The mean interval to conception: A measure of utility for the analysis of decisions involving fertility

The mean interval to conception: A measure of utility for the analysis of decisions involving fertility Emmet]. Lamb, MD, Michael Hagen, MD, and Stephen G. Pauker, MD

Boston, Massachusetts This article describes a method of assigning a utility, or relative worth, to outcomes based on the mean interval to conception (MIG). In the formula MIG = 1/fc , the subscript G is the proportion of a given cohort of women who would conceive if given an infinitely long trial and fc is the fecundability or monthly probability of conception among this subgroup. MIG and MIO (interval of observation for those who do not conceive) are used as the utilities in decision analyses of fertility treatment. This method provides a semiquantitative estimate that reflects uncertainty of cure of infertility and variation in amount of time spent in the nonpregnant state after infertility treatment. (AM J OasTET GVNECOL 1989;160:1470-8.)

Key words: Decision analysis, utility, interval to conception, fecundability Doctors make easy decisions about tests and treatment using rules of thumb or heuristics. For new and difficult decisions, they are trained to use other procedures that usually include the following steps: (1) List the available choices and the events that might result from selecting each; (2) determine the relative likelihood of each event for each choice and the consequences of each outcome in terms of the patient's values; (3) choose the option that optimizes the changes for achieving the desired outcome. Assessment of the relative worth of outcomes is common to all clinical decision making. The methods presented in this article are best understood in the framework of formal decision analysis. 1' 5 Here the decision analyst does the following: (1) explicitly structures the problem, typically as a decision tree (Fig. 1); (2) assigns a probability to each chance event; (3) assigns to each outcome a relative value, or utility, using a single consistent scale; (4) calculates the average or expected utility of each strategy by multiplying the utility value for each outcome by the probability of its occurrence; (5) evaluates the tree by "folding back" or calculating the expected utility for all branches in the tree; the strategy with the highest expected utility is deemed "best"; (6) examines the impact of various assumptions through sensitivity analysis. In sensitivity analysis the value for one or more variables is sequentially changed and the tree reevaluated to determine the level at which the preference for options changes. The utility or worth of outcomes may be measured From the DzvislOn of Climcal DecISIOn Making, Department of MediCine, Tufts University School of MediCine. Presented at the Fifty-fifth Annual Meeting of the Pacific Coast Obstetrical and Gynecologzc Society, Honolulu, HawaII, November 12-19, 1988. Reprint requests: Emmet]. Lamb, MD, Department of Gynecology and Obstetrics, Stanford Univer:,ity, Stanford, CA 94305.

1470

by an arbitrary scale. Time intervals, however, provide a utility measure that is more easily understood. The successful use of approximations of life expectancy, the mean expected duration of life, as a utility in the analysis of choices involved in the management of various diseases that reduce the duration and quality of life6 suggests that a similar concept would be useful in analysis of infertility problems. The treatment option with the shortest interval to conception, perhaps after adjustments for quality of life,7. 8 is the preferred clinical strategy. The monthly conception rate, or fecund ability, may increase with time, for example, in adolescence, or decrease with time, for example, after age 40. A simplifying assumption, that fecundability is constant, is often made in statistical analyses. To assume constant fecundability is to say that, within the range of ages of the subjects involved, pregnancies occur in a random pattern in those at risk, influenced neither by memory of past events nor by age. If data sufficient to construct a life table are available, it is easy to test whether this assumption is appropriate. Estimating fecundability, f, is one step in constructing a life table. If the hazard rate is constant, the slope of the line should not differ significantly from 0 when f is plotted against time. These notions are discussed further in the appendix and in several monographs. 6 • 9 10 Life-table methods are used to determine fecundability for single groups of infertile women. 11. 12 A quick, simple way to estimate fecund ability is to calculate the ratio of total conceptions to total patient months of exposure, an estimate that is based on the assumption of constant fecundability. I3 However, Guzick and Rock 14 demonstrated that fecund ability calculated this way for a group of women surgically treated for endometriosis was not constant but declined with time. One possible model to reconcile their findings is that some women

Mean interval to conception

Volume 160 !';umber 6

PREGNANT

162 '-.

p= .65

1? 0

SURGERY

.s:: p=.35

162 '-.

1.0

250

0

1.0 0.8

0.8

(.)

0

1ii

:§lc:

0.6 0.6

~

CHOOSE

0.4

Q.

