Arabidopsis thaliana genotypes. Using two orthogonal genetic mapping approaches, we find that between-plant allelic differences at the AtSUC8 locus are strongly associated with mixture overyielding. AtSUC8 encodes a proton-sucrose symporter and is expressed in root tissues. Genetic variation in AtSUC8 affects the biochemical activities of protein variants and natural variation at this locus is associated with different sensitivities of root growth to changes in substrate pH. We thus speculate that - in the particular case studied here - evolutionary divergence along an edaphic gradient resulted in the niche complementarity between genotypes that now drives overyielding in mixtures. Identifying genes important for ecosystem functioning may ultimately allow linking ecological processes to evolutionary drivers, help identify traits underlying positive diversity effects, and facilitate the development of high-performance crop variety mixtures.In plant communities, diversity often increases productivity and functioning, but the specific underlying drivers are difficult to identify. Most ecological theories attribute positive diversity effects to complementary niches occupied by different species or genotypes. However, the specific nature of niche complementarity often remains unclear, including how it is expressed in terms of trait differences between plants. Here, we use a gene-centred approach to study positive diversity effects in mixtures of natural  Arabidopsis thaliana genotypes, can be genetically mapped and resolved to a single gene.Biodiversity often increases the functioning and productivity of ecosystems or communities. This work shows that such a positive diversity effect, namely overyielding in mixtures of two divergent  Diversity is typically quantified as the number and abundance of species (species diversity) or the occurrence and variation of specific plant trait values 12. It is assumed that niche complementarity underlies biodiversity–ecosystem functioning (BEF) relationships, but this is difficult to demonstrate empirically because niches are difficult to quantify17. In functional ecology, trait differences are often used as an indicator of niche differences22, and are major determinants of the composition, diversity, and functioning of communities25. However, it is currently less clear which trait differences drive the positive BEF relationships in plant communities27, but see 31. The trait-based approach also has some limitations, in particular when it comes to understanding positive biodiversity effects: first, traits often co-vary because of evolutionary trade-offs between ecological strategies33, which makes it difficult to distinguish correlation from causation in trait-based analyses of BEF experiments. Second, while traits often are associated with environmental conditions34 (e.g. correlation between specific leaf area and soil moisture), it remains unclear whether differences in these traits drive the studied diversity effect, or whether other, unknown trait differences that correlate with these underlie the observed effects. Finally, it also is possible that many small phenotypic trait differences need to be considered simultaneously to adequately capture niche complementarity between plants35, making it more difficult to identify specific mechanisms that cause BEF relationships36. Here, we explore a gene-based approach to investigate the causes of positive diversity–productivity relationships in plant stands. Our new approach complements traditional trait-based methods and helps to identify causal drivers.Numerous experiments in natural and managed ecosystems including grassland, forest, and cropland have shown that productivity often increases with the diversity found in plant communities39), although there are exceptions40. The general mechanisms underlying niche complementarity and overyielding may be similar in both cases, although the potential for niche differences between species is greater than between genotypes of the same species. Here, we focus on positive genetic diversity effects in plant stands of the model plant Arabidopsis thaliana and compare genotype mixtures that contain either one or two alleles across regions of the genome. A major advantage of this approach is that the diversity of traits and alleles can not only be manipulated by assembling different plant stands from an existing pool of genotypes, but also from new genotypes created through crosses , and use association mapping to resolve it to variation at the Arabidopsis thaliana (L.) Heynh. genotypes. . This effect is relatively large for a pot-based within-species diversity experiment. For comparison, the average overyielding in field trials with crop variety mixtures typically ranges from 2 to 4%48.In order to genetically dissect the mechanisms that underly biodiversity effects on productivity, we first identified genotypes that overyielded when grown together in mixture, i.e., genotype combinations that produce more biomass than the average of their monocultures Fig. . We test52. In diallel analyses, general and specific combining abilities  gene, in which natural variation causes strong differences in root architecture between Sav-0 and Uk-153. We hypothesized that such differences could drive overyielding in genotype mixtures, which, however, turned out to be wrong. The 20 chosen genotypes were grown in all pair-wise combinations and genotype monocultures were grown in duplicates. The diallel was replicated four times at different dates , resulting in a total of 920 pots sown for this experiment. We further used two different substrates . For each genotype composition, we determined aboveground dry matter production. Comparing mixture productivities to monocultures of the mixture components, we estimated that the 190 genotype mixtures overyielded on average by 2.8% . This estimate is slightly below the overyielding observed in the parental mixture. However, if the genetic differences driving overyielding are restricted to specific genetic loci, then only part of these genotype mixtures will overyield , while the others will not. Therefore, and in order to genetically map such diversity-effect loci, we determined the average SCA across the four blocks for each of the 210 genotype compositions (190 genotype mixtures plus 20 monocultures). To adjust for differences in community productivity between substrates, and to obtain a normal distribution of residuals, we scaled the estimated SCAs by dividing these by the average community biomass on the respective substrate. SCA thus was expressed as effect relative to the mean productivity of all compositions on the substrate. Next, we tested whether variation in SCA among genotype compositions could be attributed to genetic differences at specific genomic regions. Since the published marker density for the RIL population used here was relatively low, we first constructed high-resolution genotype maps by whole-genome re-sequencing of each line . The SCA of the mono-allelic genotype mixtures  averaged 0.8% higher than the SCA of genotype monocultures, but this difference was not statistically significant. We then applied the additive partitioning method54 to test whether overyielding within genotype mixtures  was associated with the dominance of productive genotypes carrying either the Sav-0 or Uk-1 allele , or whether both allele carriers benefitted from growing in mixture . We found that overyielding was largely driven by CEs  and not by SEs . The average CE was 24.7 mg, which corresponds to 3.1% of the biomass average of all genotype compositions analyzed in the experiment. Overall, the experiment shows that genotypic mixtures overyield predominantly when component genotypes differ genetically at a single genomic region on chromosome 2, and that this overyielding is mainly