Currently, algorithms and softwares for genetic analysis of diploid organisms with bi-allelic markers are well-established, while those for polyploids are limited. Here, we present SHEsisPlus, the online algorithm toolset for both dichotomous and quantitative trait genetic analysis on polyploid species . SHEsisPlus is also optimized for handling multiple-allele datasets. It’s free, open source and also designed to perform a range of analyses, including haplotype inference, linkage disequilibrium analysis, epistasis detection, Hardy-Weinberg equilibrium and single locus association tests. Meanwhile, we developed an accurate and efficient haplotype inference algorithm for polyploids and proposed an entropy-based algorithm to detect epistasis in the context of quantitative traits. A study of both simulated and real datasets showed that our haplotype inference algorithm was much faster and more accurate than existing ones. Our epistasis detection algorithm was the first try to apply information theory to characterizing the gene interactions in quantitative trait datasets. Results showed that its statistical power was significantly higher than conventional approaches. SHEsisPlus is freely available on the web at http://shesisplus.bio-x.cn/. Source code is freely available for download at https://github.com/celaoforever/SHEsisPlus. The simplest way is to use a fixed partition size. Another way is to divide according to LD blocks. In our simulation study, we observed little difference when we tried different partitioning strategy. It is common knowledge that EM algorithm is likely to be trapped in a local optimum. To deal with this, different initial frequencies are assigned to the possible haplotypes. Loci are phased for multiple times using different initial value and the final solution is the one that is of the max likelihood. As SNPs are divided into blocks, this method ensures that optimum solution is more likely to be obtained within each block and thus, make it more likely to achieve the global optimum when all the SNPs are phased.The procedure of EM algorithmgth iteration, the posterior probabilities of genotype assignments are calculated as:In the expectation step at the Pj is the probability of the jth phenotype, given by the sum of the probabilities of each of the possible genotypes, and j. In the maximization step, the next set of estimates of haplotype probabilities are obtained by summing posterior probabilities over instances of each distinct haplotypes:where kt is present in genotype i.where The performance of this algorithm was assessed using a real dataset from tetraploid potato genotypes, which was obtained fromInteractions between SNPs are quantified by interaction information, which is the amount information bound up in a set of SNPs and is not present in any subset of these SNPs. Let’s first introduce some basic concepts in information theory.Let’s assume an attribute, A. Shannon’s entropyP(A) is the distribution of attribute A. By definition, H(A) is the amount of uncertainty about A, as estimated from its probability distribution.where 2-way interaction between 2 attributes can be quantified with mutual information:3-way interaction is measured by the intersection of all three attributes, or interaction information:Then k-way interaction information can be generalized as:According to the definition of entropy, we can define the entropy of a bi-allelic SNP as:And the entropy of two bi-allelic SNPs can be defined as:1 and SNP2 by:Then we can easily obtain the 2-way interaction between SNPGenerally, the k-way interaction of SNP set Our algorithm for epistasis analysis is the first try to apply information theory to characterizing the gene interactions in quantitative trait datasets. For quantitative traits, individuals with some particular genotype combinations might have elevated trait values. So we aim to find out that if loci from individuals with higher trait values are more or less likely to interact than from those with lower trait values. To do this, we first classify individuals into high trait value group and low trait value group. We adopt Otsu’s methodwhere,t and μi are the class means. The desired threshold corresponds to the maximum t, we can classify the individuals into two groups. Individuals with trait values higher than t are classified into the high trait value group while lower than t are classified into the low trait value group. Then interaction information is calculated in the groups respectively.After obtaining the threshold To get the k-way interaction of SNP set The difference in interaction information is:Then permutation test is performed to get the P value.As this method does not test for the individual genotype combination; instead, it evaluates the overall difference across all the genotype combinations between the high trait value group and the low trait value group in order to increase the statistical power. Therefore, to find the risk genotype combinations, we generated the counts for each combination and used the standard chi square test. We also gave the odds ratio and p values for a certain genotype combination.For case/control studies, interactions information are calculated in cases and controls respectively.The difference in interaction information is:Then permutation test is performed to get the P value.SHEsisPlus can adjust for covariates  when performing single locus association analysis. For case/control design, if no covariates are provided, SHEsisPlus gives the results of Pearson’s Chi-square test and Fisher’s exact test for alleles and genotypes, else logistic regression will be used. The regression equation is:For quantitative traits, linear regression will be used instead. The regression equation is:Y is disease status, X is genotype and In the above equations, The genotype frequencies in the Hardy-Weinberg equilibrium for a c-ploidy specie with n distinct alleles are given by individual terms in the multinomial expansion of:D’ and r are given, which can be calculated by:For linkage disequilibrium analysis, normalized AiBj, pi and qj are the frequencies of alleles Ai and Bj at locus A and B.where All the patients and controls were of Han Chinese origin and had long-term residence in the coastal areas of Shandong Province. A total of 622 unrelated cases were recruited from the gout clinic at the Affiliated Hospital of Qingdao University. All patients were diagnosed with primary gout by experienced physicians according to criteria established by the American College of Rheumatology. All 917 unrelated controls that had serum uric acid (SUA) values below 420 μmol/L, and never suffered from an acute attack of gouty arthritis were recruited. All participants with a family history of gout and severe illness, such as hepatitis or cancer, were excluded. This study was approved by the Ethics Committee of Affiliated Hospital of Qingdao University. All participants gave their written informed consent. The study was in accordance with the principles of the current version of the Declaration of Helsinki.2. All the samples were males. Systolic blood pressure (mmHg) and diastolic blood pressure (mmHg) were measured and recorded by physicians on our gout clinic. Related biochemical indicators including blood glucose, triglycerides, total cholesterol, urea nitrogen, creatinine and uric acid in the plasma were measured using an automated multichannel chemistry analyzer .Phenotype details including age, height and weight were collected in a questionnaire at the time of admission and body mass index (BMI) was calculated from the calculation formula weight (kg)/height (m)Informed consent was obtained from all subjects. All experiments were performed in accordance with relevant guidelines and regulations which were approved by Shanghai Jiaotong University.To investigate whether the known serum uric acid level associated SNPs might interact with each other, we selected eight SNPs 222324We performed linear regression to see if age and BMI might influence uric acid variability. The regression model is:−16) while the correlation between age and uric acid level was weak (p = 0.48). Thus we adjusted uric acid level with respect to BMI. Epistasis were then evaluated by SHEsisPlus.We found that BMI was significantly related to uric acid level To see if SHEsisPlus performed well when more loci were involved, we simulated larger datasets containing 100 SNPs and 1000 samples for diploids, triploids and tetraploids. We were unable to try the large datasets with polyHap and SATlotyper because of the computational burden. For large datasets, CHP was not an appropriate metric because as the length of the considered region increased, all methods would find it harder to correctly infer the entire haplotypesThis would give more weight to common haplotypes. As SHEsisPlus was designed for association study, which focused on the difference in frequencies between cases and controls, we thought this metric would be more appropriate to fit this scenario. The As the power of this entropy-based algorithm for detecting epistasis in case/control data has been well-evaluatedThe model that Plink uses to detect epistasis is linear regression. The regression model is:Test of interaction corresponds to testing whether the regression coefficient representing interaction terms . The distribution of the BMI-adjusted uric acid level was shown in −4, p = 0.001 after FDR correction) and rs12129861  showed significant association. Although rs12356193 and rs742132 didn’t show significant association with serum uric acid level, they, together with rs1183201 and rs12129861, exhibited strong interaction .We used SHEsisPlus to assess the 2-way to 8-way interactions between 8 loci . If the sample size within each group is small, the power will be limited. Therefore, it seems that the best strategy is to divide into as few groups as possible. We used the Otsu’s method to determine the threshold for sample division because it accounted for the pattern of the distribution. For the serum uric acid level dataset we used for epistasis analysis, the threshold determined by this method was approximately located in the valley of the two peaks, which was a reasonable threshold to classify the samples into high and low trait value groups. We found in our simulation study that Otsu’s method was robust because there was no extreme division generated by this method (e.g. too many samples in one group and too few in the other).http://analysis.bio-x.cn/SHEsisMain.htm.However, there are limitations of this method. It is not applicable to genome-wide datasets. It is designed for small datasets and one of its key features is that it can calculate the multi-way interaction. Multi-way interaction analysis is only applicable to small datasets because of the extremely high computational complex. However, in our previous work, we have proposed a GPU-based software .


