Why after 50 years of effective contraception do we still have unintended pregnancy? A European perspective

Abstract

Unintended pregnancy is a public health concern throughout Europe. There is no common definition and no standard way to measure unintended pregnancy. Identifying unintended births is difficult and prevalence estimates vary depending on how and when the question is asked. Abortion rates are not a proxy and are themselves notoriously inaccurate. An estimated 34% (in Western Europe) to 54% (in Eastern Europe) of pregnancies are unintended. The determinants of unintended pregnancy are the length of the reproductive span and exposure to the risk of conception; the desired number of children and contraceptive use and effectiveness. The age of sexual debut fell during the 20th century in Europe to between 15 and 18 years of age. Mean age at first birth for women is now over 30 years in most European countries and most couples want no more than two children. Thus most couples must use contraception perfectly for many years in order to avoid unintended pregnancy. Use of effective contraception is high throughout most of Europe but there is scope, through better provision of sexual health services, better formal sex education and better training of providers, to increase the uptake of the most effective contraceptives and improve use of all methods. For individual women unintended pregnancy can be a disaster and recourse to induced abortion should be freely available.

unintended pregnancy, abortion, reproduction, oral contraceptives, intrauterine devices, fertility, population

Introduction

Rates of contraceptive use throughout Europe are high, and in 2015, 90% of women in Europe who wished to avoid childbearing were using contraception. (United Nations, 2016). It is often asked why unintended pregnancy rates remain high despite the availability of effective methods of contraception. The oral contraceptive pill, still the mainstay of effective contraception, has been widely available in Europe for more than 50 years. The failure rate of presently available hormonal contraceptives and intrauterine devices is <1% per year if methods are used perfectly (Trussell, 2011) but unintended pregnancy remains a public health concern in most European countries.

Unplanned pregnancies can have a negative effect on women and their children. Many end in abortion—still a major cause of maternal mortality in countries with restrictive abortion laws. In addition unintended births are associated with an increased risk of obstetric complications; women present later for antenatal care and babies are more likely to be of low birthweight (Mohllajee et al., 2007). Children from unplanned pregnancies often have poorer mental and physical health, do less well in cognitive tests and are more likely to end up living in families in which marital relationships have broken down (Gipson et al., 2008; Carson et al., 2013).

A key objective of public health policy globally is therefore to reduce the number of unintended pregnancies. This article explores why, after more than 50 years of the availability of effective contraception, unintended pregnancy is still common in Europe.

Methods

Searches on relevant topics were performed in Medline, Popline, EMBASE, Cochrane library and the Social Sciences Citation Index databases for relevant English language publications from 1970 to 2016. Summaries were discussed by ESHRE (ESHRE Capri Workshop Group, 2017).

Results

Definition and measurement of unintended pregnancy

Not all pregnancies ending in childbirth are planned, and not all those that end in abortion are unwanted at the time they are conceived. Measurement of unintended pregnancies is therefore complicated. There is no single, validated measure or indeed definition, in common use. Many studies ask simply whether the pregnancy was planned or unplanned, intended or unintended, wanted or unwanted, often conflating these terms. The most widely used source data on unintended pregnancy in low and middle-income countries, the Demographic and Health Surveys (DHS)—used in 85 such countries since 1984—asks whether recent births, and the current pregnancy were wanted at the time, later or not at all.

Some studies categorize pregnancies occurring at a younger age, or those ending in abortion, as unintended. But some teenage pregnancies are planned, and even a few pregnancies that end in abortion were planned initially (Lakha and Glasier, 2006; Wellings et al., 2013). Furthermore, abortion estimates in countries where abortion is illegal or severely restricted are extremely unreliable. Even in high-income countries with liberal abortion laws, abortion is systematically under-reported in interviews and self-completed surveys. Retrospective questions about desired family size, timing of pregnancy or ‘wanted-ness’ are complicated by the fact that desired family size is to some extent likely to be a rationalization of actual family size, and parents are understandably reluctant to report retrospectively their existing children as unwanted or mistimed. Prospective studies have shown that a large proportion of births to women who initially reported wanting no more children, were subsequently classified by their mothers as wanted or mistimed. Likewise, births that occur after contraceptive failure are often reported as wanted (Curtis, et al., 2011; Ali et al., 2012).