due to a CE.To study the genetic basis of this effect, we established a competition diallel panel Fig. , an expeBREVIS RADIX (BRX) gene confers a fitness advantage to plants grown on acidic soil55 and alters root architecture and plant competition56. In our experiment, allelic diversity at the BRX locus was not, as originally hypothesized, associated with mixture overyielding , which we considered a candidate diversity-effect gene. The gene encodes a proton symporter that is fueled by the electrochemical gradient across the membrane. AtSUC8 is predominantly expressed in the root columella58, cells that are in direct contact with the soil, whose pH might affect its activity. To explore the idea that natural genetic variation at the AtSUC8 locus could drive functional complementarity among Arabidopsis genotypes, we re-analyzed previously published data on competition between Arabidopsis genotypes45. Single individuals of ten tester genotypes (including Sav-0 and Uk-1) each competed separately with each genotype of a panel of 98 natural accessions, in a factorial design . This SNP indeed resides in the AtSUC8 coding region. Although this is not an unequivocal proof that the identified SNP is the causal genetic polymorphism , this finding provides further evidence that genetic differences in or around the AtSUC8 gene contribute to overyielding of genotype mixtures.The Uk-1 accession was originally collected from the banks of the Dreisam river in the Schwarzwald region of southern Germany. This area is characterized by an edaphic gradient with pH ranging from neutral to strongly acidic , and the Uk-1 allele carries eleven amino acid polymorphisms (six non-conservative)  Fig. . Among tve) Fig. . To detek-1 Fig. . We nextk-1 Fig. . Relativles Fig. . HoweverArabidopsis genotypes that overyielded when grown in mixtures. We found that a large proportion of the overyielding of mixtures of the Arabidopsis accessions Sav-0 and Uk-1 was due to allelic diversity at a major-effect QTL on chromosome 2. Two aspects of this QTL mapping study are worth noting. First, our QTL mapping resolution was very high despite using only 18 recombinant lines and their parents. This was due to the competition diallel experimental design, in which genotypes with high-density marker maps were systematically combined into different genotype compositions. Second, although complex traits of individuals, such as growth, are often determined by genetic variants at many loci, each with small effect61, our results - together with findings from recent studies46 - indicate that complex community-level properties, which depend on interactions between plant individuals, can have simple genetic underpinnings. Thus, our work suggests that positive effects of plant diversity need not be irreducibly complex emergent properties but can have simple causes that are identifiable at the genetic level, even if the mixed genotypes differ at many positions in the genome. Therefore, we suggest that, in order to analyze the causes of overyielding in plant mixtures it is – at least in some cases – easier to first identify the genetic differences responsible for overyielding and then derive the responsible functional traits rather than directly search for the latter using phenotypic differences. In our study system, overyielding seems ultimately driven by allelic variation causing differences in root physiology, a fact we would very unlikely have discovered if we had started with an analysis of phenotypic trait differences used in functional ecology 19. The genetic approaches presented here, a further development of earlier approaches presented elsewhere64, may thus provide an effective way to understand the propagation of effects across different layers of biological organization, from genes to populations to communities and ecosystems.Here, we used two complementary genetic strategies, QTL- and association-mapping, to identify the genetic differences between 65. In our study we were able to associate diversity at the AtSUC8 locus with the overyielding of genotype mixtures. The respective gene encodes a proton-sucrose symporter, i.e., a membrane-associated protein that utilizes a proton gradient to transport sucrose across membranes. The gene is expressed predominantly in root tissues that are in direct contact with the soil. Genetic differences at the AtSUC8 locus affect protein function and are also associated with differences in root growth, in a substrate-pH dependent way. Soil chemistry, composition and texture, and the resulting effects on plant–plant interactions are major selective forces, but also important drivers of community structure69. Consistent with the idea that the Uk-1 genotype exhibits traits that make it better adapted to acidic soil55, plants carrying the AtSUC8Uk-1 allele showed root growth that was less sensitive so substrate acidification. Although plausible, the hypothesis that evolutionary divergence driven by local adaptation to soil conditions has shaped genetic variation at the AtSUC8 locus requires further evidence. The identification of this specific gene will facilitate future experiments or population genetic studies on the role of specific nucleotide divergence on root physiology and fitness consequences on different soil types. However, and perhaps surprisingly, genetic variation at the BRX locus itself, which had previously been shown to underlie adaptive divergence along this environmental gradient55, did not drive overyielding in our experimental plant stands. Future work should be able to establish possible reasons for these differences between AtSUC8 and BRX, and the specific physiological and morphological effects of the identified genetic variation at the AtSUC8 locus and their consequences for plant fitness under natural conditions.Identifying the genes that are important for ecosystem processes may ultimately also be useful to link ecological processes to evolutionary driversAtSUC8-mediated trait differences account for overyielding in genotype mixtures. We think that the different AtSUC8-associated responses of root growth to soil acidity promote the partitioning of the physical soil space between plants. One possibility is that pre-existing substrate heterogeneity in soil pH causes niche partitioning among the AtSUC8 variants, which then results in a more efficient use of the available soil resources72. Another possibility is that the different pH sensitivity of root growth results in a different root foraging behavior and reduced root competition among genotypes through more complex mechanisms, possibly including soil pH changes that arise from root exudates rather than pre-existing soil heterogeneity. Obviously, there are many different environmental settings or genotype and species combinations for which other traits, related to other genetic differences, may underlie niche partitioning among plants, thus causing overyielding. In each case, the trait-based approaches currently applied to the study of biodiversity effects and overyielding might benefit from gene-based approaches, ultimately not only at the within- but also at the between-species level. On the other hand, our work may offer new ways to design more sustainable cropping systems, in which species or genotype diversity can improve both yield and yield stability in the face of biotic and abiotic stress77. Here, the gene-centered approach may complement currently used trait-centered methods to facilitate the design of high-performance crop variety mixtures.A remaining question is how specific aspects of 53. The lines used for the association analysis are described in detail in Wuest et al.45.The Sav-0 and Uk-1 seeds were initially obtained from the Arabidopsis Biological Resource Center at Ohio State University. The Sav-0 × Uk-1 RIL population was described previouslyad