Abstract
Unlike a character measured at a finite set of landmark points, functionvalued traits are those that change as a function of some independent and continuous variable. These traits, also called infinitedimensional characters, can be described as the character process and include a number of biologically, economically, or biomedically important features, such as growth trajectories, allometric scalings, and norms of reaction. Here we present a new statistical infrastructure for mapping quantitative trait loci (QTL) underlying the character process. This strategy, termed functional mapping, integrates mathematical relationships of different traits or variables within the genetic mapping framework. Logistic mapping proposed in this article can be viewed as an example of functional mapping. Logistic mapping is based on a universal biological law that for each and every living organism growth over time follows an exponential growth curve (e.g., logistic or Sshaped). A maximumlikelihood approach based on a logisticmixture model, implemented with the EM algorithm, is developed to provide the estimates of QTL positions, QTL effects, and other model parameters responsible for growth trajectories. Logistic mapping displays a tremendous potential to increase the power of QTL detection, the precision of parameter estimation, and the resolution of QTL localization due to the small number of parameters to be estimated, the pleiotropic effect of a QTL on growth, and/or residual correlations of growth at different ages. More importantly, logistic mapping allows for testing numerous biologically important hypotheses concerning the genetic basis of quantitative variation, thus gaining an insight into the critical role of development in shaping plant and animal evolution and domestication. The power of logistic mapping is demonstrated by an example of a forest tree, in which one QTL affecting stem growth processes is detected on a linkage group using our method, whereas it cannot be detected using current methods. The advantages of functional mapping are also discussed.
THE theoretical principle for analyzing quantitative trait loci (QTL) dates back to Sax (1923), who first associated pattern and pigment markers with seed size in beans. However, statistical methodologies for mapping QTL on a highdensity linkage map of molecular markers had not been well established until Lander and Botstein's (1989) pioneering work. These authors employed an expectationmaximization (EM)implemented maximumlikelihood approach, proposed by Dempster et al. (1977), to map QTL on a particular chromosomal interval bracketed by two flanking markers. This socalled interval mapping method was later improved by including markers from other intervals as covariates to control the overall genetic background (Jansen and Stam 1994; Zeng 1994). The improved method, called composite interval mapping by Zeng (1994), displays increased power in QTL detection because of reduced residual variance. Kao et al. (1999) proposed using multiple marker intervals simultaneously to map multiple QTL of epistatic interactions throughout a linkage map. Currently, an upsurge of QTL mapping methodologies has been developed to consider various situations regarding different marker types (dominant or codominant), different marker spaces (sparse or dense), different experimental designs (F_{2}/backcross or fullsib family), or different mapping populations (autogamous or allogamous). The statistical methods used for QTL mapping in the literature include regression analyses (Haley and Knott 1992; Xu 1995), maximum likelihood (Lander and Botstein 1989; Zeng 1994; Kaoet al. 1999), and the Bayesian approach (Satagopanet al. 1996; Sillanpaa and Arjas 1999; Xu and Yi 2000). Many of these mapping methods have been instrumental in the identification of QTL responsible for variation in various complex traits important to agriculture, forestry, biomedicine, or biological research (Tanksley 1993; Wuet al. 2000; Mackay 2001; Mauricio 2001).
It should be noted, however, that many quantitative traits, such as body size and body shape, are inherently too complex to be described by a single value, because their phenotypes change with age, metabolic rate, or environmental stimulus. These traits, which can be expressed as a function (or stochastic process) of some independent and continuous variable, were thought of as infinitedimensional characters by Kirkpatrick and Heckman (1989) or functionvalued traits by Pletcher and Geyer (1999). The genetic determination of the character process has long intrigued students in different disciplines of biology, genetics, and breeding (e.g., Cheverudet al. 1983; Atchley 1984; Wuet al. 1992; Atchley and Zhu 1997; Rice 1997). A simple approach for mapping infinitedimensional characters is to associate markers with phenotypes separately for different ages, traits, or environments and compare the differences of QTL expression across ages, traits, or environments (Cheverudet al. 1996; Nuzhdinet al. 1997; Verhaegenet al. 1997; Emebiriet al. 1998; Wuet al. 1999). However, these separate analyses cannot provide effective estimates of genetic control over infinitedimensional characters, because they fail to capture the information about the covariances of different traits or the same trait measured at different ages or environments. Although multitrait mapping approaches can take into account simultaneously different traits or the same trait measured at different ages or environments (Jiang and Zeng 1995; Korolet al. 1995; Roninet al. 1995; Eaveset al. 1996; Knott and Haley 2000), their applications are actually limited to bivariate, or at most trivariate, analyses. As the number of traits increases these multitrait analysis approaches will have a reduced ability to produce precise estimates of genetic parameters in quantitative genetic studies (Shaw 1987).
