Effects of data imbalance on estimation of heritability

Summary Effects of data imbalance on bias, sampling variance and mean square error of heritability estimated with variance components were examined using a random two-way nested classification. Four designs, ranging from zero imbalance (balanced data) to low, medium and high imbalance, were considered for each of four combinations of heritability (h²=0.2 and 0.4) and sample size (N=120 and 600). Observations were simulated for each design by drawing independent pseudo-random deviates from normal distributions with zero means, and variances determined by heritability. There were 100 replicates of each simulation; the same design matrix was used in all replications. Variance components were estimated by analysis of variance (Henderson's Method 1) and by maximum likelihood (ML). For the design and model used in this study, bias in heritability based on Method 1 and ML estimates of variance components was negligible. Effect of imbalance on variance of heritability was smaller for ML than for Method 1 estimation, and was smaller for heritability based on estimates of sire-plus-dam variance components than for heritability based on estimates of sire or dam variance components. Mean square error for heritability based on estimates of sire-plus-dam variance components appears to be less sensitive to data imbalance than heritability based on estimates of sire or dam variance components, especially when using Method 1 estimation. Estimation of heritability from sire-plus-dam components was insensitive to differences in data imbalance, especially for the larger sample size.Supported by grants from the Illinois Agricultural Experiment Station and the University of Illinois Research Board. Charles Smith, H. W. Norton and D. Gianola contributed valuable suggestions     

Assortative mating and the genetic correlation

Summary The effect of assortative mating on the genetic correlation between traits X and Y is considered. Assortation on trait X changes the magnitude of the genetic correlation but not its sign. There are two situations depending on the signs of the correlation between mates () and of the random mating genetic correlation (): 1) if sign () = sign (), then >, where is the genetic correlation at equilibrium after continued assortation, and 2) if sign () = sign (), then < . However, negative assortative mating is virtually powerless to alter the magnitude of the genetic correlation. The consequences of a mixed assortation model, e.g., high milk production females mated to fast growing males and lesser productive females mated to slower growing sires, were also studied. Mixed positive assortation always increases the genetic correlation, but negative assortation decreases it. The implications of assortative mating on correlated responses to selection and on the equilibrium covariances between relatives for pairs of traits are discussed.     

Relationships between genotype x environment interaction and genetic correlation of the same trait measured in different environments

Summary Relationships between genotype x environment interaction and genetic correlation of the same trait measured in different fixed environments are derived by comparing the variance-covariance structures of observations between a one-way multiple-trait linear model and a two-way single-trait mixed linear model. In the latter model, heterogeneity of interaction variances among environments and non-zero covariances among interactions are assumed, in addition to the heterogeneity of error variances and non-zero covariances between genetic-group effects and interactions that were accommodated in earlier work. The results are applicable to more than two environments and to unbalanced data. This paper is a generalization and a correction of earlier works.     

The effect of simultaneous selection on the genetic correlation

The theoretical effect of simultaneous selection on the genetic correlations between two traits over 20 generations was examined using simulation. For each generation, a population of 50 male and 50 female diploid gen otypes with 15 loci, each with two alleles, was synthesized. None of the loci exhibited dominance. Five loci affected only trait 1, 5 loci only trait 2 and 5 were pleiotropic (affected both traits). Initial allelic frequencies were equal at each locus. Phenotypes were created by adding a random normal deviation for each trait to the genotype. The size of this deviation for each trait determined its heritability (h²). Index selection with h² combinations of (0.15, 0.15), (0.15,0.45) and (0.45,0.45) and relative economic weights of (1, 1) and (1, 3) for each h² combination was employed. In each generation, the highest ranking 25 genotypes of each sex were used to generate the next generation with single-pair matings, each producing two male and two female offspring. One hundred replicates were run for both negative and positive correlations. With a positive initial value, the genetic correlation tended to decline (toward zero). The rates of change were moderately affected by index weights and h². With a negative initial value, the genetic correlation tended to decrease (towards -1). However, unequal heritabilities and unequal relative economic weights slowed the rate of change with the greatest imbalances tending to hold the correlation constant or move it toward zero. These simulations illustrate that changes in parameters over time can affect the selection practiced. Under some of the conditions simulated, the use of initial genetic parameter values without change could have potentially negative effects on overall genetic gain.     

The effects of selection for gain in mice on the direct-maternal genetic correlation

Components of genetic variation for postweaning growth traits were estimated for both control and growth stocks of mice. The effect of phenotypic selection for gain, which genetically combines selection for additive direct and maternal effects, on additive genetic variance components, heritability, and additive genetic correlationsis discussed. Quantitative genetic theory predicts that simultaneous selection for two metric traits in the same direction will cause the genetic correlation between the two traits to become more negative. The results presented in this paper conflict with this theory. The direct-maternal additive genetic correlation was more negative in the control line (with 356 mice) than in the growth-selected line (with 320 mice) for the three traits analyzed (0.310 vs 0.999 for 21-day weight, 0.316 vs 1.000 for 42-day weight, and 0.506 vs 1.000 for gain from 21–42 days). Estimates were obtained by restricted maximum likelihood (REML) computed under a derivative free algorithm (DFREML).     

