Population forecasts for Bangladesh, using a Bayesian
methodology.




Abstract: 
Population projection for many developing countries could be quite
a challenging task for the demographers mostly due to lack of
availability of enough reliable data. The objective of this paper is to
present an overview of the existing methods for population forecasting
and to propose an alternative based on the Bayesian statistics,
combining the formality of inference. The analysis has been made using
Markov Chain Monte Carlo (MCMC) technique for Bayesian methodology
available with the software WinBUGS. Convergence diagnostic techniques
available with the WinBUGS software have been applied to ensure the
convergence of the chains necessary for the implementation of MCMC. The
Bayesian approach allows for the use of observed data and expert
judgements by means of appropriate priors, and a more realistic
population forecasts, along with associated uncertainty, has been
possible. Key words: Cohort component Method, Monte Carlo error; Gompertz model; Highest posterior density; Logistic model; Markov Chain Monte Carlo; Nonlinear regression model; Population projection; WinBUGS 


Article Type:  Report 
Subject: 
Markov processes
(Forecasts and trends) Monte Carlo method (Forecasts and trends) 
Authors: 
Mahsin, Md. Hossain, Syed Shahadat 
Pub Date:  12/01/2012 
Publication:  Name: Journal of Health Population and Nutrition Publisher: International Centre for Diarrhoeal Disease Research Bangladesh Audience: Academic Format: Magazine/Journal Subject: Health Copyright: COPYRIGHT 2012 International Centre for Diarrhoeal Disease Research Bangladesh ISSN: 16060997 
Issue:  Date: Dec, 2012 Source Volume: 30 Source Issue: 4 
Topic:  Event Code: 010 Forecasts, trends, outlooks Computer Subject: Market trend/market analysis 
Accession Number:  313345728 
Full Text: 
INTRODUCTION A widelyused method of forecasting the age and sexspecific population for future years, in which the initial population is stratified by age and sex and projections, is generated by application of survival ratios and birth rates, followed by an additive adjustment for net migration. To get this information, the behaviour of the related variables is analyzed based on the past data by statisticians, and then inferences are drawn from the analysis to make forecasts of the desired variable. At present, there exist two major paradigms in statistics, namely conventional (frequentist) and Bayesian statistics for the purpose of data analysis. Use of Bayesian methodology in the field of data analysis is comparatively new and has found massive support in the last two decades from the experts belonging to various disciplines. Probably, the main reason behind the increasing support is its flexibility and generality that allows it to deal with the complex situations. Besides, Bayesian method is typically preferred over classical approach in parameter estimation because of the intractable form of the likelihood function (1). There are a number of methodologies used for population projections. One of the most popular methods is cohort component method which is based on the estimates about the future levels of fertility, mortality, sex composition, migration, and other parameters. Many studies have examined the relative performance of simple mathematical models, extrapolation based on timeseries and cohortcomponent models of population forecasting. Most have found that constant growth mathematical models or standard timeseries models of population growth are as least accurate as cohor component models (24). The present study is not intended to assess the relative accuracy of various projection models. Rather, it only aims to investigate the usefulness of cohort component method in making the population projection for Bangladesh, using Bayesian approach. Bayesian analysis has been applied in cohort component model for providing a neat and transparent way of estimation. It provides probabilistic point estimates of the parameters, along with the highest posterior density interval (HPD) or Bayesian credible interval. Bayesian credible interval is a measure of uncertainty, and it is based on statistical theory and data on error distributions that provide an explicit estimate of the probability that a given range will contain the future population. This approach develops statistical prediction intervals to accompany population forecasts (57). Prediction intervals will provide extremely valuable information to datausers and will improve the quality of decisionmaking, based on population forecasts. LITERATURE