Populus trichocarpa. GWASs are often limited by the abilities of researchers to collect precise phenotype data on a high-throughput scale; to help overcome this limitation, we developed a computer vision system to measure an array of traits related to adventitious root development in poplar, including temporal measures of lateral and basal root length and area. GWAS was performed using multiple methods and significance thresholds to handle non-normal phenotype statistics and to gain statistical power. These analyses yielded a total of 277 unique associations, suggesting that genes that control rooting include regulators of hormone signaling, cell division and structure, reactive oxygen species signaling, and other processes with known roles in root development. Numerous genes with uncharacterized functions and/or cryptic roles were also identified. These candidates provide targets for functional analysis, including physiological and epistatic analyses, to better characterize the complex polygenic regulation of AR.Adventitious rooting (AR) is critical to the propagation, breeding, and genetic engineering of trees. The capacity for plants to undergo this process is highly heritable and of a polygenic nature; however, the basis of its genetic variation is largely uncharacterized. To identify genetic regulators of AR, we performed a genome-wide association study (GWAS) using 1148 genotypes of  Populus (poplar) are among the most rapidly growing trees of the northern hemisphere and have keystone roles in many natural ecosystems. They are also of major economic importance for agroforestry and as sources of wood, fiber, and biofuel [Species within the genus  biofuel . The gro biofuel . A deepeAR in poplar is a highly complex trait that is regulated by many factors, including plant age, genotype, and physiology; the many forms of plant stress; and environmental cues such as temperature, photoperiod, and nutrients. The effects of these variables are mediated via phytohormone signaling cascades that lead to differentiation and development of root tissue. Overexpression or RNAi-mediated suppression of over 20 genes, spanning phytohormone pathways, including auxin, cytokinin, gibberellin, and jasmonate (JA) signaling, has been reported to lead to increased or decreased AR and/or root growth in poplar (recently reviewed ). BioinfMeasurement of root traits to be studied by association mapping approaches such as QTL mapping or genome-wide association studies (GWASs) can be highly complex, as roots can be phenotyped with respect to not only spatial features such as their lengths and area but also their branching patterns and/or origins. These measurements can prove laborious and time-consuming if done manually. A wide array of computer vision tools have been developed to collect measurements from root images, with varying degrees of human intervention or automation across a diverse range of plants   were summarized using principal component analysis (PCA). For each of the three groups of traits used for PCA, the top two principal components (PCs) appear to explain significant portions of variance as indicated by scree plots. PCA was performed over three groups as described . The first two groups, for root area or root length traits, each across basal and lateral roots and time points, both produced first and second PCs (PC1 and PC2) representing contrasts between basal and lateral root development. The combined variance explained by PC1 and PC2 is ~92% for the group with root area traits and ~ 90% for the group with root length traits –S3. For 2hSNP > 0.10 [as computed by Genome-wide Efficient Mixed Model Association (GEMMA), P value for the associated trait, thus appearing as the peak position of a group of signals on a Manhattan plot. Only Multi-Threaded Monte Carlo SNP-set (Sequence) Kernel Association Test (MTMC-SKAT) yielded QTL peaks passing the conservative Bonferroni threshold under the assumption that each test (SKAT SNP window) is independent. We tallied the numbers of QTL peaks passing multiple-testing correction thresholds for each method, including the numbers of these QTL peaks within or outside 5 kb of the nearest gene. MTMC-SKAT using 3 kb SNP windows yielded a total of 31 unique Bonferroni-significant associations across all traits. Among these 31 associations, 22 had a window center within 5 kb of an annotated gene. With the stringency of multiple testing relaxed to a threshold for false discovery rate , a total of 164 unique associations were found to pass this threshold and/or the conservative Bonferroni threshold, and 112 of these had window centers within 5 kb of a gene. The apparent power of SKAT is in contrast to GEMMA and Generalized Mixed Model Association Test (GMMAT); GEMMA yielded only two associations passing the FDR (α = 0.10) threshold and none passing the conservative Bonferroni threshold, while GMMAT yielded none passing either. Statistical power for GEMMA and GMMAT was greatly increased by the use of Augmented Rank Truncation (ART) to combine signals across windows of SNPs, which enabled detection of over 100 associations for GEMMA and 8 for GMMAT  from our GWAS workflow. The use of MTMC-SKAT and ART proved highly valuable in improving statistical power, as 275 of our 277 associations were identified using these methods. We did not find significantly overrepresented GO terms, likely due to the heterogeneity of our candidate genes, which span diverse hormonal pathways and physiological processes.in vitro tissue cultures or at in planta wound sites  suggested that AR development depends on calcium and pH, and that hard callus may inhibit root emergence [Pinus [. These contrast with ARs growing from the sides of cuttings , which appeared to grow directly from the stem without an intermediate callus stage . This error correction was performed on ~18.7% of images. Considering our hybrid approach utilizing both computer vision and manual correction of errant labels, our method is comparable to RootReader2D [Arabidopsis and 22 in poplar [CYCLIN D3;2 , or CYCLIN D3;3  and was previously reported to be differentially expressed across seasonal stages of growth in two Populus genotypes with contrasting rates of stem growth and biomass accumulation [Arabidopsis, concurrent loss of function of all three members of the CYCD3 clade led to retarded seed development, while overexpression of CYCD3;1 led to premature, irregular, and disorganized division of the hypophysis [CYCD3;3 is negatively regulated by WUSCHEL-ASSOCIATED HOMEOBOX 5 [CYCD3 overexpression or loss of function in mature roots, several relevant root-related phenotypes have been reported for CYCD2 and CYCD4 members. CYCLIN D4;1 loss of function was reported to reduce basal meristem pericycle cell divisions as well as the number of lateral roots, while these phenotypes were rescued by exogenous auxin, perhaps due to auxin-responsiveness of D-type cyclins with overlapping functional roles [CYCD2;1 overexpression led to increased root apical meristem cell divisions [D-type cyclins regulate the G1-to-S progression of the cell cycle and comprise a family of at least 10 known proteins in n poplar . Potri.0mulation . In Arabpophysis . TranscrMEOBOX 5 , a poplaal roles . CYCD2;1ivisions  and incrivisions .NPSN11 alone yields no apparent mutant phenotype in Arabidopsis, putatively due to redundancy with a similar SNARE gene. Defects in cytokinesis and embryo development are conferred by simultaneous loss of function of NPSN11 and the functionally redundant SNAP33 [Potri.001G149200 encodes a homolog of NOVEL PLANT SNARE 11 (NPSN11), which is believed to interact with other SNARE proteins to provide energy needed for the fusion of membranes that give rise to the cell plate during cytokinesis . Loss oft SNAP33 .FASCICLIN-LIKE ARABINOGALACTAN (FLA) family, established by gene expression patterns and mutant studies to have vital roles in cell walls and root development [Populus provides support for a role of Potri.004G210600 in cell wall structure [FLA gene expression in Populus examined a subset of FLA family members and found many to be preferentially expressed in tension wood [FLA12 and FLA4, respectively [Arabidopsis FLA genes are also closely related to each of these two candidates and that specific homology is not unambiguous [Arabidopsis model supports a role for FLA4 in root development related to abscisic acid [FLA12 in the vasculature of stems [FLA4 and FLA12 side by side during root development or other contexts; further work in this area may help elucidate the extent of functional redundancy or divergence of these and other FLA genes within and across tissues.Potri.004G210600 and Potri.018G097000 encode members of the elopment . Proteomtructure . An earlion wood . Potri.0ectively , and a bmbiguous . Researcsic acid  and for of stems . We are Arabidopsis mutants of several actin genes display root-related phenotypes including altered root elongation, root morphology, and root hair formation [We report the candidate gene Potri.010G202000 encoding a putative actin-like protein. Cell expansion during root growth is influenced by bundling and rearrangements of actin filaments, regulated at least in part by auxin, cytokinin, and brassinosteroid pathways. ormation .Aux/IAA gene family believed to have at least 35 members in P. trichocarpa [Arabidopsis, Aux/IAA proteins have long been well characterized as high-level regulators of root development [Populus, at least six transgenic experiments involving the Aux/IAA-interacting TRANSPORT INHIBITOR RESPONSE 1, downstream AUXIN RESPONSE FACTOR genes, or their regulators have demonstrated important roles of Aux/IAA-dependent pathways in AR development [Populus [Potri.006G161200 encodes a member of the chocarpa . In Arabelopment . In Popuelopment , transgein planta regeneration in poplar supports a major role for JA signaling in callus and shoot regeneration [Further evidence for a role of JA signaling in rooting of poplar is indicated by an association with Potri.014G047900, encoding a homolog of DEFECTIVE IN ANTHER DEHISCHENCE 1 (DAD1), which catalyzes a key step of JA biosynthesis . Our preneration , via patS-transferase . Protein function was inferred from annotated functional domains of Arabidopsis homologs from The Arabidopsis Information Resource [Arabidopsis [Among our candidate genes, we report a notable number of genes encoding putative catalysts of PTMs, including a histidine kinase (Potri.007G056400), serine/threonine kinase (Potri.010G031800), tyrosine kinase (Potri.019G127200), dual-specificity kinases , a dual-specificity phosphatase (Potri.015G105000), and a glutathione-Resource . We are bidopsis , likely Arabidopsis mutants of a GST family gene display phenotypes including altered root growth and branching, and relative insensitivity to abiotic stress or abscisic acid treatment [Arabidopsis and is most closely related to FERRITIN 2  and FERRITIN 4  was found as an association with bud set and leaf drop [WRKY13 is likely to have roles across diverse developmental contexts including stress response [ABSCISIC ACID-INSENSITIVE 5  was similarly found as an association