Saccharomyces cerevisiae yeast strains, we applied a method for mapping heterozygous loci that non-additively contribute to heterotic growth at 37°. Among 803 candidate loci that were mapped, five were tested for their heterotic effect by analyzing backcrosses and F2 populations in a specific hybrid background. Consistently with the many mapped loci, specific analyses confirmed the minor heterotic effect of the tested candidate loci. Allele-replacement analyses of one gene, AEP3, further supported its heterotic effect. In addition to over-dominant effects, the contribution of epistasis to heterosis was evident from F2 population and allele-replacement analyses. Pairs of over-dominant genes contributed synergistically to heterosis. We show that minor over-dominant effects of multiple genes can combine to condition heterosis, similarly to loci affecting other quantitative traits. Furthermore, by finding of epistatic interactions between loci that each of them individually has an over-dominant effect on heterosis, we demonstrate how hybrid advantage could benefit from a synergistic combination of two interaction types (over-dominant and synergistic epistatic). Thus, by portraying the underlying genetic complexity, these findings advance our understanding of heterosis.Heterosis describes a phenotypic phenomenon of hybrid superiority over its homozygous parents. It is a genetically intriguing phenomenon with great importance for food production. Also called hybrid-vigor, heterosis is created by non-additive effects of genes in a heterozygous hybrid made by crossing two distinct homozygous parents. Few models have been proposed to explain how the combination of parental genes creates an exceptional hybrid performance. Over-dominant mode of inheritance is an attractive model since a single gene can potentially create the heterotic effect, but only a few such loci have been identified. To a collection of 120 hybrids, made by crossing 16 divergent  The phenomenon describing superiority of heterozygous progeny over their homozygous parents is called heterosis or hybrid vigor East, , 1911. IFrom a genetic point of view, the three main models that explain heterozygote advantage rely on dominance or deviation from additive effects in one or more loci  medium was prepared with 5-FOA . SC medium lacking uracil was used for selecting cells with intact URA3. Solid media were prepared by addition of 2% agar (Difco) to any of the liquid media listed above. The strains used in this study are diallel strains , 2% peptone (BD Bacto), 2% dextrose (J. T. Baker)] was used unless other specific medium is mentioned. For selection, YEPD was supplemented with 300 mg/L Hygromycin B (InvivoGen), 400 mg/L G418—Geneticin (Santa Cruz) or both. For selecting ΔDiploid cells were grown on YEPD plates in patches overnight and scraped off from the plates into sporulation solution [0.75% potassium acetate (Sigma)] for 2–3 days. To prepare for dissection, an aliquot of 15 μL from the sporulation culture was incubated with 15 μL of 2 mg/mL Zymolyase (Seikagaku Corporation) for 30 min at 30° to digest the spore ascus wall. Treated tetrads were dissected to individual spores using a micromanipulator microscope  on YEPD plates.2 (Applied Biosystems), 0.66 mM dNTPs mix (Larova), 2 μL buffer (Applied Biosystems), 0.5 μL Taq DNA polymerase (1.5 units), and PCR grade water to complete the volume. The PCR profile used for amplification was 94° for 10 min, followed by a touchdown profile: 94° for 30 s, annealing from the upper to the lower temperature for 1 min with a decrease of 0.5° per each of 14 cycles, and extension at 72° for 2 min. The touchdown cycles were followed by 23 cycles with the lower annealing temperature and a final elongation step in 72° for 10 min. PCR products were verified by electrophoresis on 1.5% TBE Agarose gel (Lonza) containing Ethidium Bromide (Apex BioResearch).As DNA template, either 5 μL of cells suspended in water or 50 ng of purified DNA  were used in a total reaction volume of 20 μL. The PCR mix also contained: 1.66 μM of each forward and reverse primers, 2.5 mM MgClStrains' genotype was determined either by SNP HRM analysis with fluorescent DNA intercalating dye, or by sequencing using big dye chemistry on an ABI PRISM 3730XL DNA Analyzer at the Center for Genomic Technologies, Hebrew University, Jerusalem, as described in Shapira et al. . PrimersMX4 cassette containing a resistance gene to either G418 (KanMX4) or Hygromycin B  were measured at 30° and 37° using multimode readers and the mean DT-values of two technical repeats were calculated and used for the statistical analyses.Haploid strains from SK1 and S1001 genetic backgrounds were used in the two-steps construction of allele replacement strains (see above for details on the procedure). In the first step, the original allele is deleted and replaced by a cassette containing a drug marker and in the second step the cassette is replaced with the introduced allele. For the analysis of haploid strains, gene-deleted strains from the first step and allele-replaced strains from the second step were used. For the analysis of diploids, nine genotypic groups were produced using the original and allele-replaced strains of both backgrounds. The nine groups were: three 2DT-values were used. In order to identify loci contributing to heterosis, the heterotic phenotype of hybrids was calculated based on DT-values of the hybrid and its parents in YEPD 37°. Heterotic value of each hybrid was calculated according to the following categories:Doubling time (DT)-values of strains were measured and calculated as detailed in our previous work  was faster than its faster parent (FP), its heterotic value was calculated as:If the hybrid (SP), its heterotic value was calculated as:If the hybrid was slower than its slower parent , was calculated for each hybrid and furthermore for each F2 genotypic groups, by using the following parameters: Mid-parent value (m)—Mean of homozygous parents phenotype:Another measure for level of heterosis, the degree of dominance (a)—The difference between each of the homozygous parents and the mid-parent value:Additive genetic deviation (d)— deviation of the heterozygous hybrid (Hy) from its mid-parent value:Dominant genetic deviation , and in the analyses of deletion and allele-replacement strains.P < 0.05). In every window with a significant difference, the three genotypic groups that yielded the most significant difference were chosen. Genomic windows in which the mean heterotic value of the heterozygous hybrids group was significantly larger than the means of both homozygous groups were considered as candidate loci with an over-dominant effect. Finally, since multiple comparisons were carried out, to reduce the chances for false positives identification, we applied a Benjamini–Hochberg correction at an FDR level of 0.1 to the list of significant loci.The parental strains used for making hybrids were fully sequenced and the list of SNPs for each strain that was called in that study . Parental haplotypes were deduced from the sequences and hybrids were reassigned to three genotypic groups in each window based on the haplotype combination of their parents. Due to the differences between the SNP list and the full sequence alignments, changing a haplotype call for even one parent per window, changed the genotypic grouping of the hybrids and could have changed the results of the statistical tests. In practice, grouping of hybrids in all 74 windows changed in the second analysis compared to the first scan. Therefore, using all sequence polymorphisms, not only SNPs, made this second analysis more stringent. Furthermore, instead of assigning heterotic values to hybrids as in the initial screen, in this second analysis we assigned them with continuous d/a-values. The heterotic value was set to zero in the initial scan for about half of the hybrids that grew within the parental range while d/a-values in the analysis of the 74 candidates were assigned to all hybrids. Zeroing values for half of the hybrids gave more weight to the other half (the heterotic hybrids) and favored the identification of over-dominant loci in the initial analysis. On the other hand, including continuous d/a-values for all hybrids in the second analysis, increased the variance around the mean for each genotypic group  and this larger variance made it harder to reach statistical significance. An example of the spread around the mean of each genotypic group is given in Supplementary Figure d/a-values for all hybrids is the second reason why the second analysis was more stringent.The set of hybrids in the heterozygous group varied among candidate windows. From the list of significant windows prior to the FDR correction, we chose 74 windows for which the specific SK1xS1001 hybrid was included in the group of heterozygote hybrids. This was a logical filtering step since over-dominant contribution is expected only from heterozygous loci. We then applied to these 74 windows a more stringent analysis to identify the more promising over-dominant loci in this hybrid background. For each window, we downloaded sequence alignments of the 16 parental strains from the SGRP blast server .a- and d-values in each locus. The additive genetic deviation (a) was half the difference between means of the two homozygotes and the dominant genetic deviation (d) was the difference between the mid-parent and the mean of the heterozygous segregants. Positive d-values were assigned when the mean growth of heterozygotes was faster than the mid-parent whereas negative d-values were assigned when slower. Larger d- than a-values indicate over-dominance. To compare between a- and d-values, each F2 diploid was assigned with either a- or d-values  if it was a homozygote or a heterozygote, respectively. We compared the mean a- and d-values for each locus using a t-test. For the analysis of epistasis between pairs of candidate loci, we compared the d-values of a given locus with and without heterozygotes in another locus using a t-test.The F2 population was used to evaluate the mode of inheritance of candidate loci. First, we compared the mean DT between the three genotypic groups  for each locus using a Tukey–Kramer HSD test. Secondly, we determined the mode of inheritance by comparing https://www.sanger.ac.uk/research/projects/genomeinformatics/sgrp.html). Each SNPs-containing genomic window defined a set of haplotypes for the haploid parental strains and based on the haplotypes of their parents, the 120 hybrids