libitum, and, in case of high herbivory pressure by larvae of the dark-winged fungus gnat, the insecticide ActaraG (Syngenta Agro AG) was applied according to the manufacturer’s recommendation. The date of harvesting was determined through the occurrence of 5–10 dehiscent siliques on the earliest flowering genotypes per block. The aboveground plant biomass was dried at 65 °C for at least three days and then weighed.Seeds were sown directly on soil and germinated in trays covered with plastic lids under high humidity in a growth chamber at the University of Zurich, Irchel Campus . The soil substrates are described below. After approximately two weeks, the trays were moved into a greenhouse chamber, where day-time and night-time temperatures were maintained around 20–25 °C and 16–20 °C, respectively. Additional light was provided if required to achieve a photoperiod of 14–16 h. Seedlings were thinned continuously until a single healthy seedling remained per position. The pots were watered Assessing accession pair mixtures: nine accession pairs, for which recombinant inbred line populations were publicly available, were chosen for the screen of pair-wise interactions through comparisons of monocultures and two-genotype mixtures. A further pair was chosen based on a large estimate of mixture effects in a previous study11. These genotypes of these selected pairs were grown as either monocultures or pair-wise mixtures on different soils and in pots of different sizes as follows: peat-rich soil  and in 6 × 6 × 5.5 cm or 7 × 7 × 8 cm or 9 × 9 × 10 cm pots; sandy soil, a 4:1 mixture of quartz sand:ED73, and in 7 × 7 × 8 cm pots; and Arabidopsis legacy soil, i.e., soil collected from an unrelated previous experiment on which Arabidopsis had grown , and in 7 × 7 × 8 cm pots. Each monoculture or mixture composition in each soil or pot size was grown in each of seven blocks, except those on sand-rich and legacy-soil conditions. The legacy and sandy soil conditions were included only in five of the blocks for logistical reasons. Overyielding of genotypic mixtures containing Sav-0 and Uk-1 was confirmed by growing either (i) four plants in medium sized pots (7 × 7 × 8 cm); (ii) four plants in small pots (5.5 × 5.5 × 6 cm) or (iii) two plants in small pots, all containing ED73 soil. For each pot/density type, 48 mixtures and 24 of each monoculture were sown, treated, and processed as described above. Supplementary Table QTL mapping and association study: The QTL-mapping experiment was designed as a half-diallel containing all pair-wise combinations and monocultures of 18 RILs derived from Sav-0 and Uk-153 and the two parents (190 genotype mixtures + 20 monocultures). The experiment was conducted in four sequential blocks, with monocultures grown in duplicate within a block, resulting in a total of 920 sown pots over the whole experiment. Pots that did not contain four plants (two per genotype), for example due to seedling establishment problems, were discarded. For the final analysis, data from a total of 808 pots were used. We used soil consisting of a 1:3 mixture of quartz sand:ED73 for the first two blocks. However, because seedling establishment was rather poor on this soil, we changed soil type in blocks three and four to a 3:1 mixture of quartz sand:ED73. Plants were grown and harvested as described above (42–51 days after sowing).45. In short, the experimental design of the association study consisted of a fully factorial competition treatment of growing ten tester genotypes  with each genotype of an association panel of 98 natural Arabidopsis accessions  in two replicate blocks. The design thus contained 980 different genotype mixtures and a total of 2154 pots. Each community consisted of two plants (one plant per genotype). Pots were 6 ⨯ 6 ⨯ 5.5 cm in size and contained a soil mix consisting of four parts ED73 and one part quartz sand. The raw data of the association study are available at https://zenodo.org/record/2659735#.YCt0u2Mo8mI.Experimental conditions for the genome-wide association experiment are described in detail elsewhere42. In short, reads were aligned to the Arabidopsis reference genome  using BWA version 0.7.16a79, read sorting and variant calling were performed using SAMtools version 1.5. The genome reconstruction approach broadly followed the method described by Xie and colleagues80 and is described elsewhere in detail42. Raw reads of resequencing the parental accessions Sav-0 and Uk-1 were downloaded from the NCBI SRA homepage . To genotype a wider set of RILs at the AtSUC8 locus (At2g14670 [https://www.arabidopsis.org/servlets/TairObject?id=35253&type=locus]), a Cleaved Amplified Polymorphism (CAPS)-marker assay was developed based on a EcoRV-restriction site in the AtSUC8 coding sequence that is present in the Sav-0 allele but missing in the Uk-1 allele using PCR primers 5′-GGA GAG TGT TGT TAG CCA CGT C-3′and 5′-ACG ATG TGG TAG CTG TAG ATA GAC-3′. DNA extractions for CAPS-genotyping were done using a protocol following Edwards and colleagues81. For four RIL genotypes, PCR-genotyping yielded ambiguous results, so we inferred it from flanking markers AtMSQTsnp 123: (Chr 2 pos 1798324) and AtMSQTsnp 138 (Chr 2 pos 8370574)82. We also tried to identify RILs that exhibited heterozygosity at the AtSUC8 locus to isolate heterogeneous inbred families, but failed to find any among the 101 RILs screened.For the 18 RIL genotypes used in the QTL-mapping competition diallel analysis, we performed whole-genome resequencing and genotype reconstructions before the genetic analysis. DNA extractions for genome resequencing, library preparation, sequencing and genome reconstruction were done as previously describedAtSUC8 alleles was performed by amplifying the gene body from genomic DNA using oligonucleotides 5′-ATG AGT GAC CTC CAA GCA AAA AAC GAT-3 and 5′- TTA AGG TAA CAC GGT AAA TGC CAC AAC ACT GC-3′, and further analysed using SnapGene version 6.1. The PCR fragments were then sequenced using those same oligonucleotides as well as oligonucleotide 5′-CAC AAT GAC TAA AGC ATG TGA C-3′. The C24 allele of AtSUC8 was retrieved from published sequence data83. Note that because of genomic rearrangements, the gene ID for AtSUC8 (AtC24-2G29550) in the C24 accession differs from that of the other accessions.To verify polymorphisms identified in resequencing, Sanger sequencing of the 84. Briefly, the AtSUC8 cDNAs were cloned into pOO285. cRNA was synthesized using the mMESSAGE mMACHINE kit (Lifetechnologies). Oocytes were injected with 50 nL of 150 ng/µL cRNA and incubated in Barth’s solution 2 × 4 H2O, 0.41 mM CaCl2 × 2 H2O, 0.82 mM MgSO4 × 7 H2O pH7.4) for four days. For uptake experiments, 10 oocytes were kept in 1 ml Barth solution supplemented with 14C-sucrose at a final concentration of 1 mM or substrate-free control for one hour. Afterwards, oocytes were washed twice in Barth’s solution containing gentamycin and were separated into scintillation vials. 