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Statistical comparison of Pearl rates
Statistical comparison of Pearl rates James E. Higgins, Ph.D., and Lynne R. Wilkens, M.S.P.H. Research Triangle Park, North Carolina Statistical procedures for hypothesis testing and interval estimation of the difference in a pair of Pearl rates are presented. The p value of the test statistic is evaluated exactly from the binomial distribution or approximately from the standard normal distribution. Interval estimation is provided by the test-based method of Miettinen. The procedures are applied to two published data sets, and the conceptual link between the Pearl rate and life-table methods is discussed. (AM J OBSTET GVNECOL 1985;151 :656-9.)
Key words: Pearl rate, hypothesis test, confidence interval, life table The Pearl rate is frequently used in clinical studies to summarize contraceptive efficacy. It estimates the number of pregnancies in 100 woman years of method use and is calculated as follows: N umber of pregnancies Number of woman months of pregnancy risk
X
1200
The denominator is the sum of the number of months each woman used the contraceptive method up to and including the month of conception or until termination of use. Before the widespread introduction oflife-table techniques in the early 1960s, the Pearl rate was the standard measure of contraceptive effectiveness. The Pearl rate is a good estimator of the risk of pregnancy if it can be assumed that all the women under observation have the same probability of pregnancy and of loss to follow-up and that these probabilities remain constant over the observation period. The latter assumption is often invalid, which causes the Pearl rate to be biased. I, 2 Despite this methodologic flaw, the Pearl rate has been a survivorS: recent publications include it in addition to life-table probabilities." 5 In part, the Pearl rate is appealing because of its relative simplicity both computationally and interpretatively. Additionally, since it was the standard measure for three decades, continued use of the Pearl rate provides a connection to earlier studies for which life-table probabilities cannot be calculated in retrospect. In comparative clinical trials it is routine to test for differences in life-table probabilities, and statistical tests are well defined. In studies in which Pearl rates have been compared, it was under the assumption that they
From Family Health International. Partial support for this study was provided by Family Health International with funds provided by the United States Agency for International Development, Received for publication February 9, 1984; revised September 21, 1984; accepted October 1,1984, Reprint requests: jamps E. Higgins, Ph,D, , Family Health international, Research Triangle Park, NC 27709,
656
were binomial proportions, and a statistical test of the independence of contraceptive method and pregnancy was applied. However, Pearl rates are not binomial proportions. ' ,6 As an alternative to the binomial model, Potter and Sagi ' recommended treating the Pearl rate as a ratio estimator with variance calculated by sample survey methods. This approach does not require identification of the type of distribution governing the number of pregnancies and takes into account the statistical dependence between the number of pregnancies and the number of follow-up months. The disadvantage with the approach is that in order to estimate the variance it is necessary to use essentially the same information required by life-table estimates. In 1976 Miettinen 7 attached the label "incidence density" to measures like the Pearl rate and offered a procedure for interval estimation. Recently a method was presented to test the difference in two incidence densities. 6 In the sections that follow, these new methods will be described and illustrated with the use of data from two published reports. Method
The data needed to calculate the Pearl rates in a comparative study are shown in Table l. The number of pregnancies in the study period starting at to and ending at t is n, which is the sum of n l pregnancies in L, woman months of follow-up for contraceptive C I , and n. pregnancies in L2 follow-up months for contraceptive C2 • L represents the total woman months of follow-up (L = L, + L2)' The respective Pearl rate estimates for C I and C 2 are as follows:
In epidemiologic terms the ratios d , = n/L, and d 2 = n./L 2 are the respective estimates of incidence density of pregnancy for C I and C. with the result that r, = d, X 1200 and r 2 = d 2 x 1200. Hypothesis test. The null hypothesis is Ho: R, = R2
Statistical comparison of Pearl rates
Volume 151 Number 5
657
Table I. Data summary for calculating Pearl rates for a comparative study Contraceptive method
Number of pregnancies in study period (4,-t) Woman months of risk in study period (to-t)
C2
Total
n2
n = n l + nz L = L, + L2
L2 ~ x 1200
Pearl rate
L,
C; = Contraceptive i; ni = the number of pregnancies occurring in contraceptive group i; L, = the number of woman months of risk accumulated for contraceptive group i; to = the beginning of the study period; t = the end of the study period.
