Normal linear models with genetically structured residual variance heterogeneity: a case study

Normal mixed models with different levels of heterogeneity in the residual variance are fitted to pig litter size data. Exploratory analysis and model assessment is based on examination of various posterior predictive distributions. Comparisons based on Bayes factors and related criteria favour models with a genetically structured residual variance heterogeneity. There is, moreover, strong evidence of a negative correlation between the additive genetic values affecting litter size and those affecting residual variance. The models are also compared according to the purposes for which they might be used, such as prediction of 'future' data, inference about response to selection and ranking candidates for selection. A brief discussion is given of some implications for selection of the genetically structured residual variance model.     

Litter sex ratios in Richardson’s ground squirrels: long-term data support random sex allocation and homeostasis

When costs of producing male versus female offspring differ, parents may vary allocation of resources between sons and daughters. We tested leading sex-allocation theories using an information-theoretic approach and Bayesian hierarchical models to analyse litter sex ratios (proportion males) at weaning for 1,049 litters over 24 years from a population of Richardson’s ground squirrels (Urocitellus richardsonii), a polygynandrous, annually reproducing mammal in which litter size averages from six to seven offspring and sons are significantly heavier than daughters at birth and weaning. The model representing random Mendelian sex-chromosome assortment fit the data best; a homeostatic model received similar support but other models performed poorly. Embryo resorption was rare, and 5 years of litter data in a second population revealed no differences in litter size or litter sex ratio between birth and weaning, suggesting that litter size and sex ratio are determined in early pregnancy. Sex ratio did not vary with litter size at weaning in any of 29 years, and the observed distribution of sex ratios did not differ significantly from the binomial distribution for any litter size. For 1,580 weaned litters in the two populations, average sex ratio deviated from parity in only 3 of 29 years. Heavier females made a greater reproductive investment than lighter females, weaning larger and heavier litters composed of smaller sons and daughters, but litter sex ratio was positively related to maternal mass in only 2 of 29 years. Such occasional significant patterns emphasize the importance of multi-season studies in distinguishing infrequent events from normal patterns.     

Optimal offspring sizes in small litters

Summary Numerous evolutionary models explore the trade-off between offspring size and offspring number. However, such models often fail when the number of offspring is small because optimal litter size (or litter size at optimal offspring size) may fall between the necessarily integer values for real litters. This paper extends a classic model for optimal investment per offspring to the case of small litters and predicts that range in offspring size and the largest (smallest) offspring size should decline (increase) with increased litter size. Application of the model to egg size data from a poeciliid fish,Gambusia hubbsi, reveals a surprisingly close approximation to the largest offspring size and variation in offspring size at small litter sizes.     

Mammalian reproductive strategies: A generalized relation of litter size to body size

Summary A generalized relationship of litter size to mammalian body size was predicted by a graph model. The model was used to generate hypotheses explaining specific features of variation in gestation time, relative litter weight, birth weight, and reproductive capacity. The predictions were tested by means of data from the literature.Mammals were assumed to maximize neonatal survival of offspring to the limits allowed by litter weight per female body weight. Gestation time correlated negatively with the foetal growth rate of relative litter weight. Gestation time did not correlate with the foetal growth rate of individual offspring.Relative litter weight correlated negatively with adult body weight. This relationship was explained by the higher assimilation rate per unit weight relative to metabolic rate in small mammals.Birth weight correlated positively with body weight. However, small mammals produce larger offspring than predicted by the linear relationship of birth weight to body weight in large mammals. There is obviously a minimum birth weight which cannot be decreased without special arrangements for parental care.The prediction of the relationship of litter size to body size was derived from the relations of relative litter weight and birth weight to body weight. In small mammals (less than 1 kg) litter the correlation was negative. When litter size was compared with body length, the correlation was positive in small mammals (less than 30 cm) and negative in large mammals. In both sets of data there was a negative overall correlation between litter size and body size.Reproductive capacity, defined as the number of offspring per season, correlated negatively with life-span.     

