Testing concordance of clinical characteristics in familial studies with application to inflammatory bowel diseases

The etiology of chronic Inflammatory Bowel Diseases (IBD) remains unknown, with both genetic and environmental risk factors having been implicated. A recent collaborative study of IBD provides clinical data from families with three or more affected first-degree relatives. The scientific question is whether specific clinical characteristics aggregate among affected individuals within families. Gastroenterological researchers have examined the number of concordant familial pairs in familial aggregation studies, but methods and results have been discrepant. This article investigates concepts of concordance and gives a comprehensive statistical treatment for testing concordance of various clinical traits in familial studies. For dichotomous traits, the distribution of this statistic under the null hypothesis of no familial aggregation is obtained by three methods: asymptotic, probability generating function, and permutation. The permutation method is extended to analyze aggregation for non-dichotomous traits and co-aggregations between two traits. We apply the permutation method to analyze the aforementioned multiply-affected IBD family data. Evidence is found for familial clustering of various traits, some of which are not revealed in existing studies. Such analyses provide a basis for investigating the dependence of trait aggregation upon genetic or environmental risk factors.     

A frailty model of segregation analysis: understanding the familial transmission of alcoholism

Coupled with environmental factors, genes contribute to numerous human diseases and traits. While there are many epidemiological methods to assess the familial clustering of traits, few are flexible enough to accommodate interactions between covariates and familial factors. In this paper, we propose and develop a frailty model that establishes an integrated framework to evaluate familial transmission of a disease by controlling for covariate effects and conveniently testing the interactions between covariates and familial factors. We also present a peeling algorithm that dramatically reduces the computational burden. This frailty model is employed to examine the familial transmission of major subtypes of alcoholism, namely, alcohol abuse and dependence. We conclude that alcohol dependence is strongly familial whereas alcohol abuse expresses a marginally significant pattern of familial transmission. Moreover, females manifest alcoholism at a lower threshold, and there is no sex-specific familial transmission of alcoholism after adjustment for the threshold effect.     

Analytical Relationship between Family History and Genetic and Environmental risks,with Application to Female Breast Cancer

It is well established in genetic epidemiology that family history is an important indicator of familial aggregation of disease in a family. A strong genetic risk factor or an environmental risk factor with high familial correlation can result in a strong family history. In this paper, family history refers to the number of first‐degree relatives affected with the disease. Cui and Hopper (Journal of Epidemiology and Biostatistics 2001; 6 : 331–342) proposed an analytical relationship between family history and relevant genetic parameters. In this paper we expand the relationship to both genetic and environmental risk factors. We established a closed‐form formula for family history as a function of genetic and environmental parameters which include genetic and environmental relative risks, genotype frequency, prevalence and familial correlation of the environmental risk factor. The relationship is illustrated by an example of female breast cancer in Australia. For genetic and environmental relative risks less than 10, most of the female breast cancer cases occur between the age of 40 and 60 years. A higher genetic or environmental relative risk will move the peak of the distribution to a younger age. A more common disease allele or more prevalent environmental risk factor will move the peak to an older age. For a proband with breast cancer, it is most likely (with probability ≥80%) that none of her first‐degree relatives is affected with the disease. To enable the probability of having a positive family history to reach 50%, the environmental relative risks must be extremely as high as 100, the familial correlation as high as 0.8 and the prevalence as low as 0.1. For genetic risk alone, even the relative risk is as high as 100, the probability of having a positive family history can only reach about 30%. This suggests that the environmental risk factor seems to play a more important role in determining a strong family history than the genetic risk factor. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A parametric copula model for analysis of familial binary data

Modeling the joint distribution of a binary trait (disease) within families is a tedious challenge, owing to the lack of a general statistical model with desirable properties such as the multivariate Gaussian model for a quantitative trait. Models have been proposed that either assume the existence of an underlying liability variable, the reality of which cannot be checked, or provide estimates of aggregation parameters that are dependent on the ordering of family members and on family size. We describe how a class of copula models for the analysis of exchangeable categorical data can be incorporated into a familial framework. In this class of models, the joint distribution of binary outcomes is characterized by a function of the given marginals. This function, referred to as a "copula," depends on an aggregation parameter that is weakly dependent on the marginal distributions. We propose to decompose a nuclear family into two sets of equicorrelated data (parents and offspring), each of which is characterized by an aggregation parameter (alphaFM and alphaSS, respectively). The marginal probabilities are modeled through a logistic representation. The advantage of this model is that it provides estimates of the aggregation parameters that are independent of family size and does not require any arbitrary ordering of sibs. It can be incorporated easily into segregation or combined segregation-linkage analysis and does not require extensive computer time. As an illustration, we applied this model to a combined segregation-linkage analysis of levels of plasma angiotensin I-converting enzyme (ACE) dichotomized into two classes according to the median. The conclusions of this analysis were very similar to those we had reported in an earlier familial analysis of quantitative ACE levels.     