for bud set [ABI5 roles in flowering may depend largely on transcriptional regulation of the downstream FLOWERING LOCUS C [We cross-referenced our candidate genes with those found in four association mapping studies of rooting or other developmental traits in poplar, including QTL interval mapping of AR and shoot traits in  simonii , and GWAchocarpa . Very fechocarpa . We alsoeaf drop , as wellresponse , stem deresponse , and floresponse , and we  bud set  as well  LOCUS C , while s LOCUS C , phytoho LOCUS C , and/or  LOCUS C . Statist Potri.00G086400 w LOCUS C  are giveP. trichocarpa clone bank. To facilitate the collection of quantitative measures of adventitious root development, we employed a phenotyping system tailored for our AR assay in poplar. The hundreds of candidate genes identified include regulators of cell division and structure, hormone signaling, ROS signaling, and PTMs as well as many genes of miscellaneous or unknown function. The distinct origins of basal and lateral roots were supported both by our multivariate phenotype analysis and GWAS associations. As root development is a complex and polygenic process, future research will benefit from functional studies such as through mutagenesis, in silico statistical tests for epistasis, in vitro assays to identify and confirm possible gene–gene or gene–protein interactions, and differential rooting responses to environmental treatments.We performed GWAS to identify regulators of AR capacity in 1148 genotypes from a P. trichocarpa GWAS population that was recently expanded to include a total of 1323 genotypes [P. trichocarpa spanning regions of British Columbia, Washington, Oregon, Idaho, and Northern California. Clone banks for this GWAS population were produced in multiple locations, among which a replicate in a Corvallis, OR, field location was utilized to obtain cuttings for this study. In an effort to study as many genotypes as possible given limited time and contamination of some cuttings, we studied a total of 1148 genotypes from this population. Each genotype was studied in replicates of two, as maximizing population size was prioritized over clonal replication and can be expected to provide similar benefits in reducing statistical noise. This study was performed using materials collected as described in previous work [We used clones from a enotypes , 57. Thious work . CuttingCuttings were placed in 50 mL Falcon tubes with tap water and allowed to root. The water contained no rooting hormones or other additives. As water evaporated, the tubes were refilled with fresh tap water every 3–4 days. No fungal growth was observed in the water. Beginning 2 weeks after cuttings were first placed in water, images were collected at weekly time points for 4 weeks. Prior to taking each image, plants were removed from water and placed on top of a surface with roots arranged to separate putative lateral and basal roots. To aid their recognition by our computer vision pipeline, basal roots were laid downward atop blue felt while lateral roots were laid to the side on gray felt. Each image also included a label and ruler. Plants were imaged from above using a Canon Rebel XSi DSLR camera attached to a mount and facing downward. Due to practical limitations in the number of cuttings that could be studied at once, the study was divided into eight “batches,” each of which featured up to 400 cuttings, including 2 replicates for each of up to 200 given genotypes .We adopted the DeepLab network  with back-means clustering. Images successfully segmented as such were used to produce a training set with ~500 images used to train a deep model for segmentation. Inference was performed using this model, and correct examples were used to retrain the model with an expanded dataset, resulting in a final model trained with 2239 examples (Deep Model 1). Afterward, to segment the plant into stem, root, and leaf, we performed mean-shift segmentation and applied a location threshold to produce a training set of ~900 images. These were used to train a second deep model, and the training set was again expanded by running inference on new images and selecting correct results, resulting in a final training set with 3496 training examples (Deep Model 2).Due to variations in camera settings between “portrait” and “landscape” modes, and a desire to avoid artifacts resulting from this inconsistency, we first rotated images to a uniformly landscape orientation. Next, the background was segmented based on color thresholding. As the plant, ruler, and label varied in color and were found in approximately similar positions from image to image, we first segmented these components based on their spatial positioning and colors using mean-shift segmentation and Following training of both deep segmentation models, they were applied for inference of the full dataset. All images were standardized to the same orientation, followed by deployment of both models. The final result of segmentation was separation of background, leaf, stem, root, ruler, and label for each image. Next, the segmentation results were further analyzed to produce statistics on biological traits of interest. Since the camera height varied across images, we computed the number of pixels per ruler width for each image to enable standardization via the actual size represented by a single pixel.We proceeded to compute the lengths and areas of roots on the metric scale, as well as the diameters of stems. Root statistics were computed as follows. (1) First, we further segmented roots and attempted to label individual roots by finding distinct connections to the stem or root of origin (for branch roots) and finding the longest paths of root pixels from these origins to distinguish major roots from immature branch roots and root hairs. (2) The background was classified as top background and bottom background based on the color of the felt background underneath the plant, allowing each root to be classified as lateral or basal depending on the background. (3) For each basal and lateral root, we computed root length by computing the shortest path from root tip to origin and root area by counting root pixels. LRL and total root area were computed separately for basal and lateral roots in each image. (4) All statistics were converted from pixel scale to the metric scale using estimates of pixel size obtained from the ruler in each image. RGB and false-color images for key steps of the segmentation process are shown in Source images were compared to segmented images  to identImageJ version 1.53a software and the SmartRoot version 4.21 plugin  were useWe obtained a total of 24 traits directly from the computer vision workflow. These include four time points each for three types of root area traits  and three types of LRL traits . For each of these traits, mean values of each trait were computed across two replicates for each of 1148 genotypes and used for downstream GWAS. PCA was performed using ‘stats::prcomp’ in R  to produin planta regeneration in poplar, using the same methods [Normality of traits was assessed in parallel with traits from our GWAS of  methods . In shorin planta regeneration, using methods and SNP sets detailed in this previously reported work [R2 threshold of 0.9999 (correlation between SNPs and covariates) and MAF and missing rates of 0.01 and 0.10, which were recomputed at this stage. GEMMA also provided a measure of the percentage of trait variance explained by additive genetic effects (represented by the kinship matrix) under the null model, which we consulted as a type of narrow-sense SNP heritability (h2SNP). MTMC-SKAT was run with covariates of six principal components derived from common SNPs (MAF > 0.05) to correct for population stratification, and explanatory variables from a set of ~34.0 million SNPs with missing rate below 15% and including rare SNPs (low MAF). Up to 10 million permutations were performed to compute a given empirical P value, which could thus be resolved to values as low as 1e−7. MTMC-SKAT was deployed on the high-performance cluster COMET, made available through NSF XSEDE [Association mapping of all traits was performed in parallel with traits in our published GWAS of ted work . To summted work ; (2) GMMted work ; and (3)ted work  with theted work  and prioSF XSEDE . Stem diSF XSEDE  was usedα = 0.05, N tests equal to N SNPs for GEMMA/GMMAT or N SNP windows for SKAT) and Benjamini–Hochberg FDR (α = 0.10) via ‘stats::p.adjust’ in R [P values from GEMMA or GMMAT were grouped into 1 kb windows, followed by computation of a test statistic and P value for their combined effect. A Bonferroni threshold for ART results was computed , henceforth “ART–Bonferroni” and is to be distinguished from the previously described Bonferroni threshold , henceforth “conservative Bonferroni.” To identify genes near to or encompassing QTLs, we used R scripts with InterMineR [P. trichocarpa genome [To identify QTLs from results that are statistically significant, we computed multiple testing correction thresholds using the Bonferroni method .We consulted the notation  for rela PANTHER , 67. ThiWe thank the National Science Foundation Plant Genome Research Program for support , and members of GREAT TREES Research Cooperative at OSU for its support of the Strauss laboratory. Support for the Poplar GWAS dataset is provided by the U.S. Department of Energy, Office of Science Biological and Environmental Research via the Center for Bioenergy Innovation (CBI) under Contract No. DE-PS02-06ER64304. The Poplar GWAS Project used resources of the Oak Ridge Leadership Computing Facility and the Compute and Data Environment for Science at Oak Ridge National Laboratory, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC05-00OR22725. We would like to thank the efforts of the personnel from the CBI in establishing the GWAS resource used for this study. This work used the COMET high-performance cluster at the San Diego Supercomputing Center  made available through the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562.Strauss, Fuxin, Jiang, and Muchero designed and directed the overall study and obtained funding for its execution; Ma and Peremyslova designed and/or executed the phenotypic analyses; Nagle, Yuan, and Kaur created, adapted, and executed the computer vision, computation, and data analysis pipelines; Zahl and Peremyslova performed manual inspection and correction of phenotype data; Nagle investigated the candidate genes; Niño de Rivera assisted with inspecting the results in IGV; Nagle wrote the manuscript with editing from Strauss; and all others contributed further edits and revisions.https://github.com/naglemi/mtmcskat) and the computer vision workflow .Raw data and the code used for this project are available upon request to the authors. GitHub repositories are online for MTMC-SKAT (None declared.Horticulture Research online.Web_Material_uhad125Click here for additional data file.