100 µl of 10 % SDS (w/v) was added to each scintillation vial and the samples were incubated for 10 minutes. Then, 2 mL of scintillation cocktail  was added and the vials were vortexed vigorously. Radioactivity was determined by liquid scintillation counting. Experiments were carried out using 14C-sucrose ; Hartmann Analytic, Braunschweig, Germany.Oocyte assays were performed essentially as described2O washes, then left for stratification at 4 °C overnight. Square MS plates (12 cm) were prepared with 0.8% agarose (instead of agar) and containing 1% sucrose (w/v). The pH was adjusted to 4.5 or 7 using hydrochloric acid or potassium hydroxide and the medium autoclaved. After autoclaving, the measured media pH was again determined (4.8 and 6.8). Six seeds of each of six different genotypes were sown on a plate pair  and grown in a climate chamber with long-day conditions  for seven days. Plates were scanned twice, once after 4 days and again after 7 days using an EPSON flatbed scanner (model 2450). The primary root length of seedlings was measured using the Fiji software86.Seeds were surface-sterilized with 70% ethanol, followed by 15 minutes in a solution containing 1% bleach and 0.01% Triton-X100 and three sequential HAB − 0.5mAA − 0.5mBB for the case of a monocultures and mixtures of genotypes A and B), using the glht-function of the multcomp-package87. A meta-analysis across all Sav-0/Uk-1 mixture experiments was performed using the weighted Fisher method as described previously88, and using experimental sample sizes as weights , including a block term. Overyielding of a genotype pair on a given substrate was then estimated as linear contrast between the average monoculture productivity and the mixture productivity  was then estimated by averaging across substrates. SCA outliers were removed if they differed more than two standard deviations from the population mean in their absolute value. QTL mapping of standardized mixture SCA estimates was then performed by a marker regression approach, where we first fitted a linear model predicting SCAs from allelic composition , followed by a contrast between allelic monocultures and mixtures ), again using the glht function.The mapping experiment was performed on two different substrates (two replicated blocks each), and both mean and variance of community productivities differed across substrates. The blocks with more nutrient-rich substrate also had some pots with missing plants due to seedling mortality, which were removed from the analysis. To combine all four blocks for the estimation of SCAs, we first estimated mean community biomass within substrates and calculated SCAs within substrates from average total pot biomass values (BM) as BM = Z*u + SCA whereby p-value)) of 3 was considered significant, as determined by large-scale simulations89 under the assumptions: two QTL genotypes  and an average chromosome length of 200 cM for Arabidopsis genotype pairs, where recombination events are combined in mixtures. Such a threshold is also in agreement with our previous work comparing this approach to a standard QTL mapping method and a LOD-cutoff based on re-sampling42. To estimate overyielding of each mixture and for the additive partitioning, the least square mean productivity estimate was derived for each genotype grown within a composition after adjusting for fixed block effects. A t-test was used to determine if overyielding across all genotype mixtures differed from zero (two-sided t-test with df = 189). The additive partitioning of mixture overyielding into CEs and SEs was done as described in Loreau and Hector54, and calculated for each genotype mixture separately. Subsequently, a contrast between mixtures that contained two alleles and those that contained one allele only at the QTL on chromosome 2 was tested using the glht-function as described above.A LOD score  were removed from the analysis. A genotype’s GCA was estimated as described above within each block and values were then averaged across blocks.The association study represents a factorial design in which each of ten different genotype (testers) was grown in combination with each of 98 different Pot biomass depended non-linearly on average genotype GCA (Supplementary Fig. Further information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


Many phenotypes in animal breeding are derived from incomplete measures, especially if they are challenging or expensive to measure precisely. Examples include time-dependent traits such as reproductive status, or lifespan. Incomplete measures for these traits result in phenotypes that are subject to left-, interval- and right-censoring, where phenotypes are only known to fall below an upper bound, between a lower and upper bound, or above a lower bound respectively. Here we compare three methods for deriving phenotypes from incomplete data using age at first elevation (> 1 ng/mL) in blood plasma progesterone (AGEP4), which generally coincides with onset of puberty, as an example trait.We produced AGEP4 phenotypes from three blood samples collected at about 30-day intervals from approximately 5,000 Holstein–Friesian or Holstein–Friesian × Jersey cross-bred dairy heifers managed in 54 seasonal-calving, pasture-based herds in New Zealand. We used these actual data to simulate 7 different visit scenarios, increasing the extent of censoring by disregarding data from one or two of the three visits. Three methods for deriving phenotypes from these data were explored: 1) ordinal categorical variables which were analysed using categorical threshold analysis; 2) continuous variables, with a penalty of 31 d assigned to right-censored phenotypes; and 3) continuous variables, sampled from within a lower and upper bound using a data augmentation approach.Credibility intervals for heritability estimations overlapped across all methods and visit scenarios, but estimated heritabilities tended to be higher when left censoring was reduced. For sires with at least 5 daughters, the correlations between estimated breeding values (EBVs) from our three-visit scenario and each reduced data scenario varied by method, ranging from 0.65 to 0.95. The estimated breed effects also varied by method, but breed differences were smaller as phenotype censoring increased.Our results indicate that using some methods, phenotypes derived from one observation per offspring for a time-dependent trait such as AGEP4 may provide comparable sire rankings to three observations per offspring. This has implications for the design of large-scale phenotyping initiatives where animal breeders aim to estimate variance parameters and estimated breeding values (EBVs) for phenotypes that are challenging to measure or prohibitively expensive. Time-dependent traits can be logistically challenging and expensive to measure precisely, as animals need to be observed regularly over a long period of time. Some examples include age at puberty (AGEP), mating and calving dates (particularly in the context of beef cattle) and lifespan . In the case of AGEP, indicator traits may be measured repeatedly over a period, with some pre-determined criteria to define an animal as either pubertal or non-pubertal at any given time. Possible measures include behavior monitoring to identify an estrus event, ultrasonography of ovaries to detect the presence or absence of a corpus luteum or testing for elevated blood plasma progesterone (BP4) concentrations that indicate the presence of a functioning corpus luteum –3. MeasuPast experiments which define the AGEP of individual heifers have measured animals at a range of intervals, including monthly , weekly It is difficult to directly assess the implications of censoring on the genetic analysis of a trait like AGEP, as precise phenotypes are prohibitively difficult to measure at a large-scale. Fortunately, high, positive correlations have been reported between EBVs produced using simulated phenotypes that were either uncensored, or subject to various left-, interval- or right-censoring combinations . Those fThere are several methods that can be used for deriving phenotypes from incomplete observation data which involve converting the incomplete observations into categorical or continuous variables. Researchers often convert censored AGEP observation data into a continuous variable by defining AGEP as the age of the animal when it was first observed to meet the puberty criteria within a set observation period , 3, 7. AOur primary objective was to investigate the sensitivity of estimated variance parameters and breeding values (EBVs) to varying degrees of observation censoring for a time-dependent trait, using real-life phenotype data. Our second objective was to compare