To circumvent the difficulty in manipulating a large number of correlated traits, new attempts were made by Mangin et al. (1998) and Korol et al. (2001), who transformed the initial trait space into a space of a lower dimension on the basis of principal component analysis or intervalspecific calculation of eigenvalues and eigenvectors of the residual covariance matrix. These new attempts have, to some extent, made the genetic mapping of a large number of traits more tractable, but they still treat infinitedimensional characters as discrete traits or eigenvalues and do not place the physiological mechanisms predisposing for the phenotypic variation of infinitedimensional characters in a mapping framework. In real life, an infinitedimensional character often changes its phenotype through particular physiological regulations or developmental signals in the same way that an organism tends to maximize its metabolic capacity and internal efficiency as a consequence of natural selection. Therefore, the incorporation of the underpinning physiological or developmental mechanisms of trait variation into a QTLmapping strategy may likely produce more accurate results in terms of biological reality.
The objective of this study is to propose a general theoretical framework for embedding biological mechanisms and processes in the statistical analysis of QTL mapping. A maximumlikelihoodbased method, implemented with the EM algorithm, is used to estimate QTL locations and effects on various biological processes. The newly developed method is applied in an example to map the growth of a forest tree. Compared with current mapping methods, our method incorporating growth trajectories tends to be more powerful and more precise in QTL detection and also has greater potential to increase mapping power, precision, and resolution by reducing residual variance and the number of unknown parameters to be estimated. In practice, our method is economically more feasible than previous methods because it needs a smaller sample size to obtain adequate power for QTL detection as a result of the use of multiple measurements for each individual. It can be anticipated that the method proposed in this article will have great implications for the design of an efficient early selection program and the interface of genetics, development, and evolution.
MODELING THE CHARACTER PROCESS
Many biological processes in real life are expected to arise as curves, such as growth curves, allometric scalings, hormone profiles, and norms of reaction. A growth curve or trajectory represents an individual as a function that relates the age of an individual to some measure of its size. Since there are an infinite number of ages, growth trajectories can be thought of as functionvalued traits (Pletcher and Geyer 1999; Jaffrezic and Pletcher 2000) or infinitedimensional characters (Kirkpatrick and Heckman 1989). Other examples of functionvalued traits include the continuous change of a morphological or physiological variable with body size (allometric scaling, Niklas 1994; West et al. 1997, 1999) and responsive phenotypes of a given genotype to a changing environment (reaction norm, Viaet al. 1995). The common property of these functionvalued traits is that they can be described as a function (or stochastic process) of some independent and continuous variable consisting of an infinite number of points, such as age, temperature, light intensity, or biological size.
The model for QTL mapping we developed relies on concepts from functional analysis and stochastic processes. Throughout, we use growth trajectories as a concrete example to illustrate the ideas, but allometric scalings, hormone profiles, and reaction norms can be treated in the same framework with appropriate modifications.
Growth trajectory: It is well known that there are biological laws underlying growth trajectories (Gould 1977; Alberchet al. 1979). A growth law can be visualized as the “force field” propelling a point through a phenotype space, tracing out the ontogenetic path. If the size of an organism is denoted by y, its ontogenetic trajectory y(t) can be generated through the differential dy/dt, which models the growth rate. Many differential functions have been established to describe the growth trajectory. Basically, they are sorted into three categories: (1) exponential, (2) saturating, and (3) sigmoidal (von Bertalanffy 1957; Niklas 1994). Each of these growth models has a common feature that the development of ontogenetic trajectory is regulated by a set of “control parameters” such as onset age of growth, offset signal for growth, growth rate during the period of growth, and initial size at the commencement of the growth period. Also, each of these growth models exhibits an initial phase of exponential growth due simply to the geometrically multiplying population of newly differentiated cells. This initial growth phase has the property that small perturbations in growth rate or onset age are amplified enormously during ontogeny. Thus, it is easy to find examples of how a small “mutation” in a growth parameter causes a series of developmental alterations that produce a phenotype qualitatively different from the normal one.
