Parentage analysis in Chamaelirium luteum (L.) Gray (Liliaceae): why do some males have higher reproductive contributions?


  • Smouse,

    1. Department of Ecology, Evolution & Natural Resources, Cook College, Rutgers University, New Brunswick, NJ 08901, USA
    2. Center for Theoretical and Applied Genetics, Cook College, Rutgers University, New Brunswick, NJ 08901, USA
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  • Meagher,

    1. Center for Theoretical and Applied Genetics, Cook College, Rutgers University, New Brunswick, NJ 08901, USA
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  • Kobak

    1. Center for Theoretical and Applied Genetics, Cook College, Rutgers University, New Brunswick, NJ 08901, USA
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Peter E. Smouse Dr Rm. 152A, Environmental & Natural Resource Sciences Building, 14 College Farm Road, Rutgers University, New Brunswick, NJ 08901, USA. Tel: +1 732 932 1064; fax: +1 732 932 8746; e-mail: Smouse@Aesop.Rutgers.Edu


Much of the contemporary study of adaptation in natural populations involves the regression of some component of fitness, usually survivorship or viability, on one or more characters of interest. It is difficult to apply this approach to measures of paternal reproduction, however, because paternity is typically estimated indirectly from genetic markers, rather than being measured directly from progeny counts. Here, we present maximum likelihood methods for modelling relative male reproductive success as a log-linear function of one or more potentially predictive features, as well as providing a framework for the assessment of pairwise (male:female) effects, as they affect male reproductive performance. We also provide nonparametric statistical tests for alternative models.

Using this formulation, we examine the impact of inflorescence morphology on male reproductive success in Chamaelirium luteum L., and we also assess the importance of intermate distances between males and particular females. While male reproductive success and male inflorescence morphology are both quite variable, reproductive morphology does not appear to predict male reproductive success in this study. Intermate distance is an extremely effective predictor of pairwise success, however; but averaged over females, there is almost no net effect for different males.


A basic question in studies of evolution is why some individuals are more fit than others. Measurement of fitness ultimately reduces to a determination of the number of offspring a given individual contributes to the next generation. Where parentage is uncertain, inference of parentage by means of genetic markers is a necessary part of any assessment of the relative fitnesses of the parents in question (cf. Primack & Kang, 1989; Snow & Lewis, 1993). The inference and analysis of parental contributions to progeny arrays in natural populations has led to significant advances in our understanding of breeding systems, and the methodology has now advanced to the point at which it is feasible to make statistically justifiable inferences regarding the question of why some individuals are more fit than others.

Much of the emphasis in contemporary studies has been placed on identification of the relationship between phenotypic features of organisms and their fitness, i.e. selection gradient analysis ( Lande & Arnold, 1983; Arnold & Wade, 1984; Mitchell-Olds & Shaw, 1987). This approach involves the regression of some component of fitness, usually survivorship or viability, on one or more characters of interest. It is difficult to apply such approaches to measures of paternal offspring production, however, because paternity is typically inferred from genetic data, rather than being measured directly from unambiguous progeny counts. Thus, assumptions underlying standard regression models are not met because the response variables (here the ln λk values) are estimated indirectly and with inflated error variance ( Devlin et al., 1988 ). In this paper, we describe an analytical approach to measuring the impact of phenotypic variation on reproductive fitness, for the case where male parentage is inferred using genetic data.

The basic analytical tools for assessing parental contributions in natural populations are well established ( Meagher, 1986; Thompson, 1986; Devlin et al., 1988 ; Roeder et al., 1989 ; Adams et al., 1992 ; Milligan & McMurry, 1993; Smouse & Meagher, 1994). In addressing the impact of paternal characters on reproductive performance, we can take the analytical methodology one step further. The approach outlined below integrates enumeration of paternal contributions with assessment of the impact of specific phenotypic characters on reproductive success. Our approach provides statistical tests of the impact of specific male features, and it also provides a framework for assessing pairwise (male:female) effects, as they affect reproductive performance. An example of the latter that has direct bearing in plant populations is the physical distance between particular males and females, but the approach could easily be extended to pairwise features such as phenological overlap or behavioural interactions between mates.

