Zea mays L.) breeding has resulted in a steady increase in plant performance over time, along with noticeable changes in above ground vegetative traits, but the corresponding changes in the root system are not presently known. In this study, roughly 2500 core root systems from field trials of a set of 10 diverse elite inbreds formerly protected by Plant Variety Protection plus B73 and Mo17 and the 66 diallel intercrosses among them were evaluated for root traits using high throughput image-based phenotyping. Overall root architecture was modeled by root angle (RA) and stem diameter (SD), while root complexity, the amount of root branching, was quantified using fractal analysis to obtain values for fractal dimension (FD) and fractal abundance (FA). For each trait, per se line effects were highly significant and the most important contributor to trait performance. Mid-parent heterosis and specific combining ability was also highly significant for FD, FA, and RA, while none of the traits showed significant general combining ability. The interaction between the environment and the additive line effect was also significant for all traits. Within the inbred and hybrid generations, FD and FA were highly correlated (rp ≥ 0.74), SD was moderately correlated to FD and FA (0.69 ≥ rp ≥ 0.48), while the correlation between RA and other traits was low (0.13 ≥ rp ≥ −0.40). Inbreds with contrasting effects on complexity and architecture traits were observed, suggesting that root complexity and architecture traits are inherited independently. A more comprehensive understanding of the maize root system and the way it interacts with the environment will be useful for defining adaptation to nutrient acquisition and tolerance to stress from drought and high plant densities, critical factors in the yield gains of modern hybrids.The mature root system is a vital plant organ, which is critical to plant performance. Commercial maize (The online version of this article (doi:10.1186/s40064-015-1187-0) contains supplementary material, which is available to authorized users. Diabrotica vergifera vergifera)  characterize a selected set of 12 inbreds that were key founders of modern commercial hybrids, 10 of which were formerly protected by PVP, and the 66 hybrids emanating from crossing among the inbreds for root architecture and complexity traits, (2) determine the relative importance of additive and non-additive genetic variation for root system characteristics in this set of elite maize germplasm, and (3) examine relationships between root complexity, root angle, and stalk diameter. Insights on the root system architecture of parental inbred and their F1 derived F2 populations were evaluated for shoot traits. This representative sample of modern elite germplasm was shown to be genetically and phenotypically diverse  of five consecutive plants within a plot were dug out and transported to the Agricultural Engineering Farm of the University of Illinois for cleaning. Processed roots were then phenotyped with a custom designed imaging system comprised of a cabinet box with photographic lighting, a light diffusing screen, side view and top view cameras, sample rotation device, and network connection to a desktop computer . Details on the imaging system, image processing, image analysis, and control via MatLab® code can be found in Grift et al.  of the root system and characterizes its branching pattern. For 2D images, values for FD vary between 1 (“single un-branched root”) and 2 (“highly branched system”). The logarithm of K is called fractal abundance (FA) and indicates to what degree root systems explore the soil space, i.e., the number of root branches present in a given soil volume , so only one image from the vertical axis was used for calculation of root sample FD and FA. Estimates of FD derived from orthogonal (90 degree) horizontal perspective images of root samples were highly correlated rp ≥ 0.90 and the correlation between underside and side perspectives was also high for inbreds (rp = 0.77), F1s (rp = 0.90), and check (rp = 0.93) entries. Since the fractal estimates from different perspectives are highly correlated and the side profile images contain the stem, which may reduce the accuracy of FD estimates, estimates of FD from the underside profile of root cores were used for analysis. Statistical analyses were conducted using plot means of the root traits. Root and shoot data was adjusted for maturity based on days to silking and converted to Z scores prior to multivariate analysis.The data was filtered for outliers attributable to difficulties in image processing, then averaged to produce mean values per plant. Subsampling the root system with multiple images provides better estimates for RA and SD by reducing the potential for bias from sampling 3D root structure from a single 2D perspective. The redundancy of information obtained from different perspectives of the root system samples was assessed with Pearson correlation coefficients. The correlation between one of the four estimates of RA per sample with the sample average’s RA was r1 hybrids are a selected set rather than a segregating population, genetic parameters are fixed effects. Overall significance of a genetic parameter in the model, such as the additive effects per se, is indication that variance due to that parameter is important for determining the trait phenotype within the set of material tested. The overall model accounting for genetic effects and the experimental design was:Y with intercept µ and the following effects with indexes for location (i), replication (j), entry class (k), genotypic effects , and the plot (n): ei = ith environment, i = 1, 2; rij = jth replication nested in the ith environment, j = 1, 2, 3; bk = effects accounting for the means of inbred, F1, and check classes of entries, k = 1, 2, 3; al|am = effect of the lth and mth line per se (additive effect), l, m = 1, 2, …, 12; gl|gm = general combining ability of the lth and mth line; slm = specific combining ability of the cross of the lth and mth lines; ebik = interaction of the ith environment with the kth entry class effect; rbijk = interaction of the jth replication nested in the ith environment with the kth entry class; dilm = interaction of the genetic effects al, am with the ith environment; cn = effect of the nth check entry; ecin = interaction of the nth check entry with the ith environment; εijklmn = residual error.A modified version of Eberhart and Gardner’s  general The design parameters were specified as1 hybrids block; β = 0 for the check entries. γ = 1.00 for cross between the lth and mth inbreds in the F1 hybrids block and γ = 0 for the check and parent entry classes; δ = 1.00 for the cross between the lth and mth inbreds in the F1 hybrids blocks and δ = 0 for the check and parent entry classes;β = 1.00 for the Ith parent and β = 0.50 for the cross between the lth and mth inbreds in the Fk, can be partitioned into two orthogonal contrasts, each with one degree of freedom; viz,Note that the entry class effects, b1 blocks mean − Parents blocks mean),Mean (mid-parent) heterosis = .Mean difference between diallel F1s minus the mean of the inbreds. The significances of reported model parameters and genetic effects were adjusted using the adaptive false discovery rate (Benjamini and Hochberg p) were calculated among root characteristics using plot means for each entry class.Statistical analyses were conducted with SAS/STAT software, Version 9.2 of the SAS System, using the CORR, MEANS, MIXED, MULTEST, FASTCLUS, and CANDISC procedures  and narrower trait ranges than F and narrp ≥ 0.74, Table p = −0.40). Stalk diameter showed moderately sized positive correlation coefficients with FD and FA (rp = 0.48) but small correlation coefficients with RA. The high correlation between FD and FA suggests a relationship between the level of branching and total observed root length, while their moderate correlation with SD is likely a consequence of larger plants having larger root systems.Across entry classes, FD was significantly correlated with FA (r1 hybrids differed from check hybrids for FA (q ≤ 0.05) and RA (q ≤ 0.001), but not for FD and SD. The effect of the environment was highly significant for FD, FA, and SD, but not for RA. The magnitude of the environmental effects observed for FA was similar to mean heterosis for FA, more than twice mean heterosis for FD, and about one standard deviation of the entry class means for SD. Ex-PVP F1s differed from the checks for mean FA and RA, but not FD or SD.The additive effect of the parental inbreds per se was highly significant across traits, but general combining ability was not significant for any of the phenotypes Table . Specifi1s instead of heritabilities and provided in Additional file 1s . Repeatabilities for SD among the inbreds and F1s were moderate , but the 90% confidence intervals were large, particularly for the inbreds. This variation is interpreted to be related to within-plot plant to plant variation for plant size based on observations during sampling.An overview of the temperature and precipitation in the two growing seasons is provided as supplementary information. Both environments had similar average temperatures, but the period of plant vegetative growth in the 2007 environment experienced less rainfall, more moisture evaporation, and half as many days with rain compared with the 2010 environment , while the correlation with either days to anthesis or days to flowering was high . Plant height is moderately correlated with flowering time (rp = 0.65), while the correlation of RA with plant height and flowering time was low . Overall, these observations are consistent with root and shoot development being coordinated.The additive effects of the parental inbreds are provided in Table 1 hybrids for FD and RA traits are shown in Table 1 means and mid-parent values for root traits is provided in Additional file 1s have mid-parent characteristics . A plot comparing the first principal component of the two treatments shows little change in the common variance and resembles results observed from analysis of 2007 and 2010 data  root systems are evaluated for a single developmental stage only, (2) the deep portion of the root system is not sampled, and (3) processing of each root system for imaging incurs some significant random tissue damage and loss to the root core and adversely affects the fine root structure. Despite these limitations, we were able to study the genetic basis of root architecture in our set of commercially relevant germplasm.3 root core volume. This sampling method utilizes a large proportion of the root system for estimation of complexity based on previous observations by Dwyer et al.  and growing conditions  with FD and FA among the parents, F1s, and checks, indicating that larger stemmed plants tended to have more complex root systems. As discussed previously, fractal based estimation of root branching should not be unduly biased by the overall size of the root system. Since proportional plant growth for some range of shoot to root ratios is a favorable adaptation for grain yield, the positive correlation between stem diameter and root complexity in our set of material might be expected if these traits are predictive of shoot and root mass. Evaluating the performance of elite hybrids for root system architecture and complexity under density and fertility stress as it relates to yield is also a logical extension of this work. A comparison of root effects to shoot data from normal and reduced fertility environments did not reveal any strong association with low N performance and common variance among phenotypes appeared to be related to pedigree, as might be expected in evaluation of a diallel.The distribution of plot means for SD was similar across generations and significant heterosis was not observed for this trait. This was unexpected, given the differences in plant size of hybrids compared with inbreds. PHZ51 and B73 contributed a large positive effect on SD, while only the earliest inbred, PHJ40, reduced SD significantly. Significant variation for SD among checks was not observed, though they exhibited significant variation for FD, FA, and RA. Based on our experience, estimates of SD are likely to be more sensitive to variability in plant spacing within a plot. Plants that are closer together or further apart than the target density are subject to increased or decreased stress from plant competition, which has a strong influence on plant size. Due to the shade avoidance response, higher plant densities have been associated with increased plant height and reduced stem diameter in maize . A larger population size would be required to adequately search for inbreds with divergent FD and FA effects. Instances of inbreds with significant effects increasing FD or FA while decreasing SD were not observed either, though additional data may show this is the case for PHG39. The contribution of differences in maturity, density response, and pleiotropy to this relationship are unclear at the moment, however. The direction of RA effects among the lines does not appear to be associated with those for FD, FA, or SD, suggesting the genetic control of this trait may be relatively independent by comparison. There was no apparent relationship between the heterotic pool from which a parent was derived and the contribution of positive or negative FD effects in our set of material. This suggests that the reciprocal recurrent selection and pedigree breeding in commercial germplasm has not indirectly resulted in differences in root complexity between the heterotic pools.Comparable studies investigating heterosis in maize for mature post-embryonic root architecture and complexity in field experiments appear absent. Experiments reporting data on the structure of the maize post-embryonic root system in general are rare, so this work addresses a topic with sparse information. Highly significant mean heterosis and SCA, a measure of dominance plus inter-gametic epistasis, was observed for each trait except stalk diameter. In contrast to the additive effect of the lines, significant GCA was not observed for any trait, indicating that the per se performance of the parents is sufficiently predictive of their performance in hybrid combination. The correlation structure of the root phenotypes was evident when the parents were grouped into contrasting classes based on additive effect estimates for the four traits. Our set did not include an inbred contributing effects that significantly increased FD and significantly decreased FA, or vice versa, which was expected given the size of our set and the strength of positive correlation between FD and FA (0.88 ≥ r−1 historical increase in maize grain yield. While rooting depth was the critical architectural factor in their simulation, the authors noted that enhanced water extraction capacity could also explain the improvement in performance at higher density. Changes in root architecture, but not biomass partitioning, have been implicated in improved performance of maize for yield under drought stress (Ribaut et al. Hammer et al.  simulateFrom the study of root complexity in the Ex-PVP material, there are several relationships of interest for further investigation. Inbreds and hybrids with high FD should perform better under drought conditions, due to more efficient capture of water for a given amount of root biomass. However, if root complexity is highly correlated with root biomass, increased complexity might be disadvantageous due to the importance of having a high allocation of biomass to the shoot for good performance under drought (Bänziger et al. Additional file 1.In the Supplementary Material and Results Section information characterizing the used ex-PVP inbreds and the weather conditions during field experimentation is given and additional results of the quantitative genetic analysis are presented.