Recognizing the limitations of existing measures, and concluding that the complexity of processes involved in making reproductive decisions is not well captured by single-item measures, The London Measure of Unintended Pregnancy (LMUP) asks six questions about contraceptive use, preparation for a healthy pregnancy (e.g. stopping smoking, taking folate supplements), discussions with a partner, intentions just before conception, ‘wanting a baby’ (as opposed to wanting to be pregnant) and the timing of becoming a mother (Barrett et al., 2004). Responses recorded as positive, negative or ambivalent give a total score classifying intendedness. The LMUP was developed for use in the UK and has been adapted and used in other countries but a measure which has validity across different cultural contexts is still needed.

Rates and prevalence of unintended pregnancy globally and in Europe

Considering induced abortion as a proxy measure of unintended pregnancy (although a poor proxy, as discussed earlier), abortion rates decreased throughout Europe from 52 (90% Uncertainty Interval (UI): 48–64)/1000 women of reproductive age (WRA) in 1990–1994 to 30 (90% UI: 27–38)/1000 WRA in 2010–2014, with most of the fall being due to a significant decrease in abortions in Eastern Europe after the end of non-market economies in that region and following improved access to contraception (Sedgh et al., 2016). In developing countries abortion rates fell only slightly from 39/1000 WRA (37–47) in 1990–1994 to 37/1000 WRA (34–46) in 2010–2014 (Sedgh et al., 2016).

Combining estimates of induced abortion with retrospective survey data on mistimed and unwanted births (with allowances for miscarriages), Sedgh and Hussain (2014) estimated that globally 40% of all pregnancies in 2012 were unintended. In Europe the proportion estimated to be unintended ranged from 34% in Western Europe to 54% in Eastern Europe. More precise data on the proportion of unintended pregnancies and the annual rates are available from some individual European countries. These data, illustrate not only real differences in the prevalence of unintended pregnancy but differences due to the way intendedness was measured. Using data from three national surveys undertaken in France among over 18 500 women of reproductive age in 2000, 2005 and 2010, trends in unintended pregnancy rates in relation to population shifts in contraceptive method use were calculated (Moreau et al., 2014a). Annual unintended pregnancy rates in France rose from 3.16 to 3.49% between 2000 and 2005, declining to 3.26% in 2010. In the 2010 survey Moreau et al. (2014b) randomly assigned participants to answer whether they had ‘planned’ their pregnancy or whether they had ‘wanted’ their pregnancy. The proportion of pregnancies that were ‘unplanned’, was 34% while the proportion of ‘unwanted’ pregnancies was 27%. In the Third National survey of Sexual Attitudes and Lifestyles among a general household population in the UK, the LMUP was used to explore the proportion and prevalence of unintended pregnancy (Wellings et al., 2013). Among 5686 women aged 16–44 years, 9.7% had been pregnant, with known outcome, in the year before being interviewed. Of these, the proportion of pregnancies reported as unplanned or planned were 16.2% (95% CI: 13.1–19.9) unplanned, 29.0% (25.2–33.2) ambivalent and 54.8% (50.3–59.2) planned, giving an annual prevalence estimate for unplanned pregnancy of 1.5% (1.2–1.9). Women in this survey, however, were likely to have under-reported abortions.

Reporting of pregnancy intention also depends on when the question is asked. The LMUP was used in a survey of 2908 Scottish women attending a hospital for antenatal care (at 12–16 weeks gestation) and 907 attending for abortion in 2004–2005 (Lakha and Glasier, 2006). Overall, 89.7% pregnancies among women requesting abortion were unintended, while 10% were classified as ambivalent with <1% intended at the time of conception. Among 2908 women who planned to continue pregnancy 8.6% of pregnancies were unintended, 65.6% were intended and 25.8% of women were ambivalent about their pregnancy intention. In a small study undertaken in Flanders in 2015, the LMUP was used among 517 women recruited from six maternity units (Goossens et al., 2016). The majority of the pregnancies (83%) were planned, 15% were ambivalent and 2% unplanned. The authors commented that over-representation of women of high socioeconomic status is likely to have under-estimated the rate of unintended pregnancies, which was certainly lower than in the Scottish women at similar gestations.

Finally, a DHS undertaken in Albania in 2008/09 reported that 88% of births were wanted then, 9% were mistimed (wanted later), and 4% were not wanted, while 17% of births were reported to have been unwanted at the time of conception among women who already had four or more children.