In this article, we further limit our analysis to sigmoidal, or logistic, function (Pearl 1925). The logistic curve is regarded as among the most important ones to capture the agespecific change in growth (Niklas 1994; Westet al. 2001). The logistic growth curve as a biological law can be mathematically described by
The logistic growth curve described in Equation 2 can be used to determine the coordinates of a biologically important point in the entire growth trajectory—the inflection point—where the exponential phase ends and the asymptotic phase begins (Niklas 1994). The time at the inflection point corresponds to the time point at which a maximum growth rate occurs. The time (t_{I}) and growth [g(t_{I})] at the inflection point for a QTL genotype can be derived as
STATISTICAL MODELS
Genetic design: The purpose of this article is to introduce a novel idea to QTL mapping. Hence, we suppose a simplest backcross design derived from two contrasted homozygous inbred lines. Other more complex designs, such as an F_{2} or fullsib family, can also be used. In a backcross population, there are two groups of genotypes at a locus, in which a markerbased genetic linkage map is constructed, aimed at the identification of QTL affecting an agedependent trait, such as body size or body weight. In practice, the data are observed only at a finite set of times, 1,..., m, rather than a continuum, so we have only a finite set of data on each individual i, which can be considered as a multivariate trait vector, y_{i}(1),..., y_{i}(m). This finite set of data can be modeled by a growth curve. Assume that a pleiotropic QTL of allele Q_{1} and Q_{2} affecting growth curves or trajectories is segregating in the backcross population. This QTL is bracketed by two flanking markers η and η+ 1, each with two genotypes M_{η}m_{η}, m_{η}m_{η}, and M_{η+1}m_{η+1}, m_{η+1}m_{η+1}, respectively. For a particular genotype j (j = 1 for Q_{1}Q_{2} or 2 for Q_{2}Q_{2}) of this QTL, the parameters describing its logistic curve are denoted by a_{j}, b_{j}, and r_{j}. The comparisons of these parameters between two different genotypes can determine whether and how this putative QTL affects growth trajectories.
Suppose that there are a total of N progeny in the backcross measured at each of m times. The trait phenotype of progeny i measured at time t due to the QTL located on an interval flanked by markers η and η+ 1 can be expressed by a linear statistical model (Kirkpatrick and Heckman 1989; Lander and Botstein 1989; Pletcher and Geyer 1999),
Statistical methods: The phenotypes of the trait at all time points 1,..., m for each QTL genotype group follow a multivariate normal density,
The likelihood of the backcross progeny with mdimensional measurements can be represented by a multivariate mixture model
In practical computations, the QTL position parameter θ can be viewed as a fixed parameter because a putative QTL can be searched at every 1 or 2 cM on a map interval bracketed by two markers throughout the entire linkage map. The amount of support for a QTL at a particular map position is often displayed graphically through the use of likelihood maps or profiles, which plot the likelihoodratio test statistic as a function of map position of the putative QTL.
HYPOTHESIS TESTS
After the MLEs of the parameters of interest are obtained, a number of biologically meaningful hypotheses can be tested on the basis of the logisticbased genetic model. First, the hypothesis about the existence of a QTL affecting an overall growth curve can be formulated as
Second, the hypothesis test can be performed on the time at which the detected QTL starts to exert or ceases an effect on growth trajectories, by comparing the difference of the expected means between different genotypes at various time points. At a given time t*, the hypothesis is
Third, the genotypic differences in time (t_{I}) and growth [g(t_{I})] at the inflection point of maximum growth rate (Equation 2) can be tested. The test for the genotypic difference is based on the restriction
Fourth, when there is no double “crossover” between the growth curves of the two QTL genotypes, the effect of QTL × age interaction on the overall growth curve can be tested by comparing the genotypic differences at time t = 0 and t = ∞, which is expressed by the restriction
Testing QTL × age interactions on the basis of Equations 14 and 15 can be helpful to our understanding of the way in which QTL trigger an effect on growth and development.
The test statistics for testing the hypotheses (10, 11, 12, 13, 14, 15) are calculated as the loglikelihood ratio (LR) of the full over reduced model:
EXAMPLE
The Populus map: We use an example of a forest tree to demonstrate the power of our statistical model for mapping QTL affecting growth trajectories. The study material used was derived from the triple hybridization of Populus (poplar). A Populus deltoides clone (designated I69) was used as a female parent to mate with an interspecific P. deltoides × P. nigra clone (designated I45) as a male parent (Wuet al. 1992). The hybrids between P. deltoides and P. nigra are called Euramerica poplar (P. euramericana). Both P. deltoides I69 and P. euramericana I45 were selected at the Research Institute for Poplars in Italy in the 1950s and were introduced to China in 1972. In the spring of 1988, a total of 450 1yearold rooted threeway hybrid seedlings were planted at a spacing of 4 × 5 m at a forest farm near Xuchou City, Jiangsu Province, China. The total stem heights and diameters measured at the end of each of 11 growing seasons are used in this example.