Threshold scores from repeat participants were used to calculate a test–retest reliability coefficient, which was statistically significant , indicating that the Scentroid provides reliable estimates of odor thresholds. In addition, we performed a preliminary genetic analysis evaluating the potential association of n-butanol odor thresholds to six single-nucleotide polymorphisms (SNPs) putatively involved in general olfactory sensitivity (GOS). The results of multiple linear regression analysis revealed no significant association between the SNPs tested and threshold scores. However, our sample size was relatively small, and our study was only powered to identify genetic markers with strong effects on olfactory sensitivity. Overall, we find that the Scentroid provides reliable quantitative measures of odor detection threshold and is well suited for genetic studies of olfactory sensitivity.Genetic association studies require a quantitative and reliable method for odor threshold assessment in order to examine the contribution of genetic variants to complex olfactory phenotypes. Our main goal was to assess the feasibility of a portable Scentroid air dilution olfactometer for use in such studies. Using the Scentroid SM110C and the SK5 n-butanol Sensitivity Kit (IDES Canada Inc.), n-butanol odor thresholds were determined for 182 individuals of diverse ancestry (mean age: 20.4 ± 2.5 years;  Human scent perception abilities are highly variable. It is well documented that variation exists between individuals and between sexes . SimilarA major bottleneck in the study of olfactory sensitivity has been a lack of instrumentation for accurately measuring smell. Unlike for vision and hearing, there is no standardized test for olfactory sensitivity in either the clinical or research setting . NumerouWhile similar in principle, individual threshold tests vary in their design. Common differences include the method of odor delivery, and the order in which dilutions are presented during testing. The most simple odor presentation strategy is the ascending method of limits (AML), whereby the series of diluted odorants is always presented from lowest to highest concentration. With an AML strategy, the threshold is estimated from the transition point at which the subject reports odor detection . ThreshoCommon threshold tests employed today include the Connecticut Chemosensory Clinical Research Center Test (CCCRC), Sensonic Inc.’s Smell Threshold Test (STT), and a component of the Sniffin’ Sticks test battery . These tThe main goals of our study were to (1) assess the suitability of a portable dynamic olfactometer, the Scentroid SM110C (IDES Canada Inc.), for olfactory threshold studies, and (2) carry out a pilot study to examine the potential association of a selected number of GOS genetic variants and n-butanol thresholds in a sample of diverse ancestry. We evaluated the test–retest reliability of the Scentroid SM110C using n-butanol as an odorant . The odorant n-butanol is widely used in olfaction studies, in part due to its variable odor detection threshold . We survhttp://genome.weizmann.ac.il/GOSdb): rs17712299 and rs17712299 in the ATP-binding cassette, sub-family A, member 13 gene (ABCA13); rs2889732 and rs13036385 in the BPI fold containing family B, member 4 gene (BPIFB4); and rs6746030 in the sodium channel, voltage-gated, type IX, alpha subunit gene (SCN9A) (SCN9A variant rs6746030 has previously been implicated as having an effect on olfactory function (rs6265 located in the brain-derived neurotrophic factor gene (BDNF), which has been linked to olfactory ability in previous studies (We genotyped five SNPs in three genes that had been identified in a database of candidate auxiliary olfactory genes ( (SCN9A) . The SCNfunction . In addi studies .n = 128 female; n = 54 male). 60 individuals were of East Asian ancestry , 55 were of European ancestry, 58 were of South Asian ancestry , and nine were of mixed or other ancestry. 182 participants were recruited on the University of Toronto at Mississauga campus and ranged in age from 18 to 32 years old , and also for field tests on a water treatment plant (r = 0.89).Olfactory thresholds for n-butanol were assessed using the Scentroid SM110C olfactometer and the SK5 n-butanol Sensitivity Kit (IDES Canada Inc.), shown in The odor was presented birhinally using a non-forced choice strategy that consisted of fifteen steps with dilution ratios ranging from 200 to 3,600. A fresh dilution of n-butanol was prepared daily in a stainless steel evaporation chamber by combining 0.5 µl of aqueous n-butanol with compressed air, equivalent to a concentration of 20 ppm, and was subsequently stored in a 1 L Tedlar bag . Using tA replication sample consisting of 29 participants was tested 14 to 18 weeks after their original testing date. The olfactory threshold test protocol remained unchanged.rs6265, rs2889732, rs6746030, rs13036385, rs17132289, and rs17712299.Saliva was collected with the Oragene DNA OG-500 kit (DNA Genotek Inc.) for genomic DNA extraction following recommended protocols. Samples were sent to LGC Genomics for genotyping by means of the KASP genotyping assay. Genotyping was performed for six SNPs with potential involvement in GOS phenotypes: http://ihg.gsf.de/cgi-bin/hw/hwa1.pl. Post-hoc power analysis was performed with the program Quanto (http://biostats.usc.edu/Quanto.html), using several inheritance models  and allele frequencies ranging from 0.05 to 0.50. Parameters were set based on observed n-butanol threshold scores in ppb , with a sample size of 60 and allelic effect sizes ranging from 1 to 10 beta.The olfactory threshold scores were found to be positively skewed and a log transformation was applied. Using the log-transformed data, an ANOVA was run to examine differences in n-butanol olfactory thresholds between ancestry groups. Test–retest reliability was evaluated using correlation tests and Bland-Altman plots, based on the olfactory threshold scores obtained from a subset of participants who were measured twice within an interval of 14 to 18 weeks. Associations between SNPs and olfactory sensitivity were analyzed through multiple linear regression, with age and sex included as covariates. The association tests were done individually for each ancestry group and also for the full sample, with ancestry as an additional covariate. For the association tests, we used a genotypic model, where the effects of one of the homozygous categories and the heterozygous category are reported in relation to the remaining homozygous category, which is used as a reference. The statistical analyses were carried out with IBM SPSS Statistics v.20. For all analyses the level of significance was set to alpha = 0.05. Genotype data was checked for deviations from Hardy-Weinberg equilibrium, using exact tests available at p = 0.615 age; p = 0.053 sex; p = 0.586 ancestry).The n-butanol olfactory threshold was evaluated in 182 participants of diverse ancestry using the Scentroid SM110C olfactometer. p < 0.001; n = 29). We also evaluated reliability using Bland-Altman plots, which are shown in p = 0.731, n = 29).In order to assess the reliability of the Scentroid olfactometer and our protocol for determining olfactory threshold, the thresholds from returning participants were used to calculate a test–retest reliability coefficient (Pearson correlation coefficient) . The resp > 0.05).Six SNPs that have been putatively linked with GOS were genotyped using the KASP assay. The concordance rate observed for blind duplicates was 100%. The genotype distributions of all six SNPs met the assumptions of Hardy-Weinberg equilibrium . Therefore, the Scentroid air dilution olfactometer, when used in conjunction with a non-forced choice AML strategy, produces reliable n-butanol olfactory thresholds, and compares favorably with many other n-butanol threshold tests (We measured the n-butanol olfactory threshold in 182 subjects, in order to assess the feasibility of using the Scentroid SM110C air dilution olfactometer to test olfactory sensitivity. Reliability for this instrument was established using return participant test scores, from which a Pearson correlation coefficient was calculated. The derived test–retest reliability coefficient, representing the linear relationship between the first and second test values, was found to be high (ld tests , for whild tests .Currently, the two most common methods for measuring olfactory thresholds are Sniffin’ Sticks, and versions of the CCCRC. The Sniffin’ Sticks olfactory threshold test is applied birhinally, using n-butanol as the odorant in a triple forced-choice strategy using a staircase presentation method. Staircase strategies differ from the AML in the presentation order of the dilution series, and therefore in how thresholds are estimated see . Unique Variability in the olfactory threshold scores of individual tests can be attributed to differences in experimental design and the study sample. The age of the subjects tested is a significant factor in evaluation of olfactory sensitivity due to the degradation of the sense of smell with age . The meaAs highlighted previously, the staircase odor presentation strategy of the Sniffin’ Sticks test differs from the AML presentation used in the CCCRC and our protocol. Likewise, the mode of stimulus delivery varies from felt-tipped pens, to plastic bottles, to the Scentroid air dilution olfactometer. Yet, the Scentroid as a method for odor delivery is distinctive. A single sample of the odorant needs to be prepared, and using compressed air, the instrument then further dilutes that stock sample for presentation to the test subject. The test administrator can easily switch between set dilutions, thus altering the compressed air to sample ratio.Here, we measured n-butanol olfactory thresholds using the Scentroid SM110C air dilution olfactometer. The resulting test–retest reliability coefficient indicates that the Scentroid produces reliable n-butanol olfactory thresholds. While we employed the AML strategy in a non-forced choice manner, adjustments to this protocol could improve the accuracy of odor thresholds obtained with a Scentroid. Multiple-forced choice designs are less subject to response biases, increasing the reliability of resultant threshold scores . MoreoveIn addition to testing the reliability of the Scentroid SM110C olfactometer, we carried out a pilot study in which we genotyped six SNPs in genes potentially involved in GOS phenotypes, aiming to investigate genetic associations with n-butanol olfactory thresholds. Our goal was to examine variants that may contribute to overall olfactory sensitivity.Studies have shown that individuals with heightened or decreased sensitivity to one odor tend to display similar sensitivities to multiple odorants . This haABCA13 and BPIFB4. There has been no direct evidence of these genes having an olfactory effect in humans. ABCA13 and BPIFB4 were included in the GOSdb based on RNA sequences in humans and mice, as well as rat proteome data. Conversely, SCN9A has been associated with human olfactory function in two separate studies, in addition to being