were categorized as heterozygotes or homozygotes to one or the other haplotypes  in our tests was on average over 90% of the total variance  as the phenotypic value of hybrids rather than their heterotic value (see Methods section for details). Based on the full sequence alignments, haplotype assignment changed for at least one of the 16 parental strains in each of the 74 windows compared to the assignment made based on the SNP genotypes alone. The changes in parental haplotype assignments were first because polymorphisms other than SNPs (mainly indels) were included, and in addition because the SNP content in the window changed . Consequently, grouping of hybrids into the three genotypic groups  also changed. Hence, we compared the means of the updated three genotypic groups for the 74 windows using d/a as the measure for heterosis. We note that it would have been technically very challenging to carry out the genome-wide scan in the same way as the manual analysis of the 74 windows was done.To choose loci for further analysis from the list of 74 background specific candidates, we first reanalyzed all 74 using a more stringent analysis. In this second analysis, to determine the parental haplotypes, we used the full genomic sequence of each window rather than just SNP genotypes and we used the degree of dominance (P < 0.05) based on a Kruskal–Wallis non-parametric test and after correcting for multiple testing at an FDR level of 0.1. These significant loci were scattered across the genome . Based on the comparison among means, for 53 of the 62 analyzed windows, the local heterozygous group had a tendency to grow faster than the local homozygous groups (tendency for over-dominance), and for 2 of the 62, the heterozygous group had a tendency to grow slower than the homozygous (tendency for under-dominance). In total, 55 of the 62 windows (89%) showed the expected trend among the means. For 12 of the 55 (22%), the differences were significant , which represent the effect of each locus, explained less than 10% of the total variance among hybrids Figure . TherefoAEP3 (Chr. XVI), CSR2 (Chr. XVI), RRP3 (Chr. VIII), ADR1 (Chr. IV) and one, window # 828, was located in an intergenic region . Due to large variation in d/a-values among different hybrids within each genotypic group, significant differences between the groups were found in three of the five loci, based on the more stringent Tukey–Kramer parametric test.Based on the second, more conservative, analysis that identified significant candidates in the SK1xS1001 hybrid background, five windows were selected for further analyses based on the consistency of test results between the initial scan and the background-specific analysis. These five windows showed the expected heterotic relationship, i.e., the heterozygotes group had a marked tendency to grow faster than the homozygotes groups Figure . Four wiThe first analysis of the five candidate loci was in segregating progeny. In Shapira et al. , we usedAEP3 and ADR1, local heterozygotes grew significantly faster than local homozygotes  genotyping assay was developed and used to determine which F1 segregant carried what parental allele in each candidate locus. The genotype of each diploid BC1 segregant (homozygous or heterozygous) in each candidate locus was deduced based on the combined genotype of the specific F1 and the common backcross parental strain. In this analysis, a comparison was made between means of two groups, either homozygous or heterozygous in the candidate locus, but otherwise with segregation of alleles in the rest of the genome. Thus, this comparison tested the contribution of a specific candidate locus, while averaging out the contribution of other loci. For half of the comparisons, the local heterozygotes showed a tendency to grow faster than the local homozygotes. For s Figure . AEP3 isAEP3, CSR2, and RRP3, whereas for ADR1 and Win. 828, homozygotes for the SK1 allele and heterozygotes were available. Doubling time values of F2 segregants at 37°  and dominant (d) genetic deviations as well as the degree of dominance (d/a) based on the F2 phenotypes . Furthermore, CSR2 effect was not affected by heterozygotes of other loci although both AEP3 and CSR2 locate to chr. XVI and the estimated genetic distance between them is 17 cM based on the recombination events counted in SK1xS1001 F1 segregants  as one main factor and genetic background  as another main factor , we tested the statistical significance of differences among the genotypic groups  by comparing them to the differences among hybrids within each group. In this hybrid collection, the genetic and phenotypic variations are both large. Due to the divergence that was found even in short windows, the parental strains were assigned to three or more haplotypes and consequently, per window, not all hybrids were included in the genotypic groups that were tested. Furthermore, almost half of the total phenotypic variation was attributed to differences within each genotypic group. Therefore, the power to detect significant loci by the genome-wide approach heavily depends on the number of genetic backgrounds, their genetic divergence and the phenotypic variation that their hybridization creates. The statistical power of genome-wide analyses, ours included, depends heavily on the sample size for detecting minor effect loci. Our analysis was based on 120 hybrids and tested over 50,000 loci. We identified loci with minor effects and hence with lower significance that did not pass the FDR correction for multiple testing. Nevertheless, we confirmed the minor effects of few candidates by follow up analyses. Therefore, our list of candidates probably contained some true positives with minor effects together with some false positives. Another important aspect, which ought to be considered when this association method is applied to different organisms, stems again from the number and the genetic structure of the parental strains. In other studies, constructing segregating populations originating from two parental lines limited the studied variation to that between these lines and dictated the genetic background in which further verification and analysis of specific loci would be carried out. In our case, the 803 potential loci were associated with heterozygote advantage based on the variation among all participating strains and thus, to further study the effect of specific loci, a choice of a specific hybrid and its parental strains was required. From the analysis of multiple genetic backgrounds, it was difficult to predict what the effect of a specific locus on heterosis would be in a specific hybrid background. In terms of a specific background, we chose SK1xS1001 hybrid for further analysis due to its heterotic growth at 37°. In terms of loci, out of 803 candidate windows, the ones that were heterozygous in this specific background were those that we further considered. Aside from selecting a hybrid and its heterozygous loci, moving from identifying loci based on the multi-strain analysis to testing the effect of candidate loci in a specific hybrid background represented a challenging step in this association method. The challenges were several and our workflow demonstrates the way we took to overcome these challenges. In practice, filtering the list of 74 using higher stringency analyses combining full-sequence alignments, degree of dominance values, and further statistical tests, narrowed down the list and allowed us to select five candidates to follow up further.AEP3 and ADR1, significant differences were found in the BC1 analyses, for the other three candidates, only trends were observed. Based on growth measurements in the F2 population, AEP3 had a tendency for over-dominance while CSR2 and RRP3 had tendencies for under-dominance and co-dominance, respectively. For all three genes, the dominant deviation was significantly different from the additive deviation, confirming that the trends found based on growth measurements were not just trends but rather significant effects. AEP3, CSR2, and RRP3 had a significant over-dominant, under-dominant, and co-dominant mode of inheritance, respectively. Furthermore, since CSR2 and RRP3 that were initially identified as having over-dominant effects had later been shown to have either an under-dominant or an additive effect, the F2 analysis was also the first significant indication for inter-loci interactions that could explain these changing effects. Finding that a- and d-values of these loci changed significantly after elimination of heterozygous groups in other loci confirmed the effects of intra-locus interactions.Similar to studies in crop plants, we tested the effect of the five candidates in segregating populations. In general, we found support for contribution of two loci to heterosis based on comparisons between local heterozygotes and local homozygotes in BC1 populations. However, consistent with the findings of the multi-strain scan, also in the specific background, the effects were minor. Whereas for AEP3 had the most consistent over-dominant effect, we verified its contribution by functional analyses. Based on the deletion analysis in haploids, we found that AEP3 clearly contributes to growth in both parental backgrounds. Unlike its minor heterotic effect, the simple effect of AEP3 on growth in haploids was considerable. At 37°, the original haploid grew 1.83 and 1.16 times faster than the ΔAEP3 haploid for S1001 and SK1 backgrounds, respectively. However, the effect of the different alleles was minor and statistically significant only in the S1001 background at 37°. In diploids, we found an over-dominant effect for AEP3 in the hybrid background. In the parental backgrounds, we found that the SK1 allele had either a fully dominant or a fully recessive effect. Finding support for the over-dominant effect in the hybrid background is important because this is the background in which AEP3 was identified in the first place. Furthermore, the minor contribution of this single locus supported the view that a combination of several loci underlies the heterotic phenotype in this background. Finding significant changes in the effect as a function of the genetic background was the second clear indication for epistasis.Since AEP3 turned out to be the most convincing heterosis gene, contributing to both growth and heterosis at high temperature. The protein aep3 is localized to the peripheral mitochondrial inner membrane. Its function may facilitate use of unformylated tRNA-Met in mitochondrial translation initiation and it is stabilizing the bicistronic ATP8-ATP6 mRNA transcribed from the mitochondrial genome  is another piece of evidence for the contribution of these candidates to the trait.The course of our study was directed toward identifying loci with an over-dominant contribution to heterosis. However, the F2 population and the allele-replacement analyses also demonstrated clearly the contribution of epistasis to heterosis. In the allele-replacement analysis, AEP3, had a minor effect in the SK1xS1001 background. In QTL mapping studies, loci with major effects on a phenotype are often mapped first, yet, recent advancements made it clear that for several quantitative traits of interest, many additional, or even minor-effect loci exclusively underlie most trait variance .The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