A shared response model for clustered binary data in developmental toxicity studies

Existing distributions for modeling fetal response data in developmental toxicology such as the beta-binomial distribution have a tendency of inflating the probability of no malformed fetuses, and hence understating the risk of having at least one malformed fetus within a litter. As opposed to a shared probability extra-binomial model, we advocate a shared response model that allows a random number of fetuses within the same litter to share a common response. An explicit formula is given for the probability function and graphical plots suggest that it does not suffer from the problem of assigning too much probability to the event of no malformed fetuses. The EM algorithm can be used to estimate the model parameters. Results of a simulation study show that the EM estimates are nearly unbiased and the associated confidence intervals based on the usual standard error estimates have coverage close to the nominal level. Simulation results also suggest that the shared response model estimates of the marginal malformation probabilities are robust to misspecification of the distributional form, but not so for the estimates of intralitter correlation and the litter-level probability of having at least one malformed fetus. The proposed model is fitted to a set of data from the U.S. National Toxicology Program. For the same dose-response relationship, the fit based on the shared response distribution is superior to that based on the beta-binomial, and comparable to that based on the recently proposed q-power distribution (Kuk, 2004, Applied Statistics53, 369-386). An advantage of the shared response model over the q-power distribution is that it is more interpretable and can be extended more easily to the multivariate case. To illustrate this, a bivariate shared response model is fitted to fetal response data involving visceral and skeletal malformation.     

Response to selection for litter size in Danish Landrace pigs: a Bayesian analysis

A replicated selection experiment aimed at increasing litter size (total number of pigs born per litter) in Danish Landrace pigs was conducted from 1984 to 1991. The experiment included two selection and two control lines. In each generation, 30 and 14 first litters were produced in selection and control lines, respectively, and dams produced two litters. Each replicate, consisting of one selection and one control line, was founded from 60 families chosen randomly from the population at large. Family selection was practiced, and the criterion was the predicted breeding value for litter size computed using a repeatability animal model, and taking into account all available information. The data consisted of 947 records from 523 dams (424 dams had two litters) representing five cycles of selection of increased litter size. Data were analyzed from a Bayesian perspective, based on marginal posterior distributions of genetic parameters of interest. Marginalization was achieved using Gibbs sampling, with a single chain length of 1 205 000. After discarding the first 5 000 iterations, a sample was drawn every ten iterations, so 120 000 samples in total were saved. Densities were estimated and plotted, and summary statistics were computed from the estimated densities. The posterior means (± standard error) of heritability and repeatability were 0.22 ± 0.06 and 0.32 ± 0.05, respectively. These point estimates of genetic parameters were within the range of literature values, although on the high side. The posterior mean (± standard error) of genetic response to selection, defined as the difference between the mean breeding values of the selected lines and that of the base population, was 1.37 ± 0.43 pigs after five cycles of selection. The regression (through the origin) of breeding values in the selected lines on generation was 0.25 ± 0.08 pigs. Several informative priors constructed from information obtained with field data in this population were used to examine their influence on inferences. The priors were influential because of the relatively small scale of the experiment. An analysis excluding data from one of the control lines gave smaller genetic variance and heritability, and a smaller response to selection. However, it appears that selection for litter size is effective, but that the true rate of response is probably smaller than data from this experiment suggest.     

Bio-economic modelling of sheep meat production systems with varying flock litter size using field data