included in the GOSdb. The voltage-gated sodium channel encoded by SCN9A is expressed in both nociceptive and olfactory sensory neurons, and is linked to altered pain and olfactory perception phenotypes. Individuals with loss-of-function mutations in SCN9A cannot feel pain, and exhibit general anosmia (rs6746030 and rs41268673 polymorphisms were associated with increased pain sensitivity and general olfactory acuity (From the GOS database of candidate auxiliary olfactory genes developed by  anosmia . An invey acuity  with resBDNF, rs6265, has been linked to age-related decline in olfactory sensitivity (The Val66Met polymorphism in sitivity . Initialsitivity . Our stuWe performed a small-scale genetic association test for potential GOS variants, utilizing n-butanol olfactory thresholds. No significant results were found in this pilot study. For some variants, this may be due to lack of involvement in GOS. However, several of the SNPs examined have been linked to human olfactory sensitivity previously. Our investigation was hindered by a small sample size, which was further reduced through independent analysis of each ancestry. Our post-hoc power analyses indicate that our study is only adequately powered to identify alleles with strong effects on olfactory sensitivity. For example, assuming an additive model of inheritance , we estiIn summary, in this study we show that the Scentroid SM110C provides reliable quantitative estimates of odor thresholds, and would be an adequate olfactometer to explore the genetic basis of olfactory sensitivity. Air dynamic olfactometers, which allow a continuous control of the concentration of odorant, have advantages over methods based on liquid dilutions in bottles, which may loose their smell intensity with time . A disti10.7717/peerj.643/supp-1Supplemental Information 1Click here for additional data file.10.7717/peerj.643/supp-2Figure S1Click here for additional data file.10.7717/peerj.643/supp-3Table S1Click here for additional data file.


Complex inherited phenotypes, including those for many common medical and psychiatric diseases, are most likely underpinned by multiple genes contributing to interlocking molecular biological processes, along with environmental factors . Despite We believe that multivariate approaches like parallel ICA (p-ICA) are promising, but they are not yet familiar to many investigators. Thus this review attempts to summarize their underlying principles and use, to address conceptual questions that arise regarding their interpretation and to provide practical illustrations.Just as we would not argue that complex traits at a brain level were represented by a single region or cell type, but rather by an interplay between multiple networks , similarIn order to achieve this, we first recapitulate briefly arguments regarding the genetic architecture of common complex medical disorders, difficulties encountered in applying univariate models to these illnesses and their appropriateness as targets of study for multivariate genetic strategies. To expand the latter point, we review the benefits of network-based approaches to study complex inter-related patterns, and argue that in the case of multi-model imaging and genetics data, it is significantly more informative to analyze these domains jointly rather than separately. We next provide a series of sections detailing an overview of p-ICA approaches, with examples and numerous, detailed practical instances. Finally, we address the issue of replication when employing these approaches, as well as ongoing issues in need of solution and summarize some future directions for p-ICA.To date, p-ICA approaches have been applied most often to neuropsychiatric disorders, although as we explain below, common complex medical disorders in fact cover many disease domains. As a general rule, major psychiatric disorders including schizophrenia and autism display similar inheritance patterns to common medical conditions, e.g., type-II diabetes, asthma, or inflammatory bowel disease, characterized by both fairly high heritability and genetic complexity . This viIn these conditions, cumulative evidence suggests that a common disease common variant model (CDCV) still accounts for many cases of these disorders. This model presumes multiple, [likely hundreds or thousands ] of possIn addition, uncommon, non-SNP structural variants of moderate effect size  account for a proportion of cases of schizophrenia, autism, epilepsy, and intellectual disability , albeit One reason that multivariate techniques such as p-ICA are gaining traction as alternate analytic techniques is due to limitations in univariate genetic models when applied to common complex disorders. Univariate approaches generally presume large subject populations  and comprehensive SNP sampling, representing a significant proportion of common genomic variation. If a major assumption of CDCV models is correct, i.e., that genes individually conveying modest risk for a disorder combine, , then univariate GWAS-like approaches likely detect only the “tip of the iceberg,” i.e., the very small number of individual genetic loci conveying the greatest detectable disease risk in that analysis. Multiple associated genomic markers reported for particular phenotypes to date that both transcend appropriate significance thresholds and replicate in independent samples, generally explain only a small proportion of the total genetic variability, are frequently scattered across multiple genomic regions and have unclear biologic functions. The remainder of the “iceberg” remains submerged, .Because individual genes frequently participate in multiple molecular biologic pathways and often contribute to risk for several brain-based disorders, it is difficult to infer relevant functional pathway(s) from the small number of genes typically implicated in GWAS. Univariate approaches are unlikely to detect epistasis ; none haFigure 1). This makes them ideally suited for identifying complex, but potentially weak, effects buried in a high-dimensional data set. Another way of describing a multivariate result (or an independent component) is to call it a ‘network.’ Some explanation is needed as the term network is used widely and with varying definitions across different scientific fields. For example, in the fMRI field, network is used to designate variously (1) regions correlated with a common seed-point  temed-point , or (3) ed-point , 2012a,bed-point  showed tICA  is a modIn contrast to univariate analyses, because an approach like ICA estimates all the variables jointly, by definition the voxels in the ‘network’ are functioning coherently with one another. This property of ICA methods  provides three major benefits. First, it helps with interpretation, as one can accurately assume the region (or genes) in a given component covary together. Secondly, it provides robustness to noise. For example, again to draw on the fMRI example, correlation-based approaches can be ‘tricked’ by phenomena such as phase randomized noise which can appear to represent real signal . HoweverSuch approaches are able to capture multiple links among genetic factors; this can include population effects but also weaker effects of interest or links among patterns of genetic data and patterns of phenotypic data. Finally, p-ICA enables analyses of whole-brain imaging genetics, that PLINK does not . The secFigure 2).Why should we analyze multimodal imaging and genetics data jointly, instead of just analyzing each domain separately? A useful thought experiment is to consider identifying a single relevant genetic factor and correlating it with all brain voxels across subjects to identify a putative intermediate phenotype. This is obviously informative differently than looking separately at which brain regions show group-related activity changes and which genetic factors show group-related differences. Such an analysis is a type of data fusion, because both data sets are used to estimate a result. Such approaches can provide improved ability to distinguish patients versus controls, for example, by capitalizing on the joint information  identification of maximally independent patterns within each modality, and (2) identification of the possible links among the multimodal patterns. The ICA optimization for genetics  and magnetic resonance imaging  are shown in the blue and brown boxes, respectively. W1 refers to the unmixing matrix for the genetic data and W2 refers to the unmixing matrix for the MRI data. The link among these two ICAs is described by equation F3, where Aij are the ICA loading parameters for datatype i and subject j. F1–F3 are solved for simultaneously, providing maximal independence within genetics data, MRI data, as well as the link among the two, hence the name p-ICA).ICA is an approach that takes one data set and identifies components that are maximally independent of one another. In contrast, p-ICA extends this by applying ICA to two datasets jointly, while also incorporating a term that tries to identify links among the two datasets. In the context of p-ICA, the goal is to identify maximally independent networks from two or more data sets simultaneously, as well as identifying links among them. This is done by jointly maximizing several ‘cost functions,’ one to specify the independence among networks in each of the data sets and a second that specifies the link among networks across data sets . These steps are accomplished together in a single algorithm .One of the reasons for a joint optimization in this way is it enable us to combine data types that have different ranges and properties, reducing it down to a maximization of two entropy terms  each of which controls the independence among the components within each data-type, and a cross-correlation among each data-type’s component loadings, providing a normalized measure of the strength of the relationship among each modality for each component pair,  and determine significance, typically in a patient-versus-control context embedded in the components.Consider a specific case in which we have an fMRI contrast image  and a SNP array , and aa] . In thiss,i where s indicated the SNP number (from 1 to 10) and i indicates the subjects (from 1 to 100). Consider also an fMRI contrast image with 10 voxels represented by a matrix Yv,i where v represents the voxel number (from 1 to 10) and i represented the subjects (from 1 to 100). In the case of p-ICA we want to find a representation of the data that includes SNP components, each of which can be considered a ‘pattern’ of SNPs and fMRI components, each of which can be considered a pattern of fMRI voxels. Mathematically we can write X = AS, Y = BT where A and B are called mixing matrices that indicate the degree to which each subject’s SNP and fMRI data are represented by the respective components. The components are noted as S and T for SNPs and fMRI voxels, respectively. The link, more precisely the correlation between mixing matrices A and B, reflects whether or how strongly a SNP component is associated with a fMRI component in a way that how the SNP component is expressed in subjects is related to how