REVIEW P(t+n)=P(t)+BirthsDeaths+ImmigrantsEmigrants where, t is the starting point of time; n is the projection interval; P(t) is the populationsize at time t; and P(t+n) is the population size at time t+n. If we put immigrants and emigrants together, then we get: P(t+n)=P(t)+BirthsDeaths+Net Migrants where, Net Migrants=ImmigrantEmigrants. A population grows through the addition of births and immigrants and declines through the subtraction of deaths and outmigrants. The term 'fertility' refers to the ability of an individual to give a livebirth (or births). This is equally applicable to a group or an entire population. Agespecific fertility rates are required to project the number of births in future fertility projections, which are made by projecting the course of TFR over time and translating this total fertility rate into agespecific fertility rates. In general, the projection of TFR is divided into assumptions regarding a level at which fertility eventually becomes constant in a country or a region and the path taken from current to eventual levels. Once fertility reaches its eventual level, the population will reach a stable agestructure and constant growth rate assuming that mortality and migration rates are also fixed. If the eventual fertility level is at replacement level and net migration is zero, the growth rate will eventually be zero. Both projected pace of fertility decline and the assumed eventual fertility level are important for determining trends in populationsize and agestructure. The lower the assumed eventual fertility level, the more important the pace of fertility decline becomes to projected populationsize (17). Births in cohort component models are typically projected by applying projected agespecific birth rates to projections of the female population by age. In this approach, the size and age composition of the female population of childbearing ages have a major impact on the projected number of births. Since most mothers for the first 25 years of the projection period are already alive at the time the projection is made, the size and age composition of the female population are the most predictable elements in shortterm fertility projections. Timeseries techniques have been used for projecting births or birth rates. Several authors have applied timeseries methods by themselves, using autoregressive integrated moving average (ARIMA) methods to forecast total births (1820). While these efforts yielded some insights into the use of timeseries methods on fertility, the forecasts ignored the advantage of using cohort component methods (21). This omission was partially remedied by Lee (2223) who applied timeseries methods to TFR, the sum of all agespecific rates that occur in a given year. In our study, we have applied the Gompertz model, using Bayesian methodology to TFR. The representation of mortality data via a parametric model has attracted the attention of actuaries, demographers, and statisticians for over a century. One of the most common models is that of logistic curve (13). In this paper, we adopt a Bayesian analysis to this curve, using MCMC technique to produce the posterior summaries required. For other Bayesian work relating to mortality smoothing and lifetable construction (2425), Carlin (26) used MCMC methods but not in a parametric curve modelling context. MATERIALS AND METHODS Table 1 provides TFR in Bangladesh from 1991 to 2001, which have been used for a fertility model fit to making future fertility projections. Using these data and the Gompertz growth model, a WinBUGS program has been developed to make a Bayesian analysis of the data and to provide projections of the TFR of Bangladesh. In this paper, we follow the timeseries tradition in developing a method to forecast TFR and then convert it to the agespecific fertility rates on the basis of baseyear agespecific fertility rates. Multiplying these forecasts by forecasts of the size of the agespecific female population would then yield fertility forecasts derived from both timeseries and demographic cohort component traditions. In this way, the advantages of the demographic tradition in taking account of the predictability of the size and age composition of the female population can be combined with the more statisticallyrigorous timeseries techniques of modelling the shortterm variability of the agespecific fertility rates. Let [Y.sub.i] to denote TFR in Bangladesh in the year [t.sub.i] (i=1, 2,..., 11) where i refers to successive censuses starting from 1991, for which i=1 and the data are given in Table 1. The most famous growth model is that of Gompertz (27) and is used for TFR where TFR [Y.sub.i] in the year [t.sub.i] has been assumed to follow normal distribution with respective means [h.sub.i] and common precision [tau]. Noninformative priors have been assigned to all the parameters of the model. The nonlinear regression model for TFR is described as: [Y.sub.i]=[h.sub.i]+[e.sub.i] where [h.sub.i] is the deterministic part, and [e.sub.i] is the disturbance part; assuming the disturbance to be [e.sub.i]~iid N (0, [tau]), where [tau] is the precision (=1/ variance), the fertility model and the noninformative priors might be defined as: [h.sub.i]=d+[ce.sup.