Abstract:
A method based on direct and indirect counting is disclosed for rapid and accurate linkage analysis for codominant and dominant loci. Methods for estimating gender-specific recombination frequencies are available for cases where at least one of the two loci is multi-allelic and for bi-allelic loci with mixed parental linkage phases where at least one locus is codominant. The method makes use of the full data set, yields exact estimates of the recombination frequencies when the observed and expected genotypic frequencies are equal, and are computationally efficient.

Description:
STATEMENT OF GOVERNMENT RIGHTS  
       [0001] The present invention was made, at least in part, with a grant from the Government of the United States of America (NRICGP/USDA grant# 03275). The Government may have certain rights to the invention. 
     
    
     
       FIELD  
         [0002]    The present invention relates generally to performing genetic linkage analysis, and more particularly to linkage analysis using indirect counting methods.  
         COPYRIGHT NOTICE/PERMISSION  
         [0003]    A portion of the disclosure of this patent document contains material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. The following notice applies to the software and data as described below and in the drawings hereto: Copyright ® 2002, Regents of the University of Minnesota, All Rights Reserved.  
         BACKGROUND  
         [0004]    Genetic linkage analysis is a statistical method that is used to associate functionality of genes to their location on chromosomes. It is based on the observation that genes that reside physically close on a chromosome remain linked during meiosis. Typically, markers which are found in vicinity on the chromosome have a tendency to stick together when passed on to offspring. Thus, if some disease is often passed to offspring along with specific markers, then it can be concluded that the gene(s) which are responsible for the disease are located close on the chromosome to these markers.  
           [0005]    Genetic linkage is designed to estimate the distance between genes. Normally, immediately before the gametes (sperm or eggs) are produced, there is a lining up of parental chromosomes in preparation for the separation of genetic material into gametes. An exchange of genetic material occurs between parental chromosomal pairs, which is termed recombination, or crossing over between chromosomes. The chromosomes are then separated and packaged into the gametes.  
           [0006]    Two genes that lie on separate chromosomes will be transmitted independently of each other from parent to child. The child has an equal chance of receiving the gene from his mother or from his father. This phenomenon is encapsulated in Mendel&#39;s law of independent assortment.  
           [0007]    However, two genes may also be on the same chromosome. If they are located at opposite ends, then they will once again be transmitted independently of each other. This is because they are so far away from each other that a recombination event is very likely to occur between the two loci. However, the closer the two genes lie to each other, the less likely it is that a genetic crossover will occur between them. Finally, two genes may lie so close that it is much more likely that they will remain together and be transmitted together into the forming gamete. Two examples make this clearer.  
           [0008]    If an individual has genotype A1A2 at locus A and genotype B1B2 at locus B and the loci are not linked to each other, the alleles at locus A and locus B will assort independently and four different types of gametes (A1B1, A1B2, A2B1, A2B2) will be produced in equal frequencies. This is termed independent assortment.  
           [0009]    If locus A is very close to locus B on the same chromosome, an individual will again produce four types of gametes, but now the alleles found will not be in equal frequencies. The most common types of gametes will be those that represent the alleles that occurred in each parent. The less frequent types of gametes will contain a mixture of the parental alleles that has occurred as a result of infrequent recombination events between the two loci.  
           [0010]    While computationally efficient methods are available for large scale linkage analysis for codominant loci, rapid methods are unavailable for mapping dominant loci and for the map integration of dominant and codominant loci. Most computer programs that provide linkage analysis for dominant loci such as LINKAGE implement computationally intensive likelihood analysis and generally have a limitation on the number of loci that can be analyzed jointly. A computationally efficient method for linkage analysis with codominant and dominant inheritance is needed for mapping dominant genes and for the map integration of codominant and dominant loci, because dominant inheritance mode is typical of many disease genes and many dominant markers (such as RAPD and AFLP markers) exist. Analytical formulas for maximum likelihood estimate of recombination frequency between two dominant loci in repulsion linkage phase have been developed. However, the mathematical simplicity of such an analytical formula is computationally efficient for large scale linkage analysis. However, many other cases of linkage analysis do not have a simple analytical formula for estimating recombination frequencies. The understanding of relative efficiencies of various types of genotypic data is useful for planning mapping experiments. Most results on relative efficiencies of genotypic data were based on the approximate variances and covariances of estimated recombination frequencies but the accuracy of such an approximation is unclear.  
           [0011]    Additionally, sex-influenced traits can affect linkage analysis. A sex-influenced trait has an autosomal inheritance mode that typically exhibits the pattern of “reversal dominance” in the two genders, i.e., the gene is dominant in one gender and recessive in the other. Examples of sex-influenced traits have been reported in several species. Scurs of cattle requires one scurred allele to express in males and two scurred alleles to express in females. The depth of the red color of the Ayrshire cattle is dominant in males and recessive in females. A gene affecting a chicken plumage pattern is dominant in males and recessive in females. Human baldness and short index fingers are dominant in men and recessive in women, whereas the disorder of Heberden nodes, which are bony excrescences of the phalanges of the distal interphalangeal joints of the fingers, is likely to be dominant in women and recessive in men. Another human example is the inheritance of one form of Aarskog&#39;s faciodigitogenital syndrome. Furthermore, it was recently conjectured that factors affecting the development of rheumatoid arthritis in humans show sex-influenced expression. Examples of sex-influenced traits have also been observed in mice and insects. Although methods are available for linkage analysis, a method for linkage analysis involving a sex-influenced gene is unavailable in conventional linkage analysis systems.  
           [0012]    In view of the problems discussed above, there is a need in the art for the present invention.  
         SUMMARY  
         [0013]    The above-mentioned shortcomings, disadvantages and problems are addressed by the present invention, which will be understood by reading and studying the following specification.  
           [0014]    The present invention includes systems and methods for analyzing genetic data using direct and indirect counting. One aspect of the present invention includes systems and methods that receive input data including family identification and genetic identifiers and extracting statistics regarding the genetic identifiers. The statistics may be used to compute at least one recombination frequency and LOD score for at least one locus by applying indirect counting to the statistics. In addition, the systems and methods may use the recombination frequencies and LOD scores to determining a locus order for the genetic identifiers.  
           [0015]    A further aspect of the present invention is that inheritance cases are determined that then may be used to determine an appropriate indirect counting solution.  
           [0016]    A still further aspect is that the indirect counting solution may use iterative computation to arrive at a recombination frequency.  
           [0017]    The present invention describes systems, clients, servers, methods, and computer-readable media of varying scope. In addition to the aspects and advantages of the present invention described in this summary, further aspects and advantages of the invention will become apparent by reference to the drawings and by reading the detailed description that follows. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0018]    [0018]FIG. 1 is a block diagram of a software operating environment for performing linkage analysis in which different embodiments of the invention may be practiced;  
         [0019]    FIGS.  2 A- 2 C are diagrams providing further details of input files used in the software operating environment;  
         [0020]    [0020]FIG. 3 is a diagram providing further details of screen output provided by the software operating environment;  
         [0021]    FIGS.  4 A- 4 E are diagrams providing further details of output files provided by the software operating environment;  
         [0022]    FIGS.  5 A- 5 E are flowcharts illustrating methods for performing linkage analysis using direct and indirect counting; and  
         [0023]    [0023]FIG. 6 is a diagram illustrating the major hardware components of a computer incorporating embodiments of the invention.  
     