the fMRI component is expressed in subjects. The p-ICA models suggest that analyses work most validly within certain ratios of genotype-to-sample size, limited by detection power for reasonable effect sizes  data set in a genotyped population of ∼550 psychiatric probands suffering from schizophrenia or psychotic bipolar illness, plus healthy community controls who underwent resting state MRI scans to identify imaging-genetic relationships . It is k3, limited to SNPs with potential impact on phenotypes of interest. For instance, after conducting univariate tests on individual SNPs associated with particular diagnoses in a large population sample, we select SNPs with relatively liberal association levels with diagnoses  and perform multivariate analyses with another intermediate phenotype like brain imaging, in a different, smaller sample. Though still pre-filtering, this involves relatively less bias and less weight on prior predictions. Finally, p-ICA restricted to genes previously discovered or linked to particular diseases , could discover new multivariate relationships between genes and traits without facilitating new gene discovery. The Psych Chip is an inexpensive genotyping chip manufactured by Illumina covering ∼240 K tag SNP markers and equal number of exome-focused markers, in order to augment coverage of common SNPs in psychiatrically relevant regions while simultaneously augmenting coverage across rare CNVs to focus on psychiatrically relevant very rare variants not present on the Exome chip. It contains ∼50 K common variants (GWAS) relevant to multiple common psychiatric disorders, based on information from the Psychiatric Genetics Consortium (PGC5), whose purpose is to conduct mega-analyses of genome-wide genetic data for psychiatric disorders.After initial preprocessing and quality control steps, hundreds of thousands of SNPs can remain. Different dimension reduction strategies can be applied, each with accompanying limitations. Since adjacent SNPs are likely inherited coherently forming a linkage disequilibrium (LD) block, SNPs within such blocks present redundant information from an association viewpoint and can be reduced by including one SNP within each LD block. However, we do not know which SNP is more likely to be causal, thus, one must check the removed SNPs when interpreting findings from reduced data. Otherwise, grouping and/or selection of SNPs can be based on functional locations such as exomes or promoter regions, assuming known functional active regions are a specific focus of the study. Second, based on hypotheses and known gene/pathway functions we can limit analyses to SNPs within specified pathways or genes, e.g., KEGG pathways/annotations, or restrict SNPs to those deriving from genes with known annotations/functions. Then, association analyses become partially hypothesis-driven and partially data-driven, . Limiting SNP input to biological pathways does not exclude finding new disease-related genes, but lowers the odds of discovering a true positive located in a previously unknown biological pathway. The penultimate approach selects data available from large, publicly available genetic consortia, such as the Psychiatric Genomics Consortium (PGC)The next sections reviews series of other important steps in conducting a successful p-ICA.In practice, the number of components embedded in genetic or phenotypic data is usually estimated beforehand, and associations between components then evaluated through p-ICA. Different strategies are applied to optimal estimation of component number within any observations. The information theory-based Akaike information criterion (AIC), model -selection approach  is used N-fold cross-evaluation or sub-sampling can select the number of components with highly verified (repeated) results  and by allele distribution or portions thereof. Gene-gene interaction, epistasis, and unclear inter-genic regulatory mechanisms all can contribute to such patterns. LD structure can be limited by selecting one SNP per LD block, ICA can also extract SNP patterns related to phenotypic variation or disease. Regarding epistatic (non-linear) effects, though p-ICA is a linear mixing model, it is also a data-driven approach, thus it can still pick up the impact of epistatic effects within the estimated component patterns. For example, there is no requirement that loading parameters vary linearly among subjects, thus if a set of SNPs exhibit synergistic epistatic effects we should see a superlinear relationship of the loading parameters. Future work can more explicitly  attempt to capture non-linear relationships, for example a non-linear ICA algorithm can be incorporated within p-ICA  or we coSelecting top-weighted SNPs for interpretation of the genetic component involves: (1) z transformation of weights, and (2) thresholding top weighted markers. This threshold can reflect the most significant, for instance, 1%, | Z| > 3, or an inflection point of weight distribution (which decision depends on context as to whether this is sensible for a particular study). The thresholding decision is critical, especially for further pathway enrichment, because the lower the z-threshold defined, the greater the number of genes carried forward for pathway enrichment, raising the possibility of finding false positives. One exploratory strategy is to analyze pathways at different thresholds and report/emphasize those commonly significant. Various software suites use slightly different criteria for annotating genes.6, Pathway Studio7, and Metacore (formerly GeneGo 8), and freely available programs, e.g., DAVID9, ConsensusPathDB-human10, and PANTHER11. The widely used IPA is built on its own knowledge base, a repository of expertly curated biological interactions and functional annotations created from millions of individually modeled relationships between proteins, genes, complexes, cells, tissues, drugs, and diseases. It presents enrichment tests from different aspects and biological levels, including molecular/cellular function, system development, canonical pathway, network, and diseases/disorders. DAVID is freely available and widely used; providing flexible tests, including enrichment and classification, using combined databases, e.g., Kyoto Encyclopedia of Genes and Genomes (KEGG), GOterm, BioCarta BioCarta .In brief, functional annotation is a means of identifying functional over-representation of genes associated with particular biological classifications to identify underlying biological themes. Software suites allowing such classifications have to account for the fact that genes/gene products often contribute to multiple biological pathways/systems and that hundreds or thousands of genes can act in parallel in a particular process. Gene ontologies are structured, controlled vocabularies that describe biological processes, molecular functions and cellular components associated with gene(s), i.e., the roles of genes. Such functional annotation/gene ontology pathways are relatively early “works in progress,” particular genes may participate in multiple biologic pathways, many remaining to be fully elucidated. Pathway analyses rely heavily on collected knowledge bases which have more or less complete up-to-date information on gene annotation, gene function, pathways, protein interaction, diseases association etc. Yet, knowledge we have is incomplete and rapidly changing . DiffereData-driven multivariate methods such as p-ICA have been usefully employed to assess mutual information between MRI and genetic data . Example12. One straightforward replication metric is examining results from either leave-N-out or split-sample replications, to assess derived gene components for either rank correlations or percent overlap from the most complex to the simplest level, i.e., molecular pathways, biological categories, genes and SNPs, respectively. Note, the components are data driven, so will not be identical, but quite often there are similar components, which are identified as mentioned above. If one has a need to use a component or components from one data set as a reference, that is a more formal constraint that the components should match, then one can estimate the same component from a new data set using a constrained ICA approach such as p-ICA with reference, except using the entire component as a reference instead of a smaller number of SNPs  or split-half analysis. Characterizing robustness of findings to different software settings, data and preprocessing steps is important, and is one motivation for providing p-ICA tools via a freely downloadable software package, FIT of SNPs .Though we have attempted to address many typical questions that arise when employing a multivariate imaging genetics approach, there are some ongoing issues and emerging issues that we summarize briefly below:Future research needs to confirm how best to utilize hybrid approaches (neither purely model-based nor data-driven), like p-ICA with reference ,b and toAnother important area in need of clarification, is to address what can be gained by incorporating statistical properties such as sparsity  or by liAn area of considerable interest is whether stem cell models can incorporate p-ICA-derived pathways, and more broadly, how genome biology can move more effectively toward systems biology. Additionally, p-ICA can straightforwardly accommodate omics data sets of various types, although publications exploiting this approach in those venues lie in the future.13, or the NIH postmortem gene expression database/BrainSpan14).Attention needs to be directed as to how best to utilize new databases to illuminate in what sequences genes are expressed in development and spatially localized genomic information -type model , informeDespite general agreement that many complex medical and psychiatric diseases and complex quantitative traits are underpinned by multiple genes of individually small effect, the preponderance of genetic analyses are driven by univariate strategies that fail to capture a significant percentage of the relevant genetic variants or their interactions. Such univariate approaches have led to major advances, but are limited by the need for very large sample sizes and have limitations in their ability to illuminate the underlying molecular biological pathways needed to understand etiopathology, and hence to suggest novel treatments for disorders. Multi-variate approaches such as p-ICA offer promise in addressing some the above problems, but are a “work in progress” with some practical details still being fine-tuned and where true replication ability remains to be demonstrated. Particular advantages of multi-variate approaches include statistical efficiency, , and an ability to deal with numerous and complex phenotypes in a flexible manner. Existing models range from fully data-driven approaches to “informed” hybrid models able, for example, to leverage results from GWAS. Multivariate approaches are especially at useful in identifying relationships among large, complex data sets while simultaneously performing data reduction procedures.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.