Sheep meat producers derive the majority of income from sales of weaned lambs, determined by flock conception rates, litter size, and lamb survival. Field data from commercial flocks can inform sensitivity analyses of the effect of litter size on flock productivity, feed demand, and gross margin. This study adapted an established bio-economic model of a flock of breeding ewes informed by statistical relationships (from linear models) between flock litter size (lambs born per ewe lambing) and production factors (such as flock barren rate, litter birth type and lamb birth weight) identified using 156 145 animal records from the Irish national sheep breeding database. Sensitivity analyses were undertaken to investigate the effects of flock litter size on flock production, feed demand, and gross margin. Results showed that as flock litter size increased, the proportion of lambs born as multiples increased, with 14 % of lambs born as singles when flock litter size was 2.2 lambs born per ewe lambing. Flock gross margin increased from €2 205 to €7 730 as litter size increased from 1.0 to 2.0 lambs born per ewe lambing. As litter size increased from 1.0 to 2.2 lambs born per ewe lambing, flock gross margin increased linearly by, on average, €52 per 0.01 increase in litter size. At a litter size of > 2.2 lambs born per ewe lambing, flock gross margin increased on average €12 per 0.01 increase in litter size. At a litter size of 2.2 lambs born per ewe lambing, flock efficiency (at 65.0 kg of lamb weaned per ewe presented for breeding), weaning rate (at 1.5 lambs weaned per ewe presented for breeding; not including excess lambs from large litters sold within a week after birth and thus not weaned on-farm), and gross margin (at €8 500) began to plateau. The results indicate lower marginal returns in gross margin at very high flock litter size, due to the lower value of additional lambs born as triplets and quadruplets compared with single- and twin-born lambs. However, the diminishing economic returns occurred at higher flock litter size than are currently biologically achieved in most flocks. Quantification from this analysis demonstrates how the value of increasing the number of lambs born changes at very high flock litter size, which can inform the priorities and performance benchmarking for international sheep meat production industries.     

Unit pricing: Economics and the evolution of litter size

Variability in litter size and the concept of optimality are central to our current understanding of parental investment patterns and life histories. A fundamental component of most models of optimum litter size is an apparently inescapable trade-off between litter size and size of offspring. Most previous models of litter size have focused on the evolution of an optimum litter size rather than variability in litter size. Because variability provides the raw material from which numerous optimal litter sizes are fashioned to meet prevailing conditions, approaches that specifically address adaptive patterns of variability might provide new insights into the evolution of litter size. A model based on a non-linear relationship between total parental effort and litter size reduces trade-offs between offspring size and offspring number to simple reproductive economics, and illustrates how litter size variability might be predictable under certain environmental conditions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Genetic analysis of ewe productivity traits in Moghani sheep

Genetic and environmental (co)variance components for productivity traits in Moghani sheep were estimated using data from 1344 ewes. The data were collected in the Jafarabad breeding station, north-east of Iran, during a 13-year period (1995-2008). The studied traits were litter size at birth (LSB), litter size at weaning (LSW), litter mean weight per lamb born (LMWLB), litter mean weight per lamb weaned (LMWLW), total litter weight at birth (TLWB) and total litter weight at weaning (TLWW). A model including direct additive genetic effects as well as permanent environmental effects related to repeated records of ewe was the most appropriate model for all the studied traits. Genetic parameters were estimated applying restricted maximum likelihood (REML) procedure. Direct heritability estimate for LSB, LSW, LMWLB, LMWLW, TLWB and TLWW were 0.11, 0.02, 0.15, 0.07, 0.07 and 0.06, respectively. Corresponding values for repeatability estimates were 0.16, 0.19, 0.18, 0.11, 0.13 and 0.09. Genetic correlations between the studied traits ranged from −0.99 for LSB-LMWLB and LSW-LMWLB to 0.99 for LSB-TLWB. Phenotypic and environmental correlation estimates were generally lower than those of genetic ones. Estimates of permanent environmental correlation among traits were positive and medium to high. Although low direct heritabilities were estimated for the reproductive traits, as these traits are of interest then they should be included in a breeding program.     

Behavioural differences in sub-adult female mice exposed to a murine elevated plus-maze: correlated effects of selection for high litter size

The present experiment compared the anxiety- and activity-related behaviour of sub-adult females from a mouse strain selected for over 106 generations for high litter size with that of a randomly selected control strain, to illuminate possible differences in their ability to cope with exposure to a novel environment. Selected for large litter size, the H-strain has an average litter size of 21 pups, whereas the randomly bred C-strain has an average litter size of 10 pups. The elevated plus-maze was used to measure the behaviour of the mice expressed in response to novelty. The results are described and discussed in relation to the ability to cope with novel environmental challenges. In the elevated plus-maze, the H-strain was significantly more anxious (having a lower percentage of entries into and percentage of time spent on open arms) and less active (having a lower number of entries into closed arms, and a lower total number of arm entries) than the C-strain. Thus, there were clear anxiety- and activity-related differences between the strains, which may be related to selection for large litter size. A tentative hypothesis is presented whereby selection for large litter size may accelerate adaptation to the home environment, but decrease the ability of selected animals to cope with exposure to novelty.     