The impact of contraception on fertility and unintended pregnancy

In 1978, a theoretical model developed a common metric for estimating the fertility-reducing effect of the four major direct ‘proximate’ determinants: non-marriage (assuming the women who were not married were not sexually active), postpartum infecundability, abortion and contraception (Bongaarts, 1978). In the absence of any restraint on births from these four factors, the hypothetical maximal total fertility, or total fecundity (TF), per woman was estimated to be 15.3 births. Applying the ‘proximate determinants method’ to 156 countries with data on birth or fertility rates from around 1999, Liu et al. (2008) showed that for all regions, except Africa, births averted by contraception far exceeded actual births by wide margins (Table I). An estimated 6 million births occurred in Europe in 1999, however, 15 million births were estimated to have been averted (or postponed) by the use of contraception. In contrast in Africa only 12 million births were estimated to have been averted by contraception for a total of around 30 million births that had actually occurred.

Table I

Number of births to women in a union (married or consensual) (circa 1999) and estimated number averted by contraception for the main world regions using proximate determinants methods (source: Liu et al., 2008).

Region (no. of countries)	Observed births (millions)	Births averted (millions)
Africa (49)	30	12
Asia (46)	79	174
Europe (28)	6	15
Latin America (28)	10	26
World (156)	129	242

The ‘proximate determinants’ model was later modified to include sexually active unmarried women; to adjust for overlaps between contraceptive use and postpartum infecundability, and to allow for contraceptive use-effectiveness to vary by age (Bongaarts, 2015). Selected results for four high fertility and four low fertility countries show the progressively larger impact of contraception as fertility falls (Table II). In Niger, for example, contraception reduces TF by only 6% while postpartum infecundability, due to prolonged breastfeeding, reduces it by 41%. At the other extreme, in low fertility Colombia, the fertility-reducing impact of contraception, at 60%, is much larger than impact of the other determinants. On average, a 17 percentage point increase in contraceptive use reduces the total fertility rate by one birth.

Table II

Proportionate reductions in total fertility rate (from the theoretical maximum total fecundity due to contraception, abortion, sexual inactivity and postpartum infecundability, in selected high and lower fertility non-European countries) (source: Bongaarts, 2015).

	TFR	Percentage reduction in TFR due to:
		Contraception	Abortion	Sex inactivity	PP Infecundity
Country (date)
Niger (2012)	7.6	6	4	12	41
Mali (2006)	6.6	6	5	13	38
Chad (2004)	6.3	2	6	19	44
Uganda (2011)	6.2	21	8	26	36
Peru (2000)	2.8	46	13	47	36
India (2005–2006)	2.7	42	13	38	30
Nepal (2011)	2.6	39	12	33	34
Colombia (2010)	2.1	60	18	35	36

The effect of the reductions on the theoretical maximum total fecundity (15.3) can be estimated—according to Bongaart’s model—as the product of the complement to 100 of each percentage reduction, e.g. from Colombia 15.3 × 0.40 × 0.82 × 0.65 × 0.64 = 2.1. The effects of abortion are not included in the table.

TFR, total fertility rate; PP, postpartum.

Use of the more effective methods of contraception has a bigger effect on reducing unintended pregnancies than use of less effective, traditional, methods. There is a strong negative association between abortion and the use of modern effective methods of contraception (Westoff, 2008).

The proximate determinants in Europe

The determinants of unintended pregnancy are the length of the reproductive span (i.e. first sex to the onset of permanent sterility) and exposure to the risk of conception; the desired number of children and contraceptive use and effectiveness.

The length of the reproductive life span and exposure to risk

The length of the reproductive life span is determined by age at menarche and the age of permanent sterility, which for most women occurs well before the menopause. Age of sexual debut (and therefore of first exposure to the risk of pregnancy) has fallen with time but the decline had slowed recently. In France, for example, median age at first sexual intercourse fell for both men and women between 1970 and 2010 (Bajos et al., 2010). For women the median age fell by almost 2 full years during the 1940s, from 22 to just over 20, stabilized in the 1950s, and resumed its fall in the 1960 and 1970s. In 2005, the median age of first intercourse in France was 17.6 years. A similar trend was observed in the UK where the age at sexual debut for women fell rapidly through the 1930s to the 1950s from 21 to 19 years, to age 18 in the mid-1950s, age 17 in the late 1960s (Wellings et al., 1994) and by the mid-1990s median age of first sex was 16 years (Wellings et al., 2001) where it remains (Wellings et al., 2013). In some European countries sexual debut is even earlier. In an international survey of 15-year-old girls undertaken in 2009/10 throughout Europe (Reis et al., 2012), 66% of girls in Greenland, and over 40% of girls in Romania, Greece, Ukraine, Bulgaria and the Russian Federation had sexual intercourse.