    
     DETAILED DESCRIPTION  
       [0024]    In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that other embodiments may be utilized and that logical, mechanical, electrical and other changes may be made without departing from the scope of the present invention.  
         [0025]    Some portions of the detailed descriptions which follow are presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algorithmic descriptions and representations are the ways used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like. It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the following discussions, terms such as “processing” or “computing” or “calculating” or “determining” or “displaying” or the like, refer to the action and processes of a computer system, or similar computing device, that manipulates and transforms data represented as physical (e.g., electronic) quantities within the computer system&#39;s registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices.  
         [0026]    In the Figures, the same reference number is used throughout to refer to an identical component which appears in multiple Figures. Signals and connections may be referred to by the same reference number or label, and the actual meaning will be clear from its use in the context of the description.  
         [0027]    The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims.  
       Operating Environment  
       [0028]    [0028]FIG. 1 is a block diagram of a software operating environment  100  for performing linkage analysis in which different embodiments of the invention may be practiced. In some embodiments of the invention, software environment  100  includes a linkage analysis program  110  that receives input from data file  102 , name file  104  and parameter file  106 . Note that while three input files may be used in some embodiments, the data in the files could be provided in other combinations of one or more input files or data streams. In one embodiment of the invention, linkage analysis program  110  is the Locusmap program available from the University of Minnesota. In some embodiments, linkage analysis program  110  uses the input data provided in files  102 ,  104  and  106  and the methods described in further detail below to produce screen output  108 , data errors  112 , locus info data  114 , pairwise data  116 , locus order data  118  and linkage map  120 .  
         [0029]    [0029]FIG. 2A is a diagram providing details of the information in data file  102 . As illustrated in FIG. 2A, data file  102  in some embodiments of the invention has data for a number of individuals in a number of different families. The data for each individual may include various combinations of the following:  
         [0030]    Family ID—Identifies a family to which the individual belongs.  
         [0031]    ID—Uniquely identifies the individual.  
         [0032]    Parent 1—ID for a parent of the individual.  
         [0033]    Parent 2—ID for a second parent of the individual.  
         [0034]    Sex—Gender of the individual.  
         [0035]    Genotype—one or more pairs of alleles forming loci. Phenotype information may also be included in some embodiments.  
         [0036]    Although the various values in FIG. 2A are numeric, those of skill in the art will appreciate that other non-numeric data could be substituted.  
         [0037]    [0037]FIG. 2B is a diagram providing details of the information in name file  104 . In some embodiments, the name file provides a mapping between a locus name and a chromosome number.  
         [0038]    [0038]FIG. 2C is a diagram providing details of the information in parameter file  106 . In some embodiments of the invention, parameter file  106  includes data providing the name and expected location of various input and output files. Further, the parameter file may include encoding values for gender and traits. In addition, in some embodiments of the invention, parameter file  106  includes various combinations of the following parameters:  
         [0039]    lod_threshold—LOD (logarithm of odds) score value used to determine if linkage is present.  
         [0040]    cutoff—the minimum number of offspring in a phase unknown family in order for the family to be used in calculations.  
         [0041]    brute_limit—maximum number of loci to use brute-force ordering.  
         [0042]    map_function—function used to convert recombination frequency to a genetic distance. Values include Haldane, Morgan and Kosambi.  
         [0043]    Locus_output_type—determines whether locus name or number are output.  
         [0044]    [0044]FIG. 3 is a diagram providing further details of screen output  108  provided by linkage analysis program  110 . Screen output is not required, but may be useful to determine the progress of the linkage analysis program and whether errors are being encountered.  
         [0045]    [0045]FIG. 4A is a diagram providing details of the information in data errors  112 . In some embodiments of the invention, data errors file  112  include information identifying individuals where inheritance data is missing or incorrect.  
         [0046]    [0046]FIG. 4B is a diagram providing details of the information in locus info  114 . In some embodiments, locus info  114  provides information regarding a locus name and statistical information including the percentage of heterozygous sires and dames having the named locus. Additionally, a percentage of informative meioses may be provided in some embodiments. An informative meiosis has parent allele transmission. Thus the percentage of informative meioses is a rating of how informative the data is with respect to a locus. Because both a male and a female contribute to the percentage, the percentage value can range from 0 to 200%.  
         [0047]    [0047]FIG. 4C is a diagram providing details of the information in pairwise data  116 . Pairwise data file includes linkages between loci, and statistical values such as LOD scores for the linkage.  
         [0048]    [0048]FIG. 4D is a diagram providing details of the information in locus order data  118 . In some embodiments, locus order data  118  includes a series of calculated possible loci orderings, with the most likely ordering presented first in the output data stream.  
         [0049]    [0049]FIG. 4E is a diagram providing details of the information in linkage map  120 . Linkage map  120  provides statistical data regarding the individual loci for linkage groups identified during the linkage analysis.  
         [0050]    FIGS.  5 A- 5 E are flowcharts illustrating methods for performing linkage analysis using direct and indirect counting. Direct counting is based on counting the frequencies of four haplotypes for each pair of loci and then directly computing the recombination frequency and LOD score. Indirect counting is based on counting the frequencies of genotypes for each pair of loci, and then using iterative methods to compute the recombination frequencies and LOD scores from those frequencies. The methods to be performed by the operating environment constitute one or more computer programs made up of computer-executable instructions. Describing the methods by reference to a flowchart enables one skilled in the art to develop such programs including such instructions to carry out the methods on suitable computers (the processor or processors of the computer executing the instructions from computer readable media). The methods illustrated in FIGS.  5 A- 5 E are inclusive of acts that may be taken by an operating environment executing an exemplary embodiment of the invention.  
         [0051]    [0051]FIG. 5A is a flowchart illustrating a method for performing linkage analysis according to some embodiments of the invention. The method begins by receiving input data (block  502 ). The input data typically comprises family identification data and genetic information for members of the family. Further, the input data may also include locus names data that map numeric identifiers to locus names. In addition, the input data may include parameters used to control the processing of data and for specifying the location and format for input and output data. Furthermore, control parameters may be provided on a command line for the linkage analysis program.  
         [0052]    In some embodiments of the invention the input data may be converted from an externally defined format to an internally usable format. In some embodiments, the externally defined format is the Crimap format. In alternative embodiments, the externally defined format is the “Linkage” format.  
         [0053]    Additionally, in some embodiments of the invention, the input data is scanned for sex-linked loci. If any such loci are found, they are flagged for special processing by later actions in the method.  
         [0054]    Next, a system performing the method extracts statistics from input data (block  504 ). In some embodiments, the statistics are gathered by reading through the families in the data file one by one and counting the frequencies of haplotypes and genotypes of all locus pairs. This step essentially condenses the raw genotype and phenotype data to a condensed form that can be used for further processing.  
         [0055]    [0055]FIG. 5B is a flowchart providing further details of the extract statistics processing of block  504 . The processing illustrated in FIG. 5B will be performed for each family in the input data. Statistics extraction begins by reading data for one family from the input data (block  512 ). Next, a pedigree for the family is determined (block  514 ). The grandparents (if any), parents, and offspring are identified and ordered. Half-sib families are identified and Jo split into separate families.  
         [0056]    Next, the family is prepared for processing (block  516 ). Family preparation may include all or some of the following steps:  
         [0057]    Parents and grandparents are put in the correct order.  
         [0058]    The data is scanned for dominant/recessive coded loci. Any such loci are checked for data errors and are prepped for processing, including converting the data to genotype data where possible.  
         [0059]    The data is scanned for sex-influenced coded loci. Any such loci are checked for data errors and are prepped for processing, including converting the data to genotype data where possible.  
         [0060]    The inheritance pattern of each locus is checked to make sure it is consistent across families.  
         [0061]    The data is scanned for imprinted loci. Any such loci are checked for data errors and are prepped for processing, including converting the data to genotype data where possible.  
         [0062]    All missing parental genotypes are filled in when they can be determined unequivocally.  
         [0063]    Next, the heterozygocity of the family is determined (block  518 ). Here, the number of heterozygous parents at each locus is counted. The heterozygocity data may be used as a measure of the informativeness of a family, but is not required for indirect counting.  
         [0064]    A system executing the method then proceeds to get statistics for the family (block  520 ). The statistics include genotype and haplotype frequencies that are gathered from the family data.  
         [0065]    [0065]FIG. 5C provides further details on the get statistics processing of block  520 . The system executing the methods analyzes the parent data (block  546 ). Here the parental alleles are ordered properly where possible and characteristics of each locus and locus pair are collected.  
         [0066]    Next, the case of each locus pair is determined based on an inheritance mode (block  548 ). In some embodiments of the invention, there are thirteen cases, referred to as case 0-case 12. A case may be determined by looking at the parental alleles.  
                                                       Case 0:   two multiallelic, codominant loci           Case 1:   one biallelic codominant locus, one multiallelic               codominant locus           Case 2:   two biallelic codominant loci           Case 3:   two biallelic codominant loci, mixed linkage phase           Case 4:   one multiallelic, codominant locus, one               dominant/recessive locus           Case 5:   one biallelic codominant locus, one               dominant/recessive locus           Case 6:   one biallelic codominant locus, one               dominant/recessive locus, mixed               linkage phase           Case 7:   two dominant/recessive loci, coupling phase           Case 8:   two dominant/recessive loci, mixed phase           Case 9:   two dominant/recessive loci, repulsion phase           Case 10:   one multiallelic codominant locus,               one sex-linked locus           Case 11:   one biallelic codominant locus, one sex-linked locus           Case 12:   one biallelic codominant locus, one               sex-linked locus, mixed linkage               phase                      
 