The specific objectives of this paper are to: (1) develop a formal statistical model for the estimation and testing of the impact of one or more male-specific features on relative reproductive success of those same males; (2) use that formulation to examine the impact of inflorescence morphology on male reproductive success in Chamaelirium luteum L.; (3) extend the statistical formulation to potentially predictive features that characterize male–female pairs, for example the effect of intermate distance on relative male reproductive contributions for particular females; and (4) use that formulation to assess the importance of intermate distances in C. luteum.


Chamaelirium luteum (L.) Gray

For empirical application of the methods outlined below, we will use data from a population of Chamaelirium luteum, located in the Natural Area of Duke Forest, Orange County, NC, USA. Chamaelirium luteum is a dioecious, long-lived perennial, characterized by extensive sexual dimorphism in inflorescence structure ( Meagher & Antonovics, 1982) and typically male-biased sex ratios in natural populations ( Meagher, 1981). Plants consist of a basal rosette with an upright inflorescence that appears in late spring in North Carolina; male infloresences include a leafy stem with a terminal raceme containing 180–450 flowers and female inflorescences include a leafy stem with a terminal raceme containing 25–46 flowers. Within males, raceme length is highly correlated with flower number (r = 0.72, < 0.0001) ( Meagher & Antonovics, 1982). Flower number is also correlated with inflorescence stalk leaf number in both males (r = 0.50, n = 461, < 0.0001) and females (r = 0.62, n = 281, < 0.0001) ( Meagher & Antonovics, 1982). Thus, these characters are useful measures of floral display. The population under investigation here has been extensively studied with respect to population dynamics ( Meagher, 1982) and genealogical structure ( Meagher, 1986; Meagher & Thompson, 1987; Smouse & Meagher, 1994). The population was initially mapped in 1974, and annual census records for each individual plant (and new recruits) were maintained through 1990. The present analysis will make use of genetic data from 2294 seeds collected from 70 female plants in 1981, a year in which there were 273 males and 73 females in flower, all of whose genotypes have been characterized with respect to eight allozyme loci ( Meagher, 1986). Previous paternity analyses of a subset of these data showed localized pollen dispersal ( Meagher, 1986) and variation in male reproductive success ( Meagher, 1991; Smouse & Meagher, 1994). These data have also been used to develop strategies of paternity inference for application to natural populations in general ( Meagher & Thompson, 1986; Thompson & Meagher, 1987; Chakraborty et al., 1988 ; Smouse & Meagher, 1994).

Although this population has been extensively studied, the determinants of male reproductive success remain unknown. There is a growing body of plant literature suggesting that male performance is the driving force behind floral evolution (e.g. Lloyd, 1979; Bell, 1985; Bertin, 1987). This notion is primarily based on applications of a botanical reinterpretation of sexual selection theory, as applied to hermaphroditic flowers ( Willson, 1979; Stephenson & Bertin, 1983; Charlesworth et al., 1987 ; Lyons et al., 1989 ; Arnold, 1994), but the principles also apply to a dioecious species such as C. luteum. The overall notion is that while female reproductive success is primarily limited by resources available for fruit and seed production, male reproductive success is limited by mating opportunities ( Bateman, 1948; as reinterpreted by Trivers, 1972). Thus, selection should favour reproductive effort in males, resulting in a more attractive or more conspicuous floral display, which elicits more visits by pollinators. Also, in the case of plants, proximity to females and increased opportunities for mating could be important factors governing male reproductive performance. Although relative spatial location itself may not be an object of selection, the manner in which plants respond to typical spatial patterns that occur within populations may be. The methods outlined here provide a rigorous analytical framework for addressing such predictions.

Likelihood parentage analysis

Earlier work that forms the basis for the innovations developed here establishes that there is a general likelihood formulation, allowing assessment of the unevenness of the frequency spectrum of parentage, using genetic markers ( Devlin et al., 1988 ; Smouse & Meagher, 1994). Consider a set of motheroffspring pairs (MOi: i = 1, …, N) and a set of potential fathers (Fk: k = 1, …, K); we assume that maternity is known. For each motheroffspringputative father trio, we compute

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probabilities that depend only on classic Mendelian ratios (and population gametic frequencies if there is dominance or linkage). We define a matrix, X = {xik}, of dimension 2294 (the number of MO pairs) × 273 (the number of potential fathers).