A genetic linkage map has been constructed using 90 genotypes randomly selected from the 450 hybrids with random amplified polymorphic DNAs (RAPDs), amplified fragment length polymorphisms (AFLPs), and intersimple sequence repeats (ISSRs; Yinet al. 2002). This map comprises the 19 largest linkage groups for each parental map, which represent roughly 19 pairs of chromosomes. We chose linkage group 10 from the P. deltoides parent map to detect QTL affecting diameter growth using our newly developed method.
Logistic curves: By plotting total growth against year, it is observed that each of the 90 mapped genotypes follows the Sshaped (logistic) growth curve. Figure 1 illustrates Sshaped growth curves for individual stem diameters over 11 years. A leastsquares approach was used to fit diameter growth with the logistic curve (Equation 1) for each genotype. On the basis of statistical tests, all genotypes can be well fit by a logistic curve (r^{2} > 0.95). Also, different curve shapes of these genotypes imply possible genetic control over growth trajectories. The statistical model built upon the logistic growth curve model is used to map QTL responsible for growth trajectories in diameters.
QTL detection: Using our logistic mapping model, one QTL is detected on linkage group 10 for the growth trajectory of stem diameter in the interspecific hybrids of poplar (Figure 2). The critical value for claiming the existence of QTL can be determined on the basis of the Bonferroni argument for the sparsemap case (Lander and Botstein 1989) or by permutation tests proposed by Doerge and Churchill (1996). In this example, the chromosomewide empirical estimate of the critical value is obtained from 1000 permutation tests. It is found that the critical values for declaring the existence of a QTL on the linkage group under consideration are 34.69 and 45.56 at the significance levels P = 0.05 and 0.01, respectively. The profile of the loglikelihood ratios of the full vs. reduced model across the length of linkage group 10 has a clear peak at ∼13 cM from marker CA/ CCC640R. The LR value at this peak is 51.0, well beyond the empirical critical threshold at the significance level P = 0.01.
To compare the power of our method with previous methods, the same material is subjected to interval mapping (Lander and Botstein 1989) and composite interval mapping (Zeng 1994) on the basis of the most differentiated phenotypes measured at year 11 (Figure 1). Neither of these two mapping methods can declare the existence of a significant QTL given their lower LR values. Figure 2 illustrates the result from interval mapping for diameters. No LR value from interval mapping is larger than the threshold (7.68) obtained from permutation tests at the significance level P = 0.05.
Similar conclusions about the difference of QTL detection between our method and current methods are obtained for many other linkage groups (results not shown). These suggest that our method incorporating logistic growth curves has greater power to detect a significant QTL than the current methods.
The dynamic pattern of QTL expression: Our method has an additional advantage; i.e., it can detect the dynamic change of QTL expression over time. The growth curves of diameter are drawn using the estimates of logistic parameters for two genotypes at the QTL detected on linkage group 10 (Figure 3). On the basis of the hypothesis test (11), this QTL is detected to be inactive until trees grew to ∼6 years in the field. And its effect on diameter growth increased with age. At 11 years old, genotype Q_{1}Q_{2} exhibited diameter growth 4.5 cm more than its alternative Q_{2}Q_{2}. This difference appears to increase after age 11 years, as predicted from the logistic curves estimated (Figure 3). Apparently, this QTL interacts significantly with age to affect stem diameter growth.
If two growth curves predicted by a QTL have different ages and/or growth at the inflection point, this indicates that the inflection point is under genetic determination. It is found that the QTL detected on linkage group 10 exerts strong control over the inflection point (Figure 3). The genetic control of the inflection point suggests that the growth trajectory can be genetically modified to increase a tree's capacity to effectively acquire spatial resources.
