EM.MIMv {QTLEMM}

EM Algorithm for QTL MIM with Asymptotic Variance-Covariance Matrix

Description

Expectation-maximization algorithm for QTL multiple interval mapping. It can obtain the asymptotic variance-covariance matrix of the result from the EM algorithm and the approximate solution of variances of parameters.

Usage

EM.MIMv(
  QTL,
  marker,
  geno,
  D.matrix,
  cp.matrix = NULL,
  y,
  type = "RI",
  ng = 2,
  cM = TRUE,
  E.vector0 = NULL,
  X = NULL,
  beta0 = NULL,
  variance0 = NULL,
  crit = 10^-5,
  stop = 1000,
  conv = TRUE,
  var.pos = TRUE,
  console = TRUE
)

Arguments

QTL

matrix. A q*2 matrix contains the QTL information, where the row dimension 'q' represents the number of QTLs in the chromosomes. The first column labels the chromosomes where the QTLs are located, and the second column labels the positions of QTLs (in morgan (M) or centimorgan (cM)).

marker

matrix. A k*2 matrix contains the marker information, where the row dimension 'k' represents the number of markers in the chromosomes. The first column labels the chromosomes where the markers are located, and the second column labels the positions of markers (in morgan (M) or centimorgan (cM)). It's important to note that chromosomes and positions must be sorted in order.

geno

matrix. A n*k matrix contains the genotypes of k markers for n individuals. The marker genotypes of P1 homozygote (MM), heterozygote (Mm), and P2 homozygote (mm) are coded as 2, 1, and 0, respectively, with NA indicating missing values.

D.matrix

matrix. The design matrix of QTL effects is a g*p matrix, where g is the number of possible QTL genotypes, and p is the number of effects considered in the MIM model. The design matrix can be easily generated by the function D.make().

cp.matrix

matrix. The conditional probability matrix is an n*g matrix, where n is the number of genotyped individuals, and g is the number of possible genotypes of QTLs. If cp.matrix=NULL, the function will calculate the conditional probability matrix, and markers whose positions are the same as QTLs will be skipped.

y

vector. A vector with n elements that contain the phenotype values of individuals.

type

character. The population type of the dataset. Includes backcross (type="BC"), advanced intercross population (type="AI"), and recombinant inbred population (type="RI"). The default value is "RI".

ng

integer. The generation number of the population type. For instance, in a BC1 population where type="BC", ng=1; in an AI F3 population where type="AI", ng=3.

cM

logical. Specify the unit of marker position. If cM=TRUE, it denotes centimorgan; if cM=FALSE, it denotes morgan.

E.vector0

vector. The initial value for QTL effects. The number of elements corresponds to the column dimension of the design matrix. If E.vector0=NULL, the initial value for all effects will be set to 0.

X

matrix. The design matrix of the fixed factors except for QTL effects. It is an n*k matrix, where n is the number of individuals, and k is the number of fixed factors. If X=NULL, the matrix will be an n*1 matrix where all elements are 1.

beta0

vector. The initial value for effects of the fixed factors. The number of elements corresponds to the column dimension of the fixed factor design matrix. If beta0=NULL, the initial value will be set to the average of y.

variance0

numeric. The initial value for variance. If variance0=NULL, the initial value will be the variance of phenotype values.

crit

numeric. The convergence criterion of EM algorithm. The E and M steps will iterate until a convergence criterion is met. It must be a value between 0 and 1.

stop

numeric. The stopping criterion of EM algorithm. The E and M steps will halt when the iteration number reaches the stopping criterion, treating the algorithm as having failed to converge.

conv

logical. If set to False, it will disregard the failure to converge and output the last result obtained during the EM algorithm before reaching the stopping criterion.

var.pos

logical. Determines whether the variance of the position of QTLs will be considered in the asymptotic variance-covariance matrix or not.

console

logical. Determines whether the process of the algorithm will be displayed in the R console or not.

Value

E.vector

The QTL effects are calculated by the EM algorithm.

beta

The effects of the fixed factors are calculated by the EM algorithm.

variance

The error variance is calculated by the EM algorithm.

PI.matrix

The posterior probabilities matrix after the process of the EM algorithm.

log.likelihood

The log-likelihood value of this model.

LRT

The LRT statistic of this model.

R2

The coefficient of determination of this model. This can be used as an estimate of heritability.

y.hat

The fitted values of trait values are calculated by the estimated values from the EM algorithm.

iteration.number

The iteration number of the EM algorithm.

avc.matrix

The asymptotic variance-covariance matrix contains position of QTLs, QTL effects, variance, and fix effects.

EMvar

The asymptotic approximate variances include the position of QTLs, QTL effects, variance, and fixed effects. Parameters for which the approximate variance cannot be calculated will be shown as 0. The approximate variance of the position of QTLs is calculated in morgan.

References

KAO, C.-H. and Z.-B. ZENG 1997 General formulas for obtaining the maximum likelihood estimates and the asymptotic variance-covariance matrix in QTL mapping when using the EM algorithm. Biometrics 53, 653-665. <doi: 10.2307/2533965.>

KAO, C.-H., Z.-B. ZENG and R. D. TEASDALE 1999 Multiple interval mapping for Quantitative Trait Loci. Genetics 152: 1203-1216. <doi: 10.1093/genetics/152.3.1203>

See Also

D.make Q.make EM.MIM

Examples

# load the example data
load(system.file("extdata", "exampledata.RDATA", package = "QTLEMM"))

# run and result
D.matrix <- D.make(3, type = "RI", aa = c(1, 3, 2, 3), dd = c(1, 2, 1, 3), ad = c(1, 2, 2, 3))
result <- EM.MIMv(QTL, marker, geno, D.matrix, cp.matrix = NULL, y)
result$EMvar