or equivalently Ho: DI = D2 where RI and R2 represent the true Pearl rates and DI and D2 are the true incidence densities of pregnancy for the two contraceptives. The null hypothesis of equal Pearl rates is tested by assuming that the n observed pregnancies can be attributed to one of the two methods according to an independent Bernoulli process." If the null hypothesis is true, the probability that a given pregnancy belongs to a user of contraceptive C I is estimated by p = L/L and the probability of not belonging to C I and therefore belonging to C2 is estimated by q = L,IL. Thus equal Pearl rates imply that the proportion of the total of pregnancies attributable to method C I equals the proportion of total risk time attributable to that method (n/n = L/L). The p value for the test will be established for the one-sided alternative hypothesis that the Pearl rate for C I is larger than that for C2. The p value can be calculated exactly by evaluating the binomial sum
where
(7)
=
n!/i!(n - i)!
Tables of the binomial sum for values of nand pare readily available. 8 If np <;lnd nq both exceed 5, an approximate p value can be"determined by use of the test statistic z = (nl - np)tVnpq, which has approximately a standard normal distribution under the null hypothesis. The p value is evaluated in the usual way by use of a standard normal distribution table. 9 Confidence interval. Two-sided confidence intervals for the difference RI - Rz and ratio RI/R2 of two Pearl rates are constructed by the test-based procedure. 7 The method is appropriate for use with large samples, since the estimate of standard deviation is a function of the z statistic presented earlier for the large sample hypothesis test. The test-based confidence interval for the difference in two Pearl rates is (rl - r 2)(1 ± ZI _ a/2llzl) where ZI _ a/2 is the percentile of the standard normal distribution that is used to give a (1 - u) X 100% confidence interval and lzl is the absolute value of the z statistic described in the section on hypothesis tests. The
test-based confidence interval for the ratio of two Pearl rates is (r/r,)(1 ~ ZI_a/2/(d).
Examples In a report from the Oxford Family Planning Association Contraceptive Study covering the period from May, 1968, to April, 1975,10 the Pearl rate was used exclusively as the measure of contraceptive use-effectiveness. Confidence intervals for individual method rates were calculated by assuming that the Pearl rates were binomial proportions. Differences in Pearl rates were not compared statistically. Tables HA and lIB show data on contraceptive use-effectiveness taken from Table 28 of the interim report rearranged in a convenient computational form. To test whether the Pearl rate associated with Lippes Loop A or B exceeds that of Lippes Loop CorD, we first calculated p and q from the woman months of observation as p = 4992/67,980 = 0.073 and q = 1 P = 0.927 (Table HA). Since np = 118 X 0.073 > 5 and nq = 118 X 0.927 > 5, the test statistic was calculated in three decimal-place precision as follows: z
=
[17 - (118
X
0.073)]/ V118
X
0.073
X
0.927 = 2.968
From a table of the standard normal distribution,9 the p value was determined to be 0.002. The 95% confidence interval for the ratio of Pearl rates was (4.086/ 1.924)(1 , 1.96/2968>, or l.3-3.5. Because of the few pregnancies among women in the two groups of oral contraceptive users (Table lIB), the exact method of computing the p value was used to test whether the Pearl rate for progestogen-only users is greater than that for users of combination pills. Here n l = 6 and n = 9, so P = 2436112,708 = 0.192 and q = 1 - P = 0.808. The p value was given by
~U)0.192i 0.808
9
-,
which was evaluated approximately as 0.002 by rounding the values of p to 0.19 and q to 0.81 and by referring to a table of binomial probabiliti.es.s A second example comes from a study of intrauterine
658
Higgins and Wilkens
March I, 1985 Am J Obstet Gynecol
Table IIA. Use-effectiveness of intrauterine contraceptive devices investigated in the Oxford Family Planning Association Study* Lippes Loop
Number of pregnancies Woman months of observation Pearl rate
A or B
Cor D
Total
17 4992 4.09
101 62,988 1.92
118 67,980
*Source: Table 28 of Reference 10.
Table lIB. Use-effectiveness of oral contraceptives investigated in the Oxford Family Planning Association Study* Progestogen only
Combined> 50/ J-Lg estrogen
Total
6 2436 2.96
3 10,272 0.35
9 12,708
Number of pregnancies Woman months of observation Pearl rate *Source: Table 28 of Reference 10.
