Estimate Haplotype Frequency from  Mixed  Sample  of  Pooled  and
Individual DNA


DESCRIPTION:
       Observations are genotypes of individual DNA  samples  and
       pooled  DNA  samples. Each observation is  genotypes of K-
       pooled  DNA  sample  (K=1  corresponds  to   non-pooling),
       different pools may have different pool size K.  Genotypes
       are the total number of major variant (allele) in the pool
       for  each  locus.  At  each  locus  there  is two possible
       alleles (biallelic), one  is  major  allele  of  interests
       (say,   mutation),   another   is   minor  allele. The two
       alleles at each locus is denoted by 0 and 1, genotype   is
       therefore the number of allele 1 and haplotypes are binary
       sequences.Missing information is allowed in the algorithm.
       There  are  two  types of  missing values, one is complete
       missing, another is partial missing. Partial missing means
       that  the  exact  genotype  at  one specific locus was not
       observed, but instead, it is known  that  at  this  locus,
       there are at least one major allele and at least one minor
       allele. 

USAGE:
       ehv(geno, K=2, pool.size)

REQUIRED ARGUMENTS:
geno:     n by m matrix. Genotype observations  with  rows  being
       pools  and  columns  loci. Columns are corrsponding to the
       loci considered. There are n pools/individuals and m loci.
       The  number of possible haplotypes is 2^m. The (i,j) entry
       is the genotype for the i-th pool at locus j. Genotype  is
       defined  as  the  number  of major allele in the pool. For
       example, if we observed AGGG and define arbitrarily allele
       A   as  major allele, then the genotype is 1, otherwise if
       allele G is defined as major allele, the genotype is 3  at
       this  locus.  Complete  missing genotypes are coded as -1,
       partial missing values are coded as -2.
K:       pool size when all pools are of same size. Default value
       is 1 (for individual DNA sample).
pool.size:     vector of pool size (K1, K2, ...,  Km).  The  i-th
       component,  Ki,  is the pool size for pool i. When all the
       components are 1, there is  no  pooling  of  DNA  samples.
       Default values are (1,1,...,1), i.e., each pool is in fact
       genotype for one individual.

VALUE:
h:     estimated haplotype relative frequencies.
logL:     the logarithm maximum likelihood.
V:     Variance matrix of the  estimated  haplotype  frequencies.
config.h:	estimated haplotype relative frequencies and their
       variance estimates (the last two columns). Other columns are
       their corresponding haplotype configurations of binary (0/1)
       sequences.

SIDE EFFECTS:
       the program is slow when the loci  number  exceeds 10,  or
       pool  size  exceeds  8  or  there  are  too  many  missing
       information in the data set.

DETAILS:
       The  maximum  likelihood  estimates   of   the   haplotype
       frequencis   are   obtained  by  EM  algorithm  under  the
       assumption  of  Hardy-Weinberg  equilibrium   and   random
       mating.  Ideally  the  method can be applied to cases with
       any arbitrary pool sizes (K) and locus  numbers  (m),  but
       the  compuatation burden is very heavy even for moderate K
       and m.

       If the sample size (number of pools), n (number of columns
       of genotype data), is small, the variance estimates may not
       be accurate. In this case, a simplified version of ehv, eh
       may be used and the bootstrapping method is recommended for
       estimating the variances.

       Logarithm of maximum likelihood (logL) can be used  as  in
       constructing  likelihood  ratio  test  for,  for  example,
       case-control studies.


REFERENCES:
       T.A. Louis, 1982, Finding the observed information  matrix
       when using EM algorithm, J.R.Statisti. B, 44:226-233.

       K. Lange,  1999,  Numerical  Analysis  for  Statisticians,
       Springer.

SEE ALSO:
       eh, eh1 (for individual sample only).

EXAMPLES:
       Example 1. If the genotype data are on 2 SNPs (L2 and L2)
       for 5 pools, each of the pools are respectively pools of
       size 2,1,2,1,3:
                  L1  L2
          pool-1   1   4
          pool-2  -1   0
          pool-3   2   3
          pool-4   1   2
          pool-5   2   5

       ## input data:
       > geno.data <-matrix(c(1,4,-1,0,2,3,1,2,2,5),5,2,byrow=T)
       > p.size_c(2,1,2,1,3) #pool sizes
       ## run the program:
       > ehv(geno=geno.data,  pool.size=p.size)
       ## if all the pool sizes are equal, then need only to specify K.
       > ehv(geno=geno.data, K=2) ## K=2
       > eh(geno=geno.data, K=3) ##K=3


       Example 2. Data from files. Two files are needed. The first
       is the genotype file, in which there are n rows (pools) and
       m columns (SNPs or loci). Each entry is a integer standing
       for the total number of major allele. Another file contains
       a single column of the pool sizes for each pool. For example,
       there are 4 pools and 5 SNPs, the 4 pools are pooled DNA from
       2,1,2,3 individuals, then the genotype data file, "geno.txt"
       should look like this
             4 2 3 4 0
             1 2 0 0 2
             3 2 4 4 1
             6 4 5 0 1
       and the file, "sizes.txt" of pool sizes should be
             2
             1
             2
             3
       In R, input commands:
       > geno <- as.matrix(read.table("geno.txt")) # read in genotype data.
       > size <- scan("sizes.txt") # read in the pool sizes.
       > eh.out <- ehv(geno, size)