Genetic variation in spontaneous ovulation rate and LH receptor induction in mice

Selection for litter size (Line S1) and for post-weaning body weight gain (Line G) increased spontaneous ovulation rate in mature females by 69 and 73%, respectively, over that of randomly bred control mice (Line C). Inbreeding from S1 mice with selection for litter size produced highly inbred lines with elevated ovulation rates. Inbreeding from Line C mice produced a 21% divergence among lines, but did not depress the mean ovulation rate. Crosses of these lines revealed little heterosis in ovulation rate. LH receptors were induced by treating females from 22 days of age with diethylstilboestrol for 4 days and FSH for 2 days. The in-vitro binding of 125I-labelled hCG per microgram DNA decreased 56% in response to selection for litter size and increased 57% in response to selection for body weight gain, indicating high susceptibility of this trait to genetic change. Inbreeding from Line C mice produced a 135% divergence amongst lines, but did not depress the mean LH receptor induction. Body weight had significant effects on ovulation rate and LH receptor induction. These results show that selection for litter size and for rapid post-weaning body weight gain increases ovulation rate, but we suggest that different mechanisms are involved in these responses.     

Genotype-dependent responses to levels of sibling competition over maternal resources in mice

Research on phenotypic plasticity has often focused on how a given genotype responds to the changing physical environments such as temperature or diet. However, for many species the social environment has an equally important role because of competition for resources. During early development, the level of competition for limited (maternally provided) resources will often depend critically on the number of siblings. Therefore, competition among siblings should drive the evolution of genes that allow flexible responses to realized levels of competition and maternal resource availability. However, it is unknown whether genetically based differences between individuals exist in their response to the social environment that affect their future development. Using a quantitative trait locus approach in an experimental population of mice we demonstrate that effects of sibling number on body weight depend on individual genotype at seven loci, over and above the general negative litter size effect. Overall, these litter size-by-genotype interactions considerably modified the degree to which increasing litter size caused reduced weight. For example at one locus this effect leads to a 7% difference in body weight at week 7 between individuals experiencing the extremes of the normal range of litter sizes in our population (five to nine litter mates). The observed interaction between genotype and the competitive environment can produce differences in body weight that are similar in magnitude to the main effect of litter size on weight. Our results show that different genotypes respond to the social environment differentially and that interaction effects of genotype with litter size can be as important as genotype-independent effects of litter size.     

Bayesian analysis of response to selection: a case study using litter size in Danish Yorkshire pigs

Implementation of a Bayesian analysis of a selection experiment is illustrated using litter size [total number of piglets born (TNB)] in Danish Yorkshire pigs. Other traits studied include average litter weight at birth (WTAB) and proportion of piglets born dead (PRBD). Response to selection for TNB was analyzed with a number of models, which differed in their level of hierarchy, in their prior distributions, and in the parametric form of the likelihoods. A model assessment study favored a particular form of an additive genetic model. With this model, the Monte Carlo estimate of the 95% probability interval of response to selection was (0.23; 0.60), with a posterior mean of 0.43 piglets. WTAB showed a correlated response of -7.2 g, with a 95% probability interval equal to (-33.1; 18.9). The posterior mean of the genetic correlation between TNB and WTAB was -0.23 with a 95% probability interval equal to (-0.46; -0.01). PRBD was studied informally; it increases with larger litters, when litter size is >7 piglets born. A number of methodological issues related to the Bayesian model assessment study are discussed, as well as the genetic consequences of inferring response to selection using additive genetic models.     

