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ILAR Journal V38(2) 1997
The Role of Computational Models in Animal Research
Bennett Dyke and Michael C. Mahaney
| Bennett Dyke, Ph.D., and Michael C. Mahaney, Ph.D. are in the Department of Genetics, Southwest Foundation for Biomedical Research, San Antonio, Texas. |
INTRODUCTION
The application of numerical modeling to distributions of traits as they are passed from parents to their offspring began with Mendel, who realized that proportions of offspring phenotypes produced by the various mating types followed a binomial distribution and therefore implied that an underlying binary sampling process governs inheritance. The power of this observation was confirmed repeatedly in the early years of this century, and to this day the standard test for evidence of genetic control of discrete traits is conformity to the familiar Mendelian ratios in offspring sibships--a test that is fundamentally a process of model fitting.
Although there are many discrete traits of interest in biomedical science that are amenable to simple Mendelian analysis, the great majority of common disease-related phenotypes, such as blood pressure, serum cholesterol levels, and body mass index, do not exhibit discrete states of presence or absence, but rather a degree of variability that is expressed in quantitative terms. Moreover, most of these quantitative traits do not show clear evidence of Mendelian patterns of inheritance. Much of biomedical research in the past has concentrated on the many environmental risk factors for disease, such as diet, smoking, and chemical pollutants. However, a strictly environmental approach has not been satisfactory, because degree of exposure often fails to correlate strongly with susceptibility. These apparent anomalies, combined with the observation that many diseases (or resistance to them) tend to run in families, suggest that genes, as well as environment, play a role. The same conclusion applies to quantitative phenotypes that are not directly related to disease.
The preponderance of genetic research involving nonhuman mammals has been done with highly inbred strains of rodents, where underlying genetic variability is largely removed by enforcing a mating structure of generations of close inbreeding so that individuals belonging to each strain are nearly identical both genotypically and phenotypically. Under these conditions, genetic effects may be determined experimentally by introducing or removing genetic material by selective mating between dissimilar strains, or by physical manipulation at the cellular or molecular level. These methods are not practicable with larger, relatively noninbred animals such as primates because of their longer generation times, lower reproductive rates, and expensive husbandry. Conditions for genetic research in these species contrast with highly inbred strains--individuals tend to be quite variable both phenotypically and genotypically, so that isolating the effects of particular genes must be done in a heterogeneous and often noisy background.
The larger, noninbred species, however, are often chosen as animal models of human disease, as well as for basic studies of metabolism, morphology, reproduction, and behavior, because of considerations of scale (for example, dogs and swine) or close evolutionary relationships to humans (in the case of nonhuman primates). Increasingly, the potential importance of the effects of genetic factors on phenotypes of interest in animal models is being recognized. This is because
1. The genes responsible for most of the common single gene defects are rapidly being discovered, and the human gene map is filling out. As a consequence, human geneticists are turning their attention to the search for genes that influence common diseases such as cardiovascular disease, cancer, and diabetes. The value of animal models of these diseases is increasingly a function of the extent to which effects of genes can be investigated in experimental animals.
2. It is apparent that testing drugs, vaccines, gene therapy, and diagnostic methods without making a distinction between inherited and environmental effects on the phenotypes of interest can lead to misinterpretation.
3. Genetic research is the source of much of the current progress in biology and medicine. Application of genetic analysis can make a profound contribution to our knowledge of basic physiology, morphology, reproduction, and behavior in all experimental organisms.
The challenge has been to devise methods that make it possible to determine the effects of genes on complex traits (particularly quantitative traits) in noninbred animal models. The fundamental approach has been a statistical one that looks at the distribution of phenotypes in families.
THE STATISTICAL MODELING APPROACH
Methods for statistical genetic analysis of quantitative phenotypes come originally from 2 sources. One approach, classical quantitative genetics, was developed largely for use in breeding for desirable traits in domestic animals. The other approach, segregation and linkage analysis, was originally developed to detect the influence of genetic factors in common human diseases (such as heart disease, hypertension, diabetes, and cancers) and their risk factors in human families. These are complementary approaches, appropriate for the study of quantitative phenotypes under conditions that are typical of noninbred species. They have the potential to significantly increase the value of many animal models of human disease, and to make a profound contribution to the study of basic mammalian biology.