Soil salinity can impose substantial stress on plant growth and cause significant yield losses. Crop varieties tolerant to salinity stress are needed to sustain yields in saline soils. This requires effective genotyping and phenotyping of germplasm pools to identify novel genes and QTL conferring salt tolerance that can be utilised in crop breeding schemes. We investigated a globally diverse collection of 580 wheat accessions for their growth response to salinity using automated digital phenotyping performed under controlled environmental conditions. The results show that digitally collected plant traits, including digital shoot growth rate and digital senescence rate, can be used as proxy traits for selecting salinity-tolerant accessions. A haplotype-based genome-wide association study was conducted using 58,502 linkage disequilibrium-based haplotype blocks derived from 883,300 genome-wide SNPs and identified 95 QTL for salinity tolerance component traits, of which 54 were novel and 41 overlapped with previously reported QTL. Gene ontology analysis identified a suite of candidate genes for salinity tolerance, some of which are already known to play a role in stress tolerance in other plant species. This study identified wheat accessions that utilise different tolerance mechanisms and which can be used in future studies to investigate the genetic and genic basis of salinity tolerance. Our results suggest salinity tolerance has not arisen from or been bred into accessions from specific regions or groups. Rather, they suggest salinity tolerance is widespread, with small-effect genetic variants contributing to different levels of tolerance in diverse, locally adapted germplasm. Salinity stress threatens global food security, causing annual yield losses of up to 50% in salt-affected agriculture regions . Salt-af+ to the shoot. This research identified several large-effect transporter genes, which are now utilised in wheat breeding [Nax1 and Nax2, inherited from Triticum monoccocum and mapped to the long arms of chromosomes 2A and 5A, respectively, impart salinity tolerance in wheat through the exclusion of Na+ from the xylem in roots [Kna1 gene identified in wheat on the distal arm of chromosome 4D causes Na+ exclusion from leaves [+ sequestration and balancing of K+ uptake and retention in young shoots, while cytosolic exclusion of Na+ involves its transfer from young to older tissues or sequestration to vacuoles employing Na+/H+ antiporters or efficient intracellular compartmentalization [+ accumulation have also been proposed to help combat soil salinity in various crops [Salinity tolerance in wheat is a complex phenomenon involving various physiological, biochemical, and metabolic processes ,11,12,13breeding ,17. The in roots ,19. The m leaves . Tissue lization ,22. Otheus crops ,24. In cus crops .+ concentration, Na+ compartmentalization in older leaves, shoot K+/Na+ balance, and yield impacts measured under field or controlled environment conditions [Salinity tolerance can be assessed by measuring a range of salt response traits, including plant vigour, shoot biomass, shoot Nanditions ,25. Fielnditions ,27. Consnditions ,29.+ and K+ accumulation, digital phenotyping technologies can enable a thorough investigation of the prevalence and importance of different salinity tolerance mechanisms [Digital phenotyping makes it possible to readily obtain high-accuracy, repeated, and non-destructive measures for various plant salinity response traits across large numbers of samples and time points. Trait examples include the rate of biomass accumulation, senescence rate, and various plant stress-related indices ,31. Whenchanisms ,28.Genetic analyses based on bi- or multi-parental mapping populations and genome-wide association studies (GWAS) are widely used to investigate the genetic and genic basis for trait expression ,33,34. GWheat improvement against salinity requires approaches that include different ion exclusion, sequestration, and osmotic tolerance mechanisms to efficiently surmount the limitations imposed by soil salinity. Apart from the complexity of the trait, the large genome size of bread wheat (~16 Gb)  imposes To investigate the genetic basis for salinity tolerance, we applied a high-throughput phenotyping assay, previously demonstrated to be discriminatory of salt tolerance and correlated to field responses ,28, to sDry biomass was measured by a destructive manual method at the end of experiments performed in the PPV-H to establish the relationship between dry biomass (DB) and digital shoot area extracted from images. Dry biomass estimates were highly correlated (r > 0.9) for digital shoot area in both control and salt-treated plants . The dign = 372) into three broad salinity tolerance groups: tolerant in both locations (n = 44), susceptible in both locations (n = 33), and the rest of the accessions  in the younger leaf (second leaf) at both locations. Interestingly, four accessions had Na+ concentrations of up to 350 µM Na+ g−1 DW and dSGR_SSI values close to one. For Na+ accumulation in the oldest leaf (fourth leaf), the most tolerant accessions had values < 300 µM Na+ g−1 DW. Most susceptible accessions had high Na+ accumulation in both leaves, although a few had low Na+ accumulation  and most strongly negatively correlated with Na+ accumulation in the oldest leaf , followed by 7B (10), and 7A (9). No QTL were detected on chromosomes 1D and 6D (n = 95) that did not cross the FDR threshold corresponded to 48 hapblocks and 11 QTL. Here, we reported all hapallele trait associations that were significant in more than 50 cross-validation runs at p-values < 0.00001.A five-fold cross-validation and ronments . HapalleD and 6D . Among tn = 81), followed by leaf ions (n = 64) and growth-related traits (n = 19). Seven QTL  and a moderate correlation for dSGR_SSI between both locations (rg2 = 0.25) indicated the underlying genetic architecture for this trait was similar at both locations. High rg2 values were not observed for other traits between locations due to GxE variation for most of the salinity tolerance traits. Response to salinity is growth stage-specific, and low correlations have been reported for salinity tolerance traits measured in two locations for the same accessions [Genetic correlations between the traits are reported in cessions . Previoucessions . In this+ or K+ transport, and 182 were involved in transporter activity (n = 1492) were excluded in the analysis. This included most of the genes with transporter functions. GWAS summary statistics for the genes and associated 2 KB upstream and downstream regions were used for FastBAT analysis to identify 107 genes with significant associations (p < 0.001) for different salinity tolerance traits  underlying the 95 QTL resulted in 2497 GO terms, of which 1764, 485, and 287 were related to biological processes, molecular functions, and cellular component functions. Dominant GO terms  corresponding to biological processes (n = 263), cellular components (n = 15), and molecular function (n = 110) are reported in n = 107) revealed 435 GO terms corresponding to cellular components (n = 54), molecular function (n = 55), and biological processes (n = 326), of which 39 GO terms were significantly enriched  in this gene set (Gene and gene ontology (GO) term enrichment analysis using the full list of genes . In this study, a high correlation was observed between digital shoot area and directly measured shoot biomass under both control and stress conditions (r = 0.95), giving us confidence to use these surrogate measures. The dSGR under salt stress varied significantly from control conditions, with tolerant lines maintaining higher dSGR than susceptible lines under saline conditions. The salt susceptibility index calculated with dSGR in control and salt conditions was used as the determinant trait for salt tolerance since it provides a direct measure of salt response per se and can therefore be considered a proxy trait for growth stage salinity response.+ concentration in old leaves (r = −0.52), and positively correlated with the K+/Na+ ratio in young leaves (r = 0.43). Significant but low correlations were observed for Na+ concentration in young leaves with dSGR under salt. Previous studies have reported both similar and contrasting relationships. For example, no significant relationship was found between Na+ and K+ concentrations in shoots and overall salinity tolerance [+ concentration and salinity tolerance in wheat [Our results show that under salinity, dSGR was negatively correlated to the rate of senescence (r = −0.61), followed by Naolerance ,54, and in wheat ,29 and oin wheat . Variabi+ in young leaves and low-to-moderate Na+ concentrations in old leaves compared to the other accession groups. Tolerant lines that had low Na+ concentrations in both old and new leaves could have multiple tolerance mechanisms that prevented both Na+ uptake and movement to young shoots. A few tolerant lines showed high-to-moderate Na+ concentrations in old leaves and low Na+ concentrations in young leaves  were detected on chromosomes 1A, 1B, 2B, 3A, 3B, 3D, 5A, 5B, 5D, 6B, 7A, and 7B. Most were also associated with leaf ion concentrations and respective salt tolerance indices . Among tder salt ,64, and der salt ,65. Salider salt . Similarn = 3134; Enrichment analysis of the full gene list . Enrichment analysis revealed GO terms related to senescence, ageing, organelle repair, detoxification, metabolite synthesis, heat acclimatisation, and metal ion transport functions  was associated with senescence and Naanalysis . Of thesnescence . Galactonescence ,79, and Our findings suggest salinity tolerance does not arise from or has been bred into accessions from specific regions or groups. Rather, they suggest salinity tolerance is widespread, with genetic variants contributing to different levels of tolerance in diverse, locally adapted germplasm. This observation was further supported by investigating enrichment for salinity tolerance between the tolerant and susceptible grouped accessions using the 28 hapalleles associated with dSGR_SSI and its correlated traits, which found no significant enrichment of alleles in the resistant group. This finding indicates multiple small effect QTL in germplasm contribute to tolerance in different backgrounds. Consistent with previous studies, our results demonstrate the genetic architecture underlying salinity tolerance in wheat is complex and controlled by multiple small effect QTL ,64. IncrThe association mapping panel (AMP) used to screen for salinity tolerance consisted of 580 genetically and geographically diverse wheat landrace accessions and cultivars  selectedThe AMP was screened at two locations in controlled-environment glasshouses, known as the Plant Phenomics Victoria, Bundoora (PPV-B) and Plant Phenomics Victoria, Horsham (PPV-H) facilities in Victoria, Australia. Both facilities are equipped with the Scanalyzer 3D plant-to-sensor high-throughput phenotyping system . A detailed description of the facility is provided in our previous study . ExperimEach wheat accession was sown in a white plastic pot  filled with standard potting mix  and fertiliser to ensure optimal growth. Three seeds were sown per pot and thinned to one plant at the three-leaf stage, keeping the plant of consistent