three methods for deriving phenotypes from censored observations. We hypothesized that EBVs and variance parameters would be robust to phenotype censoring, regardless of statistical method. We used AGEP4 as an example trait, but the results of this study may be applicable to many other phenotypes that are derived from incomplete observation data.n = 2,307) or admixed Holstein–Friesian × Jersey crossbred , and a small number were predominantly Jersey . Our analysis also included 50 animals who could not be assigned to the Holstein–Friesian, Jersey or Holstein–Friesian × Jersey breed categories . Animals with incomplete parentage (n = 132), incomplete observations data (n = 129), or issues with identification (n = 4) were excluded from analysis, leaving 4,745 animals remaining  approved this study and all manipulations (AE application: 14448). Data were collected from 5,010 dairy heifers, born between July and September 2018 and reared in 54 seasonal calving; pasture-based herds located across three regions  of New Zealand. The average number of heifers from each herd was 88 animals ± 45 . These 54 herds were selected based on the quality of the existing animal records and predominant breed, with a preference towards herds with mostly Holstein–Friesian animals. The breed proportions for each animal were provided by DairyNZ , and were derived using pedigree records. The resultant study animals were predominantly Holstein–Friesian  on the same day as collection. Plasma was separated and stored at −20 °C until BP4 concentration was analysed using a commercial radioimmune assay kit, as previously described [Three sampling visits at approximately 30-day intervals were conducted for each of the 54 herds between May and August 2019. The timing was chosen to meet a target of the animals in each herd being, on average, 327 days old at the second visit, when 45% were predicted to be post-pubertal based up on the stochastic model of Dennis et al. . Accordiescribed . An animescribed .We investigated three methods for deriving an AGEP4 phenotype from incomplete observation data. In the analyses presented here, every phenotype was derived from incomplete observations.Firstly, for the visit category method (CAT), we defined the phenotype for each animal as the consecutive number of the first visit it was observed with BP4 > 1 ng/mL Table . AnimalsSecondly, we used an age on visit method (AGEVISIT), with the phenotype for each animal defined as its age at the first visit that it was observed with BP4 > 1 ng/mL Table . This phThirdly, we used a data augmentation method (AUG), whereby the unobserved continuous variable representing actual age at first BP4 elevation was treated as an unknown variable whose value must fall between known upper and lower bounds Table , and play is a vector of unobserved liabilities or phenotypes (as defined for each method), b is a vector of fixed effects, u is a vector of breeding values (random effects). The vector e is a vector of residuals corresponding to each of the phenotypes. X is an incidence matrix relating each phenotype record to relevant fixed effects. All analyses included herd as a fixed effect and proportion Jersey as a fixed covariate. The incidence matrix Z relates phenotypes to their corresponding EBVs, with a row for each phenotype and a column for each animal represented in u.We fitted a linear model to these data to estimate variance parameters, fixed herd and breed effects, and to obtain EBVs. Matrix representation of the linear mixed model equation is:We produced a ‘control’ analysis for each of the three methods tested, where observations from all three visits were used. The results of these control analyses were then compared (within method) with results of seven alternate test visit scenarios. Test visit scenarios varied in timing or frequency of the observations that were retained. Each test scenario had a proportion of data selectively excluded to alter left, right, or interval censoring. In total, we defined 8 scenarios Fig. . The firWe used command line bash scripts to pre-process observation data and produce files containing the phenotypes for each method and visit scenario combination. We performed the genetic analysis and post-processing using the JWAS package  implemenWe produced 95% credibility intervals that were thresholds for the 2.5% (lower bound) and 97.5% (upper bound) percentile of all values samples within the MCMC. That is, 95% of the plausible values fell within the credibility intervals presented.We used the stability of estimated variance parameters, breeding values (EBVs) and fixed effect solutions across varying degrees of phenotype censoring to assess the sensitivity of these parameters to increased phenotype censoring. That is, if a given parameter was impervious to increased phenotype censoring, we would expect no change in estimated variance parameters, breeding values (EBVs) and fixed effect solutions as phenotype censoring was increased. To assess stability of estimated variance parameters we compared the posterior mean and 90CRI of heritability estimates. Similarly, we also compared the posterior mean and 90CRI of fixed breed covariate solutions. To assess the stability of EBVs and fixed herd effect solutions we calculated the Pearson correlation coefficient between EBVs produced using our control scenario  and the EBVs produced using each test scenario. A higher correlation demonstrated greater stability.n = 103) from the control scenario (EML) and scenarios E/M/L, E, M, L, EM, ML, and EL were positive and mostly high  .Among the methods, the AUG method generally had the highest correlations between the control scenario and the other scenarios. For scenarios E, M and L when there was a single visit to each herd, correlations ranged from We estimated the breed effect for each analysis Table  as the dn = 54) correlations between the control EML scenario and visit scenarios  were positive and generally high across methods . For all three methods, herd effect correlations decreased as phenotype censoring increased. The correlations of herd effect solutions between the control scenario and scenarios E, M and L, when only one visit was included per herd, were highest for the AUG method . The correlations of herd effect solutions between the control scenario and scenario E/M/L (which had just one randomly timed visit per herd) were lowest for the AGEVISIT method (r = 0.15), and highest for the AUG method (r = 0.87). For scenarios EM, ML and EL, when two visits were included, the correlations of herd effect solutions from the control scenario were generally very high .We also estimated herd effects for each analysis Table  based upThe credibility intervals for heritabilities overlapped by scenario (rows) and methods (columns), as presented in Table In this study, we quantified the extent of sire re-ranking based on their AGEP4 EBVs across scenarios with different phenotype censoring using a Pearson correlation coefficient Table . We deter  ≥ 0.90) for all methods between our control scenario (EML) and scenarios with two observations included . These high correlations indicate that sire selections would be similar if offspring had two or three observations, and, in general, those two observations can be any combination of the early, mid-point or late herd visits. Our results indicate that for the purpose of determining sire rankings for genetic selection, there may be limited marginal gain in a third observation per animal. This finding could be useful for large-scale phenotype collection for routine genetic evaluations or for future trial design, as AGEP and other time-dependent binary traits can be logistically and economically difficult to measure.Sire EBV rankings were robust . However, the extent of EBV re-ranking depended on the analysis method; EBVs