A more striking change with time has been seen in the age of first childbearing. In the early 1960s women had their first child in their mid-20s in most European regions and in their early 20s in Eastern Europe (Olah, 2015). By 2012 the mean age at first birth for women throughout Europe had risen significantly to reach over 30 years in most European countries (OECD, 2016).

If a woman becomes sexually active at 16 years of age, but does not have her first child until she is 30 she faces 14 years in which she needs to avoid pregnancy.

Desired family size

Fertility continues until the onset of permanent sterility so not only is the average age of first birth important but also the desired family size. Around 50% of women in Europe give ‘two children’ as their ideal family size (Breton and Prioux, 2009). However, the proportion of women remaining childless has increased, sometimes up to 25–30%, despite the fact that <5% of couples report ‘zero’ as an ideal number of children (Testa, 2012; OECD, 2016). In Europe each child takes ~2 years out of exposure to risk of unintended pregnancy—6 months to conceive, 2 months to allow for miscarriage (to occur in up to 20% of pregnancies), 9 months gestation, and a few months of lactational infertility (in Europe as a whole only 13% of women are exclusively breastfeeding). So if most women have only two children, many have only one and a not negligible proportion have none, the duration of exposure to risk is long and the use of effective contraception remains a vital determinant of unintended pregnancy for many years.

Contraceptive use and use effectiveness

Unintended pregnancy is preventable by contraception and contraceptive use is high in Europe. In Europe as a whole only 10% of couples had an unmet need for family planning (women who want to stop or delay childbearing but are not using any method of contraception) while in Northern Europe the figure was only 7% (United Nations, 2016).Unmet need for contraception in France in 2010 was only 2.2%. In a national survey of Swedish women unmet need was 8.9% (Kopp Kallner et al., 2015).

A range of highly effective methods (hormonal contraceptives pills, injections and implants, intrauterine methods and sterilization) is widely available in Europe and all have failure rates during the first year of use of 1% or less when used perfectly (Trussell, 2011). The effectiveness of a contraceptive depends on its mode of action, how easy it is to use and how easy it is to stop using it. The so-called long-acting reversible contraceptives (LARC—implants and intrauterine methods) are independent of adherence and continuation rates are much higher than those associated with all other reversible methods which can be stopped at any time the user wishes. Until very recently, the near universal finding that couples using LARC had significantly higher continuation rates and lower unintended pregnancy rates came from observational studies (Cameron et al., 2012; Peipert et al., 2012). However, a recent partially randomized trial of LARC versus short-acting reversible contraceptives showed that women who had initially asked for a less effective contraceptive method but who were randomized to receive an implant or an intrauterine device had higher contraceptive continuation rates and fewer unintended pregnancies than women randomized to receive the method that they had initially sought (Hubacher et al., 2017).

While the contraceptive methods used vary in Europe, use of effective contraception is generally high. In 2005, an estimated 58% of women in Eastern Europe, 71% in Southern Europe and 95% in Western and Northern Europe were using a modern method of contraception (Frejka et al., 2008). In France over the last 4 decades the proportion of sexually active women not wanting to conceive who rely on effective contraception has regularly increased, from 35% of women aged 18–44 years in 1978, to 52% in 1988, 62% in 2000 and was 60% in 2010. Among contraceptive users, 52% of women in France used user-dependent hormonal methods (almost always the pill), 24% used LARC and 4% relied on sterilization, while 20% were using less effective methods (barrier methods, withdrawal or periodic abstinence) (Moreau et al., 2014a). In a national survey of Swedish women (Kopp Kallner et al., 2015), 24.3% of women were using implantable or intrauterine contraception.

What can be done to reduce unintended pregnancies in Europe?