size across genotypes. The pots were kept on saucers throughout the experiment to prevent water/saline solution loss. The glasshouse was maintained at 24 °C during the day and at 15 °C during the night. At the four-leaf stage . The 33 traits observed, derived measures collected, and ratios and indices calculated are described in The plants were imaged every second or third day on the Scanalyzer 3D system using high-resolution digital top and side-mounted RGB cameras  having 28.8 megapixels, 14-bit resolution, a frame rate of 4 fps, and a cell size of 5.5 µm. The camera mounted directly above the plants was used to acquire the top-view image. Two side-view images were captured after consecutive rotations of the plant at 0° and 90°. Images were processed using LemnaGrid software  to extract total plant pixel area and relative non-green pixel area  at each time point. The full-colour images were processed by separating the region of interest  from the background. Colour classes defined as green, chlorotic, and necrotic were used to classify green and non-green plant pixels. Details of the image analysis pipeline are described in . Fresh aData analysis for all phenotypic measures was performed using R software , unless s.AverageA is the average pixel area from both side angle views and t.averageA is the average pixel area from the top view.Digital shoot area Equation (1) from the images was calculated using the formula from :(1)DigitThe digital shoot growth rate (dSGR) was calculated from the digital shoot area across time points as per Equation (2) and implemented in R package lme4 :Y = β0 +tT is the trait value under treatment (125 mM NaCl), cT is the trait value under control (0 mM NaCl), tµ is the whole population mean of the trait under treatment, and cµ is the whole population mean of the trait under control conditions.Salt susceptibility index (SSI) Equation (3) for each of the traits was calculated as an index between the control and treatment plants based on the modified formula from : (3)SSI=The digital senescence rate (dSR) was calculated with Equation (2) using yellow and necrotic pixel areas for each plant across all timepoints.All the correlation values mentioned as ‘r’ in the manuscript were analysed as Pearson’s correlation coefficient values.+ and K+ analysis, leaves were digested in 1% (v/v) nitric acid at 100 °C for 4 h in a water bath. The Na+ and K+ concentrations of the digested leaf material were determined using a flame photometer , following the protocol described in [For Naribed in .Exome SNP genotype calls reported in He et al.  for the 2 > 0.2 were used to perform PCA and kinship matrix calculation . HGWAS nk V1.9; . Traits nk V1.9;  with the10 p-value > 5.00  in more than 50 of the 100 cross-validation runs were considered to be significantly associated with the trait. The modified Bonferroni correction method proposed for GWAS studies [e), was used to correct for multiple testing. Using this method, the corrected significance p-value threshold was determined as α divided by the number of LD-based haplotype blocks (0.05/58502), resulting in a p-value significance threshold of 8.54 × 10−7.A K-fold cross-validation approach was used to confirm the significance of hapallele trait associations. The AMP accessions were randomly divided into five equal-sized groups, and HGWAS analysis was performed using four of the groups (80% of samples), with the fifth group left out each time. For each trait, five-fold cross-validation was run 20 times, amounting to 100 runs per trait. Hapallele trait associations with a log studies ,93, whicp-value for the input list of genes by considering all the SNP p-values within 2 kb of the gene and the LD r2 values between the SNPs from individual genotype data. A meta-analysis was performed for each trait separately, and genes with a trait association p-value < 0.001 were considered significant.Gene-based meta-analysis was performed using FastBAT  implemenp-value of 0.05 were reported and discussed (p-values are reported in Gene ontology enrichment analysis was performed using PANTHER  and agriiscussed . Full ge


Principal Component Analysis (PCA) and the Linear Mixed-effects Model (LMM), sometimes in combination, are the most common genetic association models. Previous PCA-LMM comparisons give mixed results, unclear guidance, and have several limitations, including not varying the number of principal components (PCs), simulating simple population structures, and inconsistent use of real data and power evaluations. We evaluate PCA and LMM both varying number of PCs in realistic genotype and complex trait simulations including admixed families, subpopulation trees, and real multiethnic human datasets with simulated traits. We find that LMM without PCs usually performs best, with the largest effects in family simulations and real human datasets and traits without environment effects. Poor PCA performance on human datasets is driven by large numbers of distant relatives more than the smaller number of closer relatives. While PCA was known to fail on family data, we report strong effects of family relatedness in genetically diverse human datasets, not avoided by pruning close relatives. Environment effects driven by geography and ethnicity are better modeled with LMM including those labels instead of PCs. This work better characterizes the severe limitations of PCA compared to LMM in modeling the complex relatedness structures of multiethnic human data for association studies. The goal of a genetic association study is to identify loci whose genotype variation is significantly correlated to given trait. Naive association tests assume that genotypes are drawn independently from a common allele frequency. This assumption does not hold for structured populations, which includes multiethnic cohorts and admixed individuals (ancient relatedness), and for family data  as fixed effects . Modern An LMM variant that incorporates PCs as fixed covariates is tested thoroughly in our work. Since PCs are the top eigenvectors of the same kinship matrix estimate used in modern LMMs , then poLMM and PCA are closely related models , so simi cohorts , we varyIn this work, we evaluate the PCA and LMM association models under various numbers of PCs, which are included in LMMs too. We use genotype simulations  and three real datasets: the 1000 Genomes Project , the HumWe use three real genotype datasets and simulated genotypes from six population structure scenarios to cover various features of interest . We intrfixed effect sizes (FES) and random coefficients (RC), which differ in how causal coefficients are constructed. The FES model captures the rough inverse relationship between coefficient and minor allele frequency that arises under strong negative and balancing selection and has been observed in numerous diseases and other traits  g λ>1.06  and thus . However, we find that both PCA and LMM (even without PCs) increase their luations . p-Valuee enough . These se enough . Additioe enough , the onls or PCA , Table 6Our evaluations conclusively determined that LMM without PCs performs better than PCA (for any number of PCs) across all scenarios without environment effects, including all real and simulated genotypes and two trait simulation models. Although the addition of a few PCs to LMM does not greatly hurt its performance , they generally did not improve it either , Table 5The presence of environment effects that are correlated to relatedness presents the only scenario where occasionally PCA and LMM with PCs outperform LMM without PCs . It is cwith PCs , Table 6with PCs , Table 6Previous studies found that PCA was better calibrated than LMM for unusually differentiated markers , which aRelative to LMM, the behavior of PCA fell between two extremes. When PCA performed well, there was a small number of PCs with both calibrated p-values and PCA assumes that genetic relatedness is restricted to a low-dimensional subspace, whereas LMM can handle high-dimensional relatedness. Thus, PCA performs well in the admixture simulation, which is explicitly low-dimensional (see Genotype simulation from the admixture model), and our subpopulation tree simulations, which are likely well approximated by a few dimensions despite the large number of subpopulations because there are few long branches. Conversely, PCA performs poorly under family structure because its kinship matrix is high-dimensional . HoweverPCA is at best underpowered relative to LMMs, and at worst miscalibrated regardless of the numbers of PCs included, in real human genotype tests. Among our simulations, such poor performance occurred only in the admixed family. Local kinship estimates reveal considerable family relatedness in the real datasets absent in the corresponding subpopulation tree simulations. Admixture is also absent in our tree simulations, but our simulations and theory show that admixture is well handled by PCA. Hundreds of close relative pairs have been identified in 1000 Genomes , but theOur tests also found PCA robust to large numbers of PCs, far beyond the optimal choice, agreeing with previous anecdotal observations , in contThere are several variants of the PCA and LMM analyses, most designed for better modeling linkage disequilibrium (LD), that we did not evaluate directly, in which PCs are no longer exactly the top eigenvectors of the kinship matrix (if estimated with different approaches), although this is not a crucial aspect of our arguments. We do not consider the case where samples are projected onto PCs estimated from an external sample , which iOne of the limitations of this work include relatively small sample sizes compared to modern association studies. However, our conclusions are not expected to change with larger sample sizes, as cryptic family relatedness will continue to be abundant in such data, if not increase in abundance, and thus give LMMs an advantage over PCA . One reaAnother limitation of this work is ignoring rare variants, a necessity given our smaller sample sizes, where rare variant association is miscalibrated and underpowered. Using simulations mimicking the UK Biobank, recent work has found that rare variants can have a more pronounced structure than common variants, and that modeling this rare variant structure (with either PCA and LMM) may better model environment confounding, reduce inflation in association studies, and ameliorate stratification in polygenic risk scores . Better The largest limitation of our work is that we only considered quantitative traits. Previous evaluations involving case-control traits tended to report PCA-LMM ties or mixed results, an observation potentially confounded by the use of low-dimensional simulations without family relatedness . An addiOverall, our results lead us to recommend LMM over PCA for association studies in general. Although PCA offer flexibility and speed compared to LMM, additional work is required to ensure that PCA is adequate, including removal of close relatives (lowering sample size and wasting resources) followed by simulations or other evaluations of statistics, and even then PCA may