[e.sup.(a[bt.sub.i])]] [Y.sup.i]~N ([h.sub.i], 1/[tau] a~N (0,0.01) b~N (0,0.1) c~N (0,0.1) d~N (0,0.1) [tau]~Ga (0.0001, 0.0001) where d is the lower asymptote, c is the upper asymptote, b is the rate at which the fertility increases, and a is the parameter that determines the shape of the Gompertz curve. For Bayesian analysis, we need to provide prior distributions to all the parameters a, b, c, d and [tau]. A massive discussion on the choice of priors is also available in the BUGS manual (28). Mortality projections are based on projecting future lifeexpectancy at birth for males and females, defined as the average lifespan of a child born today if current agespecific mortality levels were held fixed in the future. In developing countries where mortality remains high, future lifeexpectancy will be determined by the effciency of local health services, the spread of traditional (e.g. malaria) and new (e.g. AIDS) diseases, and the general standards of living and education. In this paper, we avoid the new epidemics (AIDS). The lifeexpectancy at birth (average number of years lived by a newborn baby if he/she follows the current agespecific mortality patterns) is projected on the basis of the past experience of increase in the lifeexpectancy at birth. A logistic curve has been fitted using trends in lifeexpectancy at birth, and it assumes that increase in lifeexpectancy at birth follows an Sshaped curve. The logic behind using logistic curve is that when the lifeexpectancy at birth is very low, the increase is expected to be slow due to poor health facilities. Once the health facilities are provided and with improvement in Socioeconomic conditions, the lifeexpectancy increases at a faster rate. At the higher level of lifeexpectancy, the rate of increase is slow, and it would stabilize at the biological maximum. To project the population from one year to the next, survival rates by age and sex are needed and, to obtain future survival rates, future life tables may be constructed. Model life tables developed by United Nations (29), Coale and Demeny (30), Regional life tables, and South Asian model life tables, whichever is applicable for Bangladesh, should be used. In this study, South Asian model life table has been used. Let [Q.sub.ij] be the lifeexpectancy at birth for males and females of Bangladesh in the year [t.sub.i] (i=1, 2,..., 21) where i represents time and j sex. The data were collected from office of the Bangladesh Bureau of Statistics [Sample Vital Registration System (SVRS): 2002, 2003, 200506, BBS], and a logistic growth model is used. In this model, the lifeexpectancy at birth [Q.sub.ij] in the year [t.sub.i] has been assumed to follow normal distribution with respective means [p.sub.ij] and common precision [[tau].sub.j]. Noninformative priors have been assigned to all the parameters of the model. The nonlinear regression model for the population growth is described as: [Q.sub.ij]=[p.sub.ij]+[[epsilon].sub.ij] where [p.sub.ij] is the deterministic part, and [[epsilon].sub.i] is the random error part; assuming the error to be [[epsilon].sub.ij]~N (0, [[tau].sub.j]) where [[tau].sub.j] is precision, the mortality model and the noninformative priors are: [P.sub.ij] [q.sub.1j] / 1 + [e.sup.q2j] [e.sub.q3j[t.sub.i]] + [q.sub.4j] / [Q.sub.ij]~N ([p.sub.ij], 1 / [[tau].sub.j]); J=1, 2 [Q.sub.1j]~N (0,0.01) [q.sub.2j]~N (0,0.01) [q.sub.3j]~N (0,0.01) [q.sub.4j]~N (0,0.01) [[tau].sub.j]~Ga (0.0001, 0.0001) where [q.sub.i] is the upper asymptote, q4 is the lower asymptote, [q.sub.2] and [q.sub.3] are the other parameters that define the shape of the logistic curve, and e is the base of the natural logarithm. Future international migration is more difficult to project than fertility or mortality. Migration can be volatile since shortterm changes in economic, social, or political factors often play an important role. In addition, projections are generally based on past