It is unlikely that the age of onset of first sexual intercourse will increase, that age at first childbirth will decrease or that desired family size will change much in years to come in Europe. The only realistic option for reducing unintended pregnancy further is to increase contraceptive prevalence, increase use of the most effective LARC methods, improve correct and consistent use of the less effective methods, and provide easy access to effective back-up methods (emergency contraceptives) for use when people make mistakes. Contraceptive prevalence is already high in Europe and it may be difficult to increase it by much. However, non-use contributes significantly to abortion rates. In a French study of 1525 teenagers undergoing abortion, while two-thirds of pregnancies occurred as a result of contraceptive misuse or failure, one-third followed use of no method (Moreau et al., 2012). In a Spanish survey of 2475 women attending for abortion 36% of women had used no contraceptive at the time of conception (Serrano et al., 2012), while in a similar study from Italy almost 20% of 1782 women were using no method of contraception (Cagnacci et al., 2014).

The reasons why some couples use no contraception are complex and they vary between different European countries, between couples and within couples depending on the stage of their reproductive lives. The World Health Organization lists among the barriers to contraceptive use, limited access to good quality services and to method choice; experience, or fear, of side effects; cultural or religious opposition, user’s and provider’s biases and gender-based barriers (World Health Organization 2017). Each of these barriers pertains in some parts of Europe. Access to services and to a range of methods was extremely limited in Eastern Europe during the communist era and improvements take time and commitment from governments (Jacobson, 1990; Vjatere, 1995). Condom use is still high in many eastern European countries; for example, 40% of couples in Serbia were using condoms in 2014 (United Nations, 2016) likely reflecting a limited choice of methods and services. Religious beliefs deter some couples from using effective contraception. When policy makers hold strong religious beliefs, deliberate limiting of access to contraception, particularly emergency contraception, often results (Goettig, 2018). Lack of knowledge of services, or of particular methods, is common among vulnerable groups, including immigrants who tend to have lower rates of contraceptive use and higher rates of abortion at least in the first few months after arrival in Europe (Moreau et al., 2005; Goosen et al., 2009; Omland et al., 2014). Fear of risks and side effects often leads to method discontinuation (Lindh et al., 2016) and some women, particularly young women, simply do not like the idea of using contraception or do not think about using it, especially when sex is unplanned (Brown and Guthrie, 2010).

Changes to the infrastructure would make a difference to contraceptive use. Governmental interventions to improving contraceptive services, particularly in Eastern Europe, and introducing formal sex education into primary and secondary schools should result in some improvements in patterns of contraceptive use. Access to contraceptive services and methods would likely improve if religious influences were moderated. However, unintended pregnancy rates remain high in largely protestant (and increasingly secular) countries, such as Scandinavia and the UK, where reproductive health services are universal and of good quality, where there is formal sex education in schools, and where emergency contraception is available without prescription and, in the case of the UK, contraception is free of charge (ESHRE Capri Workshop Group, 2015; Hognert et al., 2017). Given human fallibility and the disparity between intention and action, contraceptive use will always be less than perfect and unintended pregnancies will persist even in countries with the best possible sex education and services.

Improving the use of more effective methods of contraception may be more easily achieved than increasing overall use. Recent years have seen a drive to increase the uptake of LARC, and in the UK this may have contributed to the recent sustained decline in abortion rates (Morgan and Liu, 2017). Unintended pregnancy can, however, be prevented with less effective contraceptives. In Albania low fertility rates (1.6 in 2008/09) have been achieved largely through use of traditional methods of contraception: only 11% of married women use modern methods while 58% rely on withdrawal (Institute of Statistics, Institute of Public Health [Albania] and ICF Macro, 2010). Correct and consistent use of contraception is notoriously hard to achieve. Many women forget to take oral contraceptive pills and couples who use condoms do not use them at every episode of intercourse. Most interventions designed to improve correct and consistent use of these methods have, however, proved disappointing (Glasier and Shields, 2005).

The prospect of developing new contraceptive methods that more women would actually want to use, and want to use correctly and consistently, is often used by researchers in support of funding for contraceptive development. However, it takes years to develop new contraceptives and interest from funding agencies and from pharmaceutical companies has dwindled in the last 10 or 15 years as public health research priorities have switched to other health topics such as obesity and diseases of an ageing population (ESHRE Capri Workshop Group, 2012).