Table III. Clinical trials data on the use-effectiveness of two intrauterine contraceptive devices*
I Number of pregnancies Woman months of observation Pearl rate
SCS-D
TCu-200
Total
II 3492 3.78
10 4589 2.61
21 8081
*Source: Table 2 of Reference II.
contraceptive device efficacy which was analyzed by lifetable methods (Table III)." To test whether the Pearl rate for the SCS-D intrauterine contraceptive device is larger than that for the TCu-200 intrauterine contraceptive device, we calculated that p = 3492/8081 = 0.432 and q = 1 - P = 0.568. Since n = 21, both np and nq exceeded 5, and the z statistic calculated in three decimal-place precision was z = [11 - (21 x 0.432)] Y"'2-1-x-0.-4-:-32"--x-0-.-56--:-8 = 0.849 with a p value of 0.198. The 95% confidence interval for the ratio of Pearl rates was (3.780/2.615)<1 ~ 196/0849), or 0.6-3.4.
Comment In actuarial science there are links between the Pearl rate and the (cumulative) life-table pregnancy "rate"12; both are determined by a time-dependent function we will call the "force of pregnancy." The force of pregnancy generally varies with time and, for a specific point in time, gives the "instantaneous" rate of pregnancy. Figuratively, the force of pregnancy acts on the study subjects over a fixed period to cause pregnancy in some proportion of the women. The life-table pregnancy rate is not a rate but the probability or proportion of preg-
nancies in a fixed time period among a group of women who were not pregnant at the beginning of the period. It is often stated as the number of pregnancies per 100 women. By contrast, the Pearl rate measures the average force of pregnancy per fixed time period, since it is calculated as the change in the number of pregnancies per unit change in exposure months. It is expressed as pregnancies per 100 woman years of exposure. As an example, assume that 100 women not pregnant at the beginning of an efficacy study are followed for a year and that 20 become pregnant. If no women are lost from the study for reasons other than pregnancy, the I-year life-table pregnancy rate is 201100 = 0.20, or 20 pregnancies per 100 woman entering the study. Further, suppose that the pregnancies were uniformly distributed over the year. This implies that the average time of pregnancy is the midpoint of the year. Thus during the period of 1 year each woman who did not get pregnant contributed 12 months of exposure, whereas each woman who became pregnant contributed, on the average, 6 months of exposure. The total woman months of exposure is (80)(12) + (20)(6) = 1080. The Pearl rate is (2011080) X 1200 = 22, or 22 pregnancies per 100 woman years of exposure. Assume now that the 20 pregnancies in the example
Volume 151 Number 5
occurred in the first month of the study. The I-year life-table pregnancy rate would remain at 20 pregnancies per 100 women, but the Pearl rate would increase to (20/980) X 1200 = 24, since 20 of the women were observed for only 1 month. The I-month life-table pregnancy rate would also be 20 per 100 women while the Pearl rate based on the first month of follow-up would balloon to (201100) x 1200 = 240, or 240 pregnancies per 100 woman years of exposure. Even though the Pearl rate and the life-table pregnancy rate are governed by the same function, they measure different parameters and can give widely varying values. Providing a statistical comparison of Pearl rates does not replace the need for comparing pregnancy rates calculated by life-table methods. In studies that report both measures but rely on life-table rates for a statistical comparison of efficacy, it may be useful, in a descriptive sense, to present a confidence interval for the difference or the ratio of Pearl rates. Since Pearl rates usually change with time, it is important that they be compared at the same time point. If Pearl rates are the only measures of efficacy used and a comparison is desired, the methods presented here are appropriate. Most investigators are interested in comparing Pearl rates two at a time. However, the method can be extended to test simultaneously the equality of more than two Pearl rates. The extension involves assembling the data in a way similar to Table I but with as many columns as there are contraceptives to be compared. If the Pearl rates are equal, the proportion of the total pregnancies attributable to each contraceptive method equals the proportion of total exposure time attributable to the respective method, and the multinomial probability distribution can be used to evaluate the statistical significance of any differences. We have not illustrated the extension because the problem is not encountered very often and because it would add to the complexity of the paper. We would be pleased to pro-