Quantitative Genetic Analysis
The first step in statistical genetic analysis of quantitative traits is simply to determine the extent to which they are influenced by genes. The underlying assumption made in basic quantitative genetic analysis is that any phenotypic measurement P is expressed as the sum of 2 components:
where G is the component determined by genes, and E is the nongenetic (or environmental) portion. Estimates of these components are based on the assumption that the greater the effect of genes on a trait, the greater the resemblance between relatives compared to randomly selected population members. Measurements of the phenotype are made on a relatively large sample of families, and evaluation of the relative effect of the genetic and environmental components is done in terms of the variance:
where Vp is the phenotvpic variance, or simply the variance of the quantitative trait as measured in the population sample, VG is the genetic variance estimated from correlations between various classes of relatives, and VE is the residual environmental variance. The most commonly used variance-based measure of genetic contribution to a trait is the heritability (abbreviated as h2) which is simply the ratio of the genetic variance to the total phenotypic variance.
We have calculated heritability for high density lipoprotein cholesterol (HDL-C) in the pedigreed baboon colony at the Southwest Foundation for Biomedical Research (SFBR). In humans, high levels of HDL-C are thought to be protective against atherosclerosis. This work was done as part of a large project aimed at understanding genetic effects on cholesterol metabolism under contrasting high- and low-fat diets (MacCluer 1993). In this study, HDL-C values ranged from 16 to 130 mg/dl (mean 61.96 and variance 233.40) in 373 animals in the pedigree when they were measured on a normal monkey chow diet. An analysis done with the Pedigree Analysis Package (PAP) (Hasstedt 1989) gave a rough estimate for the heritability of this trait of h2 = 0.48, that is, about 48% (a substantial proportion) of the variability in the trait on this diet is attributable to genotype.
The simple model given by Equation 2 may be insensitive to the effect of genes when the variance of a quantitative trait is influenced strongly by factors such as age, sex, and environmental variables known to affect the measurements of interest. A more detailed model that partitions out "noise" due to the effects of covariates such as age and sex is often used to improve the analysis. An example of a simple extension to Equation 2 is as follows:
where the phenotypic variance is expressed as a linear function of measurements made on each individual (g i) adjusted for covariate effects (b xj), in addition to the genetic and environmental variance components as shown above in the previous equation.
These basic quantitative genetic methods have important applications in plant and animal breeding, but are relatively new to laboratory animal and biomedical science. One deficiency of these methods is that, while they can elucidate the role of genetics in a general sense, they do not evaluate the effects of individual gene loci. It is in this crucial area that statistical modeling methods become most important.
Segregation Analysis
As mentioned above, the key to inferring genetic effects on a phenotype rests on measuring the resemblance between relatives. A complicating factor, of course, is the fact that family members typically share a number of environmental features as well as genes. For example, human families typically share residence and diet, which may make them alike in the way they express a given phenotype. Such similarities are also found in nonhuman species, but often can be manipulated experimentally more readily than is the case with humans. Segregation analysis is a statistical modeling method aimed at decomposing the sources of family resemblance, with the particular aim of detecting the effects of single gene loci on a trait. It is based on the idea that there are several possible sources of family resemblance. Three basic sources are
a. Nongenetic factors, called shared environmental factors, may contribute to family resemblance.
b. Individually unidentifiable genes each having a very small effect, working together with a large number of other such genes, can influence the phenotype. Differences in the composition of this set of polygenes might explain phenotypic differences between families.
c. A single very influential gene may affect the phenotype. Such a gene may be present in some individuals in some families, but completely absent in other families, differences which might explain phenotypic variability both within and between families. A single gene of this sort is termed a major gene.
Each of these 3 sources of similarity (if operating independently) results in a unique distribution of phenotypes in a family. A phenotype influenced solely by a major gene is distributed according to Mendel's laws, which give rise to predictable ratios of individuals with different phenotypes among the members of the same family. Mendel's laws also can account for differences in phenotypic ratios between males and females, and the tendency for the phenotype sometimes to skip generations. For many qualitative traits, family members tend either to have the phenotype, or not, because they either have inherited the gene responsible for it from a parent, or they have not. Major genes can also influence quantitative traits, although their effect in such cases may require statistical analysis to detect.
A phenotype caused by polygenes tends to have a wide range of expression. Offspring of the same parents tend to be affected similarly, and the expression of their phenotypes usually lies between that of their parents. An environmentally caused phenotype is less likely than one that is inherited to show differences in incidence and expression based on family relationship. Such a phenotype is more likely to be shared by both related and unrelated members of the same household, for example.
