Estimation of Probability of Coition on Different Days of a Menstrual Cycle near the Day of Ovulation: An Application of Theory of Markov Chain

Transcription

1 Demograhy India (0) ISSN: 0970-X Vol., Issue: &, : 3-39 Research Article stimation o Probability o Coition on Dierent Days o a Menstrual Cycle near the Day o Ovulation: An Alication o Theory o Marov Chain R. C. Yadava, Shruti Verma,* & Kaushalendra K. Singh Abstract Various research studies have evidences that the day o coition with resect to the day o ovulation in a menstrual cycle may aect the robability o concetion as well as the sex ratio at birth. But the data on coition robabilities on dierent days o the menstrual cycle is rarely available and in this situation theoretical estimation may be helul or estimating these robabilities. This research wor resents a new theoretical aroach or estimating the robabilities o coition near the day o ovulation or dierent coule coital atterns. These robabilities are then utilized or estimating the robability o concetion in a menstrual cycle and the sex ratio at birth. Introduction In humans, normally there are 6 days in one menstrual cycle during which coition can result in to a concetion (the days beore ovulation and the day o ovulation itsel) and woman becomes regnant (Wilcox et al., 99; Dunson et al., 999; Wilcox et al., 00). This 6 day window is reerred to as the most ertile eriod in one menstrual cycle. Here the term `ovulation' reers to the hase o a emale's menstrual cycle in which a artially mature ovum is released rom the ovarian ollicles into the oviduct or the alloian tube. Ater ovulation, during the luteal hase the egg is available to be ertilized by serm. In humans ovulation normally occurs somewhere between the 0th and 9th day o a menstrual cycle. Many researchers have studied about the robability o concetion on dierent days or hases o menstrual cycle (Barrett and Marshall, 969; Royston, 98; Royston, 99; Dunson et al., 999; Dunson et al., 00; Stanord et al., 00). Some studies indicate that the highest robability o clinically evident concetion occurs with coition or days beore ovulation, rather than the day o ovulation itsel (Barrett and Marshall, 969; Dunson et al., 999; Dunson et al., 00; Stanord et al., 00). On the other hand, Wilcox et al. (99) conjectured about highest robability o concetion on the day o ovulation In addition to above, it is argued in many studies that the sex o the child is aected by the day o coition in relation to the day o ovulation (Guerrero, 97; James, 976; Harla, 979; France et al., 98; Perez et al., 98; France et al., 99; James, 997; James, 999; James, 000). Guerrero (97) ound that both in natural and artiicial insemination, sex ratio changes signiicantly during the menstrual interval. He obtained a more or less U-shaed distribution o the robability that the sex o child will be male or coition taing lace on dierent days in relation to ovulation. James (976; 997; 999; 000) also ound the existence o correlation between time o ertilization within menstrual cycle and the sex ratio o osring. Harla (979) and Perez et al. (98) in their studies have observed the variations in sex ratio or inseminations occurring on dierent days o the menstrual cycle. France et al. (98; 99) have ound male reerring birth sex ratio when coition receded ovulation by or more days. Although the association ound by France et al. (98; 99) is statistically signiicant, they are setical about the observed association since number o regnancies Proessor, Deartment o Statistics, Faculty o Science, Banaras Hindu University, Varanasi SRF, Deartment o Statistics, Faculty o Science, Banaras Hindu University, Varanasi *Corresonding author: Shruti Verma, mail: shrutiverma.bhu@gmail.com