Litter size at birth in purebred dogs--a retrospective study of 224 breeds

Despite the long history of purebred dogs and the large number of existing breeds, few studies of canine litter size based upon a large number of breeds exist. Previous studies are either old or include only one or a few selected breeds. The aim of this large-scale retrospective study was to estimate the mean litter size in a large population of purebred dogs and to describe some factors that might influence the litter size. A total of 10,810 litters of 224 breeds registered in the Norwegian Kennel Club from 2006 to 2007 were included in the study. The overall mean litter size at birth was 5.4 (± 0.025). A generalized linear mixed model with a random intercept for breed revealed that the litter size was significantly influenced by the size of the breed, the method of mating and the age of the bitch. A significant interaction between breed size and age was detected, in that the expected number of puppies born decreased more for older bitches of large breeds. Mean litter size increased with breed size, from 3.5 (± 0.04) puppies in miniature breeds to 7.1 (± 0.13) puppies in giant breeds. No effect on litter size was found for the season of birth or the parity of the bitch. The large number of breeds and the detail of the registered information on the litters in this study are unique. In conclusion, the size of the breed, the age of the bitch and the method of mating were found to influence litter size in purebred dogs when controlling for breed, with the size of the breed as the strongest determinant.     

Sex allocation in a polygynous mammal with large litters: the wild boar

Predictions from Trivers & Willard's (1973, Science, 179, 90-92) hypothesis of sex-biased maternal investment in polygynous species do not apply well to species where mothers produce more than one offspring per reproductive attempt. First, as litter size increases, the benefits to the mother of adjusting sex ratio decrease because (1) she could benefit more by adjusting litter size and (2) sex differences in reproductive potential are negatively related to litter size. Second, testing sex-biased investment in these species requires predictions about the simultaneous adjustment of sex ratio and litter size. The wild boar, Sus scrofa, although polygynous, produces large litters. Here we present data for 58 litters from a free-ranging wild boar population in central Spain. Maternal expenditure per individual offspring, as measured by piglet weight, was higher for male than female fetuses. In more than 81% of cases the heaviest fetus in the litter was a male regardless of the quality of the mother; this might have influenced his ranking within the 'teat order' and consequently his development and survival. Mother quality (size and weight) appeared to be related to litter size but not to the sex ratio of the litter. However, it was highly related to a variable that combined the effects of litter size and sex ratio within the litter, thus supporting Williams' (1979, Proceedings of the Royal Society of London, Series B, 205, 567-580) hypothesis that mothers should adjust both litter size and offspring sex. Copyright 1999 The Association for the Study of Animal Behaviour.     

Latitudinal variation in litter size of polar bears: ecology or methodology?

I used data reported in the scientific literature to examine latitudinal variation in litter size of polar bears in 18 different populations. No relationship was found between litter size and latitude using non-weighted regression. Regression weighted by sample size indicated a negative relationship with latitude. However, sampling biases caused by latitudinal differences in the timing of sampling and cub mortality after den emergence could produce a latitudinal cline. Stratifying analyses by sampling area, at dens or away from dens, revealed no latitudinal trend in litter size. Observed differences in litter size among populations may result from variation in demography and ecosystem productivity that is not simply related to latitude. I conclude that available data do not provide support for a biologically significant latitudinal cline in polar bear litter size. Accepted: 25 May 1999     

Exogenous steroid effects on litter size and early embryonic survival in swine

These studies confirm the efficacy of 12.5 mug estrone and 25 mg progesterone, administered on Days 16 and 17 of pregnancy, on increased litter size in swine. Substitution of estradiol (with ten-fold greater biological activity) for estrone in this regimen eliminated the positive effect on litter size and estradiol given alone at dosages of 3.1 and 6.3 mug were without effect. However, when estradiol was given in combination with progesterone (25 mg) at a dosage equivalent to 10% the effective estrone dosage (1.25 mug), a 22.6% increase in total pigs born per litter over controls was observed. Estrone alone at dosages of 6.25 and 12.5 mug did not alter litter size. At a dosage of 18.75 mug, an 8.7% decrease in total pigs born was noted. The amniotic fluid from sows treated for ten days with 12.5 mug estrone and 25 mg progesterone near the time of implantation, had significantly higher protein concentration, pH and osmolality.     