188 ,.

p= .35 IVFx3

538

:cCIl

0.4

0.2

rx:

0.2

OJ

:::)

0

CIl

0 to

Fig. 1. DecIsion tree illustrating first approach to analysis of choice of therapy for patient with bilateral hydrosalpinges. Square node indicates a choice, CIrcular nodes indicate chance events, and rectangular nodes indicate utilities. CIrcled numbers outside tree are calculated expected utility values for branch indicatd by arrow. Complete analysis considering surgical complications. ectopic pregnancy, multiple pregnancy, repeated treatment courses. etc.. would require a tree with many more branches.

are actually never going to conceive after therapy but are, in fact, covertly sterile. H . 13 The application of this concept to decision analysis is explored in this article.

Material and methods Terminology. A Iwzard rate is a measure of proneness to experience an event as a function of time. One example is the annual mortality rate; another is fecundability, the monthly conception rate. Synonyms for fecundability are fecundity, the potential fertility rate, and the force of pregnancy. In the calculation of life tables, those subjects who have not experienced the event, for example, death or conception, by the completion of a study are considered censored and are treated mathematically in a similar way. When death is the outcome studied, all subjects eventually experience the event. However, a woman may be removed from any chance of conception by becoming temporarily or permanently sterile through such censoring events as separation, contraception, or the onset of bilateral complete tubal obstruction. SurvIVorship, S" is the probability that an individual will survive, or fail to experience the event, within an interval up to and including a given time, t. If there were no losses to follow-up, survivorship could be calculated by dividing the total number of subjects surviving at least to t by the total number of subjects initially at risk, So. In practice, survivorship is estimated by life-table methods. A plot of survivorship versus time is familiar from reports of cancer therapy (Fig. 2). In fertility studies, S, describes the probability that a woman remains nonpregnant at time t and (l - S,) describes the cumulative probability of pregnancy at time t.

Q.

:CUi" CIlG> rx:'Iii 0-0 > .::;'g rx:0I :::)~ CIl

0

> ::;

a::

p=.65

1471

0

N

t1

t2

(1-C)

TIME

Fig. 2. Plot of cumulative proportion who have conceived versus time of observation. Proportions of original cohort are indicated on left axis and proportions of candidates on nght. On abscissa, 10 is start of observation, and II and 12 are arbitrary times of observation.

Consider a group of women treated for an infertility problem at time 0 and let C denote the proportion of patients who compose the group who would eventually become pregnant if they had an unlimited time of trial. We prefer to use the term candIdate rather than the term cured as used by Guzick and Rock. " Candidates may have greatly reduced hazard rates, but their fecundability, f" is not zero. At any time t, the interval of observation for the entire cohort includes the interval to conception for those who have conceived and the interval to time t for those who have not. Those who have not conceived until time t include both those who are sterile and will never conceive and those who have not yet conceived but are still candidates. Data sources and calculations. From a search of the recent medical literature, we selected articles on human fertility or infertility that contained life tables in a tabular form from studies of > 100 subjects. 'fi.21 If the report contained sufficient data to construct a life table, we entered the data into a microcomputer spreadsheet. We used several methods to estimate f, and the mean interval to conception, MIC, for the portion, C, of the cohort who would eventually conceive. First, we divided the total number of subjects who became pregnant within 12 cycles by the total number of woman months of observation before conception. In Tables I and II this column is headed "No. pregnant within 12 cycles." Second, we divided the total number of su~jects who became pregnant within 12 cycles by the total number of woman months of observation up to that point for the entire cohort. This column is headed "Oneparameter model." We calculated life tables by the Kaplan- Meier method,'2 including estimates of the cu-

1472 Lamb, Hagen, and Pauker

June 1989 Am J Obstet Gynecol

Table I. Fecundability of normal women estimated by several methods described in text Life-table estimate of cumulative proportion pregnant at n cycles

Study group*

Stop IUD I6

AID (C)17 AID (C, N)18

AID I9

No. of women

No. of cycles

n = 1

n=6

602 1303 1188 1115

12 12 12 60

0.326 0.130 0.101 0.183

0.770 0.515 0.465 0.654

* IUD, Intrauterine contraceptive device; AID, artificial insemination donor; C, cryopreserved sperm; N, nulligravid woman. Table II. Fecundability of infertile women with human menopausal gonadotropin-treated ovulatory disorders or with untreated endometriosis as estimated by methods described in test Life-table estzmate of cumulative proportion pregnant at n cycles

Study group*

No. of women

No. of cycles

n = 1

n=6

hMG-hPG treatment Amenorrhea20 Oligomenorrhea 20 Endometriosis, no therapy21

167 348 123

7 12 24

0.329 0.095 0.170

0.913 0.362 0.221

*hMG. Human menopausal gonadotropin; hPG, human pituitary gonadotropin. tCalculated at the number of cycles shown in column 3 rather than at 12 cycles. tThe proportion of candidates was determined by the authors as 0.727 using another algorithm.

mulative probability of conception (l - S) in each reported interval; only two intervals are listed in the tables. The data were also used to calculate C, the proportion of candidates, and fe, the fecundability of candidates, by use of a program that used an algorithm 22 for calculation of maximum likelihood and likelihood ratios; this program is more likely to obtain a solution than is that used by Guzick and Rock. This Fortran program ran on an IBM Personal Computer (International Business Machines, Boca Raton).