2 3 involved in their data is too small. On the other hand, Wilcox et al. (99) and Gray et al. (998) have argued in their studies that timing o insemination does not aect the sex ratio that resent contradiction to studies mentioned earlier. This suggests that the attern o coition during a menstrual cycle may aect the sex ratio at birth as well as chance o concetion in a menstrual cycle. Thus it seems desirable to examine this attern. Some studies have tried to ind the robability o coition on dierent days o the menstrual cycle relative to ovulation and also the average number o coitions in a wee (Harvey, 987; Wilcox et al., 00). However, all these studies are based on rosective data on coition on dierent days relative to ovulation in a menstrual cycle and contain a small samle o women who agreed to become an object or the study. In act, such data or general oulations are not available in literature esecially due to diiculty in determining the exact day o ovulation o a emale in the surveyed oulation. The objective o the resent study is to mae an attemt towards theoretically estimating the robability o coition on dierent days (esecially near the day o ovulation) o a menstrual cycle under dierent assumed atterns o coition during a menstrual cycle. For this urose the theory o Marov chain has been utilized. These estimated robabilities are then used to ind the robability o concetion in one menstrual cycle and sex ratio at birth or various atterns o coition in a menstrual cycle. Brie Overview o the Theory o Marov Chains In the theory o Marov chains, we consider the simlest generalization o indeendent trials which consists in ermitting the outcome o any trial to deend uon the outcome o the directly receding trial. The ossible out- comes o a trial are usually reerred to as the ossible states o the Marov chain. Here it is assumed that the trials are erormed at a uniorm rate so that the number o the trial serves as the time arameter. The outcome ( ) in any trial is no longer associated with a ixed robability ( ) but to every air o the states ( j ; ) o the two successive trials (say n th and (n + ) st trials), there corresonds a conditional robability. Here is called the robability o transition rom the state j at nth trial to state at (n+) st trial. Hence, given that j has occurred at some trial, the robability o occurrence o at the next trial is. In more general case, m-ste transition robability, i.e., the robability o transition rom state j at n th trial to the state at (n + m) th trial is denoted by. In act, Thus = ( m) m j () is also called the one-ste transition robability. Transition robabilities satisy, (m) 0, j Arranging these robabilities in the matrix orm gives the `transition robability matrix' (t..m.) o the Marov chain. For convenience the states j, etc. can be reresented as j, and so on. Now suose that a system starts with the state j. Let be the robability that it reaches the state or the irst time at the n th (n) ste and is the robability that it reaches the state at the n th ste (not necessarily or the irst time). Also let F be the robability that starting with state j, the system will ever reach, i.e., F n When F =, it is certain that starting with state j, system will reach state and in this case is a roer robability distribution and this gives the irst assage time distribution or given that system starts with j. So the mean irst assage time rom the state j to the state is given by, (n)

3 33 n n. In articular when =j, { (n), n=,, } reresents the distribution o the recurrence time o ( ) j and n F n. will imly that the return to state j is certain. In this case, n n is nown as the `mean recurrence time' or the state j. In act or studying any Marov chain and its roerties, the seciication o its states, the matrix o transition robabilities (P) and the robability distribution {s } are required to be seciied, where s is the robability that the system starts with the state at the initial trial. Now an attemt has been made to aly this theory o Marov chains to comute the robability o coition on dierent days o a menstrual cycle esecially near the day o ovulation. Here each day o the menstrual cycle is considered as a trial and each trial has two ossible outcomes: either a coition occurs on that day or coition does not occur on that day. For the urose, it is required to seciy the states and transition robability matrix. Consider that reresents the state that the last coition too lace days ( = 0,,, ) beore the seciied day. Obviously 0 reresents the state that the coition taes lace on the seciied day, reresents the state that the last coition too lace one day beore the seciied day and so on. Ater the seciication o states, it is required to seciy the transition robability matrix. In this context, it is imortant to note that on any day there are only two ossibilities: either a coition will occur or not. Thus i the system is in state on a articular day, then on the next day it will either be in state 0 (i the coition taes lace on that day) or will be in the state + (i the coition does not tae lace on that day). So rom any state, the transitions are ossible only to either 0 or + and transition robabilities to all other states will be zero. Let s j stands or the robability o the state j (j = 0,,, ) at the initial trial. The unconditional robability o entering at the n th ste is then, s j In many situations it is observed that the inluence o the initial state gradually wears o, so that or large n, the distribution in the above equation becomes nearly indeendent o the initial distribution {s j }. This is the case i converges to a limit indeendent o j, that is i the n th ower (n) n. o the transition robability matrix (P n ) converges to a matrix with identical rows. s j Method and Alication Let us assume that the initial trial starts with the day on which the irst coition occurs ater the start o menstrual cycle o woman. It is already mentioned in the revious section that reresents the state that the last coition occurred days beore the considered day, = 0,,, In the context o alication, the maximum ossible value o is to be seciied. The maximum ossible value o imlies that the number o days between two consecutive coitions cannot exceed this maximum ossible value o. Obviously this value o will deend uon the attern o coition during a menstrual cycle but or alication urose we tae the maximum ossible value o to be imlying that ater the day o a coition the next coition has to occur during the next days. This may nearly be true esecially in the case o younger emales (less than 30 years o age) in most o the oulations. O course larger values o may also be taen or emales o higher ages.