Statistical comparison of Pearl rates 659
vide the details of applying the extension to any interested reader. Although the Pearl rate is specific to pregnancy, the concept of incidence density can be applied to other risks that are of interest in clinical studies of contraceptives. The formulas given earlier for comparing Pearl rates can be used to compare incidence densities for other risks. REFERENCES 1. Potter RG ]r, Sagi PC. Some procedures for estimating the sampling fluctuations of a contraceptive failure rate. In: Kiser CV, ed. Research in family planning. Princeton, New]ersey: Princeton University Press, 1962:389. 2. Sheps MC. Characteristics of a ratio used to estimate failure rates: occurrences per person year of exposure. Biometrics 1966;22:310. 3. Shelton ]D, Taylor RN. The Pearl Pregnancy Index reexamined: still useful for clinical trials of contraceptives. AM ] OBSTET GVNECOL 1981; 139:592. 4. Nilsson CG, Allonen H, Diaz], Luukkainen T. Two years' experience with two levonorgestrel-releasing intrauterine devices and one copper-releasing intrauterine device: a randomized comparative performance study. Fertil Steril 1983;39: 187. 5. Peterson HB, Lubell I, De Stefano F, Ory HW. The safety and efficacy of tubal sterilization: an international overview. Int] Gynaecol Obstet 1983;21: 139. 6. Kleinbaum DG, Kupper LL, Morgenstern H. Epidemiologic research. Belmont, California: Lifetime Learning Publications, 1982:284. 7. Miettinen O. Estimability and estimation in case-referent studies. Am] EpidemioI1976;103:226. 8. The Staff of The Computation Laboratory. Tables of the cumulative binomial probability distribution. Cambridge, Massachusetts: Harvard University Press, 1955. 9. Beyer WH, ed. Handbook of tables for probability and statistics. Boca Raton, Florida: CRC Press, Inc., 1968: 127. 10. Vessey M, Doll R, Peto R, Johnson B, Wiggins P. A longterm follow-up of women using different methods of contraception-an interim report.] Biosoc Sci 1976;8:373. 11. Azen SP, Roy S, Pike MC, Casagrande]. Some suggested improvements to current statistical methods of analyzing contraceptive efficacy.] Chronic Dis 1976;29:649. 12. Elandt-]ohnson RC, Johnson NL. Survival models and data analysis. John Wiley & Sons, Inc., 1980:99.
Key words: Pearl rate, hypothesis test, confidence interval, life table The Pearl rate is frequently used in clinical studies to summarize contraceptive efficacy. It estimates the number of pregnancies in 100 woman years of method use and is calculated as follows: N umber of pregnancies Number of woman months of pregnancy risk
X
1200
The denominator is the sum of the number of months each woman used the contraceptive method up to and including the month of conception or until termination of use. Before the widespread introduction oflife-table techniques in the early 1960s, the Pearl rate was the standard measure of contraceptive effectiveness. The Pearl rate is a good estimator of the risk of pregnancy if it can be assumed that all the women under observation have the same probability of pregnancy and of loss to follow-up and that these probabilities remain constant over the observation period. The latter assumption is often invalid, which causes the Pearl rate to be biased. I, 2 Despite this methodologic flaw, the Pearl rate has been a survivorS: recent publications include it in addition to life-table probabilities." 5 In part, the Pearl rate is appealing because of its relative simplicity both computationally and interpretatively. Additionally, since it was the standard measure for three decades, continued use of the Pearl rate provides a connection to earlier studies for which life-table probabilities cannot be calculated in retrospect. In comparative clinical trials it is routine to test for differences in life-table probabilities, and statistical tests are well defined. In studies in which Pearl rates have been compared, it was under the assumption that they
From Family Health International. Partial support for this study was provided by Family Health International with funds provided by the United States Agency for International Development, Received for publication February 9, 1984; revised September 21, 1984; accepted October 1,1984, Reprint requests: jamps E. Higgins, Ph,D, , Family Health international, Research Triangle Park, NC 27709,