trends and current policies since no single, compelling theory of migration exists; however, data on historical migration are sparse for Bangladesh. In this work, we assumed that the population is closed, i.e. no migration takes place, or even if it does, net effect is zero. As for the sex ratio at births which divide the future number of newborns into male and female, the female to male ratio is set at 100:105 based on the results of the last five years, and it remains consistent from 2001 onward. Diagnostics Bayesian approach faces serious computational difficulties due to likely involvement of complicated mathematical expressions in the posterior distributions. Many of these have been suitably addressed with greater ease, using MCMC methods. These methods enable us to carry out analysis on a wide range of Bayesian statistical models. More details with examples of the MCMC implementation in Bayesian inference can be found elsewhere (3135). As an iterative tool, the MCMC methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution (33). MCMC tool has been used in the WinBUGS to obtain the posterior distribution of the unknown parameters in the model. In the process, we need to run a number of chains for each parameter for a long time. When the chains have run sufficiently large number of iterations and have reached the stationary distribution, the samples obtained by further running of the chains are supposed to be drawn randomly from the posterior distribution of the parameter. WinBUGS provides a number of inbuilt diagnostics to assess the convergence of chains. For a more formal approach to convergence diagnosis, the software also provides an implementation of the techniques described in Brooks and Gelman (34), and a facility for outputting monitored samples in a format that is compatible with the CODA software (36). In practice, WinBUGS allows multiple chains for each parameter to run simultaneously. Running multiple chains is a way to check the convergence of MCMC simulations. Two chains have been set in the model of this problem. When the different chains do not provide sufficient mixing of chains even after a long run, it will be an evidence of lack of convergence of the chains. Once we are convinced that chains have been converged through the diagnostics, we will need to run the simulation for a further number of iterations to obtain samples that can be used for posterior inference. The more samples we save, the more accurate will be our posterior estimates. Once we have run enough updates and are satisfied with the history of the chains, we discard the earlier samples. We obtain the summary statistics only from the samples generated afterwards. RESULTS The summary statistics of the estimated parameters of the fertility model after 10,000 initial updates were discarded and 80,000 updates were run after the initial burnin is presented in Table 2. During these updates, none of the diagnostics indicated any symptom of nonconvergence of the chains. The number of iterations required to run after the convergence of the chains is assessed on the basis of Monte Carlo error (MC error) for each parameter. MC error is an estimate of the difference between the mean of the sampled values (which we are using as our estimate of the posterior mean for each parameter) and the true posterior mean. It has been suggested by the WinBUGS manual as a rule of thumb that the simulation should be run until the MC error for each parameter of interest is less than about 5% of the sample standard deviation, and this was followed in our analysis. From Table 2, it is obvious that MC errors for each parameter were less than 5% of the sample standard deviation. [FIGURE 1 OMITTED] Figure 1 illustrates a graphical presentation of the fitting of the Gompertz model. The graph shows that the model provides a close fit (the closeness of the smoothed line representing the estimated values and the dots showing the observed values) to the observed data. Dotted blue lines provide 95% HPD (highest posterior density) region. Table 3 presents the summary statistics of estimated parameters of the mortality model on lifeexpectancy at birth for both males and females after discarding 10,000 initial updates and 70,000 updates were run after the initial burnin. During these updates, none of the diagnostics indicated a symptom of nonconvergence of the chains. While running our model with the WinBUGS, we have monitored five nodes [q.sub.1], [q.sub.2], [q.sub.3], [q.sub.4], and [tau]. The graphical presentation of the models fitted and forecasted to both males