         [0067]    Imprinted loci are handled in a similar way as sex-linked loci. The alleles of an imprinted locus can be recoded so that the locus can be analyzed using direct counting, so imprinting does not have a case of its own.  
         [0068]    Blocks  550  and  552  are executed for each individual in the family, and for each locus pair in the individual. Depending on the case, the haplotype frequencies are counted (block  552 ) and the genotype frequencies (block  550 ) are counted.  
         [0069]    For each locus pair in the family, the system compiles direct counting data for locus pairs in case 0 (block  554 ). The haplotype frequencies are condensed into counts of recombinant and non-recombinant meioses.  
         [0070]    In addition, the system compiles indirect counting data for each locus pair in the family that are in cases 1-12 (block  556 ). The list of genotype frequencies for the locus pair is sorted into proper order. If the phase can be directly determined for the locus pair, it is. Otherwise numerical methods are used to determine the phase of the locus pair. The list of genotype frequencies is reordered to compensate for the phase. The haplotype and genotype frequencies are then combined with data gathered from previous half-sib families (block  558 ).  
         [0071]    Returning to FIG. 5B, after the statistics have been gathered for each half-sib family, the system saves the family data (block  522 ). The haplotype and genotype frequencies are combined with data gathered from previous families (full-sib).  
         [0072]    Returning to FIG. 5A, after the statistics have been extracted for each family, the system then proceeds to compute recombination frequencies and LOD scores (block  506 ). The compute functions compute recombination frequencies and LOD scores for all locus pairs based on genotype frequencies and haplotype frequencies previously extracted from the raw data.  
         [0073]    [0073]FIG. 5D is a flowchart providing further details of the compute recombination frequencies and LOD scores processing of block  506 . The system computes indirect counting data for locus pairs in cases 1-12 (block  524 ). Using genotype frequency data determined above, recombination frequencies and LOD scores are computed for each locus pair using iterative functions. As noted above, for each locus pair, data has been gathered from several families. The same locus pair may fall into different cases in different families. For each case the recombination frequency and LOD score is computed using the appropriate functions, and then that data is combined together to give one recombination frequency and LOD score for each locus pair.  
         [0074]    The following tables provide the formulas for computing the recombination frequency and LOD score for each case used in some embodiments of the invention. For LOD scores, an overall LOD score (Z) and a unit LOD (u) score may be provided. The unit LOD score may be defined as the expected LOD score per offspring assuming gender-average recombination frequency.  
         [0075]    Case 1: One Biallelic Codominant Locus, One Multiallelic Codominant Locus  
                                                             TABLE 1                           Genotypic frequency, number of observations, and the       number of recombinants in the offspring from the       intercross of A 1 B/A 2 b (male) ×       A 3 B/A 4 b (female)                Number of                Number   recombinants            Genotype   Genotypic frequency a     of observations   female b     male b                 A 1 A 3 BB   q 1  = ¼(1 − x)(1 − y)   k 1     0   0       A 1 A 3 bb   q 2  = ¼xy   k 2     k 2     k 2         A 1 A 4 BB   q 3  = ¼x(1 − y)   k 3     k 3     0       A 1 A 4 bb   q 4  = ¼(1 − x)y   k 4     0   k 4         A 2 A 3 BB   q 5  = q 4     k 5     0   k 5         A 2 A 3 bb   q 6  = q 3     k 6     k 6     0       A 2 A 4 BB   q 7  = q 2     k 7     k 7     k 7         A 2 A 4 bb   q 8  = q 1     k 8     0   0       A 1 A 3 Bb   q 9  = q 3  + q 4     k 9     v 1 k 9     v 2 k 9         A 1 A 4 Bb   q 10  = q 1  + q 2         k 10     v 3 k 10     v 3 k 10         A 2 A 3 Bb   q 11  = q 1  + q 2         k 11     v 3 k 11     v 3 k 11         A 2 A 4 Bb   q 12  = q 3  + q 4         k 12     v 1 k 12     v 2 k 12         Total   1   n   n x     n y                                    
 
         [0076]    From Table 1, gender-specific recombination frequencies may be obtained by the following iterative solutions:  
                     x     (     i   +   1     )       =            a   +         bx     (   i   )            (     1   -     y     (   i   )         )             x     (   i   )            (     1   -     y     (   i   )         )       +       (     1   -     x     (   i   )         )          y     (   i   )             +                              cx     (   i   )            y     (   i   )               (     1   -     x     (   i   )         )          (     1   -     y     (   i   )         )       +       x     (   i   )            y     (   i   )                           (   1   )                       y     (     i   +   1     )       =            d   +         b        (     1   -     x     (   i   )         )            y     (   i   )               x     (   i   )            (     1   -     y     (   i   )         )       +       (     1   -     x     (   i   )         )          y     (   i   )             +                              cx     (   i   )            y     (   i   )               (     1   -     x     (   i   )         )          (     1   -     y     (   i   )         )       +       x     (   i   )            y     (   i   )                           (   2   )                               
 