656
were binomial proportions, and a statistical test of the independence of contraceptive method and pregnancy was applied. However, Pearl rates are not binomial proportions. ' ,6 As an alternative to the binomial model, Potter and Sagi ' recommended treating the Pearl rate as a ratio estimator with variance calculated by sample survey methods. This approach does not require identification of the type of distribution governing the number of pregnancies and takes into account the statistical dependence between the number of pregnancies and the number of follow-up months. The disadvantage with the approach is that in order to estimate the variance it is necessary to use essentially the same information required by life-table estimates. In 1976 Miettinen 7 attached the label "incidence density" to measures like the Pearl rate and offered a procedure for interval estimation. Recently a method was presented to test the difference in two incidence densities. 6 In the sections that follow, these new methods will be described and illustrated with the use of data from two published reports. Method
The data needed to calculate the Pearl rates in a comparative study are shown in Table l. The number of pregnancies in the study period starting at to and ending at t is n, which is the sum of n l pregnancies in L, woman months of follow-up for contraceptive C I , and n. pregnancies in L2 follow-up months for contraceptive C2 • L represents the total woman months of follow-up (L = L, + L2)' The respective Pearl rate estimates for C I and C 2 are as follows:
In epidemiologic terms the ratios d , = n/L, and d 2 = n./L 2 are the respective estimates of incidence density of pregnancy for C I and C. with the result that r, = d, X 1200 and r 2 = d 2 x 1200. Hypothesis test. The null hypothesis is Ho: R, = R2
Statistical comparison of Pearl rates
Volume 151 Number 5
657
Table I. Data summary for calculating Pearl rates for a comparative study Contraceptive method
Number of pregnancies in study period (4,-t) Woman months of risk in study period (to-t)
C2
Total
n2
n = n l + nz L = L, + L2
L2 ~ x 1200
Pearl rate
L,
C; = Contraceptive i; ni = the number of pregnancies occurring in contraceptive group i; L, = the number of woman months of risk accumulated for contraceptive group i; to = the beginning of the study period; t = the end of the study period.
or equivalently Ho: DI = D2 where RI and R2 represent the true Pearl rates and DI and D2 are the true incidence densities of pregnancy for the two contraceptives. The null hypothesis of equal Pearl rates is tested by assuming that the n observed pregnancies can be attributed to one of the two methods according to an independent Bernoulli process." If the null hypothesis is true, the probability that a given pregnancy belongs to a user of contraceptive C I is estimated by p = L/L and the probability of not belonging to C I and therefore belonging to C2 is estimated by q = L,IL. Thus equal Pearl rates imply that the proportion of the total of pregnancies attributable to method C I equals the proportion of total risk time attributable to that method (n/n = L/L). The p value for the test will be established for the one-sided alternative hypothesis that the Pearl rate for C I is larger than that for C2. The p value can be calculated exactly by evaluating the binomial sum
where
(7)
=
n!/i!(n - i)!
Tables of the binomial sum for values of nand pare readily available. 8 If np <;lnd nq both exceed 5, an approximate p value can be"determined by use of the test statistic z = (nl - np)tVnpq, which has approximately a standard normal distribution under the null hypothesis. The p value is evaluated in the usual way by use of a standard normal distribution table. 9 Confidence interval. Two-sided confidence intervals for the difference RI - Rz and ratio RI/R2 of two Pearl rates are constructed by the test-based procedure. 7 The method is appropriate for use with large samples, since the estimate of standard deviation is a function of the z statistic presented earlier for the large sample hypothesis test. The test-based confidence interval for the difference in two Pearl rates is (rl - r 2)(1 ± ZI _ a/2llzl) where ZI _ a/2 is the percentile of the standard normal distribution that is used to give a (1 - u) X 100% confidence interval and lzl is the absolute value of the z statistic described in the section on hypothesis tests. The
test-based confidence interval for the ratio of two Pearl rates is (r/r,)(1 ~ ZI_a/2/(d).