Comparison of logistic regression and linear regression in modeling percentage data

Percentage is widely used to describe different results in food microbiology, e.g., probability of microbial growth, percent inactivated, and percent of positive samples. Four sets of percentage data, percent-growth-positive, germination extent, probability for one cell to grow, and maximum fraction of positive tubes, were obtained from our own experiments and the literature. These data were modeled using linear and logistic regression. Five methods were used to compare the goodness of fit of the two models: percentage of predictions closer to observations, range of the differences (predicted value minus observed value), deviation of the model, linear regression between the observed and predicted values, and bias and accuracy factors. Logistic regression was a better predictor of at least 78% of the observations in all four data sets. In all cases, the deviation of logistic models was much smaller. The linear correlation between observations and logistic predictions was always stronger. Validation (accomplished using part of one data set) also demonstrated that the logistic model was more accurate in predicting new data points. Bias and accuracy factors were found to be less informative when evaluating models developed for percentage data, since neither of these indices can compare predictions at zero. Model simplification for the logistic model was demonstrated with one data set. The simplified model was as powerful in making predictions as the full linear model, and it also gave clearer insight in determining the key experimental factors.     

Gender difference in apolipoprotein E-associated risk for familial Alzheimer disease: a possible clue to the higher incidence of Alzheimer disease in women.

Late-onset Alzheimer disease (AD) is associated with the apolipoprotein E (APOE)-epsilon4 allele. In late-onset familial AD, women have a significantly higher risk of developing the disease than do men. The aim of this study was to determine whether the gender difference in familial AD is a function of APOE genotype. We studied 58 late-onset familial AD kindreds. Kaplan-Meier survival analysis was used to assess genotype-specific distributions of age at onset. Odds ratios were estimated by logistic regression with adjustment for age and by conditional logistic regression with stratification on families. All methods detected a significant gender difference for the epsilon4 heterozygous genotype. In women, epsilon4 heterozygotes had higher risk than those without epsilon4; there was no significant difference between epsilon4 heterozygotes and epsilon4 homozygotes. In men, epsilon4 heterozygotes had lower risk than epsilon4 homozygotes; there was not significant difference between epsilon4 heterozygotes and those without epsilon4. A direct comparison of epsilon4 heterozygous men and women revealed a significant twofold increased risk in women. We confirmed these results in 15 autopsy-confirmed AD kindreds from the National Cell Repository at Indiana University Alzheimer Disease Center. These observations are consistent with the increased incidence of familial AD in women and may be a critical clue to the role of gender in the pathogenesis of AD.     

Comparison of Logistic Regression and Linear Regression in Modeling Percentage Data

Percentage is widely used to describe different results in food microbiology, e.g., probability of microbial growth, percent inactivated, and percent of positive samples. Four sets of percentage data, percent-growth-positive, germination extent, probability for one cell to grow, and maximum fraction of positive tubes, were obtained from our own experiments and the literature. These data were modeled using linear and logistic regression. Five methods were used to compare the goodness of fit of the two models: percentage of predictions closer to observations, range of the differences (predicted value minus observed value), deviation of the model, linear regression between the observed and predicted values, and bias and accuracy factors. Logistic regression was a better predictor of at least 78% of the observations in all four data sets. In all cases, the deviation of logistic models was much smaller. The linear correlation between observations and logistic predictions was always stronger. Validation (accomplished using part of one data set) also demonstrated that the logistic model was more accurate in predicting new data points. Bias and accuracy factors were found to be less informative when evaluating models developed for percentage data, since neither of these indices can compare predictions at zero. Model simplification for the logistic model was demonstrated with one data set. The simplified model was as powerful in making predictions as the full linear model, and it also gave clearer insight in determining the key experimental factors.     

Allele frequency estimation from data on relatives.

