title: Cubic exact solutions for the estimation of pairwise haplotype frequencies: implications for linkage disequilibrium analyses and a web tool 'CubeX'.
abstract: BACKGROUND
  The frequency of a haplotype comprising one allele at each of two loci can be expressed as a cubic equation (the 'Hill equation'), the solution of which gives that frequency. Most haplotype and linkage disequilibrium analysis programs use iteration-based algorithms which substitute an estimate of haplotype frequency into the equation, producing a new estimate which is repeatedly fed back into the equation until the values converge to a maximum likelihood estimate (expectation-maximisation).
  
  
  RESULTS
  We present a program, "CubeX", which calculates the biologically possible exact solution(s) and provides estimated haplotype frequencies, D', r2 and chi2 values for each. CubeX provides a "complete" analysis of haplotype frequencies and linkage disequilibrium for a pair of biallelic markers under situations where sampling variation and genotyping errors distort sample Hardy-Weinberg equilibrium, potentially causing more than one biologically possible solution. We also present an analysis of simulations and real data using the algebraically exact solution, which indicates that under perfect sample Hardy-Weinberg equilibrium there is only one biologically possible solution, but that under other conditions there may be more.
  
  
  CONCLUSIONS
  Our analyses demonstrate that lower allele frequencies, lower sample numbers, population stratification and a possible |D'| value of 1 are particularly susceptible to distortion of sample Hardy-Weinberg equilibrium, which has significant implications for calculation of linkage disequilibrium in small sample sizes (eg HapMap) and rarer alleles (eg paucimorphisms, q < 0.05) that may have particular disease relevance and require improved approaches for meaningful evaluation.
authors.affiliation: Bristol Genetic Epidemology Laboratories (BGEL) and MRC Centre for Causal Analyses in Translational Epidemiology (CAiTE), Department of Social Medicine, University of Bristol, Canynge Hall, Whiteladies Road, Bristol, BS8 2PR, UK. tom.gaunt@bristol.ac.uk
authors.forename: Tom R
authors.initials: TR
authors.lastname: Gaunt
authors.forename: Santiago
authors.initials: S
authors.lastname: Rodríguez
authors.forename: Ian Nm
authors.initials: IN
authors.lastname: Day
mesh_terms: D000465:Algorithms; D000483:Alleles; D000818:Animals; D019295:Computational Biology / Q000379:methods*; D005787:Gene Frequency; D008040:Genetic Linkage; D020022:Genetic Predisposition to Disease; D005828:Genetics, Population; D006239:Haplotypes; D006801:Humans; D020407:Internet; D015810:Linkage Disequilibrium; D008957:Models, Genetic; D015233:Models, Statistical; D008962:Models, Theoretical; D012984:Software