Results Table I lists the results of calculations from reports used to estimate the fecundability of normal women. Table II lists results from reports of infertile women with ovulation disorders or endometriosis.

Comment Only women who would eventually conceive if given an infinitely prolonged trial period can be expected to conform to the constant hazard rate model. With time, as women in this subgroup conceive, the overall group that appears to the observer to be at risk of conception will be weighted more and more by members of the covertly sterile group. The result will be a decline of the observed hazard rate for the overall group, even if the hazard rate for those actually at risk were constant. The greater the fraction of patients who are covertly

sterile, the steeper the downward slope of the plot of fecund ability versus time. Similarly, the greater this fraction, the worse the fit between the data points calculated by life-table methods and those derived from a single-parameter model dependent on a constant hazard rate. In any group of women, of course, there is a distribution of fecund ability. The fact that women of greater fecund ability are likely to conceive earlier, leaving those with lesser fecundability remaining in the group exposed to later cycles, was noted> 50 years ago. 16. 23 The Guzick and Rock model makes the simplifying assumption that this distribution of fecundability among women can be represented by two groups, a covertly sterile group and a candidate group with a given fecundability. The fecundability of that group is constant but may vary from one group to another, depending on their disease or treatment. Thus, for a cohort of treated infertile women, the predicted cumulative proportion pregnant at time t is determined by two parameters: (1) the candidacy proportion, C, and (2) the monthly rate of pregnancy or fecundability, fe, among those candidates. Such a two-group model has been considered in analysis of survivorship from cancer and other diseases when cure is possible!' 25 We will deal only with this two-group model. To estimate a value for the utility at each branch of the decision tree for an individual patient using the

Mean interval to conception

Volume 160 !';umber 6

1473

Mean interoal to conception for those who conceived

Estimated fecundabili(v, fc By pregnancies within 12 cycles

By one-parameter model

B,v two-parameter model

Candidate ProportIOn, C

Obseroed at 12 cycles

Projected by two-parameter model

0.344 0.333 0.291 0.269

0.218 0.113 0.100 0.158

0.274 0.126 0.097 0.161

0.923 0.929 1.000 0.970

2.903 3.003 3.440 3.717

3.647 7.909 10.284 6.213

Mean znteroal to conceptwn for those who conceived

Estimated fecundabtlity, fe By pregnancies within 12 cycles

By one-parameter model

By two-parameter model

Candidate proportion, C

Obseroed at 12 cycles

Projected by two-parameter model

0.494t 0.373 0.326

0.322 0.071 0.031

0.322 0.072 0.027

1.000 1.000 1.000:1:

2.025t 2.683 3.067

3.101 13.951 37.632

quality-adjusted mean interval to conception, the decision analyst would follow these steps: (1) Obtain from the literature data, in the form of life tables, reporting studies of groups of women with the same condition or disease. These life tables can be used to estimate the proportion of candidates, C, and the fecundability of the candidates, f e . For example, in Fig. 1, the proportion of candidates, C, after surgery was set at 0.65 and the fecundability of candidates, fe, was set at 0.025. (2) Estimate MIO, the mean interval of observation, or MIC for each terminal utility node in the decision tree. In the example shown in Fig. 1, MIO, the mean interval of observation up to age 45, the end of reproductive life, for noncandidate subjects 30 years old at the start of observation would be 15 years or 540 months. MIC, the mean interval to conception for candidates after surgery, would be 110.025 or 40 months. (3) Obtain a number that increases as perceived value increases by subtracting the interval in months from 540 months. In the example, 540 - 40 = 500 months for the candidates and 540 - 540 = 0 months for the noncandidates. (4) Use the time trade-off or lottery method' 8 to adjust for the patient's perception of the quality of life during the interval and use these qualityadjusted intervals as utility values. In the example, we illustrate a case in which the patient felt that adjustment for the quality of life reduced the utility of the outcome of the top branch, pregnancy after surgery, by one half (to 250 months).' Determine the expected utility of each branch of the tree proceeding from right to left in the tree until the initial choice node is reached. The treat-