DISCUSSION
Beyond the traditional models and tools used for quantitative genetic studies, current genome technologies permit us to dissect quantitative traits into individual locus components (QTL). Through this dissection the genetic basis of quantitative traits can be better unraveled (Mackay 2001) and, ultimately, genetic improvement for these complex traits can be made more efficient (Tanksley 1993; Wuet al. 2000). Analyses and interpretations of genomic data, however, are strongly dependent upon the study material, data structure, genetic model, and statistical method used. As a result, considerable attention has been paid to the development of powerful experimental designs and analytical methodologies that can increase the power, precision, and resolution of QTL mapping. Currently, there have been many strategies proposed to increase QTL mapping. These include: (1) selecting two highly differentiated inbred lines to make a segregating generation, such as the F_{2} or backcross; (2) increasing the sample size of a mapping population by genotyping more progeny; (3) saturating the map density using informative markers, especially in genomic regions carrying QTL; (4) using composite interval mapping and multiple interval mapping (Jansen and Stam 1994; Zeng 1994; Kaoet al. 1999); and (5) developing more powerful computational technologies, such as the Bayesian approach implemented with the Markov chain Monte Carlo (MCMC) algorithm (Satagopanet al. 1996; Xu and Yi 2000). In this article, we propose that a simultaneous analysis of repeated measurements for a quantitative trait based on biological mechanisms can be used as an alternative strategy to enhance mapping power and precision.
It is well demonstrated that increased sample sizes and marker densities can almost always improve precision in QTL mapping, but they could be economically expensive in practice. Our mapping approach for repeated measurements based on growth curves can extract maximum information about QTL effects and positions contained in an arbitrary segregating family and, thus, confers an advantage for QTL detection in the situation where a limited size of genotyped samples or a limited level of marker density is used. In an example with a small sample size (N = 90) using forest tree data, our logistic mixture model offers improved power to detect a number of QTL underlying stem growth, in contrast to traditional approaches based on a single trait, which do not detect any QTL. Such differences are not surprising because a singletrait analysis approach typically cannot detect the QTL of small effect (Beaviset al. 1994).
The increased detection power of our approach results from the simultaneous use of multiple measurements that are correlated due to either the effect of pleiotropic QTL or residual covariances or both. This, in principle, is similar to the result from multitrait mapping, as shown in Jiang and Zeng (1995), Ronin et al. (1995), Eaves et al. (1996), Mangin et al. (1998), Knott and Haley (2000), and Korol et al. (1995, 2001). However, beyond these multitrait mapping approaches, our growthbased approach treats phenotypic values as a function of age, thus having the ability to analyze a quantitative trait measured at an unlimited number of time points by modeling the full, continuous growth trajectory. Moreover, instead of estimating a large number of parameters, as needed in the traditional approaches, our approach estimates a highly reduced number of model parameters, which can make an initially highdimensional mapping model more tractable and the estimates of QTL parameters more precise.
Composite interval mapping can improve mapping precision to some extent when multiple QTL are located on the same linkage group, but their use frequently depends upon many other factors, e.g., marker spacing, the choice of markers as cofactors, and genotyped sample size (Broman 2001). Multiple interval mapping, proposed by Zeng and coworkers, can simultaneously model multiple marker intervals so that multiple QTL and their epistatic interactions can be estimated (Kaoet al. 1999). Yet, a serious difficulty may be encountered when multiple interval mapping is extended to simultaneously map multiple quantitative traits or repeated measurements at different ages, because a high number of QTL effects should be modeled in these cases. Our logisticmixture model, when built upon composite interval mapping or multiple interval mapping, can make these two approaches more tractable by reducing the number of model parameters to be estimated. In fact, evidence for more than one QTL observed on some linkage groups from our approach in the poplar example (results not shown) prompts us to build the logistic mixture model upon composite interval mapping or multiple interval mapping and provide better resolution of multiple linked QTL for growth processes.
We have used the method of maximum likelihood to estimate the unknown parameters with their MLEs. The MLEs are attractive in terms of their properties of invariance, consistency, and asymptotic efficiency. Our approach, built upon the traditional maximumlikelihood method, is readily accessible to the general genetics community. Using prior information on parameters, however, we can incorporate the logisticmixture model in the Bayesian paradigm (Satagopanet al. 1996; Xu and Yi 2000). By specifying the prior density of parameters, MCMC can be used to evaluate the posterior density and provide posterior distributions of QTL effects and positions (Robert and Casella 1999).