perform poorly in terms of both type I error control and power. The large numbers of distant relatives expected of any real dataset all but ensures that PCA will perform poorly compared to LMM . Our finLet where 1 is a length-The full model of where where  locus i . Assuminwhich is readily modeled by the LMM random effect which follows from the affine transformation property of multivariate normal distributions. Therefore, the PCA term inalized . PCA hasIn practice, the kinship matrix used for PCA and LMM is estimated with variations of a method-of-moments formula applied to standardized genotypes where the unknown j=1nxij . Howeverij  and traits. Below we use the notation The basic admixture model is as described previously  and is iividuals . Random where this Beta is the Balding-Nichols distribution  with meaAs a brief aside, we prove that global ancestry proportions as covariates is equivalent in expectation to using PCs under the admixture model. Note that the latent space of m 1α in , estimatnf the number of families (paired parents) in the current generation, then the number of additional children (beyond the minimum) is drawn from We simulated a pedigree with admixed founders, no close relative pairings, assortative mating based on a 1D geography (to preserve admixture structure), random family sizes, and arbitrary numbers of generations (20 here). This simulation is implemented in the R package simfam. Generations are drawn iteratively. Generation 1 has This model draws subpopulations allele frequencies from a hierarchical model parameterized by a tree, which is also implemented in bnpsd and relies on the R package ape for general tree data structures and methods . The ancIndividuals We developed new methods to fit trees to real data based on unbiased kinship estimates from popkin, implemented in bnpsd. A tree with given inbreeding coefficients edge’ δw :(6)fSwT=These where the sum includes all To estimate population-level coancestry, first kinship  model, initial causal coefficients areTraits are simulated from the quantitative trait model of variance . Under trandom coefficients (RC) model, initial causal coefficients are drawn independently from for known stimator  v^iT=p^iSimulations optionally included multiple environment group effects, similarly to previous models , as follhigh heritability used low heritability used environment used ki is the number of groups in environment We performed trait simulations with the following variance parameters : high he Genomes .The three datasets were processed as before , and anaWe obtained the full (including non-public) Human Origins by contacting the authors and agreeing to their usage restrictions. The Pacific data  was obtaSince our evaluations assume uncorrelated loci, we filtered each real dataset with plink2 using parameters “--indep-pairwise 1000kb 0.3”, which iteratively removes loci that have a greater than 0.3 squared correlation coefficient with another locus that is within 1000 kb, stopping until no such loci remain. Since all real datasets have numerous rare variants, while PCA and LMM are not able to detect associations involving rare variants, we removed all loci with  package , and kinG-robust , and MAFeapplied .All approaches are evaluated using two complementary metrics: P-values for continuous test statistics have a uniform distribution when the null hypothesis holds, a crucial assumption for type I error and FDR control . We use where The same value with a negative sign occurs for all p-values of 1.Precision and recall are standard performance measures for binary classifiers that do not require calibrated p-values . Given tPrecision and Recall trace a curve as https://github.com/OchoaLab/pca-assoc-paper  that Linear Mixed Models generally should perform better at correcting for genetic and environmental confounding in GWAS than more commonly used Principal Components methods. public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.Our editorial process produces two outputs: (i) Decision letter after peer review:eLife. Your article has been reviewed by 4 peer reviewers, including Magnus Nordborg as the Reviewing Editor and Reviewer #1, and the evaluation has been overseen by Detlef Weigel as the Senior Editor.Thank you for submitting your article "Limitations of principal components in quantitative genetic association models for human studies" for consideration by The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission. Individual reviews are also included as they are generally helpful. As you will see, there is strong support for your work, but it is clear that improvements are in order to justify your general claims.Essential revisions:1) You must at least discuss that the scenarios you test are in a sense as unrealistic as the (much more limited) simulations previously run, both in terms of sample size and population structure. Actual human GWAS are run "within populations" to minimize environmental confounding (more on this below) as much long-range LD and involve much larger sample sizes. Does this affect your conclusions? Related to this, the historical context for several methods you cite is not given.2) Practical considerations for why PCs were/are used are not discussed either. Unbalanced designs, meta-studies, sample sizes.3) You are not discussing environmental confounding, which is not only likely more of a problem in human GWAS than any of the other things discussed here, but could potentially justify a combination of LMM and PCs. Selection could have a similar effect, as discussed in the original LMM paper by Yu et al. (2006). Ideally, this should be simulated, but it certainly must be discussed, and the conclusions must take this into account.4) You are similarly not addressing the issue of rare alleles and heterogeneity, which is a major preoccupation in human genetics. How does genetic architecture affect your conclusions? If you could address this, it would be a very nice (and timely) contribution (whereas the manuscript as currently written feels like it's fighting a 10-year old battle).5) Finally, the description of the theory is often hard to follow, relies heavily on a particular model , and appears to be wrong in a few specific cases (see comments below). The results table is likewise inscrutable, and it is difficult to relate your simulations to real-life scenarios.Reviewer #1 (Recommendations for the authors):At the very least you need to comment on the practicability and relevance issues noted above.I would also recommend adjusting the writing to make your (otherwise lucid) review of the theory more accessible to a broader audience. Not only do you assume considerable knowledge of mathematical statistics, but you are also deeply rooted in a classical quantitative genetics framework. Terms like "inbreeding edges" are likely to befuddle all but a tiny fraction of humanity!Reviewer #2 (Recommendations for the authors):1. Environmental effects are known to correlate with population structure. A silly example of this is chop-stick skills, which obviously correlate with broad levels of East-Asian ancestry, e.g. in a global population sample it would likely correlate with the first two top PCs, and additional PCs would not explain more. Hence, for such a trait the generative model could really just be the top PCs, in which case including top PCs as covariates would be optimal. An LMM would in this example probably try to account for a much more complicated structure in addition to the top PCs. Hence, in summary, I would appreciate some simulations where you have environmental contributions that are perhaps strongly correlated with individual PCs, as I believe such scenarios may exist in real data analyses.2. I really appreciate the detailed simulations in this study, but perhaps it would make sense to simulate traits given UKB genotypes? Also, and perhaps more importantly, how about also evaluating the two approaches on UKB data, to see if you really find more hits using LMMs? 3. You note that PCs are the top eigenvectors of the kinship matrix. This is usually true, but not always as when deriving PCs one should ideally apply some LD adjustment to avoid PCs capturing long-range LD regions, which can otherwise reduce power to detect variants in long-range regions. See e.g. Patterson et al. (PLoS Genet 2006) or Privé et al. (Bioinformatics 2020).4. Following up on the previous comment, I wonder if the conclusions in this paper change if PCs are derived the way they often are, i.e. excluding long-range LD regions and/or using some LD adjustments.5. Some of the LMMs cited are not classical LMMs, in the sense that they do not assume an infinitesimal genetic architecture. E.g. BOLT-LMM  assumes a mixture of two Gaussians as a prior for the effects, which they show can improve power further. Mbatchou et al. (Nat Genet 2021) also use blockwise ridge regression , which is effectively a more flexible prior. Using these, the relationship between the PCs and the model becomes yet more complicated. Similarly, the GCTA mixed model GWAS uses a LOCO approach to improve power, which I believe means that a slightly different kinship is used for each chromosome.6. Your derivation assumes that there are some ancestral allele frequencies underlying the true model, but that seems perhaps unnecessary to me because these allele frequencies only represent the "correct" weights for the variants, and we know that they are probably wrong anyway. Indeed alternative and likely better weightings can be used, see Speed and Balding . If you instead assume the sample frequencies are the right frequencies, then all the math becomes simple by standardizing the variants.7. Another publication that also provides some similar theory on the relationship between PCA and mixed models is Janss et al., (Genetics 2012). 