Oryza sativa L.): days to heading (DH) as a phenological trait and culm length (CL) as the influenced trait. This showed that CL of the cultivars that showed extremely early heading was shortened by the strong influence of DH. In a simulation study, it was shown that the NSE was able to infer the nonlinear influence and direct genetic effects with reasonable accuracy. However, the NSE failed to infer the linear influence in this study. When no influence was simulated, an ordinary bi-trait linear model (OLM) tended to infer the genetic effects more accurately. In such cases, however, by comparing the NSE and OLM using an information criterion, we could assess whether the nonlinear assumption of the NSE was appropriate for the data analyzed. This study demonstrates the usefulness of the NSE in revealing the phenotypic influence of phenological traits.Phenological traits of plants, such as flowering time, are linked to growth phase transition. Thus, phenological traits often influence other traits through the modification of the duration of growth period. This influence is a nuisance in plant breeding because it hampers genetic evaluation of the influenced traits. Genetic effects on the influenced traits have two components, one that directly affects the traits and one that indirectly affects the traits via the phenological trait. These cannot be distinguished by phenotypic evaluation and ordinary linear regression models. Consequently, if a phenological trait is modified by introgression or editing of the responsible genes, the phenotypes of the influenced traits can change unexpectedly. To uncover the influence of the phenological trait and evaluate the direct genetic effects on the influenced traits, we developed a nonlinear structural equation (NSE) incorporating a nonlinear influence of the phenological trait. We applied the NSE to real data for cultivated rice ( Phenological traits of plants, such as flowering time and days to maturity, are tightly linked to growth phase transition. It is well established that phenological traits have causal influences on other traits through the modification of the duration of growth period. These causal influences result in phenotypic correlations between the phenological traits and the influenced traits. For example, significant phenotypic correlations between the days to flowering and other agronomic traits have been reported for rice –4, wheatThe influence of the phenological trait is a nuisance in plant breeding because it hampers genetic dissection and evaluation of the influenced traits . The genyi, P and yi, A are the phenotypic values of line i of the phenological and influenced traits P and A, respectively, λ represents the magnitude of the influence of the phenological trait, ui, P and ui, A are the “direct” genetic effects on each trait, and ei, P and ei, A are the residuals. If the influence λ is not modeled explicitly as in the ordinary regression models, the genetic effect on trait A is contaminated by the effect on the phenological trait P. That is, we obtain λui, P + ui, A as the genetic effect on trait A. To date, several studies in plant and animal breeding have used structural equations to infer the causal influences between traits; for example, between yield component traits and the yield in wheat [The causal relationships between traits can be evaluated statistically using multiple trait regression models encompassed in structural equation models (SEMs). These are comprehensive regression models for complex and multivariate data. In general, SEMs consist of two components: structural equations that describe the relationships between the latent variables, and measurement equations that combine the latent variables with the observed variables . A strucin wheat  and betwin wheat –17. HoweA major drawback of a structural equation, such as illustrated in Oryza sativa L.) evaluated at four locations. Because DH of rice is linked to the growth phase transition from vegetation to reproduction, CL is often influenced by DH. However, the genetic evaluation of CL is usually conducted without considering this influence [This study developed a nonlinear structural equation (NSE) to uncover the phenotypic influence of the phenological traits on other traits and to infer the direct genetic effects on the influenced traits. The influence was modeled using B-splines to allow nonlinear relationships between the phenological traits and the influenced traits. We applied our method to days to heading (DH) and culm length (CL) of the 110 rice cultivars  were improved in Hokkaido, which is the northernmost region in Japan, and two cultivars (“Akage” and “Bozu”) were the landraces considered to be unique to this region . These cGenotypes of 3,102 genome-wide bi-allelic markers were available for the cultivars . The detSD1 had been determined  represents the phenotypic influence of DH, which is assumed to be a function of yi, DH, xi, DH and xi, CL are vectors containing the intercepts (1) and the genotypes (0 or 1) of the major gene polymorphisms of line i, βDH and βCL are vectors containing the regression coefficients, ui, DH and ui, CL are the additive genetic effects of line i, and ei, DH and ei, CL are the residuals. The phenotypic values were adjusted for their mean values such that the averages of yi, DH and yi, CL were calculated as zero. xi, DH contains the genotypes of all the heading date gene polymorphisms , whereas xi, CL contains the genotypes of both the heading date and SD1 gene polymorphisms. This NSE is illustrated graphically in βCL and ui, CL as the direct genetic effects on CL.For each cultivar L ;Pm is the weight of the mth basis function M = 7, indicating that eight basis functions were included in the model. As the number of basis functions increased, a more complex (meandering) trajectory could be depicted. However, because it was unlikely that the phenotypic influences of the phenological traits would become too meandering, we adopted a relatively small number (eight). A basis function of cubic spline is defined using five consecutive knots. Thus, the total number of knots was 12; the first and last four knots were arranged repeatedly at the points that were the lower and upper limits of DH, respectively; these repeated knots defined the boundaries of the B-splines [yi, DH value was arranged at the middle between the first point (the point where the first four knots were arranged) and the next knot. Similarly, the largest yi, DH value was arranged at the middle between the last point (the point where the last four knots were arranged) and the preceding knot. The positions of the knots and the knot interval were determined for each location. The definition of We applied the cubic spline to model -splines , and thee et al. .L  is not equal to zero, the direct genetic effects  cannot be estimated accurately because the genetic effects on DH  contaminate the effects on CL. We compared the NSE and OLM using the widely applicable information criterion (WAIC) [yi, DH and yi, CL). The NSE and OLM were each fitted to the data for each location.We also fitted an ordinary bi-trait linear model (OLM) to the data. This did not contain the term representing the phenotypic influence of DH, i.e., the first term on the right hand side of n (WAIC) ; this isn (WAIC)  to selec0, G ⊗ K), where G is the genetic variance-covariance matrix and K is the genomic relationship matrix calculated from the genome-wide markers using the A.mat function of the R package rrBLUP [0, R ⊗ I), where R is the residual variance-covariance matrix and I is an identity matrix with the size of the number of cultivars. The prior distributions of G and R were inverse Wishart distributions with four degrees of freedom and with scale matrices that were half of the phenotypic variance-covariance matrix, which resulted in the expectations for these distributions being the same as the scale matrices. We assigned non-informative prior distributions to the overall means and major gene effects. The prior distributions of the weights of the basis functions were as followsN denotes the normal distribution and Pm for m ≥2 is equivalent to assuming i, CL−Lyi, DH) as the dependent variable [6 with the first 0.1 × 106 iterations discarded as burnin, and the sampling interval was 100. The prior distributions of the OLM were the same as those of the corresponding parameters of the NSE. The parameters of the OLM were inferred using Gibbs sampling with the MCMC conditions the same as those for the NSE. Major gene effects were judged to be non-significant if 0 lay within the 0.025 and 0.975 quantiles of the MCMC samples. The calculation was performed using a program written in C language.Parameters were inferred using a Markov chain Monte Carlo (MCMC) method. The prior distribution of the additive genetic effects was a multivariate normal distribution MVN at each location were used to simulate data sets. By generating random residuals using R, 20 data sets were created for each location. We ensured that the yi, DH values created did not exceed the lower and upper limits defined by the knots in the real data analyses. In the second and third schemes, the posterior means of the parameters of the OLM  at each location were used to simulate data sets. In the second scheme, the influence was assumed to be linear; L  was set to 1.2 for all yi, DH. In the third scheme, it was assumed that DH did not influence CL, i.e., L  = 0 for all yi, DH. In both the second and third schemes, 20 data sets were created for each location by generating random residuals using R. We analyzed these simulated data sets using the NSE and OLM. The prior distributions of these methods were determined as described earlier (Parameter estimation).To assess the validity and accuracy of the inference by the NSE, we conducted simulation analyses based on the parameter values estimated in the real data analyses. We considered three simulation schemes. First, the posterior means of the parameters of the NSE . Consequently, these polymorphisms had large negative effects on CL because of the phenotypic influence of DH on CL. When the influence of DH on CL was removed using the NSE, the negative effects of these polymorphisms on CL diminished  × ui, DH). The latter component was negative because the polygenic effect on DH  was negative, and the influence of DH on CL ) was positive. Thus, after removing this latter component by modeling the influence of DH on CL using the NSE, the polygenic effects increased. However, we also observed a contrary tendency in the two landraces of the early-heading cultivars, “Akage” and “Bozu”, which showed lesser effects on CL in the NSE analyses. We were not able to find any reason to explain this phenomenon.We compared the NSE and OLM using mean log likelihood and WAIC . The meaL  = 1.2 for all yi, DH), and no influence  = 0 for all yi, DH). The inferred trajectories of the influence of DH are shown in L  values are presented in We conducted simulation analyses to assess how accurately the NSE inferred the phenotypic influence of DH and the direct major gene and polygenic effects on CL. We considered three simulation schemes that differed in the phenotypic influence of DH on CL: nonlinear influences, a linear influence . However, the NSE failed to infer the linear influence. When no phenotypic influence was simulated, the ordinary multiple-trait linear model, OLM, tended to infer genetic effects more accurately. However, the simulation study also shows the possibility of assessing whether the nonlinear assumption of NSE was