One of John Loehlin’s many contributions to the field of behavioral genetics involves gene-environment (GE) correlation. The empirical base for GE correlation was research showing that environmental measures are nearly as heritable as behavioral measures and that genetic factors mediate correlations between environment and behavior. Attempts to identify genes responsible for these phenomena will come up against the ‘missing heritability’ problem that plagues DNA research on complex traits throughout the life sciences. However, DNA can also be used for quantitative genetic analyses of unrelated individuals  to investigate genetic influence on environmental measures and their behavioral correlates. A novel feature of GCTA is that it enables genetic analysis of family-level environments  and school-level environments  that cannot be investigated using within-family designs such as the twin method. An important implication of GE correlation is its shift from a passive model of the environment imposed on individuals to an active model in which individuals actively create their own experiences in part on the basis of their genetic propensities. Annual Review of Psychology chapter on behavioral genetics  correlations of a personal as well as scientific kind. At the personal level, he introduced me to behavioral genetics in 1971 when I was a second-year graduate student in psychology at the University of Texas at Austin. He contributed to a ‘core course’ on behavioral genetics, which included the first Heredity, environment, and personality: A study of 850 sets of twins. Three quotes from this book illustrate the clarity and lack of pomposity in his writing—as well as the importance of his findings:The first clear statement about the importance of non-shared environment:“As far as personality and interests are concerned, then, it would appear that the relevant environments of a pair of twins are no more alike than those of two members of the population paired at random. Can this possibly be true? (p. 91)… Thus, a consistent – though perplexing – pattern is emerging from the data (and it is not purely idiosyncratic to our study). Environment carries substantial weight in determining personality – it appears to account for at least half the variance – but that environment is one for which twin pairs are correlated close to zero… In short, in the personality domain we seem to see environmental effects that operate almost randomly with respect to the sorts of variables that psychologists (and other people) have traditionally deemed important in personality development. What can be going on?” (p. 92).Nearly all psychological traits show moderate genetic influence : “Its message might roughly be translated: ‘Identical twins correlate about .20 higher than fraternal twins, give or take some sampling fluctuation, and it doesn’t much matter what you measure – whether the difference is between .75 and .55 on ability measures, between .50 and .30 on a personality scale, or between .35 and .15 on a self-concept composite” (p. 35).One of the earliest multivariate genetic analyses using twin data: “The motivation underlying such analyses is the hope that they may provide a powerful tool for studying how genetic and environmental influences affect phenotypic traits. The basic reasoning runs something like this: It is unlikely that our convenient phenotypic trait measures are aligned in a simple one-to-one fashion with either the genetic or the environmental sources of influence upon them. If they are not, the effects of such influences should often show up more clearly on the associations among traits than on the measures of the individual traits themselves. Thus, two genetically independent traits might be correlated because they are subject to common environmental influences, or two traits that share no important environmental inputs might both be affected by a particular gene or genes (‘pleiotropy’)” (p. 75).The beauty and clarity of John Loehlin’s writing also attracted me to behavioral genetics. It cannot be a coincidence that his undergraduate degree was English and that he is passionate about poetry. In part because of his writing and the clear thinking that underlies it, his books form part of the bedrock of behavioral genetics, bringing lucidity to difficult topics such as race differences .Another example of John Loehlin’s impact on my scientific career is my interest in GE correlation, which was sparked by John Loehlin while I was at Texas. This interest led to a paper with John Loehlin and John DeFries on GE correlation and interaction, which continues to be my most highly cited paper .In our 1977 paper, we considered the effects of GE correlation and interaction on quantitative genetic estimates, proposed three types of GE correlation , and suggested ways to assess GE correlation and interaction. In the present paper, I will briefly summarize research on GE correlation, highlighting new developments using DNA.GE correlation can be viewed as genetic influence on exposure to environments—literally, a correlation between genetic propensities and experiences. In contrast, GE interaction denotes genetic influence on response to environments, that is, a conditional relationship in which the effect of the environment on a phenotype depends on genotype , in a neighborhood , and in a school . Because these environmental variables are the same for members of a twin pair, they would be read as shared environmental influences in a twin design. However, the correlation between family-level environmental factors such as parental SES and children’s developmental outcomes could be mediated genetically but the twin design would not ‘see’ it. This is a problem primarily for research on children, but also for research that attempts to study the childhood origins of adult behavior. It is much less of a problem for twin studies investigating the effects of contemporaneous environments of adults to the extent that members of adult twin pairs live separate lives.One way to circumvent this within-family limitation of the twin design is to recast between-family factors, such as family chaos, as individual differences in children’s perceptions of their family chaos. For example, we have studied GE correlation using children’s perceptions of their experiences at home , is the first new human quantitative genetic technique in a century —can be used for quantitative genetic analyses of genetic influence. This technique, often called http://spark.rstudio.com/ctgg/gctaPower/). For example, a sample of 6,000 has 80 % power to detect a GCTA heritability estimate of 15 %. However, power declines sharply with smaller sample sizes: Samples of 4,000 and 2,000 have 80 % power to detect GCTA heritability estimates of 22 and 45 %, respectively. As discussed later, GCTA heritability estimates are limited to detecting additive effects of the common SNPs on current GWA microarrays, which results in GCTA heritability estimates often being about half of twin study estimates. GCTA can also be used to estimate genetic influence within pairs of siblings  for each pair of unrelated individuals based on that pair’s overall similarity across hundreds of thousands of SNPs; each pair’s genetic similarity is then used to predict their phenotypic similarity. Even remotely related pairs of individuals are excluded so that chance genetic similarity is used as a random effect in a mixed linear maximum likelihood model to decompose phenotypic variance into genetic variance as captured by the additive effects of causal variants in linkage disequilibrium with SNPs genotyped on DNA arrays  that have until recently been incorporated in commercially available DNA microarrays used in genome-wide association studies. In addition, GCTA is limited to detecting the additive effects of SNPs; it cannot detect gene–gene (or gene-environment) interaction. Thus, if GCTA heritability estimates are half the twin study heritability estimates, the additive effects of common SNPs can in theory account for about half of the heritability estimated from twin studies. The ‘missing GCTA heritability’, the gap between GCTA and twin study heritability estimates, could be due to nonadditive effects or the effects of rare DNA variants. In other words, GCTA estimates the lower limit of heritability from twin studies and the upper limit for genome-wide association studies. These generalizations may not apply equally to all behavioral domains. For example, childhood behavior problems and personality seem to show a greater gap between GCTA estimates and twin estimates than do other domains such as cognitive abilities; this may be due to greater assortative mating for cognitive abilities which increases additive genetic variance or to greater nonadditive effects for behavior problems and personality , a composite of parental education and occupational status, and its association with children’s intelligence (Trzaskowski et al. A strength of the child-based design using children’s genotypes in GCTA analyses rather than genotypes of their parents is that it captures the genetic influence of family SES on the children themselves. This feature of the design facilitates a bivariate GCTA that assesses the extent to which the well-known correlation between family SES and cognitive development—about 0.30 in meta-analyses Sirin —is mediaGE correlation challenges current conceptions of the environment as something ‘out there’ that happens passively to children. Finding genetic influence on measures of the environment and on their association with outcomes will make us rethink how the environment works, leading to an active model of experience in which children select, modify and create environments correlated with their genetic propensities. This active model of experience supports an educational trend in the direction of personalized learning, making the educational environment fit the pupil’s profile of strengths and weaknesses—and appetites as well as aptitudes—rather than using a one-size-fits-all curriculum (Asbury and Plomin “We may view this as a kind of cafeteria model of the acquisition of social attitudes. The individual does not inherit his ideas about fluoridation, royalty, women judges and nudist camps; he learns them from his culture. But his genes may influence which ones he elects to put on his tray. Different cultural institutions – family, church, school, books, television – like different cafeterias, serve up somewhat different menus, and the choices a person makes will reflect those offered him as well as his own biases. As he gets older, choice of cafeterias will become important, in addition to his choice of dishes within them” Loehlin , p.48.John Loehlin has written about this active model of GE correlation in relation to the development of social attitudes, which highlights the importance of choice. It is fitting for this festschrift in honor of John Loehlin to let him have the last word, especially because the environment he provided was a crucial component in the GE correlations of my life:


Manihot esculenta by the Next Generation (NEXTGEN) Cassava Breeding Project. NEXTGEN currently imputes missing genotypes in their datasets using a LASSO-penalized, linear regression method (denoted ‘glmnet’). We selected glmnet to serve as a benchmark imputation method for this reason. We obtained estimates of imputation accuracy by masking a subset of observed genotypes, imputing, and calculating the sample Pearson correlation between observed and imputed genotype dosages at the site and individual level; computation time served as a second metric for comparison. We then set out to examine factors affecting imputation accuracy, such as levels of missing data, read depth, minor allele frequency (MAF), and reference panel composition.Well-powered genomic studies require genome-wide marker coverage across many individuals. For non-model species with few genomic resources, high-throughput sequencing (HTS) methods, such as Genotyping-By-Sequencing (GBS), offer an inexpensive alternative to array-based genotyping. Although affordable, datasets derived from HTS methods suffer from sequencing error, alignment errors, and missing data, all of which introduce noise and uncertainty to variant discovery and genotype calling. Under such circumstances, meaningful analysis of the data is difficult. Our primary interest lies in the issue of how one can accurately infer or impute missing genotypes in HTS-derived datasets. Many of the existing genotype imputation algorithms and software packages were primarily developed by and optimized for the human genetics community, a field where a complete and accurate reference genome has been constructed and SNP arrays have, in large part, been the common genotyping platform. We set out to answer two questions: 1) can we use existing imputation methods developed by the human genetics community to impute missing genotypes in datasets derived from non-human species and 2) are these methods, which were developed and optimized to impute ascertained variants, amenable for imputation of missing genotypes at HTS-derived variants? We selected Beagle v.4, a widely used algorithm within the human genetics community with reportedly high accuracy, to serve as our imputation contender. We performed a series of cross-validation experiments, using GBS data collected from the species  Well-powered genomic studies require genome-wide marker coverage across many individuals. Many genotyping methods exist, and one typically selects a genotyping platform based on budgetary constraints and the available molecular tools for the species in question. Genetic variation in the human genome, for instance, has largely been captured using single-nucleotide polymorphism (SNP) arrays that can assay up to 2.5 million variants . The perGenotype imputation is a well-established statistical technique for estimating unobserved genotypes. Many genotype imputation algorithms and software packages exist, but most were primarily developed by and optimized for the human genetics community, a field where a complete and accurate reference genome has been constructed and SNP arrays have, in large part, been the common genotyping platform. These algorithms differ in their details but all essentially pool information across individuals in either a study sample or a reference panel or both to estimate haplotype frequencies from the observed genotype data, imputing missing genotypes simultaneously. Although the statistical methods for genotype imputation are now highly developed and widely used, selecting the set of haplotypes to include in the reference panel for maximum imputation accuracy in a given study population remains unclear. Selection schemes typically take one of two approaches: a ‘best match’ approach, which attempts to construct a reference panel that closely matches the ancestry of the study sample, or a ‘cosmopolitan’ approach, which makes use of all available haplotypes .Manihot esculenta . We compared Beagle v.4 to a LASSO-penalized, linear regression imputation method (denoted glmnet). We chose Beagle v.4 over other haplotype-phasing programs because the algorithm 1) scales well to large sample sizes (>1000) while other algorithms require some form of parameter space reduction to be computationally competitive, 2) requires no parameter specification, e.g. effective population size, 3) takes genotype likelihoods as input, and 4) performs genotype calling  or NA for the case where no data is observed for a given individual at a site. We converted genotype likelihoods into normalized, Phred-scaled likelihoods to use as input for Beagle.y ≥474 crosses were made, with only a subset of these crosses  producing viable progeny. The large variation in viable progeny number among attempted crosses results from the wide variation in flowering time, rate, and fertility in cassava . W. W11]. Wn = 20302). If each individual and each site in the study sample have a low proportion of missing data, no reference panel is needed to impute the missing genotypes in the sample; the almost complete data from the other individuals and the high marker density should provide sufficient information to impute with high accuracy. We tested this concept using the 694 GG individuals as our study sample. We extracted the genotype dosages and normalized, Phred-scaled likelihoods for the GG individuals at biallelic sites (n = 2490) as the target of imputation. We constructed two reference panels, one designed to represent a cosmopolitan reference panel for imputation scenario 2 and the other designed to represent a best-match reference panel for scenario 3. Variants absent from the reference panel, but present in the study sample, cannot be imputed. We, therefore, focused on the situation where the reference panel is defined on the same set of polymorphic sites as those found in the study sample, using the same set of 20205 biallelic SNPs defined in scenario 1.We assessed the impact of reference panel composition on imputation accuracy using C1 individuals  using the inferred haplotypes (imputed genotype dosages) from the 694 GG individuals; we constructed the best-match reference panel for Beagle (glmnet) using the inferred haplotypes (imputed genotype dosages) from the 80 C1 parents. Although a reference panel cannot be explicitly specified when imputing with glmnet, the algorithm can still make use of the information encoded in non-study sample individuals. The increased sample size of the training data should, in theory, increase imputation accuracy.We used the R package glmnet to fit a LASSO-penalized, linear regression model to the observed genotype data . The glmq = 1. The variable λ is a regularization parameter that controls the trade-offs between lack of fit and model complexity; λ ≥ 0 [v to the mean genotype at site v. Although the LASSO performs variable selection on its own, we performed an initial round of variable selection to decrease computation time—shrinking the variable search space to a subset of 60 markers rather than using all markers on a chromosome as potential predictors of genotype. We calculated pairwise correlations between the target marker and all intra-chromosomal markers, retaining the 60 markers that showed the strongest correlation with the target marker. We selected a maximum retention number of 60 arbitrarily. Other approaches for shrinking the variable search space exist but were not explored in this study. By default, glmnet selects a lambda value using 10-fold cross-validation, looking at 100 different lambda penalty coefficients. To decrease computation time, 5-fold cross validation was performed on 10 lambda values.We set y; λ ≥ 0 . In addiy; λ ≥ 0 . The algBeagle v.4 is an iterative algorithm for fitting a local haplotype hidden Markov model (HMM) to genotype data. The algorithm alternates between model building and sampling, using stochastic expectation maximization (EM) to converge towards the most probable solutions . There aref parameter. All genotypes in the reference panel must be non-missing and phased.The algorithm begins by imputing missing genotypes according to allele frequencies and randomly phasing heterozygous genotypes. Beagle v.4 then uses these initial haplotype estimates to obtain estimates of the HMM parameters. The algorithm constructs a directed acyclic graph (DAG) using the haplotype data and estimates the HMM parameters using observed haplotype counts and the assumption of Hardy-Weinberg Equilibrium (HWE). . Browninr, at site v, X denotes the site’s vector of observed genotype dosages and Y denotes the site’s vector of imputed genotype dosages. The sample Pearson correlation is calculated with the assumption that the genotype dosages are accurately estimated. The sample Pearson correlation is a function of two vectors, both of length L. The value of L varies across sites for two reasons: the random nature of the masking scheme and non-uniform representation of sites within the set of validation genotypes defined by Caller A and B. The Pearson correlation coefficient is undefined when either L<2 or when the vector of imputed genotype dosages is invariant.There are various metrics of imputation accuracy: imputation