4 3 Further, the transition robabilities in the transition robability matrix P will heavily deend uon the attern o coition, but it may reasonably be assumed that 0 increases as increases (It has already been argued earlier that only 0 and ;+ can be non- zero). For illustration, such a matrix P satisying the above roerties is considered, which is given below, P It is nown that normally the ovulation taes lace between 0 th to 9 th days o menstrual cycle and so, on an average th day o the menstrual cycle can be considered to be the day o ovulation. I it is assumed that irst coition in a menstrual cycle occurs 3- days ater the start o menstruation, then the most robable day o ovulation may be considered to be around th day rom the initial day o irst coition in the menstrual cycle i.e. rom initial trial in the theory o Marov chain. The sexual behavior near this day may be obtained by taing the th ower o matrix P. The comutation o 0 th, th and th owers o the matrix P is done and the obtained matrices are given below, From the matrices 0, and it is seen that the rows in these matrices are almost identical indicating that the eect o initial trial has weared o. Consequently, it can be saely assumed that the robabilities o being in the states 0,,, near the day o ovulation have become stable. Thus the robabilities in the rows o matrix reresent that ater days rom the day o irst coition o the menstrual cycle, around 3% coules will be in state 0, showing that 3 ercent emales will have coition on the day o ovulation (i th day is considered as the day o ovulation), around 8% coules will be in state i.e. there will be 8% coules whose last coition has taen lace one day beore the day o ovulation, around 0% will be in state and so on. Thus i the th day ater the initiation o irst coition in the cycle is considered to be the day o ovulation, then this matrix ( ) rovides the robability distribution o attern o coition near the day o ovulation. Further, since 0 th and th owers o P are also almost identical to th ower o P, hence even i the ovulation taes lace on 0 th or th day rom the day o irst coition, the results will be the same as discussed above. Now rom the matrix P, the distribution o irst assage time rom the state 0 to the state 0, { 00,n=,, }, can be obtained as ollows: Starting with the state 0, the robability o transition to the state 0 in one ste will be given by the irst element in the irst cell o P, robability o transition to the state 0 in two stes will be the roduct o transition robabilities corresonding to the transitions 0 and 0. Similarly robability o transition to the state 0 in 3 stes will be the roduct o transition robabilities or the transitions 0 and and 0. Similarly other robabilities can be ound. Thus the robability distribution o the irst assage time (recurrence time) or state 0 is obtained which is shown in Table. The robability distribution obtained in Table is nothing but the robability distribution o transition rom state 0 to 0 or the irst time in n stes. Here it can be observed that robability that a coule will be in state 0 ater two-stes is maximum (0.) i.e. or % coules last coition taes lace two days beore the considered day (here the th day ater the initiation o irst coition in a menstrual cycle or the day o ovulation).

5 3 P 0 P P n Now rom this robability distribution the mean recurrence time n can be calculated and it is obtained as.86 days. It reresents that on an average coules will tae around.86 days to go or the next coition. Thereore, mean number o coitions er wee will be (7/.86)=.. Similarly, some other hyothetical transition robability matrices P, P 3, P and P are considered to see the changes in mean recurrence times or various hyothetical transition atterns. Considered matrices are as ollows: P P P P