         [0077]    where x=female recombination frequency, y=male recombination frequency, superscript i=iteration number, a=(k 2 +k 3 +k 6 +k 7 )/n, b=(k 9 +k 12 )/n, c=(k 10 +k 11 )/n, and d=(k 2 +k 4 +k 5 +k 7 )/n, and where k 1  through k 12  are defined in Table 1. The gender-average recombination frequency can be estimated as θ=(x+y)/2, noting that the male and female parents have the same number of meioses. This method of estimating gender-average recombination frequency may also be used for other cases where gender-specific recombination frequencies are available.  
         [0078]    LOD scores may be determined according to the following:  
           Z   x   =N   1 log 10 [2(1 −x )]+ N   2 log 10 (2 x )+ N   3 log 10 [2 x (1 −y )+2(1 −x ) y]+N   4 log 10 [2 xy +2(1 −x )(1 −y )]  (3)  
           Z   y   =N   5 log 10 [2(1 −y )]+ N   6 log 10 (2 y )+ N   3 log 10 [2 x (1 −y )+2(1 −x ) y]+N   4 log 10 [2 xy +2(1 −x )(1 −y )]  (4)  
           Z   θ   =N   7 log 10 [4(1−θ) 2   ]+N   8 log 10 (4θ 2 )+ N   9 log 10 [4θ(1−θ)]+ N   10 log 10 {2[(1−θ) 2 +θ 2 ]}  (5)  
           u= ½(1−θ) 2 log 10 [4(1−θ) 2 ]+½θ 2 log 10 (4θ 2 )+2θ(1−θ)log 10 [4θ(1−θ)] 
         [0079]    +½[(1−θ) 2 +θ 2 ]log 10 {2[(1−θ) 2 +θ 2 ]}  (6)  
         [0080]    where N 1 =k 1 +k 4 +k 5 +k 8 , N 2 =k 2 +k 3 +k 6 +k 7 , N 3 =k 9 +k 12 , N 4 =k 10 +k 11 , N 5 =k 1 +k 3 +k 6 +k 8 , N 6 =k 2 +k 4 +k 5 +k 7 , N 7 =k 1 +k 8 , N 8 =k 2 +k 7 , N 9 =k 3 +k 4 +k 5 +k 6 +k 9 +k 12 , and N 10 =k 10 +k 11 .  
         [0081]    Case 2: Two Biallelic Codominant Loci  
                                 TABLE 2                           Genotypic frequency, number of observations, and       the number of recombinants in the offspring       from the intercross of AB/ab × AB/ab                    Number of   Number of       Genotype   Genotypic frequency a     observations   recombinants               AABB   q 1  = ¼(1 − θ) 2     k 1     0       AABb   q 2  = ½θ(1 − θ)   k 2     k 2         AAbb   q 3  = ¼θ 2     k 3     2k 3         AaBB   q 4  = q 2     k 4     k 4         AaBb   q 5  = 2(q 1  + q 3 )   k 5     2k 5 θ 2 /[(1 − θ) 2  + θ 2 ]       Aabb   q 6  = q 2     k 6     k 6         aaBB   q 7  = q 3     k 7     2k 7         aaBb   q 8  = q 2     k 8     k 8         aabb   q 9  = q 1     k 9     0       Total   1   n   n r                            
 
         [0082]    For this case, gender-specific recombination frequencies are unavailable and gender-average recombination frequency can be estimated based on Table 2. The resulting formula is:  
             θ   =         [       -   s     +         s   2     +     t   3           ]       1   3       -       [     s   +         s   2     +     t   3           ]       1   3       +       a   1     3               (   7   )                               
 
         [0083]    where s=½[a 1 a 2 /3−(2/27)a 1   3 −c], t=⅓( a   2 −a 1   2 /3), a 1 =(T+c 1 +n 4 )/T, a 2   =0.5+c   1 /T, and where T=2n, c 1 =2n 3 +n 2 , c=c 1 /(2T), n 1 =k 1 +k 9 , n 2 =k 2 +k 4 +k 6 +k 8 , n 3 =k 3 +k 7 , and n 4 =k 5 . Note that equation (7) is derived under the assumption of coupling parental linkage phases but is applicable to the repulsion linkage phases by reversing the allele definitions for one of the two loci.  
         [0084]    LOD scores may be determined according to the following:  
           Z   θ   =N   1 log 10 [4(1−θ) 2   ]+N   2 log 10 [4θ(1−θ)]+ N   3 log 10 (4θ 2 )+ N   4 log 10 {2[(1−θ) 2 +θ 2 ]}  (8)  
         [0085]    where N 1 =k 1 +k 9 , N 2 =k 2 +k 4 +k 6 +k 8 , N 3 =k 3 +k 7 , N 4 =k 5 . The unit LOD score is the same as that for the MB data type.  
         [0086]    Case 3: Two Biallelic Codominant Loci, Mixed Linkage Phase  
                                                             TABLE 3                           Offspring phenotypes and recombinants from       the mating of AB/ab (male) × Ab/aB (female)                Number of                Number   recombinants            Genotype   Genotypic frequency   of observations   female a     male a                 AABB   q 1  = ¼x(1 − y)   k 1     k 1     0       AABb   q 2  = ¼[(1 − x)(1 − y) + xy]   k 2     v 1 k 2     v 1 k 2         AAbb   q 3  = ¼(1 − x)y   k 3     0   k 3         AaBB   q 4  = q 2     k 4     v 1 k 4     v 1 k 4         AaBb   q 5  = ½[x(1 − y) + (1 − x)y]   k 5     v 3 k 5     v 2 k 5         Aabb   q 6  = q 2     k 6     v 1 k 6     v 1 k 6         aaBB   q 7  = q 3     k 7     0   k 7         aaBb   q 8  = q 2     k 8     v 1 k 8     v 1 k 8         aabb   q 9  = q 1     k 9     k 9     0       Total   1   n   n x     n y                            
 
         [0087]    From Table 3, gender-specific recombination frequencies can be obtained by the following iterative solutions:  
           x   (i+1)   =a+[bx   (i) (1 −y   (i) )]/[ x   (i) (1 −y   (i) )+(1 −x   (i) ) y   (i)   ]+cx   (i)   uy   (i) /[(1 −x   (i) )(1 −y   (i) )+ x   (i)   y   (i) ]  (9)  
           y   (i+1)   =d+[b (1 −x   (i) ) y   (i)   ]/[x   (i) (1 −y   (i) )+(1 −x   (i) ) y   (i)   ]+cx   (i)   y   (i) /[(1 −x   (i) )(1−y (i) )+ x   (i)   y   (i) ]  (10)  
         [0088]    where x=female recombination frequency, y=male recombination frequency, a=(k 1 +k 9 )/n, b=k 5 /n, c=(k 2 +k 4 +k 6 +k 8 )/n, and d=(k 3 +k 7 )/n.  
         [0089]    LOD scores may be determined according to the following:  
           Z   x =( k   1   +k   9 )log 10 (2 x )+( k   2   +k   4   +k   6   +k   8 )log 10 {2[(1 −x )(1 −y )+ xy )]}+( k   3   +k   7 )log 10 [2(1 −x )]+ k   5 log 10 {2 [x (1 −y )+ y (1 −x )]}  (11)  
           Z   y =( k   1   +k   9 )log 10 [2(1 −y )]+( k   2   +k   4   +k   6   +k   8 )log 10 {2[(1 −x )(1 −y )+ xy )]}+( k   3   +k   7 )log 10 (2 y )+ k   5 log 10 {2 [x (1 −y )+ y (1 −x )]}  (12)  
           Z   θ =( k   1   +k   3   +k   5   +k   7   +k   9 )log 10 [4θ(1−θ)]+( k   2   +k   4   +k   6   +k   8 )log 10 {2[(1 −θ)   2 +θ 2 )]}  (13)  
           u= 2θ(1−θ) log[4θ(1−θ)]+(1−2θ+2θ 2 )log[2(1−2θ+2θ 2 )]  (14)  
         [0090]    Case 4: One Multiallelic, Codominant Locus, One Dominant/Recessive Locus  
                                                       TABLE 4                           Genotypic frequency, number of observations, and the       number of recombinants in the offspring from the       intercross of A 1 B/A 2 b (male) ×       A 3 B/A 4 b (female) with B being       dominant over b                Genotypic   Number of   Number of recombinants            Genotype   frequency   observations   female   male               A 1 A 3 bb   q 1  = ¼xy   k 1     k 1     k 1         A 1 A 4 bb   q 2  = ¼(1 − x)y   k 2     0   k 2         A 2 A 3 bb   q 3  = ¼x(1 − y)   k 3     k 3     0       A 2 A 4 bb   q 4  = ¼(1 − x)(1 − y)   k 4     0   0       A 1 A 3 B-   q 5  = ¼(1 − xy)   k 5     k 5 x(1 − y)/(1 − xy)   k 5 (1 − x)y/(1 − xy)       A 1 A 4 B-   q 6  = ¼[1 − (1 − x)y]   k 6     k 6 x/[1 − (1 − x)y]   k 6 xy/[1 − (1 − x)y]       A 2 A 3 B-   q 7  = ¼[1 − x(1 − y)]   k 7     k 7 xy/[1 − x(1 − y)]   k 7 y/[1 − x(1 − y)]       A 2 A 4 B-   q 8  = ¼(x + y − xy)   k 8     k 8 x/(x + y − xy)   k 8 y/(y + x − xy)       Total   1   n   n x     n y                    
 