correlation, the Pearson correlation between observed and imputed genotypes, imputation concordance, the proportion of correctly imputed genotypes, imputation quality score (IQS), the concordance adjusted for chance agreement), etc. . We seled at site v if individual d was surveyed a minimum of seven times at site v and returns NA otherwise. The second method, Caller B, returns a genotype dosage for individual d at site v if the most likely genotype is at least 10 times more likely than the second most likely genotype and returns NA otherwise. We found that cross-validation experiments using Caller A and B validation genotypes returned similar results for imputation scenario 1 (data not shown), resulting in our decision to run scenario 2 and 3 using only Caller B validation genotypes.To calculate imputation accuracy, we masked a set of validation genotype dosages, imputed, and calculated the sample Pearson correlation between observed and imputed genotype dosages. We employed two different methods, Caller A and Caller B, to define the set of validation genotypes for cross-validation experiments. Caller A returns a genotype dosage for individual d has genotype v. We extracted each genotype’s read depth from the VCF file using VCFtools [M1, M2, …, M10. Each subset corresponds to a fold in the 10-fold cross-validation scheme. As an example, we generated the masked dataset for fold 1 by taking the original data matrix, finding the coordinates of the genotypes belonging to the set M1, and setting the elements in these coordinates to missing. This masking scheme resulted in 10 masked datasets (i.e. 10 folds). We calculated the imputation accuracy on a per-site basis for each fold and the imputation accuracy on a per-individual basis for each fold. We then calculated the median imputation accuracy per-marker and the median imputation accuracy per-individual across the 10 folds.We simulated a scenario where genotypes were missing in a random fashion across the genome and obtained estimates of imputation accuracy using 10-fold cross validation. The masking scheme is best visualized by describing the datasets as matrices, where the rows represent biallelic sites and the columns represent individuals. The elements in a matrix represent genotypes: individual VCFtools . We partWe measured computation time as the number of CPU minutes required to complete the imputation of one dataset. All jobs were submitted to the Computational Biology Service Unit at Cornell University, which uses an eight core Linux (Centos 6.2) Dell PowerEdge M600 with 16GB RAM.We imputed masked genotypes at 20205 SNPs on chromosome 5 in a sample of 694 individuals from the GG population. In this section, we report the results from cross-validation experiments where the set of validation genotype dosages was defined using Caller B see .L. The value of L varies across sites and individuals because genotype masking occurs at random and because genotype call rates vary across the 20205 sites (see v is undefined under two scenarios: when L<2 (true for 34 of the 20205 sites in the dataset) and when the vector of imputed genotype dosages at site v has a variance equal to zero. The latter occurs when imputation returns identical genotype dosages for all L masked genotypes at site v. We obtained accuracy estimates for Beagle at 13028 sites (set A) and 19933 sites (set B) for glmnet. Set A is a subset of B, i.e. every member of set A is also a member of set B. L for sites in set A imputed with Beagle. As might be expected, we observed greatest variation among accuracy estimates for small L  to analyze. We selected a filter threshold of 30 somewhat arbitrarily but opted for a moderate-stringency filter to avoid removing a large subset of sites from our analysis.The sample Pearson correlation is a function of two vectors, both of length ites see . The sam small L . We remoWe examined the effect of the proportion of missing data on imputation accuracy at the site and individual level . As mighr as a function of estimated MAF for the 9737 sites in set C. We divided the range of x into five-equally sized bins , and summarized accuracy values within each frequency bin using the mean and median  at all 20205 sites using the sample of 694 individuals from the GG population. d median . We obser when imputing with the larger reference panel . The use of information from multiple individuals and patterns of LD has been shown to lead to significant improvements in genotype-calling accuracy for low-depth sequence data  and the imputed genotype dosage. We note that to obtain true measures of imputation accuracy, the imputed genotype dosage should be correlated with the true genotype, rather than the observed genotype dosage. Unfortunately, true genotypes are not known and observed genotype dosages must be used instead. The accuracy based on correlation to the observed genotype dosages under-estimates the true imputation accuracy in two ways. First, there is error associated with the observed genotype dosage  that reduces the correlation. Second, the observed genotype dosage of individual ple, see ).Using a set of validation genotypes at biallelic SNPs on chromosome 5, we found that Beagle and glmnet impute missing variants with similar accuracies. When comparing the two methods at the site level, glmnet appears to impute with (moderately) higher accuracy relative to Beagle, regardless of levels of missing data. We, however, observe little difference between the two methods when measuring accuracy at the individual level . We obseThe unique aspects of the datasets derived from a non-human organism, such as cassava, and HTS methods, such as GBS, do not affect Beagle’s imputation accuracy in ways we do not understand or expect. For instance, we observed a decrease in imputation accuracy as MAF decreased  left, coThe modest difference in imputation accuracy between Beagle and glmnet was in some ways unexpected, largely because the two algorithms employ contrastingly different approaches to modeling genotype data. Glmnet does not attempt to directly relate observed correlation patterns to any underlying biological process, whereas Beagle specifies a statistical model for the biological aspect of the problem–namely, the haplotypes that generated the observed LD structure. Both algorithms leverage data at a subset of markers to impute missing genotypes at a particular locus, but they employ very different subset selection strategies. Glmnet selects markers solely on measures of pairwise correlation, ignoring locus order and spacing. Beagle, in contrast, focuses on a small number of nearby markers when imputing missing genotypes at a particular site . Correlation between markers is a localized phenomenon; that is, there tends to be less LD between loci that are far apart than between loci that are close together. The apparent correlations observed between distant markers are largely statistical artifacts, i.e. noise introduced by sampling variation. While glmnet and Beagle produced similar results in our cross-validation experiments, we reason that there are situations in which Beagle will outperform glmnet (e.g. when levels of spurious associations between distant markers is high relative to true levels of LD). In addition to decreased sensitivity to spurious associations between distant markers, probabilistic, phasing methods, such as Beagle, offer additional benefits, such as providing phased haplotypes and measures of imputation accuracy estimated from posterior genotype probabilities.n = 78). Reference panel B, in contrast, consists entirely of individuals listed as C1 parents in pedigree records (n = 80). The set of C1 parents in panel A is a subset of panel B. We found that imputation using reference panel A and B resulted largely in similar imputation accuracies across sites. We find this reassuring for two reasons: 1) the 617*2 haplotypes from the non-parental individuals in reference panel A appear to serve as good proxies for the haplotypes of the two C1 parents that are present in panel B but absent in panel A and 2) adding ‘extraneous’ haplotypes to the reference panel appears to introduce little error to the imputation procedure, consistent with previous observations made by those in the human genetics community [In scenario 2 and 3, we used a sample of 2490 C1 individuals to compare the accuracy of genotype imputation with a cosmopolitan reference panel (reference panel A) and a best-match panel (reference panel B). Reference panel A consists of 694 individuals, a subset of who are list as C1 parents in pedigree records  The Venn diagram shows the composition of reference panel A and B. (B) We evaluated Beagle and glmnet under three imputation scenarios: imputation guided by no reference panel (left), a reference panel with large genetic diversity , and 3) a reference panel that closely matches the ancestry of the study sample .(PDF)Click here for additional data file.S2 FigThe white and red histogram displays the distribution of all variant sites (30018) and biallelic SNPs (20302) along the length of chromosome 5, respectively.(PDF)Click here for additional data file.S3 FigNo records of genetic relatedness among the 696 reference panel individuals exist. We, therefore, performed a PCA to explore whether there is any evidence of population structure among reference panel individuals. Reference panel individuals contributing zero offspring to the C1 population appear as grey dots. Reference panel individuals contributing >0 offspring to the C1 population appear as red dots with diameters scaled proportionally to the number of offspring contributed by the individual.(PDF)Click here for additional data file.S4 FigThe proportion of missing data at a given site is measured across the 694 GG individuals. The term “missing” denotes zero reads observed at a given site for a given individual. We removed sites with >90% missing data, leaving a total of 20205 sites for cross-validation experiment 1. We used this same set of sites for scenarios 2 and 3 for reasons given in the main text.(PDF)Click here for additional data file.S5 Fig(A) The histogram shows the distribution of the mean read depth per site averaged across all 694 GG individuals. (B) The histogram shows the distribution of the mean read depth per site averaged across all 696 reference panel individuals. The red vertical line marks the mean of the distribution.(PDF)Click here for additional data file.S1 Table(XLSX)Click here for additional data file.S2 Table(XLSX)Click here for additional data file.