6 36 It is to be mentioned here that in all the matrices mentioned above robability o coition on a articular day is increasing as the duration rom the revious coition increases (see the irst column o the matrices). Again it is observed that 0 th and higher owers o the matrices P, P 3, P and P comes out to be the matrices with identical rows. Table shows the identical rows o the hyothesized matrices P, P 3, P and P. The rows in the Table are nothing but the robability distribution that coules will be in state 0,,, 3, and ater 0- days o initial oint or at the day o ovulation. Now rom the matrices P, P 3, P and P, the distribution o irst assage time or the state 0 and mean recurrence times are obtained which are shown in the Table 3. All the matrices P, P, P 3, P and P are hyothetical but looing at the calculated values o mean number o coitions er wee among coules ollowing the assumed attern o coition, it can be observed that values o mean number o coitions er wee are in the range o. to.7 which seem to be close to reality. Many revious studies have observed mean number o coitions er wee near this range. Wilcox et al. (00) have reorted the mean requency o intercourse er day as 0.9 which is equivalent to almost twice a wee. Kinsey (93) ound that among white USA women, teenage brides had intercourse on an average 3.7 times a wee which droed slightly to.6 by thirty,. at orty and. at ity years o age. Anthroologist Moni Nag (97) has comared data on coital requency in Muslim and Hindu grous rom three West Bengal villages with the indings o Kinsey or white married women in USA. Findings reveal that Hindu women o age grou 0-9 have on an average.9 coitions er wee while Sheih and Non- Sheih Muslim women o the same age grou have around. and.6 coitions er wee resectively (For details see (Potts and Selman, 979,. 7)). Yadava and Rai (989) have also reorted similar indings. Thus, the considered atterns o coition seem to be somewhat close to reality. Possible Imact o Coital Pattern on Sex Ratio at Birth The revious section was ocused towards inding the robability distribution o day o last coition near the day o ovulation in a menstrual cycle utilizing the theory o Marov chains. This was mainly done to examine the ossible imact o sexual attern on sex ratio at birth as well as chance o concetion in a menstrual cycle. It has already been mentioned that various researchers have argued that the chance o concetion as well as the sex o the child are aected by the day o coition in relation to the day o ovulation but the results are not conclusive. However, whatever be these conclusions, i the robabilities o concetion due to coition occurring on dierent days near the day o ovulation are nown, the robability o concetion in a menstrual cycle can be comuted easily under dierent assumed atterns o coition (transition robability matrices P, P, P 3 etc.). Now let a is the robability that the system will be in state on the day o ovulation and b is the robability o concetion rom coition occurring on day - ( = 0,,, 3,, ), then robability o concetion in one menstrual cycle would be, PC. () Table resents the estimate o robability o concetion in one menstrual cycle under assumed coital atterns. Here it is assumed that the robabilities o concetion rom last coition on days 0, -, -, -3, -, - (the day o ovulation taen as `0') i.e. b are aroximately 0.3, 0.3, 0., 0., 0., 0. resectively (Wilcox et al., 99). But it is observed in Wilcox et al. (99) that around 0 ercent o the concetions result in regnancy loss. In this case, under the coital atterns o matrices P, P, P 3, P and P the robability o birth in one menstrual cycle will be around 0.3. A detailed discussion on regnancy loss is also given in Potts and Selman (979),.9-. Now i c is the robability o a male birth or concetion occurring due to coition on day - ( = 0,,, 3,, ), then sex ratio at birth will be,