Selection for environmental variation: a statistical analysis and power calculations to detect response

Data from uterine capacity in rabbits (litter size) were analyzed to determine whether the environmental variance was partly genetically determined. The fit of a classical homogeneous variance mixed linear (HOM) model and that of a genetically structured heterogeneous variance mixed linear (HET) model were compared. Various methods to assess the quality of fit favor the HET model. The posterior mean (95% posterior interval) of the additive genetic variance affecting the environmental variance was 0.16 (0.10; 0.25) and the corresponding number for the coefficient of correlation between genes affecting mean and variance was -0.74 (-0.90;-0.52). It is argued that stronger support for the HET model than that derived from statistical analysis of data would be provided by a successful selection experiment designed to modify the environmental variance. A simple selection criterion is suggested (average squared deviation from the mean of repeated records within individuals) and its predicted response and variance under the HET model are derived. This is used to determine the appropriate size and length of a selection experiment designed to change the environmental variance. Results from the analytical expressions are compared with those obtained using simulation. There is good agreement provided selection intensity is not intense.     

Changes in the kinetics of follicular growth in response to selection for large litter size in mice

The present study examined a randomly selected control line (C) and a large litter size-selected line of mice (S1) to determine changes in the kinetics of follicular growth that have occurred in response to selection for large litter size. For each follicle type (FT), the number of healthy and atretic follicles, length of components of granulosa cell cycle, follicular growth rate, and follicular flux were determined microscopically from serially sectioned ovaries of Lines C and S1 mice. Selection for litter size significantly increased the number of small and medium, and some large follicle-size classes. While selection for litter size did not change the overall incidence of atresia at proestrus, it did decrease the incidence of atresia in the large Type 7 follicles by 19%. Selection for litter size also increased ovarian weight at proestrus. Selection for litter size increased the rate of growth through FT 3a, 5a, and 5b, and reduced the time required for follicles to grow from primordial to Graafian follicles from 39.1 days in Line C to 33.4 days in Line S1. Selection for large litter size also increased the flux of follicles through follicle Types 3a to 5b by 72%, and through follicle Types 6 and 7 by 21%. Genetic variation was found in many aspects of the kinetics of follicular growth.     

Simple three‐pool model accurately describes patterns of long‐term litter decomposition in diverse climates

As atmospheric CO₂ increases, ecosystem carbon sequestration will largely depend on how global changes in climate will alter the balance between net primary production and decomposition. The response of primary production to climatic change has been examined using well‐validated mechanistic models, but the same is not true for decomposition, a primary source of atmospheric CO₂. We used the Long‐term Intersite Decomposition Experiment Team (LIDET) dataset and model‐selection techniques to choose and parameterize a model that describes global patterns of litter decomposition. Mass loss was best represented by a three‐pool negative exponential model, with a rapidly decomposing labile pool, an intermediate pool representing cellulose, and a recalcitrant pool. The initial litter lignin/nitrogen ratio defined the size of labile and intermediate pools. Lignin content determined the size of the recalcitrant pool. The decomposition rate of all pools was modified by climate, but the intermediate pool's decomposition rate was also controlled by relative amounts of litter cellulose and lignin (indicative of lignin‐encrusted cellulose). The effect of climate on decomposition was best represented by a composite variable that multiplied a water‐stress function by the Lloyd and Taylor variable Q₁₀ temperature function. Although our model explained nearly 70% of the variation in LIDET data, we observed systematic deviations from model predictions. Below‐ and aboveground material decomposed at notably different rates, depending on the decomposition stage. Decomposition in certain ecosystem‐specific environmental conditions was not well represented by our model; this included roots in very wet and cold soils, and aboveground litter in N‐rich and arid sites. Despite these limitations, our model may still be extremely useful for global modeling efforts, because it accurately (R²=0.6804) described general patterns of long‐term global decomposition for a wide array of litter types, using relatively minimal climatic and litter quality data.