         [0091]    From Table 4, gender-specific recombination frequencies can be obtained by the following iterative solutions:  
                 x     (     i   +   1     )       =           ax     (   i   )            (     1   -     y     (   i   )         )         1   -       x     (   i   )            y     (   i   )             +       bx     (   i   )           (     1   -     x     (   i   )         )          y     (   i   )           +         cx     (   i   )            y     (   i   )           1   -       x     (   i   )            (     1   -     y     (   i   )         )           +       dx     (   i   )           x     (   i   )       +     y     (   i   )       -       x     (   i   )            y     (   i   )             +   e                          (   15   )                   y     (     i   +   1     )       =           a        (     1   -     x     (   i   )         )            y     (   i   )           1   -       x     (   i   )            y     (   i   )             +         bx     (   i   )            y     (   i   )             (     1   -     x     (   i   )         )          y     (   i   )           +       cy     (   i   )         1   -       x     (   i   )            (     1   -     y     (   i   )         )           +       dy     (   i   )           x     (   i   )       +     y     (   i   )       -       x     (   i   )            y     (   i   )             +   f                          (   16   )                               
 
         [0092]    where a=k 5 /n, b=k 6 /n, c=k 7 /n, d=k 9 /n, e=(k 1 +k 2 )/n, and f=(k 1 +k 3 )/n.  
         [0093]    LOD scores may be determined according to the following:  
           Z   z =( k   1   +k   3 )log 10 (2 x )+( k   2   +k   4 )log 10 [2(1 −x )]+ k   5 log 10 [[2(1 −xy )/(2 −y )] 
         [0094]    + k   6 log 10 {2[1−(1 −x ) y ]/(2 −y )}+ k   7 log 10 {2[(1 −x (1 −y )]/(1 +y )}+ k   9 log 10 [2( x+y−xy )/(1 +y )]  (17)  
           Z   y =( k   1   +k   2 )log 10 (2 y )+( k   3   +k   4 )log 10 [2(1 −y )]+ k   5 log 10 [[2(1 −xy )/(2 −x )] 
         [0095]    + k   6 log 10 {2[(1 −x (1 −y )]/(1 +x )}+ k   7 log 10 {2[1−( 1−x ) y ]/(2 −y )}+ k   8 log 10 [2( x+y−xy )/(1 +y )]  (18)  
           Z   θ   =k   1 log 10 (4θ 2 )+( k   2   +k   3 )log 10 [4θ(1−θ)]+ k   4 log 10 [4(1−θ) 2   ]+k   5 log 10 [(4/3)(1−θ 2 )] 
         [0096]    +( k   6   +k   7 )log 10 {(4/3)[1−θ(1−θ)]}+ k   8 log 10 [(4/3)θ(2−θ)]  (19)  
           u= ¼θ 2 logg 10 (4θ 2 )+ ½θ( 1−θ)log 10 [4θ(1−θ)]+¼(1−θ) 2 log 10 [4(1−θ) 2 ] 
         [0097]    +{fraction ( 1 / 4 )}(1−θ 2 )log 10 [(4/3)(1−θ 2 )]+{fraction ( 1 / 2 )}[(1−θ(1−θ)]log 10 {(4/3)[1−θ(1−θ)]}+¼θ(2−θ)log 10 [(4/3)θ(2−θ)]  (20)  
         [0098]    Case 5: One Biallelic Codominant Locus, One Dominant/Recessive Locus  
                                 TABLE 5                           Genotypic frequency, number of observations, and       the number of recombinants in the offspring from       the intercross of AB/ab × AB/ab with       allele B being dominant over allele b                Genotypic   Number of   Number of       Genotype   frequency   observations   recombinants               AAB-   q 1  = ¼(1 − θ)   k 1     2k 1 θ/(1 + θ)           (1 + θ)       AAbb   q 2  = ¼θ 2     k 2     2k 2         AaB-   q 3  =½[1 − θ(1 − θ)]   k 3     k 3 θ(1 + θ)/[1 − θ(1 − θ)]       Aabb   q 4  = ½θ(1 − θ)   k 4     k 4         aaB-   q 5  = ¼θ(2 − θ)   k 5     2k 5 /(2 − θ)       aabb   q 6  = ¼(1 − θ) 2     k 6     0       Total   1   n   n r                    
 
         [0099]    Gender-specific recombination frequencies are generally nonestimable for this case. From Table 5, the gender-average recombination frequency may be obtained using the following iterative solution:  
         [0100]    θ (i+1)   =a+bθ   (i) /(1+θ (i) )+ cθ   (i) (1+θ (i) )/[1−θ (i) (1−θ (i))]+   d /(2−θ (i) )  (21)  
         [0101]    where a=(2k 2 +k 4 )/(2n), b=k 1 /n, c=k 3 /(2n), and d=k 5 /n.  
         [0102]    LOD scores may be determined according to the following:  
           Z   θ   =k   1 log 10 [(4/3)(1−σ 2 )]+ k   2 log 10 (4θ 2 )+ k   3 log 10 {(4/3)[1−θ(1−θ)]}+ k   4 log 10 [4θ(1−θ)] 
         [0103]    + k   5 log 10 {(4/3)θ(2−θ)]}+ k   6 log 10 [4(1−θ) 2 ]  (22)  
         [0104]    The unit LOD score is the same as equation 20 above.  
         [0105]    Case 6: One Biallelic Codominant Locus, One Dominant/Recessive Locus, Mixed Linkage Phase  
                                                             TABLE 6                           Offspring phenotypes and recombinants from the mating       of AB/ab (male) × Ab/aB (female)                Number of                Number   recombinants            Genotype   Genotypic frequency   of observations   female a     male a                 AAB —     q 1  = ¼(1 − y + xy)   k 1     k 1 v 1     k 1 v 2         AAbb   q 2  = ¼(1 − x)y   k 2     0   k 2         AaB —     q 3  = ¼(1 + x + y − 2xy)   k 3     k 3 v 3     k 3 v 4         Aabb   q 4  = ¼((1 − x)(1 − y) + xy)   k 4     k 4 v 5     k 4 v 6         aaB —     q 5  = ¼(1 − x + xy)   k 5     k 5 v 7     k 5 v 8         aabb   q 6  = ¼x(1 − y)   k 6     k 6     0       Total   1   n   n x     n y                         # (1 − y) + xy], v 7  = xy/(1 − x + xy), v 8  = [(1 − x)           
 