ment option with the highest expected utility is the preferred option. In Fig. 1, the uppermost branch of the upper chance node has a value of 250 x 0.65 or 162. This expected utility, being higher than that of the other branch, is carried to the next node, in this case, the choice node. For the patient for whom these probabilities and utilities apply, the expected utility is higher for surgery than for three cycles of in vitro fertilization, and surgery is the preferred option. The first step, estimating fe and C, usually depends on finding a published study of a group comparable to the patient. If all subjects in the group have some potential for fertility after therapy, the one-parameter technique of Cramer et al. 13 would give a value for f similar to that from the two-parameter model. Note that in Tables I and II, whenever the estimate of the candidate proportion was 1.000, the estimates of fecundability by the one-parameter and two-parameter models are the same. When the candidate proportion is < l.000, fecundability estimated by the twoparameter model is higher. The procedure Guzick and Rock suggested, for using survival analysis methods to compare the effect of two treatments on fertility, includes the following steps: (1) Calculate the hazard rate, f, and the survivorship function, S" for the initial cohort with life-table methods. (2) Plot f versus t. A straight line with zero slope, or only random fluctuations about such a line, indicates that a one-parameter model should suffice. On the other hand, a downward trend in the hazard rate suggests that a fraction of the members of the group are

1474 Lamb, Hagen, and Pauker

not candidates. This indicates the need for two additional steps: (3) Estimate fe and C by the use of appropriate mathematical methods. (4) Test for statistical significance of any differences in the estimated C and fe between the treatment groups. Guzick and Rock H and Guzick et al. '5 used the leastsquares or the maximum likelihood and likelihood ratio procedures to test for a statistical difference between the results of the two treatments. The algorithm they used on a mainframe computer to determine the maximum likelihood estimate of fc and C often fails to converge. The algorithm that we used reduces this problem to a minimum and, moreover, runs on a desktop computer. Understanding the use of time-interval utilities in decision problems is helped by considering the extremes of these intervals. Very long intervals, the worst outcome, result either because of a candidacy probability near 0, immediate sterility, or because of a fecundability near 0. We consider the maximum interval to extend to the end of reproductive life, which we chose to be age 45, rather than to the end of the individual's life. As the best outcome one may use an interval of 0, immediate pregnancy, or one may use normal fecundability with a mean interval to conception equal to that of a cohort of fertile women of the same age from the general population. Determining normal fecund ability in a group of women is more difficult than determining the mortality rate for that group. Tables listing hazard rates for death at various ages, stratified by race and sex, are readily available, but this, unfortunately, is not the case for fecundability. Questionnaire surveys indicate that older women more often report difficulty achieving a desired pregnancy. However, because censoring conditions such as contraception are not tabulated, census data are of little use in quantifying the effect of age on fecundability. Other methods must be used to estimate ageadjusted fecundability, i.e., the mean interval to the first clinical pregnancy among noncontracepting populations,1O among women who have stopped using intrauterine contraceptive devices,'6 or among women having artificial insemination with donor semen. 17. '9 Data on the interval to conception from registrants in prenatal clinics can be used to estimate fe, the fecund ability of the candidate group. On the basis of these reports it appears that a reasonable baseline estimate for normal fecundability is 0.200 and that values in the range of 0.100 to 0.250, or perhaps even 0040, should be included in sensitivity analyses. The reciprocal of these provides lower bounds for the mean interval to conception, a baseline of 5 months with a range of 2.5 to 10 months. The methods in this article use data obtainable from the literature to calculate C and fe through the use of algorithms, which are very likely to obtain a solution