We denote the relative reproductive success for the kth male as 0 ≤ λk ≤ 1, averaged over the whole study, and since the list of potential fathers is exhaustive (for this closed, internally mating population), the λk values must sum to unity. Given enough genetic data, we could (in principle) assign each offspring to an unambiguous father, and could then determine the relative reproductive output of the various males by simply counting, but we have shown elsewhere ( Chakraborty et al. 1988 ) that allozyme analysis is not likely to provide this much resolution. Although categorical paternity delineation is beyond the reach of many (if not most) studies, it is nevertheless possible to estimate the λk values with likelihood analysis. For the N = 2294 offspring, we compute the likelihood of the male parentage spectrum as

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The strategy is to maximize the likelihood (L), or equivalently log-L, relative to the spectrum of male reproductive values, {λk}. Following Roeder et al. (1989 ), we used an iterative EM algorithm and found a solution set {λ*k: k = 1, …, K} of significantly heterogeneous male reproductive contributions ( Smouse & Meagher, 1994).

Male feature profiles

Analytical model

Our object here is to address the far more interesting question of why some males attain more reproductive success than others. In such an analysis, we start with the male features of possible relevance to reproductive success, and our first task is to model the relative male reproductive contributions, the λk values, as some function of the potentially interesting features. There are many ways one might model the impact of such features, but one of the more attractive methods is to use log-linear models.

In the usual log-linear model, the response variable (here ln λk) is estimated directly, and is then regressed on a set of predictor (z) variables, each measured without error. Here, the response variables cannot be measured directly, but rather are estimated indirectly. The relationship is log-linear, but a conventional log-linear regression is inappropriate, because the response variable is not measured, it is itself modelled. In addition to their analytical features, log-linear models are also a means of linearizing multiplicative contributions of different male features to overall fitness, as reflected in the λk values. Thus, log-linear models have conceptual advantages as well. For the kth male, express the collection of potentially interesting features in vector form Gk = { gjk}, and then define

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Because this is a closed population, with all the potential male parents accounted for, the λk values must sum to unity, and each is bounded by [0,1]. To ensure this, it is customary to use one of the λk values, say the last (λK) as a reference for all of the others. We define the feature differences between the kth and Kth males as

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and then rewrite eqn 3 in the form

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The final results are invariant with respect to the choice of referencing male. We can model different combinations of the J candidate features, in an attempt to extract those of major importance, anticipating a parsimonious model, with J < (K – 1).

Given a maximum likelihood solution for the vector of J regression coefficients, B*, we can back-translate to estimate the corresponding male reproductive contributions indirectly,

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As a reminder, these are ‘modelled’λk values, rather than progeny counts. We can also evaluate proper submodels (some of the βj values = 0) by using nested log-linear stepwise regression procedures, a fact we can use to explore the potentially relevant male features in some detail. The idea is to find the smallest set of predictive features that will describe the situation adequately.

Using the translation of β into λ as in eqn 6, and applying those translations to eqn 2, we maximize log-L with respect to the choice of βj. The essence of our strategy is to compute log-L with all of the β values set to zero, beginning with the null hypothesis that none of the features of interest is predictive of relative male reproductive performance. Holding all β values except the first at zero, we compute the λ values implied by alternative values of β1, and using them to compute the likelihood in eqn 2. We replace the initial estimate of β1 = 0 with that value, β*1 ≠ 0, for which log-L is maximum. Using this value of β*1 and β3 = β4 = ⋯ = βJ = 0, find that value of β2 that maximizes log-L; call it β*2. We continue in this fashion until all of the β values have been adjusted, and then cycle through them again. The process ends when further changes in the β values lead to no further improvements in log-L. In our experience, the algorithm is simple, reasonably fast for small J, and well behaved.

In the usual log-linear treatment, we extract likelihood ratio test criteria that are asymptotically χ2 distributed, but the approximations are not particularly close in our application (see below). This may be a general problem in analysis of paternity where the asymptotic conditions for χ2 approximation are not met. Thus, in these circumstances, we recommend the use of nonparametric (permutational) procedures for hypothesis testing. It is important for permutational testing to be clear on the hypothesis of interest. We are not testing here whether male reproductive contributions are homogeneous; we take the fact that they are quite heterogeneous as well established ( Meagher, 1986; Smouse & Meagher, 1994). Rather, we are testing the null hypothesis that (whatever the pattern of reproductive heterogeneity), there is no relationship between it and the profiles of candidate morphological features. Our hope, of course, is that we can disprove that null hypothesis, thereby demonstrating the predictive relevance of the features of interest. On the null premise that the feature profiles ( Zk) are irrelevant to male reproductive success, it should not matter at all which profile is attached to which male. We thus performed a nonparametric analysis by permuting feature profiles among males, and repeating the whole analysis 1000 times. The distribution of the likelihood scores obtained under randomization was then compared with the result obtained with the male feature profiles attached to the proper males.