Given genetic marker data on unrelated individuals, maximum-likelihood allele-frequency estimates and their standard errors are easily calculated from sample proportions. When marker phenotypes are observed on relatives, this method cannot be used without either discarding a subset of the data or incorrectly assuming that all individuals are unrelated. Here, I describe a method for allele frequency estimation for data on relatives that is based on standard methods of pedigree analysis. This method makes use of all available marker information while correctly taking into account the dependence between relatives. I illustrate use of the method with family data for a VNTR polymorphism near the apolipoprotein B locus.     

Clinical and genetic profile of patients with medullary thyroid cancer treated in the Cancer Centre--Institute of Oncology in Warsaw

INTRODUCTION: The aim of this study was to analyse the distribution and frequency of mutations and their correlations with clinical phenotypes of patients with MTC, to reveal the differences between sporadic and familial type of MTC, and to describe the phenotypes of patients. MATERIALS AND METHODS: 212 patients with medullary thyroid cancer (MTC) were treated in Cancer Centre in Warsaw between 1997 and 2005. In most patients, DNA isolated from peripheral blood leukocytes was tested for RET gene mutations by sequencing and accordingly MTC form was assessed. Genetic testing was performed in the relatives of patients with familial MTC in order to distinguish asymptomatic mutation carriers from noncarriers. RESULTS: RET gene mutations were identified in 46 patients (22%). The others were found noncarriers and sporadic MTC was diagnosed. MEN 2A/FMTC syndrome (multiple endocrine neoplasia type 2A/ familial type of MTC) was diagnosed in 44 patients, MEN 2B syndrome (multiple endocrine neoplasia type 2B) in 2 patients. In patients with sporadic and familial MTC, age at diagnosis and multifocal occurrence was analysed, and the results were found to be in accordance with those of other research centres. However, the distribution and frequency of mutations, as well as some clinical data, such as the frequency of pheochromocytoma occurrence as the first manifestation of MEN syndrome, differed from the published data, and further studies are necessary to reveal the reasons of these differences. CONCLUSIONS: DNA testing for RET gene mutations is reliable as a diagnostic tool and therefore it should be performed for screening of all patients with MTC or other diseases of MEN syndrome.     

Abdominal Obesity and Expression of Familial Combined Hyperlipidemia

Objective: To investigate the role of abdominal and body obesity on the prevalence of hyperlipidemia, in particular, hypertriglyceridemia, hypercholesterolemia, and high apolipoprotein B levels, in familial combined hyperlipidemia (FCHL) relatives and their spouses. Research Methods and Procedures: In FCHL relatives (n = 618) and spouses (n = 297), prevalence data of hyperlipidemia and high apolipoprotein B levels and their age and gender‐corrected odds ratios (ORs) were calculated for sex‐adjusted categories of waist‐to‐hip ratio (WHR), waist circumference, and BMI. Results: Increments of BMI, waist circumference, and WHR increased the frequency of hyperlipidemia. In the whole study population (relatives and spouses combined), frequency of hypertriglyceridemia showed a significant interaction only between WHR categories and FCHL. This was studied further after stratification of relatives by multivariable logistic regression analyses corrected for age and gender. Predominant expression of hypertriglyceridemia was observed with higher categories of WHR in FCHL relatives (prevalence up to 57.6%, OR 8.48 in highest vs. lowest WHR category, p < 0.001) but not in spouses (up to 32.9%, OR 1.05 in highest vs. lowest WHR category, not significant). Discussion: Both in spouses and FCHL relatives, increments in BMI and waist circumference increased the prevalence of hyperlipidemia. Specifically, in FCHL relatives, WHR was the most informative determinant of the expression of hyperlipidemia, in particular, hypertriglyceridemia. The data indicate that FCHL develops against a background of abdominal obesity.     