appropriate for the data analyzed by comparing the WAIC values between the NSE and OLM.L  is replaced by a scalar value λ, the parameters λ and the residual covariance are not identifiable because only a single information source, the phenotypic covariance between the traits, is available to determine these two parameters. To ensure identifiability between λ and the residual covariance, a constraint is required. A typical constraint is fixing the residual covariance to zero, which produces a sufficient rule of parameter identifiability called the “recursive rule” by Bollen [Pm and residual covariance were identifiable without constraints on the residual covariance. In the real data analysis, the correlation of the MCMC samples between the weights and the residual covariance was weak; the mean (SD) Pearson correlation coefficients were −0.19 (0.12), −0.23 (0.09), −0.17 (0.09), and −0.20 (0.12), at NICS, NIAS, FRERC, and WARC, respectively. When the parameters are not identifiable, we usually observe a high negative correlation between the MCMC samples; for example, it can reach −0.996 [L  values unique to cultivars, which would yield the phenotypic covariance between the traits unique to the cultivars. This would increase the information sources to determine the weights and residual covariance, and would make these parameters identifiable. In contrast, when the influence was linear, i.e., L  was a single value for all yi, DH, the phenotypic covariance was a constant among the cultivars. Consequently, the parameters became unidentifiable, and thus the NSE probably failed to detect the linear trend. When we simulated data sets with linear influences and no residual covariance, the NSE successfully detected the linear trajectory by imposing the constraint of being zero on the residual covariance (data not shown). The validity of assuming no residual covariance would be case-dependent. However, we should be cautious about imposing such a strong constraint. In this study, the estimated residual covariance was positive at all the locations; the posterior means (SD) were 6.68 (4.66), 6.47 (4.24), 5.23 (3.41), and 4.83 (2.59) at NICS, NIAS, FRERC, and WARC, respectively. Thus, if the residual covariance is not taken into account, the estimate of phenotypic influence will be overestimated.The reason why NSE failed to detect the linear influence is probably related to an issue of parameter identifiability. In y Bollen . In the h −0.996 . AlthougL  values was lowest in the simulation for WARC values was 0.92).In the real data analysis, the WAIC supported the superiority of the OLM only at WARC . If we fARC0.82, , suggestARC0.82, , suggestIn the real data analyses, for all the locations, the inferred trajectories of the phenotypic influence of DH on CL showed a decreasing trend as DH increased. The influence on the early-heading cultivars was positive for all the locations, suggesting that early heading shortened the CL of these cultivars. Only at NICS, the influence became negative as DH increased, suggesting that later heading was also attributable to shorter CL at this location. Although the exact reason is unclear, we speculate that the environmental conditions encountered by the later heading cultivars, such as temperature, solar radiation, and available nutrition, might not be suitable for growth and/or heading. At the other locations, the influence of DH became less prominent as DH increased, suggesting that the influence was vague for the late-heading cultivars.Hd1.4, Hd1.5, Ghd7_SNP, and Ghd7_Indel on CL decreased  alleles. When the modified genotype is cultivated at FRERC, DH would be prolonged for approximately 35 days because of this modification  tended to increase by removing the phenotypic influence of DH. Using the estimation results of the major gene and polygenic effects and the nonlinear influence of DH, the NSE can predict CL when the heading data genes are modified. For example, suppose that the non-reference alleles on fication . As a refication  would beBecause genetic resources used in plant breeding usually consist of lines/cultivars adapting to diverse environments, they often show great genetic variation in the phenological traits that have a major role in local adaptation. Consequently, field experiments at a location cannot evaluate the genetic effects of the cultivars/lines that do not adapt to the location because they are masked by the influence of the phenological traits. This phenomenon was observed for the early-heading cultivars in the present study. However, it is suggested that the genetic effects concealed by the phenological traits at the target location can be evaluated by applying the NSE. Thus, the NSE could enhance the value of genetic resource in plant breeding.Pm (0 ≤ m ≤ 7), were inferred using the following Metropolis algorithm.The weights of the basis function, Pm is the value at the current iteration and (1) Generate a proposal value P and L indicate the prior density of Pm and likelihood functions, respectively, and P−m indicates the weights except for Pm. The likelihood function is illustrated by Eq 37 in [Pm does not affect the likelihood of all the cultivars because the corresponding basis function is 0 if yi, DH is located outside the five knots that define the basis function. Thus, in the acceptance probability above, we need to evaluate the likelihood only for the cultivars that are located within the knots.(2) Accept Eq 37 in . It is nPm with (3) Replace S1 Table(XLSX)Click here for additional data file.S2 Table(PDF)Click here for additional data file.S3 Table(PDF)Click here for additional data file.S4 Table(PDF)Click here for additional data file.S5 Table(PDF)Click here for additional data file.


SBP/∆t) was defined based on the imputed phenotypes and analyzed in a genome-wide fashion. We also conducted an exhaustive genome-wide search for interaction between single-nucleotide polymorphisms (7.14 × 1010 tests) under an allelic model.We present a genome-wide association study of a quantitative trait, "progression of systolic blood pressure in time," in which 142 unrelated individuals of the Genetic Analysis Workshop 18 real genotype data were analyzed. Information on systolic blood pressure and other phenotypic covariates was missing at certain time points for a considerable part of the sample. We observed that the dropout process causing missingness is not independent of the initial systolic blood pressure; that is, the data is not missing completely at random. However, after the adjustment for age, the impact of systolic blood pressure on dropouts was no longer significant. Therefore, we decided to impute missing phenotype values by using information from individuals with complete phenotypic data. Progression of systolic blood pressure (∆p≤5 × 10−8), which was also confirmed via Monte-Carlo simulation. In view of the rather small sample size, however, the results have to be followed-up in larger studies.The suggested data imputation and the association analysis strategy proved to be valid in the sense that there was no evidence of genome-wide inflation or increased type I error in general. Furthermore, we detected 2 single-nucleotide polymorphisms (SNPs) that met the criterion for genome-wide significance ( We present a genome-wide analysis, that is, an analysis comprising all odd-numbered chromosomes, of the Genetic Analysis Workshop 18 (GAW18) real data. The structure of the GAW18 data is rather complex , making it difficult to address all dimensions of the data in a single analysis. Therefore, we restricted our analysis to 142 unrelated individuals available. Moreover, because systolic blood pressure (SBP) was analyzed extensively during the genome-wide association analysis era by large consortia, we decided to investigate the progression of SBP instead.We address the problem of missing phenotypes and impute missing values for the original sample of 939 individuals. Based on the imputed phenotype data, we conduct a genome-wide analysis of the quantitative trait progression of SBP in time and perform an exhaustive search for SNP-SNP interaction. We assess the validity of the approach in the genome-wide setting and confirm our top-ranking findings via Monte-Carlo simulation. Finally, we compare results obtained with the imputed phenotypes to the respective results obtained with the original phenotype data.In a longitudinal study repeated measurements and records related to a trait under investigation are collected at different time points for a fixed group of individuals. It is often not possible to collect complete data sets because of early dropouts, changed health conditions of subjects, or other reasons. The first step in the analysis of repeated measurements with missing values should be the verification, whether the data is missing completely at random (MCAR), missing at random (MAR), or missing not at random (MNAR) . An obseStandard methods to treat missing data, for example, complete case analysis and simple imputation methods , are not appropriate for a valid inference without assumption that the data is MCAR, or when this assumption is violated. A proper analysis of missing data requires a knowledge or estimate of the mechanism that generated the missing data . A test i AGEfor age, i SBPfor SBP, i MEDfor intake of antihypertensive medication, and i SMKfor smoking status at the initial measurement i = 0 and up to 3 follow-ups .In analyzing the GAW18 data we were dealing with quantitative, non-monotone data from the longitudinal study. For available phenotypic variables we introduced the following notations: SBP/∆t, which was defined as the difference between 2 values of SBP recorded at the last and the first examinations attended by an individual, divided by the time ∆t in years elapsed between these 2 examinations.We analyzed the quantitative trait progression of SBP in time, ∆SBP0 = β0 + β1x1 + β2x2 + β3x3, where a variable i xequals 1 if the measure of i SBPis given, and 0 if the measure of i SBPis missing, βi∈ℝ, i = 0,1,2,3. Using this model to test the hypothesis β1 = β2 = β3 = 0 resulted in the p-value 5.21 × 10−7. This led us to conclude that the data is not MCAR, given that the presence or absence of the follow-up measures is influenced by SBP0. However, when we included the covariate AGE0, the p-value became much higher (p = 0.07), which suggested that age might be a significant predictor of missingness and that the data is likely MAR.We suspected that the data was not MCAR and tried to determine whether MAR or MNAR was more proper to assume. We considered a linear regression model, i, SBPi, MEDi, SMKi AGEvariables, i = 0,1,2,3, without considering hypertension status or diastolic blood pressure. To start the imputation we extracted 187 completers, that is, individuals who had participated in all 4 examinations and for whom all measurements had been recorded. We used the information collected for completers to determine the settings of the association analysis, as well as to complete the missing phenotype values for non-completers.To impute the missing values and create complete phenotype data for each subject, we applied a kind of simple imputation method ,3, whichSBP/∆t:We suggest the following algorithm to impute the missing values in the GAW18 real phenotype data for the purposes of the analysis of ∆1. Determine the average calendar years of the 4 examinations .i , i AGE= 0,1,2,3, using the results of Step 1 and the birth year of every individual.2. Impute the missing values of 1 , α2 , α3 of ∆SBP on AGE0, MED0, SMK0, respectively, and β of ∆SBP