Male reproductive effort in Chamaelirium luteum

The relationship between reproductive effort and reproductive success is central to many theories concerning sex-specific reproductive performance; a good example of this is Bateman’s Principle (outlined above) that predicts different relationships between effort and success for males and females. In terms of reproductive effort and success, it has generally been presumed that an increase in resources allocated to attractive structures should result in an increase in reproductive success, and indeed several authors have put forward results that demonstrate this to be the case ( Schoen & Stewart, 1986; Broyles & Wyatt, 1990; Devlin et al., 1992 ).

In the context of our C. luteum example, a reasonable expectation might be that the resources a male puts into reproduction will influence his reproductive success. Meagher (1991) defined reproductive effort as the physiological resources committed to reproductive structures, and quantified the amounts of male reproductive effort for C. luteum by measuring rosette leaf number, stalk leaf number, stalk length and raceme length, all of which are correlated with pollen output and floral display size. We defined four feature variables for the kth male, relative to the values of the K = 273rd male:

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Histograms of these variables are presented in Fig. 1. Each variable is unimodally distributed, with sufficient variation among males to make the analysis worthwhile. The coefficients of variation are 0.35, 0.42, 0.32 and 0.30, respectively. That is to say, we know from earlier analyses ( Meagher, 1986; Smouse & Meagher, 1994) that we have reproductive heterogeneity among males; we also have substantial morphological variation among them; the question here is whether the two are correlated.

Figure 1.

 Histograms of highly variable morphological features, all correlated with male flower number, for K = 273 potential fathers in Chamaelirium luteum (L.): rosette leaf number; stalk leaf number; stalk length; raceme length.

Meagher (1991) showed, using a subset from the present data consisting of 102 males with unique genotypes, that the correlations between male features and reproductive success were not significant. We have analysed the model in eqn 6, for each of the four features individually, as well as for the full quartet (Table 1). Our results, using the full battery of 273 males, tell the same tale as did Meagher’s (1991) analysis. While male success is demonstrably uneven ( Meagher, 1986; Smouse & Meagher, 1994) in this data set we found no convincing evidence of a correlation between variation in male reproductive success and variation in morphological feature profile.

Table 1.   Likelihood-ratio test criteria (Λ) for the log-linear model and submodels analysed by text eqn 6, describing the impact of male morphology on male reproductive success, using morphological features; P-values for test criteria were evaluated via permutational procedures. Thumbnail image of

Permutational shuffling of male feature profiles shows the same result in graphic fashion ( Fig. 2). Since these permutations were based on random rearrangement of the data, the random distribution of each estimated β-coefficient is centred on zero, though not symmetrically so in all cases. The lack of symmetry around zero suggests that the log-likelihood ratio is a better test statistic for evaluating permutation results than the β-values themselves. Also, as anticipated above, the asymptotic approximation to a χ2 distribution does not appear to be a reliable basis for significance tests, based on the observed log-likelihood ratios generated by this analysis ( Fig. 2).

Figure 2.

 Permutation analysis of the impact of male morphology on male reproductive contributions for Chamaelirium luteum (L.); each panel shows likelihood ratio score as a function of parameter estimate value for random permutations of feature scores among males; the horizontal line in each panel represents the 5% critical value in a two-tailed test of significance; the parameter estimate from the original data set is indicated.

The lack of fit of our log-likelihood ratios to a χ2 distribution is a striking outcome of our analysis. When a test statistic is asymptotically χ2 distributed, the asymptote in question refers to the sample size or information content of the underlying data. In our case, we are using an apparently quite large data set, with 2294 progeny and 273 male parents. However, in this analysis, information content of the data is derived from other factors as well, such as the number of genetic loci and the number and frequencies of alleles within loci ( Meagher & Thompson, 1986). Thus, the axis along which the ‘asymptote’ should be evaluated in the case of a paternity analysis is complex, and likely to be multidimensional. In most genealogical inference applications, a permutation test or some other resampling scheme should be used to evaluate statistical significance. The log-likelihood ratio, while apparently not a close approximation to a χ2 distribution, is a useful test statistic for evaluating the results of permutation-based significance testing. Therefore, we recommend that hypothesis testing for future analyses such as this should be based on the permutation test used here.