produced by the AUG method were the most robust to this degree of phenotype censoring based upon a single observation. Furthermore, the timing of the single visit affected sire re-ranking. For the CAT and AUG methods, scenario E based on a single ‘early’ visit per herd resulted in the least re-ranking relative to our control scenario. In general, the AGEVISIT method resulted in the most re-ranking when only a single visit was included, but the timing of a single visit did not appear to be as important. These results suggest that EBVs calculated using the AUG method are robust using a single observation per animal, potentially reducing the data measurement effort required and improving the scalability of phenotype collection. For example, a single observation per animal could enable a larger number of animals to be measured, which would improve the accuracy of sire EBVs, increasing the number of selection candidates, and thus the selection intensity for the trait.Furthermore, the E/M/L scenario represents a likely practical measurement regime should a phenotype like AGEP4 be measured at scale. Seasonal calving dates tend to be aligned within regions of New Zealand and other countries that have pasture-based dairy systems, which means that the average ages of birth year groups are similar across herds. Hence, it may be infeasible to collect heifer BP4 status across large numbers of herds at a specific average age. Instead, it would be more reasonable to recommend that animals are measured on a single visit within a defined age window. We were not able to fully test this scenario using our current data, as our visits were scheduled to occur at only three average herd ages . However, if this phenotype was measured at scale, we would likely obtain BP4 status for a sire’s offspring across a more diverse range of ages, which may minimize sensitivity to censoring. Nevertheless, it will be important to fully quantify the implications of random visit times, as it is also possible that under this constraint, a single observation does not provide adequate data to inform sire ranking. That is, two or more observations may be required if the timing of observations cannot be aligned across herds.The breed effects, which represent the estimated difference between breeds, depended on analysis method and scenario for phenotyping. The negative Jersey breed effect solutions when all three observation visits were included (scenario EML) indicated that Jersey animals experienced BP4 elevation before Holstein–Friesian animals, although the size of this difference varied by analysis method. Furthermore, within each analysis method, the estimated breed difference was smallest when the scenario did not include the ‘early’ visit . It has been previously reported that Jersey animals attained puberty 70 d earlier than Holstein–Friesian animals in a New Zealand system . A breedIt is important that differences between breeds are well estimated. Firstly, in a multi-breed analysis, the accuracy of breed effects will influence the dispersion parameters, and thus affect estimates of heritability. When breed is fitted as a fixed effect in a genetic analysis, the heritability should represent the proportion of phenotypic variance that is due to within breed additive genetic variance . If the breed effect is poorly estimated, then variance due to breed can be incorrectly attributed to additive genetic variance, thus inflating the estimated heritability. Secondly, the inaccuracy of breed differences will systematically effect EBV rankings when animals are compared across breed. In dairy sectors with mixed-breed populations, such as in New Zealand, farmers are often provided with ‘across breed’ EBVs . This allows farmers to select the highest-ranking animals on an index, or for a given trait, irrespective of breed composition. If the breed difference for a trait is not accurate, this will result in systematic under- or over-estimation of a certain breed, leading to suboptimal genetic selection decisions. Third, the estimated breed differences may be useful in optimizing future phenotype collection. For example, our results indicate that the model solutions are most sensitive to left censoring of the phenotype. The estimated breed difference can provide useful insight for reducing left censoring when measuring this trait across breed.The Jersey breed was not well represented by pedigree Jerseys, as in this study as there were only 24 animals that were > 90% Jersey, and they were all in the same contemporary group. However, around half of the animals included in this study were admixed crosses between the Holstein–Friesian and Jersey breeds, and these cross-bred animals were present in 53 of the 54 herds. The large number of cross-bred animals provided a reasonable basis for estimating a breed difference between Holstein-Friesians and Jerseys, as around half of the phenotypes included in the analysis will contribute to the solution for this fixed covariate. As indicated, we were not able to separate the effects of heterosis and breed, as heterosis coefficients and breed fractions were correlated in this population. It is possible that if we were able to repeat this study in a population with greater representation of 100% Jersey animals, we would find that the estimation of breed differences behaved differently across method and across scenarios. Hence, further investigations including data from this breed are required.We quantified the extent that herds re-rank based on mean AGEP4 across the phenotype censoring scenarios using a Pearson correlation coefficient. Our results indicated a large degree of re-ranking of herds between our control scenario and the test scenarios with increased censoring for the CAT and AGEVISIT method. In contrast, the herd rankings under the AUG method were remarkably consistent across various visit scenarios. Similar to the estimation of breed differences, the accuracy of the herd effects are important for the accurate estimation of dispersion parameters and EBVs. The stability of herd effects between scenarios for the AUG method will contribute to the stability of EBVs between scenarios.Estimated heritabilities for AGEP4 varied little between different methods or scenarios. We did, however, observe a tendency for higher heritabilities when the ‘early’ visit was included in the analysis. This aligns with indications from this research that breed and herd effects appear better estimated under scenarios that minimize left censoring. Including the early visit in the analysis differentiates animals with left-censored phenotypes and provides separation between their phenotypes and those of the remaining animals in the study. It would seem that this is an important distinction to make when estimating variance parameters, although the importance of including the early visit may be relative to the extent of left-censoring in the dataset. Across the methods and regimes analysed, the heritability for AGEP4 was estimated between 0.05 and 0.60, with a mean of around 0.25. This spread in heritabilities is comparable to those reported in current literature, which range from 0.10 to 0.56 , 20.Using our method of AGEP4 phenotyping based on one to three herd visits to collect BP4, there will be some animals with false negatives for BP4 status. Concentrations of BP4 are cyclic in post-pubertal heifers and are not elevated for about one week of the three-week estrous cycle. Hence, these periods of naturally low BP4 in post-pubertal animals will have produced false negatives in our data . The four-week interval between herd visits means that natural BP4 depression in post-pubertal animals can only create a maximum of one false negative record per animal. False negatives will mean that some animals will be penalized incorrectly. False negatives may increase