO RATES
2083
12. Tango T. Equivalence test and condence interval for the dierence in proportions for the paired-sample design.
Statistics in Medicine 1998; 17:891–908.
13. Minitab Inc. Minitab Release 12.1. Available from www.minitab.com, 1998.
14. Rao PV. Statistical Research Methods in the Life Sciences. Duxbury Press: California, 1998.
15. Hald A. Statistical Tables and Formulas. Wiley: New York, 1952.
16. Lentner C (ed.). Geigy Scientic Tables: Volume 2. 8th edn. CIBA-GEIGY: Basle, 1982; 89 – 102.
Copyright ? 2003 John Wiley & Sons, Ltd.
Statist. Med. 2003; 22:2071–2083