on SBP0 for completers.3. Calculate the correlation coefficients αC and a non-completer N, define a function where wide hat denotes the normalized value of the corresponding variable for instance, and where (·) Iequals 1 if the corresponding measurement for a non-completer is given, or equals 0 otherwise.4. Given a completer d on the list of all completers.5. For every fixed non-completer determine the "closest" completer by finding the minimal value of the function i, MEDi , SMKi , i SBP= 0,1,2,3, for a given non-completer by those from the "closest" completer.6. Impute the missing values of Imputation was performed for 752 non-completers. Table First, we filtered 142 individuals from the GAW18 data set who were indicated as being nonrelated. To check for unobserved relatedness among these individuals, we computed a pair-wise identity-by-state (IBS) matrix from the genotype data. For a pair of individuals, their IBS-value was computed as the portion of alleles identical by state divided by the number of alleles genotyped in both individuals. The mean IBS-value over all pairs of individuals was 0.74; its SD (standard deviation) was 0.0063. Seven pairs had IBS-values higher than 0.84, which was more than 14 SD above the mean, indicating evidence of relatedness. For each of the 7 pairs, the individual with the lower genotyping rate was removed from the analysis. We also excluded 8 other individuals who had genotype missing rates exceeding 40%, leaving 127 individuals for analysis.p<1 × 10−6), or low minor allele frequency (MAF<1%). That left 429,516 quality controlled SNPs for analysis, and the overall genotyping rate of the remaining data was 99.9%.Finally, we removed 42,519 SNPs because of either low genotyping rate (<99%), deviations from Hardy-Weinberg equilibrium  for quantitative traits and used the implementation in INTERSNP .A,IM= β0 + β1x1 + β2x2 + β1,2x1x2 against the model A M = β0 + β1x1 + β2x2, where β0 is the intercept parameter i xis the number of copies of minor alleles of SNP i, i = 1,2; for a given individual, x1x2 is the interaction term; and βs are the coefficients to be estimated. Let A,I SSEand A SSEbe SSEs of the respective models, and let n be the number of individuals in the analysis. Then, the test statistic t = /A SSEis F-distributed with  DF and yields a test for allelic interaction. Analysis was performed by a parallelized version of INTERSNP, which uses the OpenMP API specification for parallel programming (http://openmp.org/wp). In the interaction analysis, we applied a stronger MAF criterion of 5% and analyzed 7.14 × 1010 SNP pairs.For 2-marker analysis, we used a test for allelic interaction. We compared the sum of standard errors (SSE) of the model F-distribution might lead to inflated p-values. To overcome this issue, we validated the top-ranking p-values of the single-marker analysis via Monte-Carlo simulation. In each permutation replicate, the quantitative trait values were randomly distributed over the individuals of the sample. In total, 108 permutation replicates were conducted and for a given SNP, its point-wise Monte-Carlo p-value was computed as k/108, where k is the number of permutation replicates with a test statistic t larger than that of the real data. For computation time reasons, it was not possible to validate the results of the interaction analysis via simulation, since at least 1013 replicates would have been needed.When the sample size is small, low MAF can lead to a small number of "effective" analysis observations and the application of the SBP/∆t yielded a genome-wide inflation factor of λ = 0.979, indicating that, overall, our analysis can be judged as valid in the sense that we do not have evidence for inflated p-values caused either by population stratification or by our phenotype imputation method. Table p = 1.35 × 10−9) and rs13157168 (p = 2.45 × 10−8), met the criterion for genome-wide significance. These SNPs have rather low MAF, and the results might be artifacts. However, Monte-Carlo simulation with 108 permutation replicates confirmed the findings; none of the replicates had a more extreme test statistic than the real data. Nevertheless, the top SNP rs3093642 should be treated with caution because its p-value for deviation from Hardy-Weinberg equilibrium was rather close to the predefined exclusion criterion. The second SNP rs13157168 is an intronic variant in the RASA1 gene. Association of RASA1 with variable capillary and arteriovenous malformations was reported by Boon et al [p-values r pof the single-marker analysis performed on SNPs on the real phenotypes are markedly higher than the corresponding p-values i pfor the imputed phenotypes. SNPs with an at least suggestive level of evidence for association, unfortunately, were not found among the top ranks.The analysis of ∆p = 7.9 × 10−6) [COL4A1 gene. Variants in COL4A1 are significantly associated with brain small vessel disease, in which stiffness of blood vessels is affected [Table  × 10−6) . SNP rs7In longitudinal studies, repeated measurements related to a trait under investigation are performed subsequently within a particular time period. The attractiveness of longitudinal studies is the ability to capture the dynamics of traits. Investigating the progression of disease by using this additional dimension of time might reveal insights on underlying disease mechanisms. Given the nature and the running time of a longitudinal study, it is expected that some measurements might not be available, whereas others might be missed. Imputation of missing values helps to enrich data and to use maximum information, although it has own weaknesses and it is not self-evident that it leads to improved power.SBP/∆t). In particular, the imputation reduced the impact of outlier SBP values, and made it possible to define the values of the trait in a more uniform time period. This, in turn, helped to reduce the potentially negative impact of short time periods on the values of the trait.In our setting, the imputation of missing values in the GAW18 phenotype data set helped to streamline the definition of the trait under investigation (∆The method of choice to check the validity of imputation is to carry out a sensitivity analysis to investigate the dependence of the results on the particular imputation method. In our case, the sensitivity analysis could have been carried out by the multiple imputation technique. Unfortunately, this technique is time-consuming and requires software implementation; consequently, it was not applied in our work. This is a potential limitation of our approach. An argument in favor of our imputation method is that the mean of SBP values within an examination is preserved by the imputation.Our analysis strategy helped to identify some suggestive association findings. At this point, however, the results do not prove that the analysis using imputed phenotype data is superior to that without imputation. To prove this statement, it would be necessary to know whether some of the SNPs we have identified represent "true" associations. Those have to be investigated in studies with independent data. In summary, our work can be seen as a suggestion for longitudinal data analysis, which was quite reasonable in the sense that it did not show genome-wide inflation or increased type I error in general.The authors declare that they have no competing interests.TV designed and performed the imputation of the missing phenotypes. TV and DD carried out the analysis of unrelated individuals. MA worked on the pre-processing of the data. CH provided support in using INTERSNP software. TB conceived of the study and suggested main lines of the analysis. AL coordinated the study, participated in its design and improvement. All authors read and approved the final manuscript.


Prediction of genetic values has been a focus of applied quantitative genetics since the beginning of the 20th century, with renewed interest following the advent of the era of whole genome-enabled prediction. Opportunities offered by the emergence of high-dimensional genomic data fueled by post-Sanger sequencing technologies, especially molecular markers, have driven researchers to extend Ronald Fisher and Sewall Wright's models to confront new challenges. In particular, kernel methods are gaining consideration as a regression method of choice for genome-enabled prediction. Complex traits are presumably influenced by many genomic regions working in concert with others (clearly so when considering pathways), thus generating interactions. Motivated by this view, a growing number of statistical approaches based on kernels attempt to capture non-additive effects, either parametrically or non-parametrically. This review centers on whole-genome regression using kernel methods applied to a wide range of quantitative traits of agricultural importance in animals and plants. We discuss various kernel-based approaches tailored to capturing total genetic variation, with the aim of arriving at an enhanced predictive performance in the light of available genome annotation information. Connections between prediction machines born in animal breeding, statistics, and machine learning are revisited, and their empirical prediction performance is discussed. Overall, while some encouraging results have been obtained with non-parametric kernels, recovering non-additive genetic variation in a validation dataset remains a challenge in quantitative genetics. Six years after the rediscovery of Mendel's laws of heredity, Toyama Kametaro's experimental work on silkworm breeding showed the first case of Mendelian inheritance in animals Onaga, . Yule 1 made a fConcurrently, whole-genome prediction of “total” genetic effects has been motivated by the fact that phenotypes and genotypes may not be linearly related and that the additivity assumption, even though useful, is violated  that maps these genotypes to responses (y). The data generating model is then y = f(X) + ϵ, where y is the vector of phenotypes and ϵ is a vector of residuals. In general, all kernel methods differ from each other in the choice of the mapping function f(.) and the type of regularization used to balance goodness of fit and complexity, as discussed later.We first review kernel-based prediction models being used for prediction using genomic data. Our aim is to approximate an unknown “true” genetic signal f(X) there are a few things that we need to set up prior to the search. One is whether we should impose a restriction on the search space or not. The parameter space is a space where all models are characterized by parameters. In a linear model, where response values have linear relationship with respect to the parameters, though, it could be non-linear on covariates such as in the case of polynomial regression. Once the parameters are given, all models are distinct. For example, the two linear models would beOur main interest is to identify the model that gives best prediction among a set of candidate models. To find the best predictive function ai, bi, and ci then Models 1 and 2 can be differentiable. In other words, we just need to fill out the unknown parameters of the given models. The best linear unbiased predictor (BLUP) is a procedure for filling the unknown values , where Γ is a “true” genomic relationship matrix among animals, i.e., at the quantitative trait loci affecting the trait. Since the latter are unknown, we approximate the vector of genetic values g with a linear function such that y = Xβ + ϵ, where X is an n  × m (number of markers) matrix of additive marker genotypes potentially