and females are depicted in Figure 2 and 3. The graphs show that the model provides a close fit (the closeness of the smoothed line representing the estimated values and the dots showing the observed values) to the observed data. Dotted blue lines provide 95% HPD (highest posterior density) region. The graph of the projections approaches to Sshape, indicating the stabilization of the lifeexpectancy at birth for males as well as females. [FIGURE 2 OMITTED] [FIGURE 3 OMITTED] DISCUSSION The final calculations of cohort component method combine the results from the mortality, migration, and fertility modules. On the basis of the future forecasts of population growth components, the forecasted population of Bangladesh from 2006 to 2051 has been presented in Appendix. The present study was an attempt to show the application and suitability of the MCMC tool in Bayesian data analysis for fitting population data and making projection of the future population, using cohort component model. The use of Bayesian approach in fitting the components of growth models allows for further extensions over classical estimation methods, leading to a more realistic forecasts and associated uncertainty measures. The cohort component population projection method follows the process of demographic change and is viewed as a more reliable projection method than those that primarily rely on census data or information that reflect population change. In this paper, we had been presenting the basics of the Implementation of the Bayesian data analysis with an illustration of the population projection. We have not performed the sensitivity analysis taking different prior distributions mainly because the selected priors were noninformative. These priors did not provide substantial information to the posterior distribution. However, they were necessary for the implementation of the Bayesian data analysis. Limitations In this study, we are unable to provide future forecasts for the component of migration because of sparse data for Bangladesh. To overcome this problem, we have used a strong assumption, and this is the major drawback of our study. Apart from this shortcoming, the total fertility rate has declined to replacement level in 2010 and afterwards, which is unrealistic for Bangladesh but it is evident from Figure 2 and 3 that the mortality component has fitted very well. In both fertility and mortality models, we have applied noninformative priors, and it is also a limitation of this study. We hope to further explore these areas in future, using Bayesian methods motivated by the augments provided throughout this paper. Conclusions Utilizing Bayesian methods to the growth components, a more realistic summary in population forecasts has been produced because it allows formal incorporation of expert judgement embodied in priors and, hence, alter the forecasted population characteristics and their levels of uncertainty. In this paper, we have applied noninformative priors to fertility and mortality models and, thus, a large level of uncertainty in the forecasted population is resulted. This level of uncertainty could be reduced through the inclusion of informative priors. Moreover, informative priors based purely on expert opinions regarding the future of population growth rates could have been included. Such prior information would result in further reductions in the estimated uncertainty due to added information in the parameter estimation and modelchoice procedures. 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Mahsin Lecturer Applied Statistics University of Dhaka Dhaka 1000 Bangladesh Email: mahsin@isrt.ac.bd Appendix. Age and sexstructure of the projected population (in thousands), 20062051 Age (years) Sex 2001 2006 2011 (base) All ages Persons 123,851 133,436 143,515 Males 63,895 68,699 73,700 Females 59,956 64,737 69,815 05 Males 8,362 6,711 6,722 Females 7,724 6,326 6,361 >510 Males 8,822 8,189 6,623 Females 7,956 7,533 6,234 >1015 Males 8,421 8,779 8,163 Females 7,432 7,914 7,510 >1520 Males 6,292 8,391 8,759 Females 5,672 7,404 7,897 >2025 Males 4,859 6,265 8,368 Females 6,057 5,644 7,384 >2530 Males 4,895 4,834 6,243 Females 5,865 6,023 5,626 >3035 Males 4,313 4,863 4,812 Females 4,436 5,825 5,999 >3540 Males 4,204 4,276 4,835 Females 3,795 4,397 5,794 >4045 Males 3,426 4,149 4,235 Females 2,774 3,749 4,362 >4550 Males 2,610 3,356 4,085 Females 1,991 2,727 3,705 >5055 Males 2,175 2,521 3,266 Females 1,826 1,937 2,673 >5560 Males 1,309 2,055 2,408 Females 1,047 1,746 1,872 >6065 Males 1,529 1,194 1,902 Females 1,299 971 1,646 >6570 