# Perform model fitting for all ordered pairs of markers in a genomic region of interest `scan_pvl` calculates log likelihood for d-variate phenotype model fits. Inputted parameter `start_snp` indicates where in the `probs` object to start the scan. ```r scan_pvl( probs, pheno, kinship = NULL, addcovar = NULL, start_snp = 1, n_snp, max_iter = 10000, max_prec = 1/1e+08, cores = 1 ) ``` ## Arguments - `probs`: an array of founder allele probabilities for a single chromosome - `pheno`: a matrix of phenotypes - `kinship`: a kinship matrix for one chromosome - `addcovar`: a matrix, n subjects by c additive covariates - `start_snp`: index of where to start the scan within probs - `n_snp`: the number of (consecutive) markers to include in the scan - `max_iter`: maximum number of iterations for EM algorithm - `max_prec`: stepwise precision for EM algorithm. EM stops once incremental difference in log likelihood is less than max_prec - `cores`: number of cores to use when parallelizing via parallel::mclapply. Set to 1 for no parallelization. ## Returns a tibble with d + 1 columns. First d columns indicate the genetic data (by listing the marker ids) used in the design matrix; last is log10 likelihood ## Details The function first discards individuals with one or more missing phenotypes or missing covariates. It then infers variance components, Vg and Ve. Both Vg and Ve are d by d covariance matrices. It uses an expectation maximization algorithm, as implemented in the `gemma2` R package. `gemma2` R package is an R implementation of the GEMMA algorithm for multivariate variance component estimation (Zhou & Stephens 2014 Nature methods). Note that variance components are fitted on a model that uses the d-variate phenotype but contains no genetic information. This model does, however, use the specified covariates (after dropping dependent columns in the covariates matrix). These inferred covariance matrices, $\hat{Vg}$ and $\hat{Ve}$, are then used in subsequent model fitting via generalized least squares. Generalized least squares model fitting is applied to every d-tuple of markers within the specified genomic region for `scan_pvl`. For a single d-tuple of markers, we fit the model: $$ vec(Y) = Xvec(B) + vec(G) + vec(E) $$ where $$ G \sim MN(0, K, \hat{Vg}) $$ and $$ E \sim MN(0, I, \hat{Ve}) $$ where $MN$ denotes the matrix-variate normal distribution with three parameters: mean matrix, covariance among rows, and covariance among columns. $vec$ denotes the vectorization operation, ie, stacking by columns. $K$ is a kinship matrix, typically calculated by leave-one-chromosome-out methods. $Y$ is the n by d phenotypes matrix. $X$ is a block-diagonal nd by fd matrix consisting of d blocks each of dimension n by f. Each n by f block (on the diagonal) contains a matrix of founder allele probabilities for the n subjects at a single marker. The off-diagonal blocks have only zero entries. The log-likelihood is returned for each model. The outputted object is a tibble with d + 1 columns. The first d columns specify the markers used in the corresponding model fit, while the last column specifies the log-likelihood value at that d-tuple of markers. ## Examples ```r # read data n <- 50 pheno <- matrix(rnorm(2 * n), ncol = 2) rownames(pheno) <- paste0("s", 1:n) colnames(pheno) <- paste0("tr", 1:2) probs <- array(dim = c(n, 2, 5)) probs[ , 1, ] <- rbinom(n * 5, size = 1, prob = 0.2) probs[ , 2, ] <- 1 - probs[ , 1, ] rownames(probs) <- paste0("s", 1:n) colnames(probs) <- LETTERS[1:2] dimnames(probs)[[3]] <- paste0("m", 1:5) scan_pvl(probs = probs, pheno = pheno, kinship = NULL, start_snp = 1, n_snp = 5, cores = 1) ``` ## References Knott SA, Haley CS (2000) Multitrait least squares for quantitative trait loci detection. Genetics 156: 899–911. Jiang C, Zeng ZB (1995) Multiple trait analysis of genetic mapping for quantitative trait loci. Genetics 140: 1111-1127. Zhou X, Stephens M (2014) Efficient multivariate linear mixed model algorithms for genome-wide association studies. Nature methods 11:407-409. Broman KW, Gatti DM, Simecek P, Furlotte NA, Prins P, Sen S, Yandell BS, Churchill GA (2019) R/qtl2: software for mapping quantitative trait loci with high-dimensional data and multi-parent populations. GENETICS https://www.genetics.org/content/211/2/495.

scan_pvl function