With statistical modeling techniques (Elston and others 1986; Hasstedt 1989; Lange and others 1988), the presence of major genes can be detected, even when the pattern that reveals the gene is partially hidden by polygenes and environmental factors. Building on the basic quantitative genetic model of Equation (2), segregation analysis further partitions the genetic variance component into a portion (VMG) due to a major locus and a portion (VPG) due to polygenes as follows:
This simple model can be extended in a fashion analogous to Equation (3) so that the phenotypic distribution is modeled as a linear function of these 3 variance components, plus the effects of selected covariates.
In operation, the technique of segregation analysis is basically one of systematic trial and error: models consisting of various major gene/polygene/environmental combinations are tried until one is found that best fits the phenotype pattern observed in the sample of families being analyzed. An understanding of the contributions of polygenes and environment is important, but discovery of a major gene is by far the most significant and exciting result of these analyses, because it is a first step toward the application of molecular genetic analysis and the potential for sophisticated new methods of diagnosis and treatment.
We again use the example of HDL-C in baboons to show the application of segregation analysis. In this analysis, MacCluer and others (1988) tested a series of models using PAP (Hasstedt 1989). Underlying the method is the assumption that the distribution of the trait in the population can be decomposed into the sum of 3 separate phenotypic distributions, each with its own mean (m1, m2, and m3). Associated with each of these phenotypic distributions is a putative genotype for a major locus (A1A1, A1A2, and A2A2). Likewise, associated with each genotype is a transmission parameter tau (t1, t2, and t3), which is the probability that allele A1 is passed from parent to offspring. The phenotypic distributions share a common variance and a common polygenic heritability, or the proportion of the variability due to polygenes for each distribution. Transmission models to be tested are constructed by defining various combinations of parameters, some of which may be set at fixed values, while others are estimated, that is, they are free to change during the operation of the program until they reach values that make the model fit most closely to the observed phenotypic distribution.
In this study, 9 models were tested, of which we show 5 that best illustrate the method. The fullest, or general transmission model used here incorporates 9 parameters. These are the means of the 3 distributions, their single common variance and polygenic heritability, the 3 transmission probabilities of the underlying putative genotypes, and the frequency of allele A1 (which is all that is needed to define the 3 genotype frequencies under assumptions of random mating). Because it incorporates the greater number of parameters, we expect this model to fit the observed phenotypic distribution better than models based on fewer parameters.
Although the general transmission model is the one against which all others are evaluated, it is usually easier in practice to begin estimating parameters in simpler models, and to work up to more complex models with more parameters. Following this practice, a sporadic model was constructed by constraining the 3 distribution means to be equal to each other (but otherwise free to vary), the transmission probabilities to 1.0, the polygenic heritability to 0, and the frequency of A1 to 1.0. This means that parameters of a single distribution are estimated without the influence of any genetic factors whatsoever, making all phenotypic variation a function of unmeasured environmental factors alone. In this study, the estimated values shown in Table 1 (mean 66.4, variance 323.28) recovered the observed phenotypic mean and variance. Also shown are the natural log likelihood (-2738.9) by which the model is scored, and the c2 values resulting from comparison with the general model.
A polygenic model was tested next. Parameters were set in the same way as in the sporadic model, except that in addition to the mean and variance of the single distribution, the polygenic heritability was estimated (that is, unfixed and allowed to vary). The log likelihood of this model (-2660.0) is greater than that for the sporadic model, and a chi square test (c2 = 157.8, 1df, P < .0001) indicates a significantly better fit to the observed distribution when genetics is taken into account. This analysis yielded the estimate of heritability (h2 = 0.48) reported above. This is only an approximate estimate of the total heritability, since it is based on the assumption that the genetic contribution to the trait is entirely polygenic, which may not be the case if evidence for a major gene emerges from subsequent stages of the segregation analysis.
A codominant mixed model, tested next, adds a codominant major gene to the polygenic model (a codominant gene locus is one in which the heterozygous genotype has an expression intermediate between the homozygotes). Heritability is estimated as in the polygenic case, but now phenotypic values are no longer constrained to a single distribution, and 3 separate means corresponding to genotypes A1A1, A1A2, and A2A2 are estimated, as is the frequency of allele A1. Transmission probabilities for this allele are set to the Mendelian ratios 1, 1/2 and 0. Respectively, these values represent the probabilities of transmitting the A1 allele to an offspring when the parental genotype is A1A1 (t1 = 1), A1A2 (t2 = 1/2) or A2A2 (t3 = 0). The heritability shown here (h2 = 0.18) is an estimate of the effect of polygenes on each of the three separate distributions, rather than on the phenotypic distribution as a whole. Comparison of log likelihoods indicates a significantly better fit of this model than the polygenic model alone (c2 = 81.2, 1df, P < .0001). Although this result is suggestive of the presence of a major gene, convincing evidence requires 2 more steps.