Distance effects

Analytical model

Many workers have demonstrated that mating success drops off with distance between mates ( Handel, 1983; Ellstrand & Marshall, 1985; Meagher, 1986; Schoen & Stewart, 1987; Devlin & Ellstrand, 1990; Adams & Birkes, 1991; Adams et al., 1992 ; Devlin et al., 1992 ; Godt & Hamrick, 1993; Kaufman et al., 1998 ). Meagher (1991), using a subset of 102 of these males and 575 of the progeny whose male parentage was inferred using likelihood criteria, showed that distance between mates is important for C. luteum. For the distance model, Adams & Birkes (1991) use an equation of the general form

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a modified log-linear model for the (pairwise) productivity of the kth male, in combination with the ith female. Adams & Birkes (1991) have shown that a formulation similar to this distance model can be used to model the effects of phenological overlap. In principle, there is nothing to prevent us from examining other male:female pairwise features, as predictors of performance.

Following Adams & Birkes (1991), a pairwise distance model takes the form of a likelihood specification for the male parentage spectrum of the hth offspring in the ith sibship

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The parentage spectrum (the array of λk values) depends on the parameter γ, as well as on the array of physical distances from the ith female to each of the K = 273 males. The likelihood of 70 female sibships, each with a different male parentage spectrum, is

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As before, we used a search routine, but this time we found a single coefficient, γ, that maximized the likelihood of the entire data set. Also as before, this is an indirect log-linear estimation scheme for the case where the λik values cannot be directly estimated.

To provide a test of the distance hypothesis, we have again adopted the nonparametric permutational approach. Whatever the array of overall male reproductive contributions, we ask whether the relative distance between each of the males and a particular female is predictive. On the null premise that the array of intermate distances bears no relationship to relative male contributions, we shuffled the positions of the males within the plot, and extracted an estimated value of γ for each random shuffle. We repeated the process 1000 times, and plotted the distribution of random γ, for comparison with the value obtained from the males, in their actual physical locations.

Mating distances in Chamaelirium luteum

Meagher (1991) mapped the sites of the 273 males and 70 females of this study. The positions of the kth potential father and ith mother are defined by their respective x and y field coordinates

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For the kth potential father and ith mother, we calculated a Euclidean distance

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We tried various monotonic transformations of the Euclidean distance, and they all showed the same general trend of decreasing male reproductive contribution with increasing distance from the female. Logarithmic distance

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was numerically the best predictor. Logarithmic distance is also compatible with the biological idea that pollen dispersal exhibits an exponential decay with increasing distance from the source. When we standardized the distance of the kth father from the ith mother, relative to that of the Kth (273rd) father and the ith mother, that modelling choice comes down to

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We obtained an estimate of γ = –0.90; P ≤ 0.001. The likelihood-ratio test criterion is 556.85 (nominally, 1 degree of freedom), and nonparametric analysis of the distance model, with the physical locations of males randomly permuted, shows conclusively that the impact of intermate distances is important in determining the male reproductive spectrum for individual females ( Fig. 3). As mentioned above for the β-analyses, the χ2 approximation is much inflated for this analysis, so we strongly encourage the use of permutational procedures for determining significance levels.

Figure 3.

 Permutation analysis of the impact of intermate distances on male reproductive contributions for particular females of Chamaelirium luteum (L.); likelihood ratio score is plotted as a function of parameter estimate value for random permutations of male spatial locations; the horizontal line represents the 5% critical value in a two-tailed test of significance; the parameter estimate from the original data set is indicated.

A biological interpretation of our distance estimate is illustrated in Fig. 4. With a parameter value of γ = –0.90, male reproductive success drops off rapidly at distances greater than 10 cm to nearly zero at distances of 100 cm or greater ( Fig. 4a). However, this is only part of the story. The number of male gametes that are presented to a given female, from a surrounding circle of a given radius, depends not only on the relationship of λ to distance (as indicated by γ), but also on the number of males within that radius. Thus, the cumulative male reproductive contribution at a distance of 100 cm might be relatively high, even though the probability of an individual pollen transfer event is small, because the number of opportunities for pollen transfer is large. This can be seen by considering the cumulative expected paternal contributions, given the actual spatial distribution of mating opportunities in this population ( Fig. 4b), which shows a diminishing return of male gamete capture by females over a range of 100 cm to 1000 cm. The shape of this diminishing return of male gamete capture depends on the precise spatial geometry of the population.