Examples In a report from the Oxford Family Planning Association Contraceptive Study covering the period from May, 1968, to April, 1975,10 the Pearl rate was used exclusively as the measure of contraceptive use-effectiveness. Confidence intervals for individual method rates were calculated by assuming that the Pearl rates were binomial proportions. Differences in Pearl rates were not compared statistically. Tables HA and lIB show data on contraceptive use-effectiveness taken from Table 28 of the interim report rearranged in a convenient computational form. To test whether the Pearl rate associated with Lippes Loop A or B exceeds that of Lippes Loop CorD, we first calculated p and q from the woman months of observation as p = 4992/67,980 = 0.073 and q = 1 P = 0.927 (Table HA). Since np = 118 X 0.073 > 5 and nq = 118 X 0.927 > 5, the test statistic was calculated in three decimal-place precision as follows: z
=
[17 - (118
X
0.073)]/ V118
X
0.073
X
0.927 = 2.968
From a table of the standard normal distribution,9 the p value was determined to be 0.002. The 95% confidence interval for the ratio of Pearl rates was (4.086/ 1.924)(1 , 1.96/2968>, or l.3-3.5. Because of the few pregnancies among women in the two groups of oral contraceptive users (Table lIB), the exact method of computing the p value was used to test whether the Pearl rate for progestogen-only users is greater than that for users of combination pills. Here n l = 6 and n = 9, so P = 2436112,708 = 0.192 and q = 1 - P = 0.808. The p value was given by
~U)0.192i 0.808
9
-,
which was evaluated approximately as 0.002 by rounding the values of p to 0.19 and q to 0.81 and by referring to a table of binomial probabiliti.es.s A second example comes from a study of intrauterine
658
Higgins and Wilkens
March I, 1985 Am J Obstet Gynecol
Table IIA. Use-effectiveness of intrauterine contraceptive devices investigated in the Oxford Family Planning Association Study* Lippes Loop
Number of pregnancies Woman months of observation Pearl rate
A or B
Cor D
Total
17 4992 4.09
101 62,988 1.92
118 67,980
*Source: Table 28 of Reference 10.
Table lIB. Use-effectiveness of oral contraceptives investigated in the Oxford Family Planning Association Study* Progestogen only
Combined> 50/ J-Lg estrogen
Total
6 2436 2.96
3 10,272 0.35
9 12,708
Number of pregnancies Woman months of observation Pearl rate *Source: Table 28 of Reference 10.
Table III. Clinical trials data on the use-effectiveness of two intrauterine contraceptive devices*
I Number of pregnancies Woman months of observation Pearl rate
SCS-D
TCu-200
Total
II 3492 3.78
10 4589 2.61
21 8081
*Source: Table 2 of Reference II.
contraceptive device efficacy which was analyzed by lifetable methods (Table III)." To test whether the Pearl rate for the SCS-D intrauterine contraceptive device is larger than that for the TCu-200 intrauterine contraceptive device, we calculated that p = 3492/8081 = 0.432 and q = 1 - P = 0.568. Since n = 21, both np and nq exceeded 5, and the z statistic calculated in three decimal-place precision was z = [11 - (21 x 0.432)] Y"'2-1-x-0.-4-:-32"--x-0-.-56--:-8 = 0.849 with a p value of 0.198. The 95% confidence interval for the ratio of Pearl rates was (3.780/2.615)<1 ~ 196/0849), or 0.6-3.4.
Comment In actuarial science there are links between the Pearl rate and the (cumulative) life-table pregnancy "rate"12; both are determined by a time-dependent function we will call the "force of pregnancy." The force of pregnancy generally varies with time and, for a specific point in time, gives the "instantaneous" rate of pregnancy. Figuratively, the force of pregnancy acts on the study subjects over a fixed period to cause pregnancy in some proportion of the women. The life-table pregnancy rate is not a rate but the probability or proportion of preg-
nancies in a fixed time period among a group of women who were not pregnant at the beginning of the period. It is often stated as the number of pregnancies per 100 women. By contrast, the Pearl rate measures the average force of pregnancy per fixed time period, since it is calculated as the change in the number of pregnancies per unit change in exposure months. It is expressed as pregnancies per 100 woman years of exposure. As an example, assume that 100 women not pregnant at the beginning of an efficacy study are followed for a year and that 20 become pregnant. If no women are lost from the study for reasons other than pregnancy, the I-year life-table pregnancy rate is 201100 = 0.20, or 20 pregnancies per 100 woman entering the study. Further, suppose that the pregnancies were uniformly distributed over the year. This implies that the average time of pregnancy is the midpoint of the year. Thus during the period of 1 year each woman who did not get pregnant contributed 12 months of exposure, whereas each woman who became pregnant contributed, on the average, 6 months of exposure. The total woman months of exposure is (80)(12) + (20)(6) = 1080. The Pearl rate is (2011080) X 1200 = 22, or 22 pregnancies per 100 woman years of exposure. Assume now that the 20 pregnancies in the example