June 1989 Am J Obstet Gynecol

and can be run on a microcomputer. Application of these methods to formal decision analysis allows for several possible outcomes, each with an intermediate value for the mean interval to conception or interval of observation. Moreover, the two-parameter model allows the decision analyst to model the value, or utility, of perfect knowledge of the patient's candidacy status, C or not-Co This value could be used in modeling the decision to do a laparoscopy after tuboplasty. We thank Jerry Halpern, PhD, of the Department of Health Research and Policy, Stanford University, for his major help in the development of the mathematical and statistical concepts used in this paper. REFERENCES 1. Kassirer JP, Moskowitz AJ, Lau J, Pauker SG. Decision analysis: a progress report. Ann Intern Med 1987; 106:275. 2. Pauker SG, Kassirer JP. Decision analysis. N Engl J Med 1987;316:250. 3. Weinstein MC. Fineberg HV, Elstein AS, et al. Clinical decision analysis. Philadelphia: WB Saunders. 1980. 4. Sox H, Blatt M, Higgins M, Merton K. Medical decision making. Boston: Butterworths, 1988. 5. Holzman S. Intelligent decision systems. Addison Wesley, 1989. 6. Beck JR, Pauker SG. Gottlieb JE, Klein K, Kassirer JP. A convenient approximation of life expectancy (the "DEALE"). II. Use in medical decision-making. AmJ Med 1982;73:889. 7. McNeil BJ, Weichselbaum R, Pauker SG. Speech and survival. Trade-offs between quality and quantity of life in laryngeal cancer. N EnglJ Med 1981;305;982. 8. Read JL, Quinn RJ, Berwick DM, Fineberg HV, Weinstein MC. Preferences for health outcomes. Comparison of assessment methods. Med Decis Making 1984;4:315. 9. Lee E.T. Statistical methods for survival data analysis. Belmont, California: Lifetime Learning Publications, 1980. 10. Leridon H. Human fertility, the basic components. Chicago: University of Chicago Press. 1977. 11. Lamb EJ, Cruz AL. Data collection and analysis in an infertility practice. Fertil Steril 1972;23:310. 12. Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. J Am Stat Assoc 1970;53:579. 13. Cramer DW, Walker AM, Schiff I. Statistical methods in evaluating the outcome of infertility therapy. Fertil Steril 1979;32:80. 14. Guzick DS, Rock JA. Estimation of a model of cumulative pregnancy following infertility therapy. AM J OBSTET GyNECOL 1981;140:573. 15. Guzick DS, Bross DS, RockJA. A parametric method for comparing cumulative pregnancy curves following infertility therapy. Fertil Steril 1982;37:503. 16. Tietze C. Fertility after discontinuation of intrauterine and oral contraception. Int J Fertil 1968; 13:385. 17. Trounson AO, Matthews CD, Kovacs GT, et al. Artificial insemination by frozen donor semen: results of multicentre Australian experience. ImJ Androl 1981:4:227. 18. David G, Czyglik F. Results of AID for first and succeeding pregnancies. In: David G, Price WS. Artificial insemination and semen preservation. New York: Plenum Press, 1980:211. 19. Schoysman-Deboeck A, Merckx M, Luc S, Vekemans M. Results of AID in 865 couples. In: David G, Price WS. Artificial insemination and semen preservation. New York: Plenum Press, 1980:231. 20. Dor J, Itzkowic DJ, Mashiach S, Lunenfeld B, Serr DM. Cumulative conception rates following gonadotropin therapy. AMJ OBSTET GYNECOL 1980;136:102.

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Volume 160 Number 6

21. Olive DL, Martin DC. Treatment of endometriosisassociated infertility with C02 laser laparoscopy: the use of one- and two-parameter models. Ferti! Steril 1987: 48:18. 22. Gill PE, Murray W. Numerical methods for constrained optimization. Academic Press, 1974. 23. Gini C. Decline in the birthrate and the "fecundability" of woman. Eugenics Rev 1925; 17:258. 24. Goldman AI. Survivorship analysis when cure is a possibility: a Monte Carlo study. Stat Med 1984;3:153. 25. Berkson J, Gage RP. Survival curve for cancer patients following treatment. J Am Stat Assoc 1952;47:501.

Appendix The assumption of a constant hazard rate in survival analysis results in mathematical features that make its use appealing. For instance, when the hazard rate is constant, the survivorship function has an exponential distribution and survivorship has a smooth exponential decline when plotted against time. Moreover, if all subjects eventually suffer the event, the mean interval to the event corresponds to the area under the curve of the survivorship function plotted against time. The median interval is related to the mean interval by the formula: Mean = Medianlln(2). Furthermore, when the hazard rate is constant, the area under the curve has a simple reciprocal relationship with the hazrd rate. Assuming that all women in the cohort would eventually conceive and using f to represent their fecund ability, the hazard rate for conception, and MIC to represent the mean interval to conception: 1

f

MIC

MIC 1

f

0)

Pt

=

(1 - C)· P"nut.( + C· Pt,t

(3)

Since the probability of pregnancy in the noncandidate group remains 0 regardless of time, the first term in equation 3 drops out. For the group of candidates with a given, constant fecundability, fe, the probability of pregnancy at time t, can be estimated by the equation for an exponential decline: Pt" = 1 - exp( - fe . t)

Referring back to the assumption that Pt,note = 0 and substituting equation 4 into equation 3, we arrive at a simple expression for the cumulative proportion of pregnancy for the original cohort: Pt = C . (1 - exp( - fc . t)

(4)

Thus, for a cohort of treated infertile women, the predicted cumulative proportion pregnant at time t is determined by two parameters: (1) the candidacy proportion, C, and (2) the monthly rate of pregnancy or fecundability, fe, among those candidates. As illustrated in Fig. 2, one can also consider C to be a scaling factor determined by that fraction, C/1.000, of the original cohort who can possibly conceive. Multiplying this fraction by the values of the scale for the entire cohort on the left produces the scale tor candidates alone shown on the right.