Although the results of our approach are quantified by differences in the parameters controlling the overall shapes of different logistic curves, they can also be interpreted as regular genetic parameters, i.e., the additive or dominant effect of a QTL on growth at an arbitrary time point and the percentage of the total phenotypic variance explained by this QTL. According to classical quantitative genetics theory, the expected genetic values for QTL genotypes Q_{1}Q_{2} and Q_{2}Q_{2} at time t can be expressed, respectively, as
Our method can be extended to incorporate a general biological process of an organism into a QTL mapping framework. Such a process can be allometric scalings (West et al. 1997, 1999), growth models (Gould 1977; Alberchet al. 1979), or continuous responses to the environment in which an organism is reared (Viaet al. 1995). However, for clarity of description, we based our analysis on growth curves only. For growth models, we further limited our analysis to sigmoidal or logistic curves. Logistic growth curves are now regarded as one of the ubiquitous phenomena in biology, holding for every cell, organ, tissue, organism, or population, in a range from microbes (10^{–13} g) to blue whales (10^{8} g), no matter what species it is derived from (Westet al. 2001). The pattern of the logistic growth curve can be different among species, populations, and genotypes (Hofet al. 1999; Robertet al. 1999). However, it is also worthwhile to incorporate other biologically meaningful models (reviewed in Niklas 1994) into our analysis, as long as they fit well a dataset for particular species, environments, or developmental stages.
To incorporate a general biological process, we should first have a descriptive mathematical function that is expressed as
The method proposed in this study can be extended to other situations, such as partially informative markers or dominant markers, to deal with linked QTL of epistasis or to combine it with selective genotyping. In this study, it is assumed that residual variances and covariances among different ages are stationary. This assumption simplifies the mathematical manipulation of the residual variancecovariance matrix (inversion, factorization, etc.), but may be deviate from reality. The extension of our analysis to nonstationary variancecovariance structures is possible, as proposed by NunezAnton (1997) and NunezAnton and Zimmerman (2000) in their structured antedependent models. Also, Kirkpatrick and coworkers proposed Legendre polynomials to model the dynamic changes of genetic or residual variance and covariance with age (Kirkpatrick and Heckman 1989; Kirkpatrick et al. 1990, 1994). These parametric models for covariance function were improved by Pletcher and Geyer (1999) to assure the positive definite property of the functions. These different models for covariance function with some modifications can be incorporated into our mapping strategy.
Functional mapping: Since Lander and Botstein's (1989) interval mapping, there has been a wealth of literature reporting on the development of statistical methods for QTL mapping. The transition from a usual single or twotrait analysis to treatment of multiple measurements from different traits significantly improves all aspects of utilization of the mapping information contained in the data. In traditional mapping strategies, the combination of statistics and molecular genetics makes it possible to identify QTL that contribute to complex traits. However, in this study we attempt to combine powerful statistics and molecular genetics with developmental mechanisms underlying biological features, relationships, and processes to shed light on the genetic basis of complex, or quantitative, traits. This new strategy, which is called functional mapping due to the implementation of different mathematical functions of biological means, offers four significant advantages over previous strategies when applied to QTL mapping: (1) Results from functional mapping are closer to biological reality because the underlying biological mechanisms are considered; (2) smaller sample sizes may be used to achieve adequate power and precision for QTL detection because multiple measurements on the same individuals increase precision for mapping; (3) a large number of variables can be analyzed simultaneously by treating growth or a process as a smooth curve, and also the estimates of a small number of parameters can increase the precision of parameter estimation and the flexibility of the model; and (4) functional mapping allows for the testing of different biological hypotheses and this has a direct impact on applied breeding and the developmental studies of genetics and evolution.
Acknowledgments
We thank Dr. Alan Agresti, Dr. Myron Chang, Dr. Ramon Littell and Dr. Sam Wu for their helpful discussions on this study and three anonymous referees for their constructive comments on the earlier version of this manuscript. This work is partially supported by grants from the National Science Foundation to G.C. (DMS9971586) and an Outstanding Young Investigator Award of the National Natural Science Foundation of China to R.W. (30128017). The publication of this manuscript is approved as journal series R08640 by the Florida Agricultural Experiment Station.
APPENDIX A
In what follows, we derive the loglikelihood functions used to estimate the parameters in Ω = (a_{j} b_{j} r_{j} ρσ^{2}). The symbol ′ denotes the estimates of parameters from the previous step.
APPENDIX B
Below, we describe a mathematical procedure for calculating the integral of a logistic curve,
Footnotes

Communicating editor: C. Haley
 Received September 10, 2001.
 Accepted May 6, 2002.
 Copyright © 2002 by the Genetics Society of America