Meta-analysis of studies with missing data

Summary .  Consider a meta-analysis of studies with varying proportions of patient-level missing data, and assume that each primary study has made certain missing data adjustments so that the reported estimates of treatment effect size and variance are valid. These estimates of treatment effects can be combined across studies by standard meta-analytic methods, employing a random-effects model to account for heterogeneity across studies. However, we note that a meta-analysis based on the standard random-effects model will lead to biased estimates when the attrition rates of primary studies depend on the size of the underlying study-level treatment effect. Perhaps ignorable within each study, these types of missing data are in fact not ignorable in a meta-analysis. We propose three methods to correct the bias resulting from such missing data in a meta-analysis: reweighting the DerSimonian–Laird estimate by the completion rate; incorporating the completion rate into a Bayesian random-effects model; and inference based on a Bayesian shared-parameter model that includes the completion rate. We illustrate these methods through a meta-analysis of 16 published randomized trials that examined combined pharmacotherapy and psychological treatment for depression.     

A note on the measurement of redundancy

Plant Ecology - The present paper points out a simpler method for computing the redundancy coefficient proposed by Orlóci (1975) using the inverse of the variance-covariance matrix. It is...     

Effect of lactation length adjustment procedures on genetic parameter estimates for buffalo milk yield

The objectives of this study were to estimate the genetic parameters for milk yield unadjusted and adjusted for days in milk and, subsequently, to assess the influence of adjusting for days in milk on sire rank. Complete lactations from 90 or 150 days of lactation to 270 or 350 days in milk were considered in these analyses. Milk yield was adjusted for days in milk by multiplicative correction factors, or by including lactation length as a covariable in the model. Milk yields adjusted by different procedures were considered as different traits. Heritability estimates varied from 0.17 to 0.28. Genetic correlation estimates between milk yields unadjusted and adjusted for days in milk were greater than 0.82. Adjusting for days in milk affected the parameter estimates. Multiplicative correction factors produced the highest heritability estimates. More reliable breeding value estimates can be expected by including short length lactation records in the analyses and adjusting the milk yields for days in milk, regardless of the method used for the adjustment. High selection intensity coupled to the inclusion of short length lactations and adjustment with multiplicative factors can change the sire rank..     

New features in the dendroTools R package: Bootstrapped and partial correlation coefficients for monthly and daily climate data

Climate-growth relationships are usually analysed using monthly climate data. The dendroTools R package also provides methodological approaches that enable climate-growth analysis for daily climate data. Such analysis reveals more complete climate signal patterns. In this article, new functions of the dendroTools R package are presented. Partial correlation coefficients are now implemented and can be used to calculate the strength of a linear relationship between two variables, while controlling for a third variable. Bootstrapped correlations can then be used to provide insights into the confidence intervals of statistical estimates. The calculation of partial and bootstrapped correlations is available for daily and monthly data. Finally, data transformation, S3 generic plotting and summary functions are also presented here.     

Impact of scoring error and reproducibility RAPD data on RAPD based estimates of genetic distance

RAPD band reproducibility and scoring error were evaluated for RAPDs generated by 50 RAPD primers among ten snap bean (Phaseolus vulgaris L.) genotypes. Genetic distances based on different sets of RAPD bands were compared to evaluate the impact of scoring error, reproducibility, and differences in relative amplification strength on the reproducibility of RAPD based genetic distance estimates. The measured RAPD data scoring error was 2%. Reproducibility, expressed as the percentage of RAPD bands scored that are also scored in replicate data, was 76%. The results indicate that the probability of a scored RAPD band being scored in replicate data is strongly dependent on the uniformity of amplification conditions between experiments, as well as the relative amplification strength of the RAPD band. Significant improvement in the reproducibility of scored bands and some reduction in scoring error was achieved by reducing differences in reaction conditions between replicates. Observed primer variability for the reproducibility of scored RAPDs may also facilitate the selection of primers, resulting in dramatic improvements in the reproducibility of RAPD data used in germplasm studies. Variance of genetic distances across replicates due to sampling error was found to be more than six times greater than that due to scoring error for a set of 192 RAPD bands. Genetic distance matrices computed from the RAPD bands scored in replicated data and RAPD bands that failed to be scored in replicated data were not significantly different. Differences in the ethidium bromide staining intensity of RAPD bands were not associated with significant differences in resulting genetic distance matrices. The assumption of sampling error as the only source of error was sufficient to account for the observed variation in genetic distance estimates across independent sets of RAPD bands.     