First, parameter estimates of the candidate model must be close to those of the general transmission model. The general model differs from the codominant mixed model in that it estimates all parameters, including transmission probabilities, which are no longer constrained to the Mendelian ratios. Comparison of the entries for the 2 models in Table 1 shows parameters to be in remarkably good agreement. A formal statistical requirement is that the fit of the candidate model must not be significantly worse than the general transmission model. That this is so can be seen from the c2 value of 2.99 (2df, P = .2238) for the codominant mixed model.
Second, to be certain that the better fit was not simply the result of changing the number of phenotypic distributions from 1 to 3, a so-called environmental mixed model is run. This model is parameterized like the codominant mixed model, except that transmission probabilities are estimated, although they are all constrained to be equal to the frequency of allele A1. This allows 3 phenotypic distributions, but assumes that they are the result of some major non-Mendelian ("environmental") factor rather than a major gene. As in the codominant mixed model, variation around the means of each of the distributions can be influenced by polygenes. The fit of the environmental model must be significantly worse than the general transmission model; the c2 value of 31.03 (1df, P < .0001) indicates that this is the case.
Thus, we have evidence for the existence of a major codominant gene locus (not dominant/recessive) plus polygenes contributing to HDL-C levels on a chow diet in baboons. Table 2 gives parameter estimates derived from the model. Genotype frequencies are Hardy-Weinberg proportions computed from the frequency of allele A1, and the number of animals is based on the 373 members of the study population. Although the calculations are not given here, from the information in Table 1 it is quite simple to estimate the relative contributions of the major gene, polygenes, and environment to the total phenotypic variance. In this case, the major gene accounts for 34.9% of the total variance in HDL-C, polygenes for 11.7%, and the remaining 45.6% is attributed to environmental factors that are random with respect to genotype.
A useful byproduct of this analysis is that genotypes of the major gene can be predicted on a probabilistic basis for each individual in the pedigree. In animal models, this makes it possible to choose individuals for purposes of experiment or breeding that are likely to be carrying genes that influence the trait in question, even though the function of the actual gene has not been identified. For example, in comparative physiologic and molecular studies of high and low HDL-C, choosing matched samples of animals from the 15 presumptive A1A1 high HDL-C, and the 239 A2A2 low HDL-C distributions increases the probability that metabolic differences are endogenous, rather than due to exogenous environmental causes.
This example is a deliberately simplified one. As pointed out above, for purposes of illustration we have abbreviated our description of the full sequence of models actually tested in the search for an HDL-C major gene. Furthermore, the models developed in this study were relatively simple compared to many other segregation analyses, which may include more than one trait or major locus, interaction effects, and correlates such as age and sex. The ability to model such biological complexity makes the method a powerful and flexible tool for analysis of any biological or behavioral trait that can be expressed as a quantitative measure. Segregation analysis can detect the presence of a major gene, but cannot identify its function or chromosomal location in the genome. Nonetheless, the method has important applications in noninbred species for which a detailed gene map has not yet been developed.
Linkage Analysis
Quantitative trait linkage analysis is the next logical step in characterizing genetic influences on continuously distributed traits. Classical genetic linkage analysis uses statistical methods to determine the likelihood that 2 or more genetic loci exhibiting simple Mendelian inheritance are located on the same chromosome. These may be either genes with known function, or "anonymous" markers whose function is unknown. Quantitative trait linkage analysis extends these methods, making it possible to determine linkage relationships between discrete loci and a quantitative trait locus (or QTL).
There are 2 general approaches to quantitative trait linkage analysis: penetrance model-based and nonpenetrance model-based methods. Penetrance model-based linkage analysis is an extension of segregation analysis, and usually begins with detection of a major gene in the manner described above. The best estimates of the segregation model parameters (such as genotypic means and variances, polygenic heritability, major gene frequency, and covariate effects) are then used as starting values for a new combined segregation and linkage analysis that includes parameters such as the marker gene frequency and the recombination frequency (a measure of the relative distance between the marker locus and the major locus). In the simplest test for evidence of linkage, a model in which the recombination fraction is constrained to 0.50 (indicative of no linkage) is compared to that of a model in which the recombination is estimated. A significant difference between the likelihoods of these models is interpreted as evidence for linkage between the 2 loci.