Males 814 1,320 1,052 Females 629 1,148 880 >7075 Males 926 650 1,086 Females 699 515 974 >7580 Males 358 663 484 Females 258 511 396 >80 Males 580 483 657 Females 496 367 502 Age (years) Sex 2016 2021 2026 All ages Persons 154,212 164,899 174,494 Males 78,957 84,267 88,942 Females 75,255 80,632 85,552 05 Males 7,158 7,408 7,173 Females 6,805 7,008 6,817 >510 Males 6,646 7,088 7,335 Females 6,290 6,728 6,940 >1015 Males 6,605 6,629 7,070 Females 6,220 6,276 6,715 >1520 Males 8,145 6,593 6,617 Females 7,498 6,210 6,267 >2025 Males 8,737 8,127 6,578 Females 7,880 7,482 6,199 >2530 Males 8,340 8,711 8,103 Females 7,366 7,861 7,466 >3035 Males 6,218 8,310 8,679 Females 5,609 7,343 7,839 >3540 Males 4,786 6,188 8,269 Females 5,973 5,585 7,315 >4045 Males 4,793 4,749 6,140 Females 5,756 5,935 5,552 >4550 Males 4,175 4,731 4,688 Females 4,318 5,698 5,880 >5055 Males 3,984 4,079 4,622 Females 3,642 4,244 5,608 >5560 Males 3,128 3,825 3,916 Females 2,595 3,534 4,128 >6065 Males 2,238 2,917 3,567 Females 1,777 2,464 3,366 >6570 Males 1,687 1,995 2,601 Females 1,505 1,625 2,263 >7075 Males 872 1,409 1,666 Females 757 1,296 1,409 >7580 Males 816 662 1,070 Females 764 593 1,028 >80 Males 629 846 848 Females 500 750 760 Age (years) Sex 2031 2036 2041 All ages Persons 182,384 188,577 193,432 Males 92,739 95,687 97,989 Females 89,645 92,890 95,443 05 Males 6,717 6,386 6,272 Females 6,383 6,068 5,960 >510 Males 7,102 6,651 6,323 Females 6,750 6,322 6,009 >1015 Males 7,318 7,084 6,634 Females 6,926 6,736 6,308 >1520 Males 7,057 7,304 7,072 Females 6,705 6,916 6,727 >2025 Males 6,602 7,041 7,287 Females 6,256 6,693 6,905 >2530 Males 6,559 6,583 7,020 Females 6,185 6,242 6,679 >3035 Males 8,074 6,535 6,559 Females 7,445 6,168 6,224 >3540 Males 8,637 8,034 6,503 Females 7,809 7,416 6,145 >4045 Males 8,206 8,570 7,973 Females 7,273 7,764 7,373 >4550 Males 6,060 8,099 8,459 Females 5,502 7,206 7,692 >5055 Males 4,580 5,921 7,913 Females 5,787 5,415 7,093 >5560 Males 4,437 4,397 5,685 Females 5,455 5,629 5,267 >6065 Males 3,652 4,138 4,100 Females 3,931 5,195 5,360 >6570 Males 3,180 3,257 3,690 Females 3,093 3,612 4,773 >7075 Males 2,173 2,657 2,720 Females 1,963 2,682 3,133 >7580 Males 1,266 1,650 2,018 Females 1,118 1,557 2,128 >80 Males 1,119 1,380 1,761 Females 1,064 1,269 1,667 Age (years) Sex 2046 2051 All ages Persons 196,933 198,964 Males 99,671 100,682 Females 97,262 98,282 05 Males 6,210 6,099 Females 5,902 5,796 >510 Males 6,210 6,149 Females 5,902 5,844 >1015 Males 6,307 6,194 Females 5,997 5,890 >1520 Males 6,622 6,296 Females 6,299 5,989 >2025 Males 7,056 6,608 Females 6,715 6,288 >2530 Males 7,266 7,035 Females 6,888 6,700 >3035 Males 6,995 7,239 Females 6,660 6,869 >3540 Males 6,527 6,961 Females 6,201 6,634 >4045 Males 6,453 6,477 Females 6,109 6,165 >4550 Males 7,869 6,369 Females 7,306 6,053 >5055 Males 8,265 7,688 Females 7,571 7,191 >5560 Males 7,597 7,935 Females 6,898 7,364 >6065 Males 5,301 7,084 Females 5,015 6,569 >6570 Males 3,656 4,727 Females 4,926 4,609 >7075 Males 3,082 3,054 Females 4,140 4,272 >7580 Males 2,066 2,341 Females 2,486 3,285 >80 Males 2,189 2,426 Females 2,247 2,765 Table 1. Total fertility rate in Bangladesh from 1991 to 2001 Year (ti) 1991 1992 1993 1994 1995 1996 TFR (Yi) 4.24 4.18 3.84 3.58 3.45 3.41 Year (ti) 1997 1998 1999 2000 2001 TFR (Yi) 3.1 2.98 2.64 2.59 2.56 Source: Statistical Pocket Book; 20012007, Bangladesh Bureau of Statistics (BBS) Table 2. Summary statistics of the node of fertility model Node Mean SD MC error a 3.625 2.095 0.0831 b 0.1878 0.07114 0.00247 c 3.488 1.23 0.04591 d 5.058 0.7275 0.02837 [sigma] 0.1011 0.03013 3.61E04 HPD region Node 2.50% Median 97.50% a 1.13 3.9 7.496 b 0.08302 0.1769 0.3508 c 6.537 3.226 1.857 d 4.168 4.871 6.952 [sigma] 0.06131 0.09513 0.1768 Table 3. Summary statistics of the node for lifeexpectancy at birth for both males and females Node Mean SD MC error [q.sub.1] Male 16.35 4.579 0.1705 Female 18.21 4.442 0.1735 [q.sub.2] Male 4.873 .9117 0.03781 Female 5.308 0.6038 0.02583 [q.sub.3] Male 0.2561 .06799 .002848 Female 0.268 0.0471 0.00204 [q.sub.4] Male 54.8 .5047 0.01926 Female 54.46 0.3151 0.01172 [sigma] Male 0.6012 .1136 .001698 Female 0.5063 0.09371 0.00126 HPD region Node 0.025 Median 0.975 [q.sub.1] Male 9.589 15.65 27.05 Female 11.59 17.56 29.1 [q.sub.2] Male 3.459 4.725 7.185 Female 4.27 5.252 6.593 [q.sub.3] Male 0.1583 0.2424 0.4299 Female 0.1887 0.2622 0.372 [q.sub.4] Male 53.68 54.85 55.61 Female 53.77 54.48 55.0 [sigma] Male 0.4279 0.5849 0.8674 Female 0.3617 0.4934 0.7248 
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