Polymorphism of melanic ladybirds maintained by frequency-dependent sexual selection

Sexual selection, whether by female preference or male competition, is almost inevitably frequency-dependent. Female preference gives rise to a 'rare male effect', by which the rarer male phenotypes gain a relatively greater selective advantage. In addition to this effect, the proportion of females expressing a preference may also be frequency-dependent.
Frequency-dependent expression of mating preference can arise in at least two ways: (1) when females encounter a succession of courting males while searching for a male they prefer; (2) when females chose a male from within a lek. Models of mating behaviour reveal a clear distinction between the frequency dependence in the expression of female preference and the frequency dependence in the consequent selection of the males. When expression of preference is highly dependent on frequency, the selection of males is constant or only slightly frequency-dependent: constant expression of preference produces high frequency dependence of selection. Analysis of general models shows that genetic polymorphisms can be maintained under a wide range of conditions.
The ladybird, Adalia bipunctata , is polymorphic for several melanic and non-melanic phenotypes. Females have a genetically determined preference for melanic males. Non-melanic phenotypes mate assortatively. By estimating the parameters of a detailed model of natural selection, sexual selection and assortative mating, it has been shown that the Adalia bipunctata polymorphism will be maintained at frequencies observed in the wild.     

Hsp90 at the crossroads of genetics and epigenetics

Hsp90 is a specialized molecular chaperone that is capable of buffering the expression of abnormal phenotypes.Inhi-bition of Hsp90 activity results in the expression of these phenotypes that are otherwise masked.Selection of offspringfrom the crossing of affected progenies results in inheritance and enrichment of these phenotypes,which can becomeindependent of their original stimuli.The current combined evidence favours a model involving the interplay betweengenetics and epigenetics.The recent proteomics efforts to characterize the Hsp90 interaction networks provide further cluesinto the molecular mechanisms behind this complex phenomenon.This review summarizes the most recent experimentalobservations and briefly discusses the genetic and epigenetic views used in explaining the different observations.     

Simulation of Mendelism revisited: the recessive gene for attending medical school.

Much of the recent confusion concerning studies of complex phenotypes such as neuropsychiatric disorders may derive from the inappropriate assumption of simple Mendelian transmission. This has sometimes led to unrealistic expectations regarding the potential benefits of linkage studies. To investigate how Mendelism may be simulated, we collected data on a common familial behavioral trait, attendance at medical school, among the relatives of 249 preclinical medical students. The "risk" of first-degree relatives going to medical school was approximately 61 times that of the general population. Complex segregation analysis carried out under a unified model provided strong evidence of vertical transmission. The results were compatible with transmission of a major effect, and a recessive model provided as satisfactory a fit as a general single-locus model. Moreover, a commonly applied test, allowing the transmission probability parameter (tau 2) to deviate from its Mendelian value, did not give a significant improvement of fit. Only a more general model where all three transmission probabilities (tau 1, tau 2, and tau 3) were unrestricted resulted in a significantly better fit than did the recessive model.     

Familial identification: population structure and relationship distinguishability

With the expansion of offender/arrestee DNA profile databases, genetic forensic identification has become commonplace in the United States criminal justice system. Implementation of familial searching has been proposed to extend forensic identification to family members of individuals with profiles in offender/arrestee DNA databases. In familial searching, a partial genetic profile match between a database entrant and a crime scene sample is used to implicate genetic relatives of the database entrant as potential sources of the crime scene sample. In addition to concerns regarding civil liberties, familial searching poses unanswered statistical questions. In this study, we define confidence intervals on estimated likelihood ratios for familial identification. Using these confidence intervals, we consider familial searching in a structured population. We show that relatives and unrelated individuals from population samples with lower gene diversity over the loci considered are less distinguishable. We also consider cases where the most appropriate population sample for individuals considered is unknown. We find that as a less appropriate population sample, and thus allele frequency distribution, is assumed, relatives and unrelated individuals become more difficult to distinguish. In addition, we show that relationship distinguishability increases with the number of markers considered, but decreases for more distant genetic familial relationships. All of these results indicate that caution is warranted in the application of familial searching in structured populations, such as in the United States.     

Assessing the role of HLA-linked and unlinked determinants of disease.