         [0106]    From Table 6, gender-specific recombination frequencies may be obtained by the following iterative solutions:  
           x   (i+1)   =av   1   (i)   +cv   3   (i)   +dv   5   (i)   +ev   7   (i)   +f   (23)  
           y   (i+1)   =av   2   (i)   +b+cv   4   (i)   +dv   6   (i)   +ev   8   (i)   (24)  
         [0107]    where a=k 1 /n, b=k 2 /n, c=k 3 /n, d=k 4 /n, e=k 5 /n, f=k 6 /n,  a v 1 =[x(1−y)+xy]/(1−y+xy), v 2 =xy/(1−y+xy), v 3 =2[x(1−y)+xy]/(1+x+y−2xy), v 4 =2[(1−x)y+xy]/(1+x+y−2xy), v 5 =xy/[(1−x)(1−y)+xy], v 6 =[x+(1−x)y]/[(1−x)(1−y)+xy], v 7 =xy/(1−x+xy), v 8 =[(1−x)y+xy]/(1−x+xy).  
         [0108]    LOD scores may be determined according to the following:  
           Z   x   =k   1 log 10 [2(1 −y+xy )/(2 −y )]+ k   2 log 10 [2(1 −x )]+ k   3 log 10 [(2/3)(1 +x+y− 2 xy )] 
         [0109]    + k   4 log 10 {2[(1 −x )(1 −y )+ xy )]}+ k   5 log 10 [2(1 −x+xy )/(1 +y )]+ k   6 log 10 (2 x )  (25)  
           Z   y   =k   1 log 10 [2(1 −y+xy )/(1 +x )]+ k   2 log 10 (2 y )+ k   3 log 10 [(2/3)(1 +x+y− 2 xy )] 
         [0110]    + k   4 log 10 {2[(1 −x )(1 −y )+ xy )]}+ k   5 log 10 [2(1 −x+xy )/(2 −x )]+ k   6 log 10 [2(1 −y )]  (26)  
           Z   θ =( k   1   +k   5 )log 10 [(4/3)(1−θ+θ 2 )]+( k   2   +k   6 )log 10 [4θ(1−θ)]+ k   3 log 10 [(2/3)(1+2θ−2θ 2 )] 
         [0111]    + k   4 log 10 {2[(1−θ)2+θ 2 )]}  (27)  
           u=[ ½(1−θ+θ 2 )]log 10 [(4/3)(1−θ+θ 2 )]+[½θ(1−θ)]log 10 [4θ(1−θ)] 
         [0112]    +[¼(1+2−2θ 2 )]log 10 [(2/3)(1+2θ−2θ 2 )]+{[(1−θ)2+θ 2 )]}log 10 {2[(1−θ)2+θ 2 )]}  (28)  
         [0113]    Case 7: Two Dominant/Recessive Loci, Coupling Phase  
                                 TABLE 7                           Genotypic frequency, number of observations, and the       number of recombinants in the offspring from the       intercross of AB/ab × AB/ab with allele A being       dominant over a and B being dominant over b                Genotypic   Number of           Genotype   frequency a     observations   Number of recombinants               A-B-   q 1  = ¼[2 + (1 − θ) 2 ]   k 1     4k 1 θ(1 + θ)/[2 +                   (1 − θ) 2 ]       A-bb   q 2  = ¼θ(2 − θ)   k 2     2k 2 /(2 − θ)       aaB-   q 3  = ¼θ(2 − θ)   k 3     2k 3 /(2 − θ)       aabb   q 4  = ¼(1 − θ) 2     k 4     0       Total   1   n   n r                    
 
         [0114]    In this case, both parents are assumed to have coupling linkage phase (Table 7). The gender-average recombination frequency can be obtained from the following iterative solution:  
         θ (i+1) =4 aθ   (i) (1+θ (i) )/[2+(1−θ (i) ) 2 ]+2 b /(2−θ (i) )  (29)  
         [0115]    where a=k 1 /(2n), and b=(k 2 +k 3 )/(2n).  
         [0116]    LOD scores may be determined according to the following:  
           Z   0   =k   1 log 10 {(8/9)[1+0.5(1−θ) 2 ]}+( k   2   +k   3 )log 10 [(4/3)θ(2−θ)]+ k   4 log 10 [4(1−θ) 2 ]  (30)  
           u=q   1 log 10 {(8/9)[1+0.5(1−θ) 2 ]}+( q   2   +q   3 )log 10 [(4/3)θ(2−θ)]+ q   4 log 10 [4(1−θ) 2 ]  (31)  
         [0117]    Case 8: Two Dominant/Recessive Loci, Mixed Phase  
                                 TABLE 8                           Genotypic frequency, number of observations, and the       number of recombinants in the offspring from the       intercross of AB/ab × Ab/aB with allele A being       dominant over a and B being dominant over b            Gen-       Number of           otype   Genotypic frequency a     observations   Number of recombinants               A-B-   q 1  = ¼[2 + θ(1 − θ)]   k 1     k 1 θ(5 − θ)/[2 + θ(1 − θ)]       A-bb   q 2  = ¼[1 − θ(1 − θ)]   k 2     k 2 θ(1 + θ)/[1 − θ(1 − θ)]       aaB-   q 3  = ¼[1 − θ(1 − θ)]   k 3     k 3 θ(1 + θ)/[1 − θ(1 − θ)]       aabb   q 4  = ¼θ(1 − θ)   k 4     k 4         Total   1   n   n r                    
 
         [0118]    In this case, one parent is assumed to have coupling phase and the other repulsion phase. The gender-average recombination frequency can be obtained from the following iterative solution:  
         θ (i+1)   =a θ (i) (5−θ (i) )/[2+θ (i) (1−θ (i) )]+ bθ   (i) (1+θ (i) )/[1−θ (i) (1−θ (i) )]+ c   (32)  
         [0119]    where a=k 1 /(2n), b=(k 2 +k 3 )/(2n), and c=k 4 /(2n). For the case when the two loci are dominant and both parents have repulsion linkage phase (DD-RR data type), an analytical formula for maximum likelihood estimation of recombination frequency may be used.  
         [0120]    LOD scores may be determined according to the following:  
           Z   θ   =k   1 log 10 {(8/9)[1+½θ(1−θ)]}+( k   2   +k   3 )log 10 {(4/3)[1−θ(1−θ)]}+ k   4 log 10 [4θ(1−θ)]  (33)  
           u=q   1 log 10 {(8/9)[1+½θ(1−θ)]}+( q   2   +q   3 )log 10 {(4/3)[1−θ(1−θ)]}+ q   4 log 0 [4θ(1−θ)]  (34)  
         [0121]    Case 9: Two Dominant/Recessive Loci, Repulsion Phase  
                                 TABLE 9                           Genotypic frequency, number of observations, and the       number of recombinants in the offspring from the       intercross of Ab/aB Ab/aB with allele A being dominant       over a and B being dominant over b.                    Number of           Genotype   Genotypic frequency   Observations   Expected recombinants               A_B —     p 1  = ¼(2 + θ 2 )   k 1     k 1  θ(2 + θ)/(2 + θ 2 )       A_bb   p 2  = ¼(1 − θ 2 )   k 2     k 2  θ/(1 + θ)       aaB —     p 3  = ¼(1 − θ 2 )   k 3     k 3  θ/(1 + θ)       aabb   p 4  = ¼θ 2     k 4     k 4         Total   1   n   n r                    
 