Figure 4.

 Biological consequences of the estimated distance effect (γ=–0.90) on male reproductive success in Chamaelirium luteum: (a) regression estimates of male reproductive performance λ (based on ML estimates of γ and eqn 9) as a function of log10 distance to females; (b) cumulative paternal contributions as a function of distances between males and females in the C. luteum population.


The morphological profile is not predictive

By extending the established likelihood framework for assessing unequal paternal contributions, we can address several new questions. The use of log-linear methods allows us to test hypotheses about the predicates of that unequal success. In the first extension, we modelled male fertility values as functions of candidate male features, thought to be involved in differential reproductive success. The regression equation involved is analogous to that widely used to describe the covariance between fitness and phenotype ( Lande & Arnold, 1983; Arnold & Wade, 1984; Mitchell-Olds & Shaw, 1987). Thus, the estimates of βj obtained with our approach can be interpreted as directional selection gradients. It should be noted that this same formulation could easily be adapted to quadratic and cross-product terms of the feature profile, allowing one to investigate other aspects of the selection gradient associated with the male feature profile, such as stabilizing or disruptive selection. Although non log-linear models require more elaborate manipulation with multinomial data ( Smouse, 1974), we could (in principle) accommodate them. Given our difficulty with finding even a suggestion of a pattern in the present data, further elaboration of the feature profile hardly seems profitable here.

Our failure to demonstrate any predictive utility for the male feature profile is disappointing, but it should not be attributed to a lack of statistical resolution. Our sample sizes are substantial, and the demonstrable heterogeneity of both male reproductive performance ( Meagher, 1986; Smouse & Meagher, 1994) and male inflorescence size ( Fig. 1) is substantial. Unless a predictive connection between them were to be extremely subtle, we should easily have been able to detect it. Suffice it to say that we detected no pattern, corroborating earlier analyses of a subset of these same data ( Meagher, 1991). How far we dare extrapolate from that failure is unclear; but, at least for the case at hand, there is nothing to extrapolate.

Isolation by distance

In the second extension, we incorporated dyadic features of the females and the targeted males. This extension adds considerable reality to paternity inference because sexual reproduction of one sex cannot be viewed in isolation from what is happening with the other sex. This approach, with the addition of a single parameter (the regression coefficient for distance effects), greatly improves the power of male fertility assessment because every intermate distance is separately and explicitly accounted for in the analysis. We demonstrated a profound ‘isolation by distance’ effect, with male reproductive contributions to particular females declining sharply with increasing intermate distance.

This is hardly a surprising result, but it has interesting ecological implications. It is not likely that selection could be acting on spatial proximity between sexes, per se, given the many random factors associated with dispersal and establishment, but it is plausible that selection might act on a male’s ability to make the most of proximity. That is to say, the parameter γ might itself be subject to selection, and specific attraction of different pollinators could result in different pollen dispersal profiles ( Stanton et al., 1989 ; Stanton et al., 1991 ).

It is possible to integrate morphological and distance effects into one analysis using the approaches outlined above. Morphological feature analysis and distance analysis both use the same log-linear model. In principle, a large distance effect, such as was found with Chamaelirium luteum, could confound or mask the effects of morphological variation. In fact, we applied our approach to the data set, combining distance and morphological features, and found that morphological features were still unimportant as predictors of male reproductive success even when distance effects were taken into account.


There is a highly uneven distribution of parentage in both natural and captive populations. Paternity inference has now advanced to the point at which simply demonstrating uneven reproduction is no longer sufficient. We have outlined an estimation and testing framework that permits wide exploration of the potential causes of differential paternal success. Our application to C. luteum has been a mixed success; we have made more complete use of the information content of the data and developed a more clear understanding of distance effects, but there are still some open questions. In general, however, deployment of these methods should increase our understanding of the basic biology of reproductive success as a component of fitness.


We thank C. Chevillon, D. Costich, E. Elle, S. Handel, S. Kaufman, W. G. Hill, R. Shaw and three anonymous reviewers for comments on the manuscript. This paper has also benefited from our ongoing discussion with E. Thompson. P.E.S. was supported by NJAES/USDA-32106 and NSF-BSR-90-06589, T.R.M. by NSF-BSR-83-14887 and NSF-BSR-90-06589, and C.J.K. by NJAES/USDA-32106.