A combined segregation and linkage analysis was used in a recent study of the sources of variation in low density lipoprotein cholesterol (LDL-C) in 1183 pedigreed baboons at SFBR (C.M. Kammerer, personal communication). Segregation analysis found evidence for a major gene for this phenotype (Konigsberg and others 1991), and the combined analysis tested for linkage between the major locus and a DNA marker in the LDL receptor locus. A likelihood ratio test comparing a model with tight linkage against one with no linkage (recombination fraction equal to 0.50) rejected the no linkage model at p = 0.0009, equivalent to a log10 odds ratio (lod score) of 2.4, which is highly suggestive of linkage in this chromosomal region.
Nonpenetrance model-based linkage analysis methods have been developed to test for linkage between marker loci and QTLs in the absence of prior statistical evidence for a major gene. These approaches are often referred to as "allele-sharing'' methods because they rely on the probability that any 2 individuals will share alleles that come from a common ancestor (that is, alleles that are identical by descent or 1BD), which is a function of their kinship. For example, full sibling pairs have a higher probability of sharing alleles that are IBD than do half siblings. The best known application of allele-sharing methods to quantitative trait linkage analysis is the sib-pair test (Haseman and Elston 1972). This test estimates the correlation of the squared difference in trait values between pairs of sibs, and the probability that the pairs are IBD at a marker locus. A correlation significantly less than zero implies linkage of the marker and a QTL influencing the trait.
The sib-pair method has been popular in studies of human linkage because it requires collection of data only from pairs of sibs and their parents, rather than from extended pedigrees. The power of the method depends on the availability of large numbers of sib pairs, however, and the savings in data collection may be offset by loss of statistical power to detect linkage. When extended pedigrees are readily available for study (as is the case in many laboratory animal colonies), limiting an allele-sharing analysis to sibs alone wastes the allele-sharing information inherent in all other classes of relatives. A more promising allele-sharing approach to quantitative trait linkage analysis is an extension of the variance component methods used in quantitative genetic analysis. Unlike combined linkage and segregation analysis, these methods (Goldgar 1990; Amos 1994; Blangero and Almasy 1996) do not estimate the parameters of a major gene (allele frequencies and transmission probabilities), but they do require calculation of IBD probabilities for relationship pairs beyond the nuclear family (such as grandparent-grandchild; uncle-nephew; and 1st, 2nd, 3rd cousins). Recently developed algorithms (Duggirala and others 1996) have reduced the computational burden of applying this approach to complex pedigrees that typify animal colonies. At SFBR we have begun to map and screen the baboon genome in search of genes influencing bone mineral density and other phenotypes related to osteoporosis using variance component linkage methods.
Data requirements for quantitative trait linkage analysis are greater than those of segregation or quantitative genetic analysis, since it requires knowledge of marker loci that are mapped to specific chromosomes, as well as measurements on phenotypes of interest. As of October 1996, slightly more than one sixth of the approximately 100,000 loci in the human genome had been mapped, and it is anticipated that a complete map will be done by the year 2005 (Schuler and others 1996). Clearly, gene maps of this resolution are not likely to be available for most animal models of human disease in the near future. Nonetheless, candidate gene libraries have already been established for at least 30 nonhuman animal species, and concerted gene mapping efforts are under way for a variety of species (including mice, pigs, rats, chickens, baboons and laboratory opossums). Moreover, chromosomal homologies and the fact that many marker typing reagents developed for humans work well with closely related species make it possible to map interesting genes to small chromosomal regions without the need for full-scale gene maps, as we have been doing with bone density genes in baboons at SFBR.
STATISTICAL GENETIC MODELING AND THE USE OF ANIMALS
Managing the production of animals used in genetic research may require more attention to colony breeding and demographic structure than is common for animals used in other kinds of biomedical research. Relatively large samples of animals (N³200) with both parents known are often required for analysis, with phenotypic measurements made on as many of these individuals possible. Nuclear families should be structured so that relatively large full sibships (offspring sharing both parents) are available (4-6 individuals per sibship). Large half-sibships (offspring sharing one parent) are acceptable, but to simplify analyses, an animal that has bred with more than one mate should not be mated to an animal that also has bred with more than one mate. Since for many species there are usually more breeding age females than males in a colony, in practice it may be simpler to avoid producing maternal half-sibships. Pedigrees of 3 to 4 generations in depth are desirable, although extremely deep pedigrees in most colonies tend to become overly complex and may require simplification by ignoring links between families prior to analysis.