(5)

Consider now an infertile woman, age 30, who states that she desires pregnancy but, for various reasons, would not want to conceive after reaching age 37. Let us see how one can calculate the probability that she will conceive during this 7-year interval. Let us first consider the case in which we know that she is a candidate, a member of the group that would eventually conceive. Assuming that the hazard rate is constant for candidates, then the probability that a pregnancy will occur before a given time is determined by equation 4. This calculation may be carried out for any two times, for example, tl and t2 in Fig. 2, to calculate the probability that pregnancy will occur in the interval between the times: Pmt = pt2 - P,), We are interested in a conditional probability, the probability of a conception in the interval conditional on the subject being at risk of conception at the start of the interval. To be at risk of conception during the current menstrual cycle, one must not have been pregnant at the start of the cycle. If 70% of the original cohort had already conceived by tlo then only 30% of the original cohort would be at risk. Therefore the probability of conception during an interval conditional on not being pregnant at the start of the interval, regardless of when times t) and t2 occur, is given by the formula:

(2)

Guzick and Rock l4 noted that the cumulative probability of conception at time t is equal to a weighted average of the cumulative probabilities of conception in the two groups that we have called the noncandidates and the candidates:

1475

P(tntlnotpreg)

=

Pt' - ptl I - Ptl

Substituting from equation 4 and reducing: p(mtlnotpreg)

==

[1 - exp( - fc . t2)] - [1 - exp( - fc . tl)]

P(mtlnotpreg)

1 - [(1 - exp( -fc . tl)]

exp( -fc . tl) - exp( -fc . t2) = exp( - fc . t1)

P(mtlnotpreg) = 1 - exp[ - fC
(6)

Equation 6 can be used to estimate the probability of conception for a woman of age, AI, before she reaches any other age, A2 , at which she would no longer want to become pregnant. If pregnancy has not occurred by t) and the woman is a candidate, that is, she would eventually become pregnant if given an infinitely long trial, the expected additional time to pregnancy is given by equation 2, MIC = life, regardless of when tl is. Now, let us consider the entire cohort, not just the candidates. At the beginning of the observation period, to, the proportion of the cohort at risk of conception is indicated by C. Later, at time t, the proportion of the original cohort still at risk is indicated by C . (1 - p), where p is the proportion of those at risk in the cohort who have conceived before time t. However, the proportion of the current cohort, the cohort at time t, at risk is different than this but less than C. For example, if C, the original proportion at risk is 0.80 and

1476

Lamb, Hagen, and Pauker

June 1989 Am J Obstet Gvnecol

p = 0.30, then the proportion of the original cohort at risk at t would be 0.8.0.7 = 0.56, but the proportion of the current cohort at t that is at risk is: C . (1 - p) C . (1 - p)

+

0.56 0.56 + 0.2

(1 - C)

0.74

In general, the mean interval of observation for the entire cohort starting at any time t is given by the weighted average of the interval to conception for those who will conceive after t and the interval of observation for those who will not: MIO = [ (

,

C . (1 - PI)

C . 0 - PI)

+

(1 - C)

)

.1.J + fc

C· (1 - P ). ) (540 - Ao) [( 1 - C . 0 - PI) + (1' - C)

J

0.2' (0.186)

(7)

1 ]

+ 0.8 . 0.02 +

0.2' (0.186) ) [( 1 - 0.2 . (0.186) + 0.8 . (540 - 360) MIO, = 2.22

J

+ 172.00

= 174.22

By first calculating MIO for the entire cohort at t2 , then repeating the calculation at t" and subtracting one result from the other, one can estimte the MIO for the entire cohort for any given interval. Since the second term cancels out, it does not matter which age one selects for the end of reproductive life. For the interval between t, and t 2 , the mean interval of observation, MIO, is represented by the area of the polygon LMNO in Fig. 2. The area of this polygon, the area under the curve between t, and t2 , is estimated by: MIO".

'2

=

[0 -

C) .

(t2

-

t,)] + t2

or MIOti •

,2

=

[0 ~

C) .