Joint model selection of marginal mean regression and correlation structure for longitudinal data with missing outcome and covariates

This work develops a joint model selection criterion for simultaneously selecting the marginal mean regression and the correlation/covariance structure in longitudinal data analysis where both the outcome and the covariate variables may be subject to general intermittent patterns of missingness under the missing at random mechanism. The new proposal, termed “joint longitudinal information criterion” (JLIC), is based on the expected quadratic error for assessing model adequacy, and the second‐order weighted generalized estimating equation (WGEE) estimation for mean and covariance models. Simulation results reveal that JLIC outperforms existing methods performing model selection for the mean regression and the correlation structure in a two stage and hence separate manner. We apply the proposal to a longitudinal study to identify factors associated with life satisfaction in the elderly of Taiwan.     

A note on the tests for clustered matched-pair binary data

McNemar's test is used to assess the difference between two different procedures (treatments) using independent matched-pair data. For matched-pair data collected in clusters, the tests proposed by Durkalski et al. and Obuchowski are popular and commonly used in practice since these tests do not require distributional assumptions or assumptions on the structure of the within-cluster correlation of the data. Motivated by these tests, this note proposes a modified Obuchowski test and illustrates comparisons of the proposed test with the extant methods. An extensive Monte Carlo simulation study suggests that the proposed test performs well with respect to the nominal size, and has higher power; Obuchowski's test is most conservative, and the performance of the Durkalski's test varies between the modified Obuchowski test and the original Obuchowski's test. These results form the basis for our recommendation that (i) for equal cluster size, the modified Obuchowski test is always preferred; (ii) for varying cluster size Durkalski's test can be used for a small number of clusters (e.g. K < 50), whereas for a large number of clusters (e.g. K ≥ 50) the modified Obuchowski test is preferred. Finally, to illustrate practical application of the competing tests, two real collections of clustered matched-pair data are analyzed.     

Accounting for missing data in the estimation of contemporary genetic effective population size (Ne)

Theoretical models are often applied to population genetic data sets without fully considering the effect of missing data. Researchers can deal with missing data by removing individuals that have failed to yield genotypes and/or by removing loci that have failed to yield allelic determinations, but despite their best efforts, most data sets still contain some missing data. As a consequence, realized sample size differs among loci, and this poses a problem for unbiased methods that must explicitly account for random sampling error. One commonly used solution for the calculation of contemporary effective population size (N_e) is to calculate the effective sample size as an unweighted mean or harmonic mean across loci. This is not ideal because it fails to account for the fact that loci with different numbers of alleles have different information content. Here we consider this problem for genetic estimators of contemporary effective population size (N_e). To evaluate bias and precision of several statistical approaches for dealing with missing data, we simulated populations with known N_e and various degrees of missing data. Across all scenarios, one method of correcting for missing data (fixed‐inverse variance‐weighted harmonic mean) consistently performed the best for both single‐sample and two‐sample (temporal) methods of estimating N_e and outperformed some methods currently in widespread use. The approach adopted here may be a starting point to adjust other population genetics methods that include per‐locus sample size components.     

Experimental study of the influence of errors of genetic correlation estimates on unrestricted,optimum, and desired gains selection indices

Summary Effects of errors in estimates of the genetic correlation on the accuracy of unrestricted, optimum, and desired gains selection indices were examined experimentally in Tribolium castaneum. Three lines were selected for three generations for pupal weight at 21 days and adult weight at 31 days, using unrestricted (I9), optimum (O9), and desired gains (G9) index selection methods. The genetic correlation between pupal and adult weights in the base population was 0.95. The optimum index was designed to set the response of pupal weight by a fixed amount, while in the desired gains index the responses of pupal and adult weights were specified as being equal to 31. Three other indices were constructed using a deliberately incorrect genetic correlation (0.25), i.e., unrestricted (I2), optimum (O2), and desired gains (G2). Responses observed in unrestricted index lines (I9 versus I2) and optimum index lines (O9 versus O2) did not differ significantly, even though lines I9 and I2 differed in a practical sense. Responses in desired gains index lines (G9 versus G2) differed significantly. Responses obtained for aggregate genotype (pupal weight + adult weight) and for the component traits were greater in line I9 than those obtained in line I2. Responses obtained in the O9 and O2 lines for pupal and adult weights were similar, while those obtained in the G9 and G2 lines were similar for pupal weight but not (P<0.05) for adult weight. Therefore, underestimation of the genetic correlation seems to affect the efficiency of a desired gains index more than that of unrestricted or optimum indices.