Emergency contraceptives are useful for individual women but have not yet been shown to be effective at the public health level, probably because most women who could use them after unprotected sex to prevent a pregnancy fail to do so (Polis et al., 2007). The currently available oral emergency contraceptives work mainly by preventing ovulation (Glasier et al., 2014). Over the counter access to new methods of emergency contraception deliberately designed to inhibit implantation, and thus be effective for a longer period of time after intercourse, would give women a better chance of preventing pregnancy after unprotected sex.

Better training of providers to educate women and men about the risks of pregnancy and help them to choose the most effective method best suited to their individual circumstances is also required

Conclusions

Unintended pregnancies are highest in societies where small families are desired, because of the increase in the duration of exposure to the risk. If a couple want only one or two children they face years of needing to use contraception perfectly, and that is not easy. The incidence of unintended pregnancy can be reduced by raising contraceptive prevalence but in Europe the scope for this is limited. Understanding the reasons for patterns of contraceptive use among populations and individuals is complex and interventions can prove disappointing. In many social contexts prevailing social norms restrict women’s agency in controlling their own lives; religious beliefs may influence the extent to which contraception is encouraged; provision of sex education is often controversial and governments may lack both the resources and the will to provide an adequate infrastructure to reproductive health provision. Given patterns of family formation, sexual behaviour and contraceptive use in Europe, and because of ambivalence about the big decisions of life where there are always pluses and minuses, there will always be unintended pregnancies. The public health consequences of unintended pregnancy are much greater in parts of the world where maternal mortality and morbidity are high, and although there are some measurable consequences for both the mother and the child when unintended pregnancy results in birth, at a public health level in Europe these too may be minor. However, for individual women unintended pregnancy can be a disaster and, however, distasteful to some, recourse to induced abortion should be freely available. For women who do not want to continue an unintended pregnancy abortion should be available as early in gestation as possible. A pill which women could use within 1 or 2 days of missing a menstrual period is perfectly feasible (Li et al., 2015) and should be further developed.

Acknowledgements

The secretarial assistance of Mrs Simonetta Vassallo is gratefully acknowledged.

Authors’ roles

All lecturers and Chairmen contributed to the preparation of the final article.

Funding

The meeting was organized by the European Society of Human Reproduction and Embryology with an unrestricted educational grant from Institut Biochimique S.A. (Switzerland).

Conflict of interest

None.

APPENDIX

A meeting was organized by ESHRE [29–30 August 2017] to discuss why after 50 years of modern contraception do we still have unintended pregnancy? The lecturers included: D.T. Baird (Centre for Reproductive Biology, University of Edinburgh, UK), N. Bajos (INSERM, Paris, France), J. Cleland (Department of Population Health, London School of Hygiene & Tropical Medicine, London, UK), A. Glasier (Simpson Centre for Reproductive Health, University of Edinburgh, Edinburgh, UK), C. La Vecchia (Department of Clinical Sciences and Community Health, Università degli Studi di Milano, Milan, Italy), H. Leridon (Directeur de recherche émérite, INED, / French Institute for Demographic Studies, Paris cedex, France), I. Milsom (Department of Obstetrics and Gynaecologist, Sahlgrenska Academy at the University of Gothenburg, Gothenburg, Sweden), K. Wellings (London School of Hygiene & Tropical Medicine, London, UK). The chairs included: G. Benagiano (Gynaecology, Obstetrics and Urology, Sapienza University of Rome, Policlinico Umberto I, Roma, Italy), S. Bhattacharya (Professor of Reproductive Medicine, Head of Division of Applied Health Sciences and Director Institute of Applied Health Sciences, School of Medicine and Dentistry, University of Aberdeen, Aberdeen Maternity Hospital, Foresterhill, Aberdeen, UK), P.G. Crosignani (IRCCS Ca’ Granda Foundation, Maggiore Policlinico Hospital, Milano, Italy), J.L.H. Evers, (Maastricht University and Academisch ziekenhuis Maastricht, Dept. Obstetrics & Gynaecology, Maastricht, The Netherlands), E. Negri (Department of Biomedical and Clinical Sciences, Università degli Studi di Milano, Milano, Italy), A. Volpe (Dipartimento Integrato Materno Infantile, Università di Modena, Modena, Italy).

Author notes

†

The list of the ESHRE Capri Workshop Group contributors is given in the Appendix.