         [0122]    The recombination frequency may be obtained from the following:  
         θ={[−(2 k 1−4( k 2 +k 3)−2 k 4)±{square root}{[2 k 1−4( k 2 +k 3)−2 k 4] 2 +8[2( k 1 +k 2 +k 3)+2 k 4]2 k 4}]/[−2[2( k 1 +k 2 +k 3)+2 k 4]]} 2   (35)  
         [0123]    LOD scores may be determined according to the following:  
           Z   x   =n log 10 (2)+( k   2   +k   4 )log 10 ( x )+( k   1   +k   3 )log 10 (1 −x )  (36)  
           Z   y   =n log 10 (2)+( k   3   +k   4 )log 10 ( y )+( k   1   +k   2 )log 10 (1 −y )  (37)  
           Z   θ =2 n log 10 (2)+( k   2   +k   3 +2 k   4 )log 10 (θ)+(2 k   1   +k   2   +k   3 )log 10 (1−θ)  (38)  
           u =2[log 10 (2)+θlog 10 (θ)+(1−θ)log 10 (1−θ)]  (39)  
         [0124]    Case 10: One Multiallelic Codominant Locus, One Sex-Linked Locus  
                                                                               TABLE 10                           Offspring phenotypes and recombinants from the mating of A 1 B/A 2 b × A 3 B/A 4 b.            Genotype and   Number of       Male   Female       phenotype   offspring   Frequency   recombinants   recombinants            Marker   Trait   M   F   M   F   M a     F   M a     F               A 1 A 3     expressed   m 1     f 1     p 1  = ¼(1 − xy)   p 8     q 1  = ¼y(1 − x)/p 1     0   q 5  = ¼x(1 − y)/p 1     0       A 1 A 4     expressed   m 2     f 2     p 2  = ¼(1 − y + xy)   p 7     q 2  = ¼xy/p 2     0   q 6  = ¼β/p 2     1       A 2 A 3     expressed   m 3     f 3     p 3  = ¼(1 − x + xy)   p 6     q 3  = ¼α/p 3     1   q 7  = ¼xy/p 3     0       A 2 A 4     expressed   m 4     f 4     p 4  = ¼(x + y − xy)   p 5     q 4  = ¼α/p 4     1   q 8  = ¼β/p 4     1       A 1 A 3     unexpressed   m 5     f 5     p 5  = ¼xy   p 4     1   q 4     1   q 8         A 1 A 4     unexpressed   m 6     f 6     p 6  = ¼(1 − x)y   p 3     1   q 3     0   q 7         A 2 A 3     unexpressed   m 7     f 7     p 7  = ¼x(1 − y)   p 2     0   q 2     1   q 6         A 2 A 4     unexpressed   m 8     f 8     p 8  = ¼(1 − x)(1 − y)   p 1     0   q 1     0   q 5                            
 
         [0125]    The recombination frequency may be obtained from the following:  
         θ (i+1)   =aλ   3   (i)   +bλ   2   (i) +2 cλ   1   (i) +2 g+e  for  Ab/aB×Ab/aB   (40)  
         [0126]    where λ 1 =θ/(1+θ), λ 2 =θ(1+θ)/(1−θ+θ 2 ), λ 3 =1/(1−½θ), λ 4 =θ/(1+2θ−2θ 2 ), λ 5 =θ/(1−2θ+2θ 2 ), a=(m 1 +f 6 )/2n, b=(m 2 +f 5 )/2n, c=(m 3 +f 4 )/2n, d=(m 4 +f 3 )/2n, e=(m 5 +f 2 )/2n, g=(m 6 +f 1 )/2n, and where m 1  and f 1  are defined in Table 10.  
         [0127]    LOD scores may be determined according to the following:  
           U   F =¼(1−θ 2 )log[4(1−θ 2 )/3]+½(1−θ+θ 2 )log[4(1−θ+θ 2 )/3]+¼θ(2−θ)log[4θ(2−θ)/3] 
         [0128]    +{fraction ( 1 / 4 )}θ 2 log(4θ 2 )+½θ(1−θ)log[4θ(1−θ)]+¼(1−θ) 2 log[4(1−θ) 2 ]  (41)  
         [0129]    Case 11: One Biallelic Codominant Locus, One Sex-Linked Locus  
                                                                   TABLE 11                           Offspring phenotypes and recombinants from the mating of AB/ab × AB/ab.            Genotype and   Number of       Observed and expected       phenotype   offspring   Frequency   recombinants            Marker   Trait   M a     F a     M a     F a     M a     F a                 AA   expressed   m 1     f 1     p 1  = ¼(1 − θ 2 )   p 6     ½θ(1 − θ)/p 1     0       Aa   expressed   m 2     f 2     p 2  = ½(1 − θ + θ 2 )   p 5     ½θ(1 + θ)/p 2     1       aa   expressed   m 3     f 3     p 3  = ¼θ(2 − θ)   p 4     ½θ/p 3     2       AA   unexpressed   m 4     f 4     p 4  = ¼θ 2     p 3     2   ½θ/p 4         Aa   unexpressed   m 5     f 5     p 5  = ½θ(1 − θ)   p 2     1   ½θ(1 + θ)/p 5         aa   unexpressed   m 6     f 6     p 6  = ¼(1 − θ) 2     p 1     0   ½θ(1 − θ)/p 6                            
 
         [0130]    The recombination frequency may be obtained from the following:  
         θ (i+1) =2 aλ   1   (i)   +bλ   2   (i)   +cλ   3   (i) +2 d+e   (42)  
         [0131]    where λ 1 =θ/(1+θ), λ 2 =θ(1+θ)/(1−θ+θ 2 ), λ 3 =1/(1−½θ), λ 4 =θ/(1+2θ−2θ 2 ), λ 5=θ/( 1−2θ+2θ 2 ), a=(m 1 +f 6 )/2n, b=(m 2 +f 5 )/2n, c=(m 3 +f 4 )/2n, d=(m 4 +f 3 )/2n, e=(m 5 +f 2 )/2n, g=(m 6 +f 1 )/2n, and where m i  and f i  are defined in Table 11.  
         [0132]    LOD scores may be determined according to formula 41 above.  
         [0133]    Case 12: One Biallelic Codominant Locus, One Sex-Linked Locus, Mixed Linkage Phase  
                                                               TABLE 12                           Offspring phenotypes and recombinants from the mating of AB/ab × aB/Ab.            Genotype/               Phenotype   Number Frequency   Recombinants            Mark   Trait   M a     F a     M a     F a     M a     F a                 AA   expressed   m 1     f 1     p 1  = ¼(1 − θ) 2  + ¼θ (1 − θ) + ¼θ 2     p 6     (¼θ (1 − θ) + ½θ 2 )/p1   (¼θ (1 − θ))/p 6         Aa   expressed   m 2     f 2     p 2  = ¼(1 − θ) 2  + θ (1 − θ) + ¼θ 2     p 5     (θ (1 − θ) + ½θ 2 )/p2   (½θ 2 )/p5       aa   expressed   m 3     f 3     p 3  = ¼(1 − θ) 2  + ¼θ (1 − θ) + ¼θ 2     p 4     (¼θ (1 − θ) + ½θ 2 )/p 3     (¼θ (1 − θ))/p4       AA   unexpressed   m 4     f 4     p 4  = ¼(1 − θ)   p 3     (¼θ (1 − θ))/p4   (¼θ (1 − θ) + ½θ 2 )/p3       Aa   unexpressed   m 5     f 5     p 5  = ¼(1 − θ) 2  + ¼θ 2     p 2     (½θ 2 )/p5   (θ (1 − θ) + ½θ 2 )/p2       aa   unexpressed   m 6     f 6     p 6  = ¼θ (1 − θ)   p 1     (¼θ (1 − θ))/p6   (¼θ (1 − θ) + ½θ 2 )/p1                          
 