7 abc SRB PC () Guerrero (97) has obtained robabilities o male birth or concetions occurring due to coition on days 0, -, -, -3, -, - i.e. c as 0.3, 0., 0., 0., 0.60, 0.68 resectively. In addition, we have also calculated the robabilities o male birth or concetions occurring on days 0, -, -, -3, - i.e. c as 0.7, 0.9, 0.60, 0.8, 0.8 resectively using the igure in Wilcox et al. (99),(Note that in Wilcox et al. (99) robabilities c or = 3 and are based on very small number o women, so we have clubbed the two grou o observations and considered the value o c 3 and c as 0.8). Table shows the obtained sex ratio at birth or assumed coital atterns. Discussion The resent aer resents a new aroach or theoretical estimation o the robabilities o coition on dierent days o a menstrual cycle o a woman which may be helul in estimating the robability o concetion in a menstrual cycle as well as the sex ratio at birth in the oulation. Although the issue o robability o concetion on dierent days near the day o ovulation as well as its imact on sex o the baby are yet not ully resolved, still i these are nown, the resent aer would be helul in throwing new lights on these asects. Acnowledgement Shruti Verma is thanul to University Grants Commission (U.G.C.) or roviding inancial assistance during this research. 37 Reerences Barrett, J. C. & Marshall, J. (969) The ris o concetion on dierent days o the menstrual cycle, Poulation Studies, 3(3),. -6. Dunson, D. B., Baird, D. D., Wilcox, A. J., & Weinberg, C. R. (999) Day- seciic robabilities o clinical regnancy based on two studies with imerect measures o ovulation, Human Reroduction, (7), Dunson, D. B.,Weinberg, C. R., Baird, D. D., Kesner, J. S., & Wilcox, A. J. (00) Assessing human ertility using several marers o ovulation, Stat Med, 0(6), France, J. T., Graham, F. M., Gosling, L., Hair, P., & Knox, B. S. (99) Characteristics o natural concetual cycles occurring in a rosective study o sex reselection: ertility awareness symtoms, hormone levels, serm survival, and regnancy outcome, International Journal o Fertility, 37(),. -. France, J. T., Graham, F. M., Hair, P., & Gosling, L. (98) A rosective study o the reselection o the sex o osring by timing intercourse relative to ovulation, Fertility and Sterility, (6), Guerrero, R. (97) Association o the tye and time o insemination within the menstrual cycle with the human sex ratio at birth, The New ngland Journal o Medicine. Harla, S. (979) Gender o inants conceived on dierent days o the menstrual cycle, The New ngland Journal o Medicine, 300,. -8. Harvey, S. M. (987) Female sexual behavior: luctuations during the menstrual cycle, J Psychosom Res, 3(), James, W. H. (976) Timing o ertilization and sex ratio o osring: a review, Annals o Human Biology, 3(6), James, W. H. (997) Sex ratio, coital rate, hormones and time o ertilization within the cycle, Annals o Human Biology, (), James, W. H. (999) The status o the hyothesis that the human sex ratio at birth is associated with the cycle day o concetion, Human Reroduction, (8), James, W. H. (000) Analysing data on the sex ratio o human births by cycle day o concetion, Annals o Human Biology, (),

8 Kinsey, A., Pomeroy, W., Martin, C., & Gebhard, P. (93) Sexual behaviour in the human emale, W. B. Saunders and Co., Philadelhia. Nag, M. (97) Sex, culture and human ertility: India and the United States, Current Anthroology, 3(3). Perez, A., ger, R., Domenichini, V., Kambic, R., & Gray, R. (98) Sex ratio associated with natural amily lanning, Fertility and Sterility, 3(),. -3. Potts, M. & Selman, P. (979) Society and ertility, Macdonald and vans Ltd. Royston, J. P. (98) Basal body temerature, ovulation and the ris o concetion, with secial reerence to the lietimes o serm and egg., Biometrics, 38(), Royston, P. (99) Identiying the ertile hase o the human menstrual cycle, Stat Med, 0(),. -0. Stanord, J. B., White, G. L., & Hatasaa, H. (00) Timing intercourse to achieve regnancy : current evidence, Obstetrics and Gynecology, 00(6). Wilcox, A. J., Baird, D. D., Dunson, D. B., McConnaughey, D. R., Kesner, J. S., & Weinberg, C. R. (00) Day- seciic robabilities o clinical regnancy based on two studies with imerect measures o ovulation, Human Reroduction, 9(7), Wilcox, A. J., Weinberg, C. R., & Baird, D. D. (99) Timing o sexual intercourse in relation to ovulation: eects on the robability o concetion, survival o the regnancy, and sex o the baby, The New ngland Journal o Medicine, 333(3),. 7-. Yadava, R. C. & Rai, M. K. (989) Some results on requency o sexual intercourse among urban coules, Journal o Family Welare, 3(6), Table : Distribution o the First Passage Time or the State 0 n 00 * *Distribution o irst assage time rom state 0 to state 0 Table : ntries o the identical rows or the 0 th and higher owers o the matrices P, P 3, P and P P P P P Table 3: Distributions o the First Passage Time and Mean Recurrence Times or the State 0 or the Transition Probability Matrices P, P 3, P and P 00 * or n P P 3 P P Mean Recurrence Time Mean number o coitions er wee *Distribution o irst assage time rom state 0 to state 0

9 Table : Probabilities o Concetion in One Menstrual Cycle under Assumed Coital Patterns PC* *Probability o concetion in one menstrual cycle 39 Table : Sex ratio at birth under assumed coital atterns a SRB G b SRB W P P P P P a SRB calculated or c o Guerrero (97), b SRB calculated or c o Wilcox et al. (99)