The relationship between increased risk in relatives over population prevalence (lambda R = KR/K) and probability of sharing zero marker alleles identical by descent (ibd) at a linked locus (such as HLA) by an affected relative pair is examined. For a model assuming a single disease-susceptibility locus or group of loci tightly linked to a marker locus, the relationship is remarkably simple and general. Namely, if phi R is the prior probability for the relative pair to share zero marker alleles identical by descent, then P (sharing 0 markers/both relatives are affected) is just phi R/lambda R. Alternatively, lambda AR, the increased risk over population prevalence to a relative R due to a disease locus tightly linked to marker locus A, equals the prior probability that the relative pair share zero A alleles ibd divided by the posterior probability that they share zero alleles ibd, given that they are both affected. For example, for affected sib pairs, P (sharing 0 markers/both sibs are affected) = .25/lambda S. This formula holds true for any number of alleles at the disease locus and for their frequencies, penetrances, and population prevalence. Similar formulas are derived for sharing one and two markers. Application of these formulas to several well-studied HLA-associated diseases yields the following results: For multiple sclerosis, insulin-dependent diabetes mellitus, and coeliac disease, a single-locus model of disease susceptibility is rejected, implying the existence of additional unlinked familial determinants. For all three diseases, the effect of the HLA-linked locus on familiality is minor: for multiple sclerosis, it accounts for only a 2.5-fold increased risk to sibs over the population prevalence, compared to an observed value of 20; for coeliac disease, it accounts for approximately a 5.25-fold increased risk to sibs, while the observed value is on the order of 60; for insulin-dependent diabetes mellitus, it accounts for a 3.42-fold increased risk in sibs, while the observed value is 15. In all cases, the secondary determinants must be outside the HLA region. For tuberculoid leprosy, an unlinked familial determinant is also implicated (increased risk to sibs due to HLA = 1.49; observed value = 2.38). For hemochromatosis and Hodgkin's disease, there is little evidence for HLA-unlinked familial determinants. With this formula, it is also possible to examine the hypothesis of pleiotropy versus linkage dis-equilibrium by comparing lambda AS with the increased risk to sibs due to the associated allele(s).(ABSTRACT TRUNCATED AT 400 WORDS)     

Approaches to familial aggregation: hypothesis testing and estimation when families are selected through parent probands under a variant of single ascertainment

Epidemiologic approaches to testing and estimating familial aggregation of a disease consist of comparing rates of disease in relatives of individuals with the disease (known as case probands) with rates of disease in relatives of individuals without the disease (known as control probands). Gold et al. (J Am Stat Ass 1967;62: 409-420) derived an explicit mathematical model and sampling methods, under which this approach is equivalent to testing the null hypotheses that the disease risk in families is homogenous. A basic assumption of this model is that every family member has the same risk of disease and that disease status is independent among family members, although the disease risk may vary between families. When the disease is suspected of having a genetic component, rather than being purely environmental, this model has been shown to be appropriate for detecting disease aggregation in siblings, when relatives are siblings of probands. This model however is unrealistic for use in nuclear families when the affected status of offspring is not independent of the affected status of parents, and these families are selected through an affected or an unaffected parent, so that a parent is the proband and relatives are offspring of probands. We extend the Gold et al. model to allow for the disease risk in offspring to vary with the affected status of the parent. We assume that families are selected through affected and unaffected parents, under a variation of single ascertainment. Under this study design, we show that the usual test of association between affected status of probands and relatives, performed by comparing sample proportions of affected relatives of affected and unaffected probands, respectively, is no longer equivalent to a test of homogeneity of disease risk in offspring. Instead, it is equivalent to testing that the disease risk in offspring is independent of the number of affected parents. This test reduces to a test of homogeneity if and only if one assumes that the variation in disease risk in offspring, between families, is solely due to the variation in the number of affected parents. As a result, we show that under this study design, the standard chi2 test must be modified in order to obtain a valid test of familial aggregation. In addition the sample proportions of affected relatives of case and control probands, respectively, are shown to provide unbiased estimates of the expected risk of disease in an offspring given an affected/unaffected parent. We apply these results to methods of sample selection and discuss the practical implications of these findings.     

Regressive logistic models for familial diseases: a formulation assuming an underlying liability model

Statistical models have been developed to delineate the major-gene and non-major-gene factors accounting for the familial aggregation of complex diseases. The mixed model assumes an underlying liability to the disease, to which a major gene, a multifactorial component, and random environment contribute independently. Affection is defined by a threshold on the liability scale. The regressive logistic models assume that the logarithm of the odds of being affected is a linear function of major genotype, phenotypes of antecedents and other covariates. An equivalence between these two approaches cannot be derived analytically. I propose a formulation of the regressive logistic models on the supposition of an underlying liability model of disease. Relatives are assumed to have correlated liabilities to the disease; affected persons have liabilities exceeding an estimable threshold. Under the assumption that the correlation structure of the relatives' liabilities follows a regressive model, the regression coefficients on antecedents are expressed in terms of the relevant familial correlations. A parsimonious parameterization is a consequence of the assumed liability model, and a one-to-one correspondence with the parameters of the mixed model can be established. The logits, derived under the class A regressive model and under the class D regressive model, can be extended to include a large variety of patterns of family dependence, as well as gene-environment interactions.     