(t2

-

exp( - fc . t

2) . 0

fc . t exp( - fc . t) dt

t,)] +

[exP( - fc . t,) . (1

+ fc

. t,)

+ fc . t2)]

Editors' note: This manuscript was revised after these discussions were presented. Discussion

where p, is the proportion of the candidates who have conceived by time t as determined by equation 4 and Ao is the mean age in months for all members of the cohort at the start of treatment. Ao = 360 for age 30; A ~ 540 at age 45. For example, let C = 0.80 and fc ~ 0.20. At the end of 7 years of observation, the probability of pregnancy for the candidates would be 0.814 and the mean interval of observations would be: MIO, = [ 0.2' (0.186)

utility value assigned to it to obtain the expected utility is a much easier calculation to understand and to perform. For utility values one can use the mean interval to conception in the branches for those who conceive and the mean interval of observation in the branches for those who have not conceived. The calculations for folding back the tree are easily done for multiple branches with a microcomputer program.

(8)

This calculation gives a value that could be used to evaluate the first branches after the decision node of the decision tree shown in Fig. 1. In a decision analysis, however, multiplying the probability of an event by the

DR. PAUL KIRK, Portland, Oregon. At first reading this article seems somewhat intimidating, but on second reading it clearly presents a challenge to our specialty in general and to our colleagues specializing in infertility in particular. The intimidation is not a fault of the authors but an admission of inadequacy by this reviewer, an indaequacy born of working in a discipline that appreciates the importance of statistical proof but largely shuns the mathematical concepts involved in obtaining the proof. The authors have done four things for us. First, they have introduced us to a definition of formal decision analysis systems that have long been used in business and engineering schools. Second, they have developed a model using the analogy of life expectancy with which we are already familiar. Third, they have reviewed the literature for the last five years and, where possible, applied this model to already published materials. Fourth, they have implicitly challenged us to take a look at how we examine, analyze, publish, and interpret the results of the many and varied therapies for infertility. We are all aware of the great challenge that our specialty faces in terms of the ethical principles with which we underpin the care we provide our patients. Dr. Louis Vontver and I recently attended a workshop for curriculum development for students and residents sponsored by the Association of Professors of Gynecology and Obstetrics: "Medical Ethics, the Law, and Professional Liability." Among the 47 issues that were listed, many related to infertility and its management. We are all aware of the truly commercial aspects of infertility treatment including some in vitro fertilization programs and "department store" types of infertility clinics. We are aware as well of great impatience for results. We are impatient, the media relations (marketing) departments of our hospitals are impatient, our patients and their families are impatient. I understand Dr. Lamb's challenge as a plea for our specialty to establish some consistency in the way we report and analyze the results of different treatments. This would improve our ability to predict the likelihood of pregnancy for any given patient. Dr. Lamb, how close are you to developing the ideal tool-is MIC it? Is it reasonable to expect a society such as the American Fertility Society to take the lead in establishing guidelines and standards of reporting of infertility treatment? DR. FERDERICK HANSON, Davis, California. I have

Volume 160 !';umber 6

several questions. First, how well does this model fit the data when compared with models in which the fecundability varies in a continuous distribution from woman to woman while at the same time there exists the possibility of a sterile subpopulation? Second, how would the authors clinically use this method of decision analysis? Would not alternative measures of fertility be preferable in practice? For example, the chance of conceiving within 2 years would include both sterile and nonsterile couples. Because the effect of permanent sterility might differ by treatment, it would possibly be more informative than just knowing the chance of conceiving among couples who will conceive. How useful to couples is the mean interval to conception compared with the chance of conceiving at a fixed time interval? Your model uses two numbers. Rather than trying to explain the separate concepts of the mean time to conception and the chance that a couple will be sterile, would it not be better to give them the choice of conception over a fixed time period? DR. EUGENE C. SANDBERG, Portola Valley, California. From my limited exposure to decision analysis, it does not appear to me that it has made many inroads into medical practice in its 15 years or so of existence. This is not because of any serious deficiency in decision science so much as it is that the medical world is not ready for, nor does it yet sense a need for, clinical decision analysis. It is one of numerous techniques being developed simultaneously on parallel tracks in various fields without the anticipation that they will ultimately interact. There has been considerable movement in the past 10 or 20 years toward quantification and categorization of information in our field, for example, Apgar scores, Bishop scores, various scores for high-risk pregnancy, and ICDA, the International Classification of Disease, Adapted. We can now make diagnoses by artificial intelligence computer systems and therapeutic decisions by using decision analysis. The availability of desktop computers to store and manipulate data in large amounts points to a greater and greater potential for mechanization of thought and decision making. The future I see embodies a great deal of robotics and decision-making machinery. I see that machinery in a spacecraft, two generations out from Earth, regularly receiving information from Earth. The information, constantly evolving decision tress and estimates of utilities and probabilities, is used to make diagnostic and therapeutic medical decisions. The new information, obtainable only by study of large populations, will have to be obtained on Earth. To keep the medical care on the spacecraft current, the information will have to be transmitted to the decision-making machinery. The physicians aboard, especially those in the second generation in space, will not have the opportunity for specialty training and will never have experience with a large group of patients to develop clinical intuition, the art of medicine. They must, perforce, be technicians, not experienced thinkers. Clinical thinking and decision making must be performed for them. The refinement of data, the establishment of computer algorithms