Other aspects of breeding structure may depend on the kind of analysis being done. Segregation analysis requires a normal range of variability in the population. This means that animals should not be bred selectively for affected or extreme phenotypes, particularly if these are the phenotypes to be analyzed. For segregation analysis it is best that inbreeding be avoided, although mating between distant relatives can be handled by breaking pedigree links. On the other hand, the power of linkage analysis can be improved by the appropriate use of inbreeding, as well as by selecting for extreme phenotypes.
On balance, a breeding plan that minimizes inbreeding and avoids selection for extreme phenotypes is probably the safest strategy for the moment. This is because segregation analysis is likely to be an important analytic method of genetic analysis in the absence of detailed gene maps, and because such a strategy increases the probability that a given breeding colony will be useful for the analysis of a broad variety of phenotypes. The situation may change in the future, however, as gene maps for experimental animals become more extensive, and linkage analysis becomes the method of choice for identifying genes underlying disease-related phenotypes.
The way in which pedigreed animals are used as research subjects also depends to some extent on the life history of the species in question. With rapid and prolific breeders having large litters and short generation times (such as rodents) it may be possible to replace entire pedigrees quickly enough that sacrifice of key family members is not a serious concern. In contrast, with species (such as nonhuman primates) that reproduce relatively slowly, key family members often must be held for breeding purposes, making it difficult to select subjects for invasive or life-threatening experiments, or for long periods of experimental isolation, which can seriously disrupt a breeding program. In the latter case especially, a premium is usually placed on maintaining animals in the pedigree throughout their lives in a healthy condition with minimal experimental intervention. Phenotypes are preferably derived from venipuncture or other innocuous tissue sampling, morphometrics, radiation and sonic imaging, and other such methods. It is important to note, however, that virtually all breeding colonies can produce more offspring than are required to replace those lost through natural mortality. Pedigreed colonies frequently serve as an excellent source of production for animals used in non-genetic research, with a core of infants carefully selected on the basis of family relationships and reserved for breeding and genetic research, and the remainder (often a substantial majority) supplied to the biomedical community at large.
REFERENCES
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ACKNOWLEDGMENTS
Supported by NIH Grants HL28972, RR09950 (BD) and HL54141 (MCM). Comments by Drs. J. Blangero, S. Williams-Blangero, J. MacCluer, and B. Mitchell of the Southwest Foundation are gratefully acknowledged.
TABLE 1 Segregation analysis parameters for 5 transmission models
| m1 | m2 | m3 | s2 | h2 | f(A1) | t1 | t2 | t3 | InL | c2 | df | |
| Sporadic | 66.4 | 323.28 | [0] | [0] | [0] | [0] | [0] | -2738.9 | 243.99 | 6 | ||
| Polygenic | 64.0 | 277.22 | .48 | [0] | [0] | [0] | [0] | -2660.0 | 86.16 | 5 | ||
| Codominant mixed | 59.3 | 67.9 | 104.5 | 159.52 | .18 | .80 | [1] | [.5] | [0] | -2619.4 | 2.99 | 2 |
| General | 59.3 | 68.1 | 105.4 | 156.00 | .20 | .79 | 1.0 | .53 | .08 | -2617.9 | 0 | |
| Environmental | 59.3 | 74.7 | 109.5 | 136.89 | .84 | .81 | .81 | -2633.4 | 31.03 | 1 |
Key:
For putative genotypes A1A1, A1A2, A2A2
m1, m2, and m3 are the means of the associated phenotypic distributions
s2 is the common variance of the distributions
h2 is the shared polygonic heritability of the distributions
f(A1) is the frequency of allele A1 in the population
t1, t2, and t3 (tau values) are the associated probabilities that allele A1 is passed from parent to offspring
For each model
numbers in square brackets indicate fixed values
InL is the natural log likelihood of the model tested
TABLE 2 Parameter estimates from the codominant mixed model
| Genotype | A1 A1 | A1A2 | A2 A2 |
| Frequency | .64 | .32 | .04 |
| Number of animals | 239 | 119 | 15 |
| Mean HDL-C levels | 59.3 | 67.9 | 104.5 |
| Variance = 159.52 h2 =0.18 |
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