A modeling exercise to show why population models should incorporate distinct life histories of dispersers

Dispersal is an important form of movement influencing population dynamics, species distribution and gene flow between populations. In population models, dispersal is often included in a simplified manner by removing a random proportion of the population. Many ecologists now argue that models should be formulated at the level of individuals instead of the population level. To fully understand the effects of dispersal on natural systems, it is therefore necessary to incorporate individual-level differences in dispersal behavior in population models. Here, we parameterized an integral projection model, which allows for studying how individual life histories determine population-level processes, using bulb mites, Rhizoglyphus robini, to assess to what extent dispersal expression (frequency of individuals in the dispersal stage) and dispersal probability affect the proportion of successful dispersers and natal population growth rate. We find that allowing for life-history differences between resident phenotypes and disperser phenotypes shows that multiple combinations of dispersal probability and dispersal expression can produce the same proportion of leaving individuals. Additionally, a given proportion of successful dispersing individuals result in different natal population growth rates. The results highlight that dispersal life histories, and the frequency with which disperser phenotypes occur in the natal population, significantly affect population-level processes. Thus, biological realism of dispersal population models can be increased by incorporating the typically observed life-history differences between resident phenotypes and disperser phenotypes, and we here present a methodology to do so.     

Segregation analysis of cryptogenic epilepsy and an empirical test of the validity of the results.

We used POINTER to perform segregation analysis of cryptogenic epilepsy in 1,557 three-generation families (probands and their parents, siblings, and offspring) ascertained from voluntary organizations. Analysis of the full data set indicated that the data were most consistent with an autosomal dominant (AD) model with 61% penetrance of the susceptibility gene. However, subsequent analyses revealed that the patterns of familial aggregation differed markedly between siblings and offspring of the probands. Risks in siblings were consistent with an autosomal recessive (AR) model and inconsistent with an AD model, whereas risks in offspring were inconsistent with an AR model and more consistent with an AD model. As a further test of the validity of the AD model, we used sequential ascertainment to extend the family history information in the subset of families judged likely to carry the putative susceptibility gene because they contained at least three affected individuals. Prevalence of idiopathic/cryptogenic epilepsy was only 3.7% in newly identified relatives expected to have a 50% probability of carrying the susceptibility gene under an AD model. Approximately 30% (i.e., 50% x 61%) were expected to be affected under the AD model resulting from the segregation analysis. These results suggest that the familial distribution of cryptogenic epilepsy is inconsistent with any conventional genetic model. The differences between siblings and offspring in the patterns of familial risk are intriguing and should be investigated further.     

Frequency-dependent sexual selection

Sexual selection by female choice is expected to give rise to a frequency-dependent sexual advantage in favour of preferred male phenotypes: the rarer the preferred phenotypes, the more often they are chosen as mates. This 'rare-male advantage' can maintain a polymorphism when two or more phenotypes are mated preferentially: each phenotype gains an advantage when it is rarer than the others; no preferred phenotype can then be lost from the population. Expression of preference may be complete or partial. In models of complete preference, females with a preference always mate preferentially. Models of partial preference are more realistic: in these models, the probability that a female mates preferentially depends on the frequency with which she encounters the males she prefers. Two different 'encounter models' of partial preference have been derived: the O'Donald model and the Charlesworth model. The encounter models contain the complete preference model as a limiting case. In this paper, the Charlesworth model is generalized to allow for female preference of more than one male phenotype. Levels of frequency dependence can then be compared in the O'Donald and Charlesworth models. The complete preference model and both encounter models are formulated in the same genetical terms of preferences for dominant and recessive male phenotypes. Polymorphic equilibria and conditions for stability are derived for each of the three models. The models are then fitted to data of frequencies of matings observed in experiments with the two-spot ladybird. The complete preference model gives as good a fit as the encounter models to the data of these and other experiments. The O'Donald and Charlesworth encounter models are shown to produce a very similar frequency-dependent relation. Generally, as females become less choosy, they express their preference with more dependence on male frequency, whereas the resulting selection of the males becomes less frequency dependent. More choosy females are more constant in expressing their preference, producing greater frequency dependence in the selection of the males.