# EM Computation of Haplotype Probabilities, with Progressive Insertion of Loci

### Description

For genetic marker phenotypes measured on unrelated subjects, with linkage phase unknown, compute maximum likelihood estimates of haplotype probabilities. Because linkage phase is unknown, there may be more than one pair of haplotypes that are consistent with the oberved marker phenotypes, so posterior probabilities of pairs of haplotypes for each subject are also computed. Unlike the usual EM which attempts to enumerate all possible pairs of haplotypes before iterating over the EM steps, this "progressive insertion" algorithm progressively inserts batches of loci into haplotypes of growing lengths, runs the EM steps, trims off pairs of haplotypes per subject when the posterior probability of the pair is below a specified threshold, and then continues these insertion, EM, and trimming steps until all loci are inserted into the haplotype. The user can choose the batch size. If the batch size is chosen to be all loci, and the threshold for trimming is set to 0, then this algorithm reduces to the usual EM algorithm.

### Usage

1 2 | ```
haplo.em(geno, locus.label=NA, miss.val=c(0, NA), weight, control=
haplo.em.control())
``` |

### Arguments

`geno` |
matrix of alleles, such that each locus has a pair of adjacent columns of alleles, and the order of columns corresponds to the order of loci on a chromosome. If there are K loci, then ncol(geno) = 2*K. Rows represent the alleles for each subject. |

`locus.label ` |
vector of labels for loci. |

`miss.val ` |
vector of values that represent missing alleles in geno. |

`weight` |
weights for observations (rows of geno matrix). |

`control` |
list of control parameters. The default is constructed by the function haplo.em.control. The default behavior of this function results in the following parameter settings: loci.insert.order=1:n.loci, insert.batch.size=min(4,n.loci), min.posterior= 0.0001, tol=0.00001, max.iter=500, random.start=0 (no random start), iseed=NULL (no saved seed to start random start), verbose=0 (no printout during EM iterations). See haplo.em.control for more details. |

### Details

The basis of this progressive insertion algorithm is from the sofware snphap by David Clayton. Although some of the features and control parameters of this S-PLUS version are modeled after snphap, there are substantial differences, such as extension to allow for more than two alleles per locus, and some other nuances on how the alogrithm is implemented.

### Value

list with components:

`converge` |
indicator of convergence of the EM algorithm (1 = converge, 0 = failed). |

`lnlike` |
value of lnlike at last EM iteration (maximum lnlike if converged). |

`lnlike.noLD` |
value of lnlike under the null of linkage equilibrium at all loci. |

`lr` |
likelihood ratio statistic to test the final lnlike against the lnlike that assumes complete linkage equilibrium among all loci (i.e., haplotype frequencies are products of allele frequencies). |

`df.lr` |
degrees of freedom for likelihood ratio statistic. The df for the unconstrained final model is the number of non-zero haplotype frequencies minus 1, and the df for the null model of complete linkage equilibrium is the sum, over all loci, of (number of alleles - 1). The df for the lr statistic is df[unconstrained] - df[null]. This can result in negative df, if many haplotypes are estimated to have zero frequency, or if a large amount of trimming occurs, when using large values of min.posterior in the list of control parameters. |

`hap.prob` |
vector of mle's of haplotype probabilities. The ith element of hap.prob corresponds to the ith row of haplotype. |

`locus.label` |
vector of labels for loci, of length K (see definition of input values). |

`subj.id` |
vector of id's for subjects used in the analysis, based on row number of input geno matrix. If subjects are removed, then their id will be missing from subj.id. |

`rows.rem` |
now defunct, but set equal to a vector of length 0, to be compatible with other functions that check for rows.rem. |

`indx.subj` |
vector for row index of subjects after expanding to all possible pairs of haplotypes for each person. If indx.subj=i, then i is the ith row of geno. If the ith subject has n possible pairs of haplotypes that correspond to their marker genotype, then i is repeated n times. |

`nreps` |
vector for the count of haplotype pairs that map to each subject's marker genotypes. |

`max.pairs` |
vector of maximum number of pairs of haplotypes per subject that are consistent with their marker data in the matrix geno. The length of max.pairs = nrow(geno). This vector is computed by geno.count.pairs. |

`hap1code` |
vector of codes for each subject's first haplotype. The values in hap1code are the row numbers of the unique haplotypes in the returned matrix haplotype. |

`hap2code` |
similar to hap1code, but for each subject's second haplotype. |

`post` |
vector of posterior probabilities of pairs of haplotypes for a person, given their marker phenotypes. |

`haplotype` |
matrix of unique haplotypes. Each row represents a unique haplotype, and the number of columns is the number of loci. |

`control` |
list of control parameters for algorithm. See haplo.em.control |

### Note

Sorted order of haplotypes with character alleles is system-dependent, and can be controlled via the LC_COLLATE locale environment variable. Different locale settings can cause results to be non-reproducible even when controlling the random seed.

### See Also

`setupGeno`

,
`haplo.em.control`

### Examples

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