the residual variance associated with our AGEP4 phenotypes, and therefore reduce the estimated heritability of our phenotype. We would expect false negatives to occur at random across our population, without bias towards the daughters of any particular sire. Therefore, we would not expect false negatives to have implications for sire rankings. This theory is supported by the findings  where thThe ‘control’ scenario presented here used only monthly measures and therefore was already censored, and this limits our ability to make inference about the implications of censoring. Our results indicate that the marginal gain of a third BP4 observation is not likely to justify the cost or effort associated with this measurement. The least censored version of this phenotype would be daily observations for each animal over the complete window where it was biologically possible for them to attain puberty; however, we were not able to test the implications of moving from no censoring (daily observations) to three observations per daughter. That said, Stephen et al.  recentlyIt may be feasible for animal breeders with a given budget to increase the number of animals measured in a progeny test by reducing the number of observations per animal when collecting performance data for time-dependent traits such as AGEP. The three main findings of our study are, first, using the three analysis methods investigated, two observations per offspring may provide comparable sire rankings to three observations per offspring. Second, using a data augmentation approach, one observation per offspring may provide comparable sire rankings to three observations per offspring. Third, using any of the three methods investigated, one or two observations per offspring may provide comparable estimated heritabilities. However, it is worth noting that breed differences tended to decrease as phenotype censoring increased, and so care should be taken when applying these findings to an across-breed evaluation system.Our findings have implications for the design of large-scale phenotyping initiatives as reducing the number of observations per animal for time-dependent traits could lead to a larger number of animals phenotyped, potentially improving accuracy and intensity of selection.


Genome-wide tests, including genome-wide association studies (GWAS) of germ-line genetic variants, driver tests of cancer somatic mutations, and transcriptome-wide association tests of RNAseq data, carry a high multiple testing burden. This burden can be overcome by enrolling larger cohorts or alleviated by using prior biological knowledge to favor some hypotheses over others. Here we compare these two methods in terms of their abilities to boost the power of hypothesis testing.We provide a quantitative estimate for progress in cohort sizes and present a theoretical analysis of the power of oracular hard priors: priors that select a subset of hypotheses for testing, with an oracular guarantee that all true positives are within the tested subset. This theory demonstrates that for GWAS, strong priors that limit testing to 100–1000 genes provide less power than typical annual 20–40% increases in cohort sizes. Furthermore, non-oracular priors that exclude even a small fraction of true positives from the tested set can perform worse than not using a prior at all.Our results provide a theoretical explanation for the continued dominance of simple, unbiased univariate hypothesis tests for GWAS: if a statistical question can be answered by larger cohort sizes, it should be answered by larger cohort sizes rather than by more complicated biased methods involving priors. We suggest that priors are better suited for non-statistical aspects of biology, such as pathway structure and causality, that are not yet easily captured by standard hypothesis tests. This multiple-testing correction from this stringent approach is a burden for identifying genome-wide significant findings.Genomics experiments involve testing thousands to millions of hypotheses. In functional genomics and proteomics, each gene or protein usually corresponds to a single test, with 20,000 or more tests required for an RNAseq or proteomics experiment. In human genetics, the number of independent tests accounting for linkage disequilibrium in a single ethnicity is usually assumed to be about 1 million for all but the rarest variants. To maintain a family-wise error rate (FWER) controlled at 0.05, a long-standing approach has been to apply a Bonferroni correction, requiring a single-testA robust solution to this problem has been to gather large cohorts of unrelated individuals, particularly for GWAS . While tRather than increasing cohort sizes, an alternative approach is to incorporate prior knowledge about functional effects of genes or SNPs. In GWAS, this may increase the power to detect SNPs with true associations or to identify which SNP in a linkage disequilibrium (LD) region is most likely to be the causal variant –7. OtherA representative approach incorporated 450 different annotations into GWAS analysis of 18 human traits; the number of loci with high-confidence associations was increased by around 5mentclasspt{minimaIn this paper, we use theoretical models and derivations to investigate the dependency of power on population size and incorporating priors. We consider an oracular hard prior, which tests a subset of the hypothesis that is guaranteed to include all the true positives. We show analytically that in the limit of small effect sizes and most relevant to genomics studies, population sizes are exponentially more important than priors in determining the power. We then show that given historical trends in cohort sizes, it is nearly impossible for new analytical methods to improve power faster than larger studies that use conventional methods.To ground our analysis in improvements in GWAS power that can be anticipated simply due to increases in cohort sizes enabled by increases in DNA analysis efficiencies, we characterized the doubling times for GWAS cohorts and the number of loci detected at genome-wide significance. We then determined analytically the strength of a prior required to give an increase in power equivalent to the power gain from an increased cohort size. We show that an oracular hard prior, which restricts testing to a fraction of tests that still includes every true positive, has difficulty matching the power gain from a typical one-year increase in cohort size. We then show that a non-oracular hard prior, which excludes some true positives from the tested set, often performs worse than tests without a prior at all.Exponential increases in genotyping and sequencing efficiency have enabled similar increases in GWAS cohort sizes. Larger sample sizes in turn have greater power to detect SNPs with smaller and smaller effects. We quantified increases in cohort sizes and genome-wide significant SNPs through a systematic analysis of GWAS cohorts, traits, and loci as compiled by the GWAS Catalog . Quantifp value The GWAS Catalog contains results for 5123 total studies describing 3034 traits and 126,788 associations that are genome-wide significant, defined as Results for a well-studied case-control disease trait, breast cancer susceptibility, demonstrate the progress in identifying genome-significant loci Fig. . As of 2y is the cohort size or the number of genome-wide significant loci, t is the publication date in fractional years, and p = 0.384) or for loci doubling time (p = 0.652), justifying the use of an exponential growth model.Regression fits for the five breast cancer susceptibility studies estimate the cohort size doubling time to be entclass1pt{minimaAnalogous results for a well-studied quantitative trait, blood triglyceride levels, show similar progress from 2010 to 2020 Fig. . As of 2ars Fig. .Fig. 