Volume 151 Number 5
occurred in the first month of the study. The I-year life-table pregnancy rate would remain at 20 pregnancies per 100 women, but the Pearl rate would increase to (20/980) X 1200 = 24, since 20 of the women were observed for only 1 month. The I-month life-table pregnancy rate would also be 20 per 100 women while the Pearl rate based on the first month of follow-up would balloon to (201100) x 1200 = 240, or 240 pregnancies per 100 woman years of exposure. Even though the Pearl rate and the life-table pregnancy rate are governed by the same function, they measure different parameters and can give widely varying values. Providing a statistical comparison of Pearl rates does not replace the need for comparing pregnancy rates calculated by life-table methods. In studies that report both measures but rely on life-table rates for a statistical comparison of efficacy, it may be useful, in a descriptive sense, to present a confidence interval for the difference or the ratio of Pearl rates. Since Pearl rates usually change with time, it is important that they be compared at the same time point. If Pearl rates are the only measures of efficacy used and a comparison is desired, the methods presented here are appropriate. Most investigators are interested in comparing Pearl rates two at a time. However, the method can be extended to test simultaneously the equality of more than two Pearl rates. The extension involves assembling the data in a way similar to Table I but with as many columns as there are contraceptives to be compared. If the Pearl rates are equal, the proportion of the total pregnancies attributable to each contraceptive method equals the proportion of total exposure time attributable to the respective method, and the multinomial probability distribution can be used to evaluate the statistical significance of any differences. We have not illustrated the extension because the problem is not encountered very often and because it would add to the complexity of the paper. We would be pleased to pro-
Statistical comparison of Pearl rates 659
vide the details of applying the extension to any interested reader. Although the Pearl rate is specific to pregnancy, the concept of incidence density can be applied to other risks that are of interest in clinical studies of contraceptives. The formulas given earlier for comparing Pearl rates can be used to compare incidence densities for other risks. REFERENCES 1. Potter RG ]r, Sagi PC. Some procedures for estimating the sampling fluctuations of a contraceptive failure rate. In: Kiser CV, ed. Research in family planning. Princeton, New]ersey: Princeton University Press, 1962:389. 2. Sheps MC. Characteristics of a ratio used to estimate failure rates: occurrences per person year of exposure. Biometrics 1966;22:310. 3. Shelton ]D, Taylor RN. The Pearl Pregnancy Index reexamined: still useful for clinical trials of contraceptives. AM ] OBSTET GVNECOL 1981; 139:592. 4. Nilsson CG, Allonen H, Diaz], Luukkainen T. Two years' experience with two levonorgestrel-releasing intrauterine devices and one copper-releasing intrauterine device: a randomized comparative performance study. Fertil Steril 1983;39: 187. 5. Peterson HB, Lubell I, De Stefano F, Ory HW. The safety and efficacy of tubal sterilization: an international overview. Int] Gynaecol Obstet 1983;21: 139. 6. Kleinbaum DG, Kupper LL, Morgenstern H. Epidemiologic research. Belmont, California: Lifetime Learning Publications, 1982:284. 7. Miettinen O. Estimability and estimation in case-referent studies. Am] EpidemioI1976;103:226. 8. The Staff of The Computation Laboratory. Tables of the cumulative binomial probability distribution. Cambridge, Massachusetts: Harvard University Press, 1955. 9. Beyer WH, ed. Handbook of tables for probability and statistics. Boca Raton, Florida: CRC Press, Inc., 1968: 127. 10. Vessey M, Doll R, Peto R, Johnson B, Wiggins P. A longterm follow-up of women using different methods of contraception-an interim report.] Biosoc Sci 1976;8:373. 11. Azen SP, Roy S, Pike MC, Casagrande]. Some suggested improvements to current statistical methods of analyzing contraceptive efficacy.] Chronic Dis 1976;29:649. 12. Elandt-]ohnson RC, Johnson NL. Survival models and data analysis. John Wiley & Sons, Inc., 1980:99.