8. The authors mention that LMMs are not well suited for unbalanced samples, e.g. where the case-control ratio in the sample is <5%, and cite Zhou et al., Nat Genet 2018 as a solution. However, before Zhou et al. there were no computationally efficient generalized mixed models capable of being able to be applied to UKB sample sizes, which could help explain why LMMs haven't been adopted as quickly as one might have expected.9. I believe there is some concern regarding meta-analyzing LMM GWAS summary statistics, e.g. whether one should use the actual effective sample sizes or the effective sample sizes (which I believe is the best approach). I believe this is probably the main reason why LMMs are not as widely used as we would like. I would appreciate some discussion or reflection on this.Reviewer #3 (Recommendations for the authors):The presentation of the results in Table 3 is nearly incomprehensible.The authors are incorrect about no other troublesome cases on line 48. The differential variance between populations can induce bias (see papers from Xihong Lin).The author's discussion of the benefits of including sample PCs in LMM (eg 468-472 on p28) is misinformed. In particular, it is a critical step for methods like BOLT.The authors say "low-dimensional" when they mean "low-rank".The Sul and Eskin paper does not result in a tie. It states that creating a second kinship matrix from the PCs can address the issue presented in Price et al. entirely with LMMs.It would also be interesting to know to what extent the singular value distribution matters versus the rank? The authors have some discussion of this in 5b but it is not well developed.Reviewer #4 (Recommendations for the authors):I have the following comments for consideration:Historical context: The overall reasoning for controlling population stratification and relative kinship for different types of populations was given in [26]. It would be helpful to revise the introduction part to focus more on the original research papers and their reasonings: [5] (PCA), [9] (Q), [26] , and [16] . The later studies with LMM from the groups behind GCTA should be mentioned as the (emerging/emerged) trend, in addition to those comparison studies.A couple of relevant research may be examined and discussed, in terms of different types of populations, determining the number of PCs to be included in the LMM, model comparison, and modeling fitting (redundancy). https://doi.org/10.1534/genetics.108.098863 https://doi.org/10.1038/hdy.2010.11One question is the justification for examining a single heritability of 0.8. This level is generally regarded very high. It would be more convincing to see the results when h2 = 0.3-0.6, particularly if this is set for a large population with different subgroups.The choice of m1=n/10 is not adequately justified. Even though we typically assume that there are many loci and the detection power increases with larger sample sizes, the actual detected numbers of loci in empirical studies are lower than that. With the current set m1 (Table 2), what were the power values, and are they close to the empirical studies?"The largest limitation of our work is that we only considered quantitative traits". This may be expanded to include that with the overall simulation scheme, you assume the causal variants are functioning across different groups of a large population (2.2.5 Trait Simulation)? Real data with measured traits may be different. This can also be different for different types of complex diseases. Some clear context information should be given at the beginning too.Among these simulated population and real data, have you examined whether a small number of PCs are indeed significant to explain some trait variation? I think this was the original thinking of applying PCs to correct the population stratification.L46 and L487. Assume the low-dimension part of the relatedness matters in terms of trait differences among groups, which need to be adjusted.L53-57. This is not an accurate summary of the relevant papers. Earlier papers set up the general LMM framework with different components that may be included or included and individual components that can have varied forms or reported the first actual association study with the LMM framework. These two are earlier papers that (re-)introduce LMM to the association studies, and markers were used for kinship, structure, and PCA.L64-70. Redundancy is not an issue since the objective is to control false positives using two types of covariates.L166. Large "and"(?) Family.L282. Theoretical and empirical evidence of these two needs to be provided.Table 3. Not clear about the asterisk. Essential revisions:1) You must at least discuss that the scenarios you test are in a sense as unrealistic as the (much more limited) simulations previously run, both in terms of sample size and population structure. Actual human GWAS are run "within populations" to minimize environmental confounding (more on this below) as much long-range LD and involve much larger sample sizes. Does this affect your conclusions? Related to this, the historical context for several methods you cite is not given.We added to our discussion comments on sample size, which is smaller than many of the largest modern studies. However, we do not expect our key conclusion to change for larger sample sizes, since cryptic family relatedness is not only present there but it is expected to be even more abundant compared to smaller sample sizes.Many modern GWAS are multiethnic or include admixed individuals, so this is not an artificial setting that has no bearing on real analyses. We now cite 21 recent papers on this topic to back up this assertion. We focused on these more extreme cases as this is where challenges to PCA and LMM performance are most expected. We agree that environmental confounding is an important issue for multiethnic studies, and have now included numerous simulations with considerable environment effects.Our simulated genotypes do not have LD, and our real data was pruned to exclude short range LD in order to simplify evaluations. The reviewers point out several variants to PC and kinship estimation that can improve handling of LD, and we now incorporate those in our discussion. We agree LD should be handled with appropriate estimators when present, but overall we do not think the presence of LD changes our key conclusion, namely that LMM outperforms PCA and that this is mostly due to the presence of cryptic family relatedness.For clarity, we opted to focus on describing the most common modern versions of PCA and LMM which are tested, as opposed to numerous older variants we did not test here, which modeled population and family structure in somewhat different ways. We provide historical context strictly as needed to avoid confusion and keep our already lengthy work as brief as possible.2) Practical considerations for why PCs were/are used are not discussed either. Unbalanced designs, meta-studies, sample sizes.We added a discussion paragraph to sample sizes and related questions. Indeed, PCA scales better than classic LMMs with sample size, although many recent LMMs are now scaling better, although some do so by changing the inference model somewhat , so those new models deserve further study. However, cryptic family relatedness is expected to become more abundant at larger sample sizes, so PCA should perform worse there.We also briefly discuss unbalanced case-control designs. However, quantitative trait studies do not typically exhibit unbalanced designs, which is the sole focus of this work, so this interesting question is out of scope.Meta-analysis does not give either PCA or LMM an advantage, as it can be applied to summary statistics from either or even a combination of models.3) You are not discussing environmental confounding, which is not only likely more of a problem in human GWAS than any of the other things discussed here, but could potentially justify a combination of LMM and PCs. Selection could have a similar effect, as discussed in the original LMM paper by Yu et al. (2006). Ideally, this should be simulated, but it certainly must be discussed, and the conclusions must take this into account.We added numerous simulations with large environment effects. Although an LMM with PCs can occasionally significantly outperform an LMM without PCs, in these cases the difference in performance tended to be small, and including environment variables as fixed effects in our regression improves performance much more than using PCs. Therefore, we are hesitant to recommend always including PCs in an LMM since this tends to be a poor solution to the problem of environmental confounding, and more explicit modeling of environment effects is instead recommended.4) You are similarly not addressing the issue of rare alleles and heterogeneity, which is a major preoccupation in human genetics. How does genetic architecture affect your conclusions? If you could address this, it would be a very nice (and timely) contribution (whereas the manuscript as currently written feels like it's fighting a 10-year old battle).We now include rare variants in our discussion. In our evaluations we had to ignore rare variants because at our small sample sizes we have no power to detect association at these variants with low counts . However, we agree that modeling rare variants is important in biobank-scale analyses, and their covariance structure is different than for common variants , which raises very interesting challenges we hope to tackle in future work.By heterogeneity, one reviewer clarifies below to mean effect size heterogeneity across populations, which is the focus of much recent work. However, this questions is not relevant to our LMM vs PCA comparisons, as neither of these basic models is equipped to model effect size heterogeneity, neither is expected a priori to perform better than the other, and both models can be extended to model ancestry-specific effects in the way it is done in TRACTOR, for example.5) Finally, the description of the theory is often hard to follow, relies heavily on a particular model , and appears to be wrong in a few specific cases (see comments below). The results table is likewise inscrutable, and it is difficult to relate your simulations to real-life scenarios.We updated the descriptions of our theory as suggested in the more specific comments below, and there we also address the particulars of our model. We improved Table 3 with the main statistical evaluations.Reviewer #1 (Recommendations for the authors):At the very least you need to comment on the practicability and relevance issues noted above.I would also recommend adjusting the writing to make your (otherwise lucid) review of the theory more accessible to a broader audience. Not only do you assume considerable knowledge of mathematical statistics, but you are also deeply rooted in a classical quantitative genetics framework. Terms like "inbreeding edges" are likely to befuddle all but a tiny fraction of humanity!We clarified the concepts highlighted and additionally edited based on feedback from additional members of our lab who reviewed the revised manuscript.Reviewer #2 (Recommendations for the authors):1. Environmental effects are known to correlate with population structure. A silly example of this is chop-stick skills, which obviously correlate with broad levels of East-Asian ancestry, e.g. in a global population sample it would likely correlate with the first two top PCs, and additional PCs would not explain more. Hence, for such a trait the generative model could really just be the top PCs, in which case including top PCs as covariates would be optimal. An LMM would in this example probably try to account for a much more complicated structure in addition to the top PCs. Hence, in summary, I would appreciate some simulations where you have environmental contributions that are perhaps strongly correlated with individual PCs, as I believe such scenarios may exist in real data analyses.We added numerous simulations with random environment effects generated from the population labels given by the three real datasets . These environment effects are strongly correlated with ancestry and are also low-dimensional, which could give PCA an advantage based on those properties as you and many others expect. By necessity, the environment simulations have a lower heritability, to accommodate for large environment effects. Thus, for