3GWp value = 0.05, two-sided t-test). The only trait for which the number of loci is doubling more slowly than the cohort size is age-related macular degeneration, with a cohort doubling time of 1.2 years and a loci doubling time of 1.8 years. Early studies of this phenotype with small cohorts nevertheless discovered significant loci with large effects. This may explain the subsequent slower rate of discovery.We then restricted attention to traits where doubling times could be calculated robustly, with a standard error of 0.5 or smaller, and with at least 10 SNP associations, yielding 33 traits comprising 187 studies and 15,051 associations  regularly apply hard priors to reduce the testing burden from passenger mutations Fig. . The uniHard priors for cancer studies are easier to formulate, given the strong functional effects assumed for cancer driver mutations. Similarly, positional cloning studies to identify Mendelian disease loci successfully rely on sequencing to identify germ line mutations with strong functional effects and high penetrance. For GWAS of complex disorders, genetic variants predisposing to disease are less readily identified by strong effects on protein structure or function. Nevertheless, priors that focus attention on genes expressed in a tissue of interest at an appropriate developmental stage, or priors that focus attention on SNPs that are within gene-based boundaries or coincide with regulatory regions, have been discussed and applied.S of a hard prior as the fold-reduction in the number of hypotheses tested: rather than testing all A hypotheses, only a subset of size A/S is tested. The stronger the prior, the larger the value of S, and the smaller the multiple testing burden. For a GWAS, a p value cutoff of p value cutoff to For a general mathematical framework, we define the strength p value of p-value cutoff for GWAS studies. This type of prior could suffer from false negatives, however, in excluding most of the genome from testing.This type of approach can identify some variants that are not identified using a conventional threshold. For example, a breast cancer susceptibility GWAS investigating breast cancer risk identified the SNP rs11571833 in BRCA2 with S required to give the same boost in power as simply waiting one to two years for a larger study.Instead of devising and applying priors, an alternative strategy would be to accumulate larger cohorts for greater power to detect real effects. Larger cohorts and improved priors could of course be pursued simultaneously. Nevertheless, given the 18–24 month doubling time for cohort size and the similar amount of time required to develop and benchmark a new computational method, it is worthwhile to consider the prior strength p value of S is the prior strength  and decreasing the accuracy. As the accuracy is decreased, the performance of the prior degrades and eventually falls below the performance without a prior. This transition point between using a non-oracular prior and avoiding a prior altogether is readily computed. A second transition occurs when the prior accuracy f falls below the requested power For a requested power of ates Eq. . This reOur analyses of non-oracular priors in GWAS are based on a type I error of While our main focus has been GWAS, similar analyses are readily performed for RNAseq studies. Here, assuming tests of 25,000 genes, the type I error single-test threshold is For studies with power requested closer to 50%, priors with accuracy of 60% can be effective. Power requests this low are unusual, however, because of potentially large numbers of false negatives. These results suggest that prior accuracies in general must be 90% or higher for non-oracular priors to be considered in practice for genomics studies.Despite long-standing efforts to exploit prior knowledge to boost the power of GWAS, RNAseq, and other genome-wide tests, including our own efforts to develop and apply gene-based priors for GWAS , 9, tradWhat, then, is the value of priors in GWAS? We are ourselves convinced that developing priors to boost statistical significance for GWAS is not a productive area for future research. We do think, however, that they have an important role in bridging between statistical significance and biological mechanism. While larger cohorts will provide stronger statistical associations, they will not necessarily identify which SNP or SNPs in a linkage disequilibrium block is most likely to be responsible for the observed effect; which gene is affected by the causal SNP; or how many independent effects or different genes are involved. These types of problems are excellent opportunities to develop and use of priors for inference over multiple types of data and existing biomedical knowledge.In the context of GWAS, mathematical analyses of standard statistical tests demonstrate that doubling the size of a cohort improves test power more than is possible with any reasonable prior, including oracular priors that are guaranteed to include every true positive in the set of tested hypotheses. In reality, priors are non-oracular and inaccurate, with true positives inevitably left out of the tested set. Whether for GWAS, RNAseq, or other genome-scale hypothesis testing, when even moderate power is a goal, priors can require an unrealistically high accuracy of 90% or more, making it nearly impossible to improve upon performance without a prior.Datasets were collected from the GWAS Catalog  [Consider a study with N. The variance is minimized with N according toThe estimated allele frequency Each GWAS study may be related to multiple records, each of which demonstrates the significant association between the trait or phenotype investigated by the study and a SNP. These records generally account for linkage disequilibrium by reporting the most significant SNP in a linkage region. The number of associations for each study was counted as the number of associations passing the genome-wide significance threshold of The GWAS Catalog assigns a specific trait to each study . We usedy representing either the cohort size or the number of associations, t the publication date with months and days converted to fractional years, and regression coefficients Doubling times for cohort sizes and number of associations for each phenotype were estimated as a linear model, p value of p value of p value was then calculated by the F-statistic.Significance of the exponential fit was assessed by three nested regression models denoted p values were calculated for a two-sided test of the null hypothesis Doubling times for cohort size , priors were much less effective than increased population sizes in improving power. We developed analytical approximations that helped us understand the reasons for our results. A steepest descents approximation relates the quantile N and S with equal critical effect size can be estimated by returning to the approximate resultContours of f defined as the fraction of true positives contained within the tested set. In the limit S retains its definition as the ratio of the total number of possible tests, A, to the tested set with size A/S. We again define z and A non-oracular prior has an accuracy f. Therefore, the statistical test must have power f (which would correspond to a lower-tail area N, the effect size With overall false-negative rate Additional file 1. GWAS Catalog summary: A summary of the traits and studies from the GWAS Catalog that were analyzed.Additional file 2. GWAS trait summary: A summary of the 49 traitsin the GWAS cCatalog with at least 3 effective studies, permittingestimation of doubling times for cohorts and loci.Additional file 3. Breast cancer GWAS significantassociations: Data table used to generate Fig. 1.Additional file 4. Triglycerides GWAS significantassociations: Data table used to generate Fig. 3.