Passer domesticus) population with known pedigree. We fit pedigree‐based  linear mixed models , with and without additive genetic and individual maternal effects, and use deviance information criterion (DIC) for choosing between these models. Pedigree and R‐code for simulations are available. For this study system, the simulation studies show that only large maternal effects can be identified. The genetic maternal effect  has to be at least half of the total genetic variance to be identified. The consequences of omitting a maternal effect when it is present are explored. Our results indicate that the total  variance are accounted for. When an individual  maternal effect is omitted from the model, this only influences the estimated (direct) individual  variance. When a genetic maternal effect is omitted from the model, both (direct) genetic and (direct) individual variance estimates are overestimated.In this paper, we demonstrate how simulation studies can be used to answer questions about identifiability and consequences of omitting effects from a model. The methodology is presented through a case study where identifiability of genetic and/or individual  maternal effects is explored. Our study system is a wild house sparrow ( I have a biological hypothesis I want to test for my favorite study system. Is my data set large enough and does it have enough structure to verify my hypothesis? Is the test I use appropriate? And if I leave out important terms in my models what happens with the estimates of the other parameters?Most quantitative biologists working with natural populations are in this situation of doubt regularly. In this paper, we describe through a case study how simulation studies can be used to answer some of these questions.Even though simulations are an established methodology Ripley , it is nPasser domesticus)  and individual  effects, but also directly by its mother's genes and/or individual effects. These effects can be interpreted as a case of social effects  Willham . If therA pedigree‐based  mixed model often called the animal model can be used to identify different genetic and environmental  effects, including maternal effects  .The aim of the paper was to propose a protocol for testing whether a given study system is structured enough to accurately identify random effects, for example, maternal effects. The R‐code is available (see Passer domesticus) populations on six islands off the coast of Helgeland, Northern Norway . The pedigree spans over seven generation, it consists of np=3574 individuals and observations are available for our focus trait (bill depth) for nd=1025 of the individuals in the pedigree (nd<np). The study system was used in Holand et al.  genetic effect for individual i, mm(i) is the additive maternal genetic effect of individual i's mother (m(i) is the index of the mother of individual i). ai and mm(i) are modeled as random structured effects. pm(i) is the maternal individual effect that affect individual i and ϵi is the (direct) individual effect for individual i. pm(i) and ϵi are modeled as an independent identical distributed (i.i.d.) effects. Further, each of them (ai and mm(i), and ϵi and pm(i)) are assumed to be independent. For the population, the additive genetic effect is assumed to follow a Gaussian distribution A is the relationship matrix , where I is the identity matrix and σp2 is the maternal individual variance, and ϵ∼N, where σϵ2 is the individual (direct) variance. Individual effects are often referred to as environmental effects. To complete the Bayesian modeling, priors are assigned to parameters. The variances σa2, σm2, σp2, and σϵ2 are given InvGamma priors, and β0 is assigned a flat prior.To model direct additive genetic and maternal effects, we use an animal model: In this paper, we use four different animal models, denoted M1–M4. The first model, M1, is an animal model without any maternal effects. M2 and M3 are extensions of M1, including only one of the maternal effects; the genetic maternal effect is included in M2 and the individual maternal effect in M3. M4 is the full model specified in eq. . These mα, we reject the null hypothesis if the probability of getting a more extreme test statistic is less than α.Traditionally, hypothesis tests are performed for a parameter. Under the null hypothesis, the parameter has a specific value, often zero. Based on a test statistic (often the estimator of the parameter), its sampling distribution and a chosen significant level s, we sample a data set from the model when H0 is true, and calculate the test statistics. We repeat this S times, and the  distribution of the corresponding S test statistics is an approximation to our samplings distribution from which we can find critical values of interest. An important property of a test is its power. The power of a test is the probability to reject the null hypothesis when it is not true, that is, to correctly reject H0. The power function can be obtained by performing simulations studies for a set of values for the parameters. The proportion of the simulated data sets that are rejected is an estimate of the corresponding power.If we do not know the sampling distribution, we can find the critical value by simulations: For each simulation There are two basic questions we want to answer by simulation studies: (1) Are maternal effects identifiable for our study system? and (2) What is the effect of not including a maternal effect in the model when it is present?The question about identifiability can be set up in a hypothesis setting. For example, the null hypothesis is the model without maternal effects (M1 in Table H0 and conclude with H1 based on (simulated) data. To compare the models, we use difference in DIC, ΔDIC. If the difference is above a critical value CΔDIC we conclude that the H1 is true, and we have identified a maternal effect.We want to find when we can reject Now several choices arise, and we provide suggestions for how to approach these using simulations. Different versions of the generic simulation algorithm below will be used.Step 1: Set parameters.Step 2: Simulate S new data sets, ys, s = 1, 2,…S according to H0 and/or H1.    • Set NA values according to the missing structure of the data set.Step 3: For each data set ys, fit model(s) according to H0 and/or H1, and calculate and/or store relevant quantities.ΔDIC=DIC(H0)−DIC(H1). We suggest that CΔDIC is based on a chosen significance level α and calculated as the corresponding quantile CΔDIC of ΔDIC based on simulations. In step 2 in the simulation algorithm, we sample according to the model under H0. In step 3 models, under both H0 and H1 are fitted, and ΔDICs=DIC(H0)−DIC(H1) is calculated and stored.The difference in DIC between the two models is calculated as α = 0.05. The appropriate limit for rejecting H0 (CΔDIC) is then calculated from the 95% quantile of the empirical simulated distribution of ΔDIC.We choose a significance level, for example, H0 is rejected when H1 is true. If H0 is rejected, we have identified a maternal effect. The power is a function of the magnitude of the effect, that is, σm2 for the test in , we can use simulations. In step 2 of the algorithm data sets are simulated from H1, and in step 3 both models (H0 and H1) are fitted and ΔDICs calculated and stored. The power is then estimated by the proportions of the ΔDICs that are larger than CΔDIC. The power function can be estimated by doing similar simulations for a range of values for the maternal variance.The power of a test is the probability that the t in eq. . Hence, H1 (step 2), and in step 3 the model under H0 is fitted. The posterior of the parameters  are stored. From the posterior means, we have the sampling distribution of the (other) variance parameters when maternal effects are present in the system, but not in the model.To answer this question, we can do simulations similar to those we used to find the power function. For a specific value of the maternal variance, data sets are simulated from the model under H1. In step 3, the same model is fitted, that is, H1, and posterior quantities such as that posterior mean, median, and the 95% credibility interval are stored. Afterward we can compare the mean of the posterior means with the true parameter values. Further, the coverage can be found by finding the proportions of the credibility intervals that contain the true parameter value.If we fit the same model as we have simulated from several times, we want our estimates  to be close to centered around the true parameter and that the credibility interval is a good quantification of the uncertainty. In statistical terms, we want our estimator to be (close to) unbiased, that is, when the experiment is repeated (many simulations), the mean of the estimates approaches the true parameter. Further, we want the credibility interval to have the right coverage, that is, to be (close to) having the property of a frequentist confidence interval: In the long run, the true parameter should be in the 95% confidence interval in 95% of the simulations. We can check our estimators by simulations. In step 2, we simulate from a model, say σ^a,M12 and σ^ϵ,M12. These estimates can be used to guide the parameter choices in the simulation studies. Both because of computational cost, and to be able to interpret results, it is practical to make some restrictions. We suggest that the genetic and individual effects should be hold constant and equal to σ^a,M12 and σ^ϵ,M12, respectively. For model M1 and M4 that means that values for σa2 and σm2 are chosen such that σa2+σm2=σ^a,M12 and for model M3 and M4 σϵ2+σp2=σ^ϵ,M12.The results obtained from the simulation study depend on the values chosen for the parameters. How should we choose the values for the parameters so that they are realistic to the trait in question? We suggest that the true data set first is fitted to the simplest model we consider, that is, an animal model without maternal effects (M1) for our study. Denote the estimated variances from this model H0:M1 versus H1:M2 , and simulation study 2 (S2) is according to the test H0:M1 versus H1:M3 . See Figure M2 and M3. We also explored what happens when we go from one to two maternal effects, that is, H0:M2 versus H1:M4 and H0:M3 versus H1:M4, but these results are not presented. Only Gaussian traits were considered, and all simulations and inference were performed using AnimalINLA R package (Holand and Martino ∑ai=0 and ∑mi=0 (Steinsland and Jensen S = 1000 data sets.We show results for two simulation studies. Simulation study 1 (S1) is according to the tests; How should I set the parameter values such that my simulation study is relevant?, and set σ^a,M12=0.6 and σ^ϵ,M12=0.4, giving us a phenotypic variance σP2=1, and β0=0. The phenotypic data are only used for this purpose. Further, a discretization of 0.1 was used for the variance parameters. The parameter values used in the simulation studies S1 and S2 are summarized in Table We set parameters according to our suggestions for setting parameters in section H0:M1 versus H1:M2, that is, an animal model without maternal effects versus one with maternal genetic effects.In the first simulation study (S1), we consider CΔDIC. The sampling distribution for ΔDIC when M1 is true is given by the boxplot for σm2=0 in Figure CΔDIC=110 and is marked by a horizontal dotted line.We first find the critical value σm2) are visualized by boxplots in Figure σm2=0.3 is needed to get a power of 0.8. Hence, a substantial portion of the genetic effects needs to be maternal (remember σa2+σm2=0.6) to have a high probability to identify it.Next, we want to find how large the genetic maternal effect has to be to be identified. The sampling distributions for different values of the maternal genetic variances σm are visuσa2) than the model including additive maternal effects (M2). But not all the genetic variance (they sum to 0.6) are taken up by σa2. Indeed, we see from Figure σϵ2 increases when the maternal effect is high. That is, it seems that excluding the maternal genetic effect from the model causes both the additive (direct) genetic variance estimate (σa2) and (direct) individual variance estimate (σϵ2) to increase.The effect of not including the maternal effects in the model when they are present can be explored in Figure σm2 or σa2 is a relative small part of the total genetic variance (σa2+σm2=0.6). This is also found in Holand et al. , we consider CΔDIC=7.The results are summarized in Figure σp2=0.1 and close to 1 for σp2=0.2. But, this is still a substantial proportion of the individual variances (remember σϵ2+σp2=0.4).From Figure σϵ2) when fitting the model M1  is approximately 0.4 (σϵ2+σp2=0.4) independently of the true maternal individual variance (σp2). Further, from Figure σa2) does not change systematically as a function of the individual maternal variance σm2. In this situation, it seems as the omitted maternal individual effects are taken up by the direct individual effects only.To evaluate the consequences of not including the maternal individual effects in the model when they are present, we study Figure σϵ2. This might also cause the poor coverage of σϵ2 and σp2. As for (S1), the coverages are poor when one of the variances is small  is good for all values of σϵ2. On the other hand, when fitting the correct model (M3) coverage is poor for σa2 for small values of σϵ2. From this we learn that for our study system it is not possible to estimate variance parameters in model M3 precisely, the estimates are biased, and the confidence intervals do not have the right coverage.To explore the properties of the variance estimators, we look at the coverage Fig. B and theall Fig. B–D. ThisThe ability to identify and distinguish both genetic and individual maternal effects (M4) can be explored similarly. The results (not presented here) show that this study system neither has the right properties to identify both individual and genetic maternal effects, nor to estimate the maternal variances precisely.We have used simulation studies to explore and gain understanding of our study system's ability to identify maternal effects as well as the consequences of omitting maternal effects from the model when they are present. We have learned that maternal effects can (only) be identified if they are substantial. For our study system, the genetic maternal effect has to be about half of the  genetic to be identified. We also get similar result for individual maternal effect.We have considered a Gaussian trait and maternal effects. The same methodology can also be used for non‐Gaussian traits, and other effects, for example, additive genetic effects or sex‐linked effects or the consequences of missing not at random . This indicates that using one fixed critical value independent of the models compared  gave very different critical values  might gFitting a model without maternal effects when maternal effects are present will affect the estimated variance parameters. Generally, the total true variance seems to be accounted for. When the maternal individual  effect is omitted from the model, only the (direct) individual variance seems to be affected, and it is estimated to be the sum of direct and maternal individual effects. This result is in accordance with the findings in Larsen et al. . They peWhen a maternal genetic effect is omitted from the model, it influences estimates of both direct genetic and direct individual variances. The total  variance are accounted for in the results, but the total amount of genetic variance  is not always picked up by the estimated (direct) genetic variance. Hence, it is not generally true that all genetic effects are accounted for by a (direct) additive genetic effect.The simulation studies showed that when at least one of the variance parameters are close to zero, we might have biased estimates, and poor coverage for several of the variance parameters. This indicates prior sensitivity, and we have learned that priors needs to be carefully chosen, and prior sensitivity should be checked.For complex systems, it is difficult to have an intuition for our study system's ability to identify effects of interest and how omitting effects that are present influence parameter. Simulations studies are a powerful tool in this situation. Fast simulation and inference make simulations studies more attractable. We are able to explore identifiability properties and the consequences of omitting effects from the model for our study system with the models (including priors) and inference method we have chosen.AnimalINLA R package (available at www.r-inla.org).The house sparrow pedigree and R‐code for performing the simulation studies are archived in the None declared.