         [0134]    where λ 1 =θ/(1+θ), λ 2 =θ(1+θ)/(1−θ+θ 2 ), λ 3 =1/(1−½θ), λ 4 =θ/(1+2θ−2θ 2 ), λ 5 =θ/(1−2θ+2θ 2 ), a=(m 1 +f 6 )/2n, b=(m 2 +f 5 )/2n, c=(m 3 +f 4 )/2n, d=(M 4 +f 3 )/2n, e=(m 5 +f 2 )/2n, g=(m 6 +f 1 )/2n, and where m i  and f i  are defined in Table 12.  
         [0135]    LOD scores may be determined according to formula 41 above.  
         [0136]    The system also computes direct counting data for locus pairs in case 0 (block  526 ). Using haplotype frequency data, the recombination frequencies and LOD scores are directly computed for each locus pair. Direct counting methods for determining recombination frequencies and LOD scores are known in the art.  
         [0137]    Next, the computed indirect counting data and direct counting data are combined (block  528 ). Recombination frequencies and LOD scores based on both direct and indirect counting methods are combined to compute a single recombination frequency and LOD score for each locus pair.  
         [0138]    Returning to FIG. 5A, the loci are ordered (block  508 ). The order loci functions split the loci into linkage groups and orders each linkage group, based on recombination frequencies and LOD scores previously computed.  
         [0139]    [0139]FIG. 5E is a flowchart providing further details of the order loci processing of block  508 . A system executing the method begins by determining linkage groups (block  530 ). All of the loci are divided into distinct linkage groups.  
         [0140]    Next, for each linkage group the system computes Two-point Likelihoods (block  534 ) A likelihood is computed for each locus pair in the linkage group, this is used for ordering the loci. The most likely orders of the loci in the linkage group are computed using one of three different ordering methods, quick order (block  536 ), brute force order (block  538 ), or 3-point order (block  540 ). The most likely orders for the linkage group may then be placed in an output data stream (block  542 ). In addition, the most likely orders for the linkage groups, may be placed to an output data stream.  
         [0141]    Next, a linkage map is computed for the most likely order for the linkage group and printed to an output file (block  544 ).  
         [0142]    Returning to FIG. 5A, a system executing the invention may output additional data (block  510 ) In some embodiments, this additional data comprises pairwise data comprising pairwise recombination frequencies and LOD scores and locus info. In further embodiments, locus info comprising information about the informativeness of each locus is computed and placed on an output data stream.  
         [0143]    [0143]FIG. 6 is a diagram of the hardware and operating environment in conjunction with which embodiments of the invention maybe practiced. The description of FIG. 6 is intended to provide a brief, general description of suitable computer hardware and a suitable computing environment in conjunction with which the invention may be implemented. Although not required, the invention is described in the general context of computer-executable instructions, such as program modules, being executed by a computer, such as a personal computer or a server computer. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types.  
         [0144]    Moreover, those skilled in the art will appreciate that the invention may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.  
         [0145]    As shown in FIG. 6, the computing system  600  includes a processor. The invention can be implemented on computers based upon microprocessors such as the PENTIUM® family of microprocessors manufactured by the Intel Corporation, the MIPS® family of microprocessors from the Silicon Graphics Corporation, the POWERPC® family of microprocessors from both the Motorola Corporation and the IBM Corporation, the PRECISION ARCHITECTURE® family of microprocessors from the Hewlett-Packard Company, the SPARC® family of microprocessors from the Sun Microsystems Corporation, or the ALPHA® family of microprocessors from the Compaq Computer Corporation. Computing system  600  represents any personal computer, laptop, server, or even a battery-powered, pocket-sized, mobile computer known as a hand-held PC.  
         [0146]    The computing system  600  includes system memory  613  (including read-only memory (ROM)  614  and random access memory (RAM)  615 ), which is connected to the processor  612  by a system data/address bus  616 . ROM  614  represents any device that is primarily read-only including electrically erasable programmable read-only memory (EEPROM), flash memory, etc. RAM  615  represents any random access memory such as Synchronous Dynamic Random Access Memory.  
         [0147]    Within the computing system  600 , input/output bus  618  is connected to the data/address bus  616  via bus controller  619 . In one embodiment, input/output bus  618  is implemented as a standard Peripheral Component Interconnect (PCI) bus. The bus controller  619  examines all signals from the processor  612  to route the signals to the appropriate bus. Signals between the processor  612  and the system memory  613  are merely passed through the bus controller  619 . However, signals from the processor  612  intended for devices other than system memory  613  are routed onto the input/output bus  618 .  
         [0148]    Various devices are connected to the input/output bus  618  including hard disk drive  620 , floppy drive  621  that is used to read floppy disk  651 , and optical drive  622 , such as a CD-ROM drive that is used to read an optical disk  652 . The video display  624  or other kind of display device is connected to the input/output bus  618  via a video adapter  625 .  
         [0149]    A user enters commands and information into the computing system  600  by using a keyboard  40  and/or pointing device, such as a mouse  42 , which are connected to bus  618  via input/output ports  628 . Other types of pointing devices (not shown in FIG. 6) include track pads, track balls, joy sticks, data gloves, head trackers, and other devices suitable for positioning a cursor on the video display  624 .  
         [0150]    As shown in FIG. 6, the computing system  600  also includes a modem  629 . Although illustrated in FIG. 6 as external to the computing system  600 , those of ordinary skill in the art will quickly recognize that the modem  629  may also be internal to the computing system  600 . The modem  629  is typically used to communicate over wide area networks (not shown), such as the global Internet. The computing system may also contain a network interface card  53 , as is known in the art, for communication over a network.  
         [0151]    Software applications  636  and data are typically stored via one of the memory storage devices, which may include the hard disk  620 , floppy disk  651 , CD-ROM  652  and are copied to RAM  615  for execution. In one embodiment, however, software applications  636  are stored in ROM  614  and are copied to RAM  615  for execution or are executed directly from ROM  614 .  
         [0152]    In general, the operating system  635  executes software applications  636  and carries out instructions issued by the user. For example, when the user wants to load a software application  636 , the operating system  635  interprets the instruction and causes the processor  612  to load software application  636  into RAM  615  from either the hard disk  620  or the optical disk  652 . Once software application  636  is loaded into the RAM  615 , it can be used by the processor  612 . In case of large software applications  636 , processor  612  loads various portions of program modules into RAM  615  as needed.  
         [0153]    The Basic Input/Output System (BIOS)  617  for the computing system  600  is stored in ROM  614  and is loaded into RAM  615  upon booting. Those skilled in the art will recognize that the BIOS  617  is a set of basic executable routines that have conventionally helped to transfer information between the computing resources within the computing system  600 . These low-level service routines are used by operating system  635  or other software applications  636 .  
         [0154]    In one embodiment computing system  600  includes a registry (not shown) which is a system database that holds configuration information for computing system  600 . For example, Windows® 95, Windows 98®, Windows® NT, Windows 2000® and Windows XP® by Microsoft maintain the registry in two hidden files, called USER.DAT and SYSTEM.DAT, located on a permanent storage device such as an internal disk.  
       CONCLUSION  
       [0155]    Systems and methods for performing linkage analysis using direct and indirect counting methods have been disclosed. The systems and methods described provide advantages over previous systems. For all the cases, direct and indirect counting typically yield the same results as maximum likelihood analysis. The inventive method of direct and indirect counting is therefore a useful addition or alternative to current methods available for linkage analysis including complex maximum likelihood analysis due to its mathematical simplicity and computational efficiency. When combined with the strategy of two-point analysis for linkage detection, the method of direct and indirect counting can provide rapid large scale joint linkage analysis of codominant and dominant loci, which is useful to facilitate mapping dominant loci using codominant markers and the map integration of codominant and dominant loci. The estimates of recombination frequencies from direct and indirect counting are the expected fraction of recombinants whether the estimates are within or out of the parameter space. This is helpful in interpreting the estimates in situations where the meanings of the estimates are not easily interpretable. For example, if a maximum likelihood using numerical maximization yielded an estimate out of the parameter space, the estimate itself could tell whether the problem was due to the algorithm of numerical maximization or due to a wrong model or sampling. A wrong inheritance model can result in a serious bias in estimating recombination frequencies (including estimates out of the parameter space) and such a bias can be evaluated conveniently using the method of direct and indirect counting.  
         [0156]    The systems and methods of the present invention therefore provide simple solutions for linkage analysis to facilitate large scale joint linkage analysis with codominant and dominant loci, and for designing mapping experiments.  
         [0157]    Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement which is calculated to achieve the same purpose may be substituted for the specific embodiments shown. This application is intended to cover any adaptations or variations of the present invention.  
         [0158]    The terminology used in this application is meant to include all of these environments. It is to be understood that the above description is intended to be illustrative, and not restrictive. Many other embodiments will be apparent to those of skill in the art upon reviewing the above description. Therefore, it is manifestly intended that this invention be limited only by the following claims and equivalents thereof.