comparison, we also simulated numerous low heritability traits without environment, whose results are comparable to the original high heritability simulations. Surprisingly, we find that LMMs fit environment effects nearly as well as PCA, since both increase their AUCs substantially compared to the low heritability simulations . We do find that with environment, occasionally, an LMM with PCs slightly but significantly outperforms an LMM without PCs, and in other cases PCA similarly outperforms LMM. But we do want to emphasize that the effect is often not there, and that in general LMM without PCs performs nearly as well as the best competing method, and significant differences are invariably small. Lastly, we included an LMM with the true group labels as fixed covariates, and in most cases that was the best method and its AUCs were greater by considerable amounts. Overall, we do see this potential advantage to including PCs, but it is in no way a reliable solution, and it is no replacement for more direct modeling of environment effects.2. I really appreciate the detailed simulations in this study, but perhaps it would make sense to simulate traits given UKB genotypes? Also, and perhaps more importantly, how about also evaluating the two approaches on UKB data, to see if you really find more hits using LMMs? n = 1000 individuals (unpublished results), so this is not a merely hypothetical concern. A fair and complete evaluation that considers all of these variables is an entire line of work separate from the current evaluation, but it is definitely the next step.We are interested in this question of larger datasets, but it is out of scope of the current work. One immediate concern is that the LMM used in our paper (GCTA) cannot be used in these larger datasets. GCTA is in a sense a classic LMM that in our evaluations performs as well as older methods  but is faster than those, so we have no concerns that either of those would have similar power if we were able to run them on UKB (but this is not possible). In contrast, the scalable LMM approaches applicable to the UKB  do so by making additional approximations that older LMMs do not use, which worry us as they may potentially reduce power and could confound the results. BOLT-LMM in particularly has performed poorly in our hands in simulations with That said, we believe the main conclusion will not change on UKB or other large datasets. In particular, we expect cryptic family relatedness to remain a serious problem, and perhaps have a larger impact as the abundance of distant relatives only increases with sample size.3. You note that PCs are the top eigenvectors of the kinship matrix. This is usually true, but not always as when deriving PCs one should ideally apply some LD adjustment to avoid PCs capturing long-range LD regions, which can otherwise reduce power to detect variants in long-range regions. See e.g. Patterson et al. (PLoS Genet 2006) or Privé et al. (Bioinformatics 2020).4. Following up on the previous comment, I wonder if the conclusions in this paper change if PCs are derived the way they often are, i.e. excluding long-range LD regions and/or using some LD adjustments.We incorporated these variants of the PCA approach into our discussion, and in particular added the following conclusion: “These improved PCs remain inadequate models of family relatedness, so an LMM will continue to outperform them in that setting.”5. Some of the LMMs cited are not classical LMMs, in the sense that they do not assume an infinitesimal genetic architecture. E.g. BOLT-LMM  assumes a mixture of two Gaussians as a prior for the effects, which they show can improve power further. Mbatchou et al. (Nat Genet 2021) also use blockwise ridge regression , which is effectively a more flexible prior. Using these, the relationship between the PCs and the model becomes yet more complicated. Similarly, the GCTA mixed model GWAS uses a LOCO approach to improve power, which I believe means that a slightly different kinship is used for each chromosome.We incorporated the LOCO approach into our discussion, and added the following conclusion: “While LOCO kinship estimates vary for each chromosome, they continue to model family relatedness, thus maintaining their key advantage over PCA.”We also incorporated BOLT-LMM and REGENIE into a different part of our discussion, and agreed with your conclusion: “Many of these newer LMMs also no longer follow the infinitesimal model of the basic LMM , and employ novel approximations, which are features not evaluated in this work and worthy of future study.”6. Your derivation assumes that there are some ancestral allele frequencies underlying the true model, but that seems perhaps unnecessary to me because these allele frequencies only represent the "correct" weights for the variants, and we know that they are probably wrong anyway. Indeed alternative and likely better weightings can be used, see Speed and Balding . If you instead assume the sample frequencies are the right frequencies, then all the math becomes simple by standardizing the variants.pˆTi . Nevertheless, in PCA and LMM these biased estimates perform as well as unbiased ones .”Our methods contained early on the following explanation: “However, this kinship estimator has a complex bias that differs for every individual pair, which arises due to the use of this estimated To further explain, our recent work  has shown that sample allele frequencies are not the right frequencies, although that is a pervasive assumption, and in particular kinship estimates are biased when making this precise assumption. In that work we developed a new estimator, popkin, that is unbiased and overcomes the problem that the true ancestral allele frequencies are unknown, and estimates kinship without bias without having to estimate such allele frequencies at all. We use popkin to visualize population structure in Figure 1. and to calculate eigenvalues later. However, our followup work found that PCA and LMM work just fine with standard biased kinship estimates, because the intercept compensates (in a complicated way) for the bias in PCs or random effects . Nevertheless, it is very important to distinguish between true and estimated allele frequencies, as biases creep in when such distinctions are ignored, and regardless of whether this is ultimately the correct model or not .Since Speed and Balding (2012) is relevant to a new related discussion about PCA and LMM variants, all of which attempt to improve LD modeling, we added this paper to that discussion as well.7. Another publication that also provides some similar theory on the relationship between PCA and mixed models is Janss et al., (Genetics 2012). We incorporated Janss et al., (2012) in our paper, cited in introduction, methods, and discussion.8. The authors mention that LMMs are not well suited for unbalanced samples, e.g. where the case-control ratio in the sample is <5%, and cite Zhou et al., Nat Genet 2018 as a solution. However, before Zhou et al. there were no computationally efficient generalized mixed models capable of being able to be applied to UKB sample sizes, which could help explain why LMMs haven't been adopted as quickly as one might have expected.We agree, and added a discussion paragraph concerning the sample size limitations of the present work that echo these points.9. I believe there is some concern regarding meta-analyzing LMM GWAS summary statistics, e.g. whether one should use the actual effective sample sizes or the effective sample sizes (which I believe is the best approach). I believe this is probably the main reason why LMMs are not as widely used as we would like. I would appreciate some discussion or reflection on this.We agree these are important questions to be analyzed in future work, but meta-analysis is a very different kind of model that falls outside the scope of this present work.Reviewer #3 (Recommendations for the authors):The presentation of the results in Table 3 is nearly incomprehensible.We modified Table 3 considerably for greater clarity. We simplified the statistical tests to center on the hypothesis that LMM with no PCs is better than all other cases . For additional clarity we now include all p-values. Now we use a Bonferroni threshold to declare significance .The authors are incorrect about no other troublesome cases on line 48. The differential variance between populations can induce bias (see papers from Xihong Lin).We apologize, we were unable to identify the precise papers from Xihong Lin you are referring to, using your keywords of PCA and differential variance between populations. However, we stand by our comment, in that numerous papers have proposed various explanations for poor PCA performance that have not stood the test of time, and family relatedness is the only explanation that recurs and that numerous papers/authors agree upon.The author's discussion of the benefits of including sample PCs in LMM (eg 468-472 on p28) is misinformed. In particular, it is a critical step for methods like BOLT.We added the following to the discussion (appears a few paragraphs after the lines cited above): “A different benefit for including PCs were recently reported for BOLT-LMM, which does not result in greater power but rather in reduced runtime, a property that may be specific to its use of scalable algorithms such as conjugate gradient and variational Bayes .”To further clarify, the BOLT-LMM authors make no claim that there are gains in performance  due to inclusion of PCs, it is solely a runtime advantage. Further, this appears to be a property of the specific algorithm used by BOLT-LMM, and not a general property of LMMs. Thus, our original statement stands.The authors say "low-dimensional" when they mean "low-rank".Low-rank is more correct mathematically, so we included a clarification in the first mention of “low-dimensional” relatedness. We also replaced some mentions of dimension with “matrix rank” when that is more precise, and in other cases added the adjective “model dimension” when that was appropriate. We also entirely replaced the vague term “dimensionality” with more precise terms, which varied depending on context. However, we thought “low-rank” may be too technical to use exclusively, so we did not fully excise the more intuitive “low-dimensional” expression from our work.The Sul and Eskin paper does not result in a tie. It states that creating a second kinship matrix from the PCs can address the issue presented in Price et al. entirely with LMMs.You are correct. However, we sought to indicate whether a standard LMM  outperformed PCA, and in that regard it was a tie. LMMs with multiple kinship matrices are not commonly used and are not the subject of our investigation.It would also be interesting to know to what extent the singular value distribution matters versus the rank? The authors have some discussion of this in 5b but it is not well developed.We indeed consider this in what is now Figure 6 Figure supp 1, which shows the cumulative distribution of eigenvalues. However, that figure shows that none of the datasets stand out in having exceptional distributions