centered or scaled; β is a vector of regression coefficients on marker genotypes; and ϵ is a residual that includes model misspecification and environmental effects not considered in this analysis. Under the independence assumption between g and ϵ, the variance-covariance matrix of y isSuppose underlying signal is given by β ~ N and ϵ ~ N. Here, Vg = XXTσ2β is the covariance matrix “due to” markers. The problem is to predict g such that two conditions are met: (1) E(ĝ) = E(g) = 0, and (2) var(ĝi − gi) is minimum for i over all linear functions that satisfy the unbiasedness condition (1). If normality is assumed, the BLUP of g(ĝ) is the conditional mean of g given the data, and takes the formoften assuming that XXT is invertible. Here, Cov(X) = XXT is a covariance matrix of marker genotypes (provided that X is centered), often considered to be the simplest form of additive genomic relationship kernel, G. We can refine this kernel by relating genetic variance σ2g and marker variance σ2β under the following assumptions. Again, assume genetic value is parameterized as gi = ∑xijβj, where both x and β are treated as random and independent. Under Hardy-Weinberg equilibrium, E(xij) = 2pj and Var(xij) = 2pj(1 − pj), where pj is the minor allele frequency of locus j, and assuming linkage equilibrium of markers ,assuming that Under the homogeneous marker variance assumption, one obtains the relationship2β in Equation (1) with (2), we getReplacing σG matrix introduced in VanRaden  so the marked genetic variance is given by the sum of individual marker variances. With this, σ2β = σ2g/m, and G in Equation (3) becomes the second additive genomic relationship kernel of VanRaden G, were 0 < λ < 1 is the weight placed on A relative to G , contrary to assuming g = Xβ as in the case of GBLUP. This g(xi) function is viewed as the conditional expectation E(yi|x = xi), that is, the average phenotypic value of individuals possessing marker genotype xi without restricting the form of gi. RKHS regression proceeds by searching a function and uses the residual sum of squares as a loss function, and assigns the squared norm of g under a Hilbert space as a penalty. The objective function to be minimized with respect to g isGenomic BLUP (GBLUP) is a linear model characterized by parameters that relate to additive quantitative genetics theory. We now extend the search space by eliminating some restrictions. For instance, semiparametric regressions identify functional forms “before as well as in the midst” of the fitting process Berk, . A repro represents a Hilbert space, very rich class of functions. While there are countless candidates for g in non-parametric regression, with this setting, the representer theorem developed by Kimeldorf and Wahba  is minimized by taking its derivative with respect to ĝ = Kso that the predicted genetic value is given by This regression procedure is also known as kernel ridge regression in machine learning, and was first introduced in quantitative genetics by Gianola et al.  and GianL2 norm (Euclidean norm) to regularize the regressions α. This regularizer assures that optimal solutions lie in a finite-dimensional rather than in an infinite-dimensional space. In practical situations, sparsity induced by the L1 norm (Manhattan norm) may be preferable. This is equivalent to replacing α′Kα in Equation (5) with L2 may deliver better predictions.Note that the representer theorem requires assigning the The important connection between RKHS regression and BLUP was brought up first by Harville ; Robinsoy is the response variable, X is an incidence matrix linking the response to some nuisance effects; β and α are regression coefficients and ϵ is a residual that includes model misspecification and environmental effects not considered in this analysis. The two random components of the model follow the distribution α ~ N and ϵ ~ N, where σ2α is the additive genetic variance and A is the numerator relationship matrix between individuals (relationships in the absence of inbreeding). Henderson's mixed model equations (MME) arewhere α* = A−1α (assuming that A−1 exists) so that Equation (7) is reexpressed asNow, transform additive genetic effects into α* ~ N, and the corresponding MME arewhere A is symmetric. By multiplying the α* equation by A−1, one obtainsbecause β = 0) a priori, then Equation (10) reduces toIf phenotypes are pre-corrected for systematic effects −1y. This is equivalent to Equation (6) and is the Bayesian kernel ridge regression employed in de los Campos et al.  isGBLUP is also linked to BLUP of marker regression coefficients. Here, we show how BLUP of regression on markers and BLUP of additive genetic values are related to each other. This relationship was first shown by Henderson  in the cUsing Equation (3), we getĝ from GBLUP, BLUP of marker coefficients is given by XT (XXT)−1ĝ. We arrive at the same prediction regardless of whether we start from the genotype matrix X or from the g. Note that marker-based regressions can also be “kernelized” when the squared norm of β is assigned as penalty function. Ridge regression with markers treated as random effects is known to be equivalent to BLUP, i.e., RR-BLUP  for “aa,” “Aa,” and “AA,” respectively, two vertices are adjacent if and only if codes at just one SNP locus differ by 1.Suppose that all individuals in a sample have been genotyped for biallelic SNPs spanned over the whole genome, with these SNPs used for genome-enabled prediction of quantitative traits. The number of possible genotypic configurations with a et al.  used SNPA or the genomic relationship matrix G among individuals are employed. With this, the set of additive effects in a sample of individuals is assumed to follow a multivariate normal distribution with a zero mean vector and a covariance matrix that is proportional to A or G, instead of the identity matrix I. On the other hand, kriging explicitly assumes at the onset that observed values are spatially correlated.In standard regression analysis, independent and identically distributed (i.i.d.) observations is a common assumption while, in kriging, spatially correlated random variables are considered  + ei having the measurement on individual i  at point xi. A random field g is a random function on a space p-dimensional grid graph ℤp3 of all SNP genotypes, and xi and ei are the ith observed vertex and residual. In other words, yi is a response value associated with a spatial point location x. which is a vertex. We assume second order stationarity in the observations, that is, the mean is constant but unknown and the covariance depends only on the “distance” between any of two vertices (∥xi − xj∥), but not on the locations per se. We wish to predict the phenotype of an individual having vertex x0 using all available data. Kriging is the best predictor ,E[yi] = E[y0] = μ holds because of the stationarity assumption made. Setting Equation (13) to zero gives the unbiasedness conditionwhere (assuming no nuisance parameters) μ is the mean values of the responses; ŷ0 is an unbiased predictor if Thus, As defined by Henderson , BLUP isn + 1 partial first derivatives with respect w and λ to 0, one obtains the kriging system of linear equations shown below.Adding the unbiasedness conditions to the above equation via a Lagrange multiplier (λ) and setting the V = {Cov(g(xi), g(xj))}, the residual covariance matrix Iσ2e, the vector of ones 1. The right-hand side contains covariances between the unsampled location to be predicted and sampled locations Ci0 = {Cov(g(x0), g(xi))}. We predict a value at some unsampled vertices by replacing the unknown weights with ŵ. Although derived from a different perspective, the predicted value ŷ(x0) is essentially BLUP of Henderson .The genomic relationship matrix of VanRaden  represeni, j) is given byIn a Gaussian kernel, we embed individuals in the Euclidean space, and the corresponding metric is a squared Euclidean norm. For example, the spatial genetic distance between individuals  is the SNP genotype code for individual i at SNP k. The smaller the Euclidean distance is, the stronger the similarity in state between two genotype vectors. Taking exponentiation of the negative Euclidean distance changes the direction of relatedness, that is, a larger distance produces a smaller value of the Gaussian kernel and a smaller spatial genetic similarity. Gianola et al. . Some applications of this in the context of genome-enabled prediction are in Long et al. , so the squared norm is dropped. The Matérn covariance function is a generalization of an Euclidean distance-based kernel that contains the Gaussian and the exponential kernels as particular cases. An advantage of this function is that the actual form of a kernel can be inferred from the data directly, permitting flexible kernel selection. Based on simulated data, the Gaussian kernel appeared to give the best fit within this specific class of kernel . A diffusion kernel for binary genotypes (presence or absence) is likewise constructed aswhere d is the Hamming distance, that is, number of coordinates at which x and x′ differ. Morota et al. . The authors used a diagonal matrix for Σ−1, with its kth element equal to the heterozygosity at the kth SNP locus: 2pk (1 − pk), and evaluated the kernel performance over a fixed grid of values of ν. This kernel aims to expand the underlying metric space with its heavier tails. The t kernel resulted in a similar performance as the Gaussian kernel ) = 0. Then the covariance function becomes Cov(g(x..), g(x′..)) = E(g(x..), g(x′..)) = K(g(x..), g(x′..)), namely, a kernel.Covariance functions over the prediction grid need to be specified for interpolation when applying kriging and Gaussian process regression. In this setting, the kernel matrix It is worth keeping in mind that the kernel defines the the inner product, the inner product defines the covariance, and the covariance brings up a new metric called Hilbert space (distance). While some simply structured kernels such as the IBS kernel , dominance (D), and additive by dominance (G#D) kernels as in Morota et al.  involving hundreds of thousands of SNPs was reported by Ozaki et al. . The autGenome-enabled prediction of traits outside of animal and plant domains has been mainly carried out by targeting specific genes or variants identified from inference procedures  as opposed to using all available markers. This indicates that deciding which genetic variants to include in the prediction equation is a crucial component. In this line, Morota et al.  classifip-values derived from a series of single marker regressions or with inclusion of a genomic relationship matrix to correct for false positive associations due to population structure or relatedness. A new approach, kernel-based GWAS, appears to be emerging. For example, Maity et al.  additive effects, (b) total genetic effects, and (c) raw phenotypes. The first type is mainstream in genome-enabled selection schemes, and we have given an overview of kernel methods in which their predominant aim is to capture total genetic effects. Although currently receiving less attention, predicting raw phenotypes is especially vital for assessing health or medical outcomes. Arguably, many non-genetic factors affect raw phenotypes, and these cannot be well predicted without environmental information, so imperfect information on environmental variables hinders achieving greater predictive performance (Heslot et al., Gota Morota and Daniel Gianola assembled relevant literature and wrote the manuscript.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.