This method can improve statistical power by incorporating independent but relevant information about the research objective. Using the real genetic and phenotype data of chromosome 3 from Genetic Analysis Workshop 18, 1 SNP from gene CACNA2D3 was found to have significant association with mean arterial pressure.In a genome-wide association study, association between disease trait and hundreds of thousands of genetic markers are tested. Several methods have been proposed to control the false discovery rate in such high-throughput data to adjust for multiple hypotheses testing. For Genetic Analysis Workshop 18, we applied the method of false discovery rate control with  For genome-wide association studies, the p values for multiple testing can be adjusted by the results from previous genetic linkage study with improved power [Recent developments in technologies have made it possible to collect a large amount of data and perform thousands of statistical tests on the data. A lot of methods have been proposed to control multiple testing. The family-wise error rate control method is too stringent. The false discovery rate (FDR) method proposed by Benjamini and Hochberg  is more ed power .p values from family-based association test by independent information from population based association. We find this method can improve statistical power by conducting a simulation study.In Genetic Analysis Workshop 18 (GAW18), a family-based association test study was conducted to identify single-nucleotide polymorphisms (SNPs) that have significant association with blood pressure, a major cardiovascular and heart disease risk factor. Current genome-wide association studies have identified several genes that are associated with blood pressure or hypertension ,5. With The GAW18 data set consists of 1043 individuals from 20 Mexican American pedigrees from Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) Project 2. Real genetic data for 472,049 SNPs on odd-numbered autosomes are available for 959 individuals. A total of 932 individuals have real phenotype data, including sex, age at examination, year of examination, systolic blood pressure (SBP), diastolic blood pressure (DBP), antihypertensive medications, hypertension diagnosis, and tobacco smoking for up to 4 time points. The maximal set of genetically unrelated individuals consists of 157 individuals, among which 142 individuals have real genotypes. Our study focused on real genetic data for chromosome 3 only.i, we define the quantitative phenotype as mean arterial pressure (MAP) [As a result of the high missing rate in real phenotype data of GAW18 for second or later examinations, all the analysis was based on phenotype information for the first examination only. For each individual re (MAP) , which cMAP is a term often used in medicine to describe an average blood pressure by combining DBP and SBP in an individual.p values from the population-based association test as weights to control the FDR of family-based association tests. The following material describes our analysis steps.We applied adjusted linear regression models to each of the SNPs in chromosome 3 within unrelated individuals to conduct a population-based association test. Family-based association tests for this quantitative trait were applied to all the trio families using the quantitative transmission disequilibrium test (QTDT). We then applied the WFDR control by incorporating Apply the adjusted linear regression model to the unrelated subjects. The adjusted covariates are sex, age at baseline examination, use of medications, and principal component from population stratification. Thus the model fitted for each SNP is as follows:jth SNP. We denote p value for jth SNP as p values from family-based association tests in the later step.where p value for jth SNP as p values from m hypothesis tests, then Benjamini-Hochberg procedures rejects any null hypothesis for any Break the large pedigrees into trios. A trio family consists of 2 genotyped parents and 1 offspring. The offspring should have both genotype and phenotype information. Apply family-based association tests for quantitative trait by PLINK  on MAP aApply WFDR control to adjust j, we compute α to each p values.For each test To evaluate the performance of the WFDR approach, simulations were implemented. First, we simulated 250 trio families with a single quantitative phenotype, 49,999 SNPs were simulated under the null hypothesis in which each SNP is independent to the phenotype; 1 SNP was simulated under the alternative hypothesis to be associated with the phenotype. The association of the phenotype and genotype was based on the linear relationshipα is a constant, genotype is an ordinal variable of the SNP genotype , ε is random error, which follows normal distribution, and β is the coefficient that reflects the effect size of relationship between genotype on phenotype. We also simulated 250 independent individuals for each SNP; each SNP was used to test population-based association.where p values were calculated by counting how many replications were significant after applying the ordinary FDR or WFDR approach under different effect sizes.Different effect sizes were simulated for the associated SNP. We applied 1000 replications. We also applied both the ordinary FDR approach and the WFDR approach on the simulated 250 trio families. Empirical Among all family members, 58% of are females and the median MAP is 88, ranging from 58 to 154. Among unrelated individuals, 58% are females and the median MAP of all unrelated individuals is 89 (70 to 126).For genetic data, we started quality control with 65,460 SNPs of chromosome 3 using PLINK  was usedp Values of each SNP and corresponding positions were provided in the Manhattan plots  to the raw p values from the family-based association test. Table After applying FDR control on the weighted β to assess the power under different effect sizes. We chose βs as 0.1, 0.3, 0.4, 0.5, 0.6, 0.8, 0.9, 1.0, 1.5 and 2.0. Figure We applied simulation with 10 different effect sizes of p values from the family-based association test by incorporating the population-based association test. A simulation study was conducted to examine the improvement of statistical power of WFDR comparing FDR under different effect sizes.In this study, we extended the WFDR to adjust According to the simulation results, we believe that the weighted method for FDR control improves statistical power if there is a correct assignment of weights as we can see a big increase in the statistical power from FDR to WFDR when the effect size between phenotype and genotype is moderate. Although our study only focused on continuous outcome, the method can also be applied to other disease outcomes, such as binary outcome. Further analysis and simulation could be explored to assess the efficiency of the WFDR method for binary outcome. We are further developing our method for family-based association tests on the nuclear family.p values for family-based association test, which we believe we will more powerful than FDR. Based on GAW18 data, we evaluated the genetic association between each single SNP from chromosome 3 and MAP using the family-based association test for quantitative trait. Using ordinary FDR to control for multiple testing, we found no significant association. However, applying WFDR, we found 1 SNP rs9828485 in gene CACNA2D3 that is strongly associated with MAP. CACNA2D3 has been reported to be associated with cardiac disease [CACNA2D3 could also lead to blood pressure.We propose a WFDR method to adjust The authors declare that they have no competing interests.XQ and WX designed the analysis plan. XQ carried out the analysis and drafted the manuscript. XS generated genetic quality control. All authors participated in discussions of the analysis plan and helped revise the manuscript. All authors read and approved the final manuscript.