except for the admixed family simulation. Since these distributions do not clearly separate the cases where PCA performed poorly from the rest, we decided that a brief mention of this analysis in the main text was enough to state that it was done but it was inconclusive. We think that spending additional time on this tangent is distracting from the stronger results we obtained from looking at the local kinship distributions.Reviewer #4 (Recommendations for the authors):I have the following comments for consideration:Historical context: The overall reasoning for controlling population stratification and relative kinship for different types of populations was given in [26]. It would be helpful to revise the introduction part to focus more on the original research papers and their reasonings: [5] (PCA), [9] (Q), [26] , and [16] . The later studies with LMM from the groups behind GCTA should be mentioned as the (emerging/emerged) trend, in addition to those comparison studies.For reference, these are the citations (as numbers may have changed in revision):[5]: Price et al., (2006): PCA[9]: Pritchard et al., (2000): Q[16]: Zhao et al., (2007): PC in LMM and actual association[26]: Yu et al., (2006): LMM, Q in MMEarlier draft versions of our introduction presented the historical context in more detail, similarly to what you are suggesting, but after much consideration we converged onto the current version, which we believe is appropriately more focused on the current approaches that were tested. In particular, our paper does not test the Q association model directly, and only tests one newer LMM that uses the standard kinship estimator (unlike the above early LMMs), so we thought spending more time explaining previous approaches that we are not testing can be distracting and confusing to readers. We thought the text flowed better the way it is written, not only primarily describing current approaches, but also describing PCA and LMM papers separately rather than chronologically (the timelines overlap considerably). We chose to focus on explaining the methods and their properties clearly instead. We do cover the conclusions of those early papers and highlight the differences between those and current approaches, but we do so strictly as needed  and as concisely as possible for clarity.A couple of relevant research may be examined and discussed, in terms of different types of populations, determining the number of PCs to be included in the LMM, model comparison, and modeling fitting (redundancy). https://doi.org/10.1534/genetics.108.098863 https://doi.org/10.1038/hdy.2010.11Zhu and Yu, 2009: Thank you for making us aware of this excellent evaluation, we included it in Table 1, and cited it in numerous locations in the introduction and discussion.LRR2 in all of our simulations, for the baseline PCA and LMM models without loci, it comes at a considerable computational expense for LMMs, especially for the largest datasets, so we decided not to pursue it for this work. Furthermore, as variance components are estimated in GCTA using REML, we are also unsure if those estimates are appropriate for use in LRR2 .Sun et al., 2010: We now cite this paper in the discussion: “Additionally, PCA and LMM goodness of fit could be compared using the coefficient of determination generalized for LMMs .” Although we would have liked to try calculating these One question is the justification for examining a single heritability of 0.8. This level is generally regarded very high. It would be more convincing to see the results when h2 = 0.3-0.6, particularly if this is set for a large population with different subgroups.We repeated most of our simulations with a heritability of 0.3, and all of our conclusions hold.The choice of m1=n/10 is not adequately justified. Even though we typically assume that there are many loci and the detection power increases with larger sample sizes, the actual detected numbers of loci in empirical studies are lower than that. With the current set m1 (Table 2), what were the power values, and are they close to the empirical studies?m1 = round(nh2/8), which empirically balances power well with varying n and h2.” Note that for the original heritability of h2 = 0.8 the above formula reduces to the original one. This heuristic formula aims to maintain similar power (or AUCs) across studies with different parameters, and it works well enough empirically, though it is not exact and we did observe that AUCs vary somewhat between studies. More parameters surely matter for determining AUC.Since we now consider varying heritability, we extended our heuristic formula to “We do not recover most of the causal loci we are simulating, reflected in the fact that AUCs are near 0.1 or 0.2 . We added a new figure that compares AUC to power, and find rough agreement between AUC and calibrated power calculated at an empirical type I error rate of 1e-4 . Therefore, our power at that level is roughly 0.1 to 0.2. In contrast, other empirical studies try to select parameters so that power is near 50%, although the p-value thresholds at which power is measured varies considerably in those previous studies. We think our simulation is more realistic in the sense that most causal loci are not recovered by GWAS, but their presence is important as they add to the background polygenic effect, so it makes sense to model a large number of causal loci even if we can only hope to recover a small proportion of them."The largest limitation of our work is that we only considered quantitative traits". This may be expanded to include that with the overall simulation scheme, you assume the causal variants are functioning across different groups of a large population (2.2.5 Trait Simulation)? Real data with measured traits may be different. This can also be different for different types of complex diseases. Some clear context information should be given at the beginning too.By this we assume you mean that we did not simulate ancestry-specific effect size heterogeneity . Although some work has suggested that effect sizes can vary across ancestries, other recent work shows that this is not common . However, this is not a limitation in the context of our paper, as neither the basic PCA or LMM are equipped to handle effect size heterogeneity, so adding such details to our simulation is not expected to benefit either approach. There are more recent approaches that extend that model effect size heterogeneity, such as TRACTOR, but they are not the focus of this evaluation. Additionally, both PCA and LMM can, in principle, be extended to model effect heterogeneity in exactly the same way as TRACTOR (which is a fixed effects model most like PCA).Among these simulated population and real data, have you examined whether a small number of PCs are indeed significant to explain some trait variation? I think this was the original thinking of applying PCs to correct the population stratification.We now performed these tests  and they always yielded small numbers of PCs that were significant: minimum 1 observed for the small sample size admixture simulation, most other cases had several significant PCs, maximum below 20 observed for the Admixed Family simulation and for Human Origins . Overall they were roughly consistent with the numbers of PCs needed for the PCA method to maximize its performance in each of these datasets.We introduced the following text in the section where we discuss the latent dimension of datasets and whether they could be used to predict when PCA would perform poorly (they don’t): “We also calculated the number of PCs that are significantly associated with the trait, and observed similar results, namely that while the family simulation has more significant PCs than the non-family admixture simulations, the real datasets and their tree simulated counterparts have similar numbers of significant PCs .”L46 and L487. Assume the low-dimension part of the relatedness matters in terms of trait differences among groups, which need to be adjusted.It is unnecessary to state that PCA assumes that the effect on the trait is also low-dimensional. Under non-zero heritability, high-dimensional relatedness (family structure) always affects the trait. It is impossible for the effect to be restricted to a low-dimensional component, because close relatives will share an outsize proportion of variants (whether they are common or rare in different groups) and thus have much more similar traits than random people (in the same group or otherwise).L53-57. This is not an accurate summary of the relevant papers. Earlier papers set up the general LMM framework with different components that may be included or included and individual components that can have varied forms or reported the first actual association study with the LMM framework. These two are earlier papers that (re-)introduce LMM to the association studies, and markers were used for kinship, structure, and PCA.We reworded the first sentence to: “Early LMMs used kinship matrices estimated from known pedigrees or using methods that captured recent relatedness only, and modeled population structure (ancestry) as fixed effects .” The intent of that sentence was not to describe early LMMs entirely, or as abstract frameworks, but to point out that the specific kinship estimates used then in practice were different than the ones commonly used now, though we agree the original version was misleading.L64-70. Redundancy is not an issue since the objective is to control false positives using two types of covariates.We agree, redundancy does not often cause problems in practice, but without a clear benefit it’s not worth the clear risk of loss of power (which we demonstrate for large numbers of PCs) and increased computational burden.L166. Large "and"(?) Family.We reworded to say “Both Large and Family simulations …”.L282. Theoretical and empirical evidence of these two needs to be provided.We previously provided theoretical justification for our SRMSDp and AUC metrics in Appendix B, showing that SRMSDp reflects null p-value calibration and agrees with the inflation factor and type I error rate, while AUC corresponds to calibrated power. We also provided Figure 2 Figure supp 1, which showed an excellent monotonic correspondence between SRMSDp and the inflation factor. We added two supplementary figures that provide additional empirical justification. Figure 2 Figure supp 2 shows that SRMSDp and type I error are monotonically related, and that they agree as expected for calibrated models. Figure 2 Figure supp 3 shows a positive correlation between AUC and calibrated power, which holds well separately per dataset, although the slope of this relationship varies between datasets because they have different proportions of loci where the alternative hypothesis holds.Table 3. Not clear about the asterisk.This table has been reorganized at the request of another reviewer. The asterisk denoted that model with that number of PCs was calibrated (determined by whether the mean absolute SRMSDp was below 0.01). In the new table there is instead a column that denotes if each model is calibrated as a boolean .