Mean interval to conception

1477

and flow charts, and the development of decisionmaking programs will eventually allow man in that era to carry up-to-date medical practice to essentially any point in space. The sciences that will make this possible, including decision analysis, are in their infancy. The step described by the authors, in my view, is a moment, certainly not more than that, but just as certainly a necessary bridge to the next moment. DR. MERLE ROBBOY, Newport Beach, California. Dr. Lamb, is the effect of confounding variables, conditions like endometriosis, tubal disease, and oligospermia, additive or exponential in the calculation of the value of the mean interval to conception? DR. SIMON HENDERSON, San Francisco, California. While thinking about my question on this very erudite paper, I felt rather like Pooh Bear, a bear with very little brain. This bear asks whether the literature is adequate to give us the necessary basic data. We know that research in infertility is often very unscientific and usually uncontrolled. I would therefore ask Dr. Lamb whether these most important inputs for the equations are not still unknown rather than the firm data he seems to suggest. DR. LAMB (Closing). Dr. Kirk, I believe that we are quite close to applying decision analysis in the everyday clinical practice of obstetrics and gynecology. Medical decision analysis is much more applicable to our clinical practices today than it is to the science fiction practice of which Dr. Sandberg spoke. Listening to the papers preceding mine today, I envisioned decision trees with choice nodes representing several questions of evident current interest to clinicians: Should screening laparoscopy be done for patients with chronic pelvic pain? Should gonadotropin-releasing hormone analog replace danocrine in the treatment of endometriosis? Should patients seen in private offices be screened for cervical chlamydia? These questions could be fruitfully analyzed by formal decision analysis and, indeed, some closely related questions have been. Today, however, decision analysis is more widely used to address questions in the settings of epidemiology and public policy than it is in the setting of the practitioner'S office. However, this is not always the case. At Tufts, where this work was initiated, the medical decision-making unit provides a consultation service patterned after consultation services in other subspecialties, which use technology in the solution of clinical problems for individual patients. Decision technology will become more widely known in medicine because of its inclusion in undergraduate medical curricula and in continuing medical education programs. Desktop computers and software to handle the calculations needed for formal decision analysis are now easily available. I think that decision analysis soon will be used on a fairly wide basis. The principles on which formal decision analysis is based form a basic science for clinical medicine. Just as we all now apply in our practices principles we learned in courses in anatomy, physiology, and biochemistry, doctors in the near future will apply the rules of probability and Bayesian analysis to the care of their patients.

1478 Lamb, Hagen, and Pauker

Dr. Hanson, I know of no way to use previously published data in a model in which there are a variable hazard rate and also a not-at-risk group. To use the Cox regression techniques, 1 one needs access to more data than journals usually publish. I hope to explore the use of Cox's methods in decision analysis in the future but this will necessitate using my own data base. The questions will be limited to ones involving those variables previously tabulated and other investigators will not have easy access to the same data. It is for these reasons, primarily, that we chose the model reported. The editors of journals could facilitate formal analysis of clinical questions relating to fertility by encouraging publication of full tabular data in addition to graphs of life tables. Graphs are certainly easier to understand but do not provide the information needed for the

June 1989 Am J Obstet Gynecol

type of subsequent analysis that we report in this article. Dr. Robboy, I think that a technique such as the Cox multiple regression analysis will be required to deal with questions involving more than one disorder that reduces fecund ability. Multiple diseases increase the likelihood of the event death but decrease the likelihood of the event pregnancy. Therefore the techniques used by Beck et at. to account for the reduction of life expectancy as a result of several diseases by the addition of exponents cannot be translated directly for use in the analysis of problems of infertility. REFERENCE 1. Cox DR. Regression models and life tables.

[B]1972;34:187.

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