Journal of Heredity 2003:94(1)
© 2003 The American Genetic Association 94:1-7
Establishing Appropriate Genome-Wide Significance Levels for Canine Linkage Analyses
From the Laboratory of Statistical Genetics, Rockefeller University, 1230 York Ave., New York, NY, 10021 (Gordon, Corwin, and Ott), and the Divisions of Human Biology and Clinical Research, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave. N, D4-100, Seattle, WA 98109-1024 (Mellersh and Ostrander). We gratefully acknowledge grants K01-HG00055 (to D.G.), HG00008 (to J.O.), K05 CA90754 (to E.A.O.), and R01 CA92167 (to E.A.O.).
Address correspondence to Derek Gordon, Laboratory of Statistical Genetics, Rockefeller University, Box 192, 1230 York Ave., New York, NY 10021, or e-mail: gordon{at}linkage.rockefeller.edu.
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Abstract |
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A threshold of 3.3 for a genome-wide maximum LOD score (MAXLOD) has been demonstrated in human linkage studies as corresponding to a type I error rate of 5%. Generalization of this work to other species assumes the presence of an infinitely dense marker map. While this assumption is increasingly realistic for the human genome, it may be unrealistic for the dog genome. In this study we establish the analytic and empirical thresholds for MAXLOD in canine linkage studies corresponding to type I error rates of 5% and 1% for autosomal traits. Empirical thresholds are computed via simulation assuming a 10 cM map with no fine mapping performed. Pedigree structures for simulations were drawn from two canine disease studies. Five thousand replicates of genome-wide null genotype data were simulated and analyzed for each disease. We determined that MAXLOD thresholds of 3.2 and 2.7 correspond to analytic and empirical type I error rates of 5%, respectively. In all cases, the MAXLOD thresholds from simulations were always at least 0.5 LOD units below the corresponding analytic thresholds. We therefore recommend that a threshold of 3.2 be used for canine linkage studies when fine mapping is performed, and that researchers perform their own simulation studies to assess genome-wide empirical significance levels when no fine mapping is performed.
One of the most important questions in the field of statistical inference is the establishment of thresholds for a particular test statistic that guarantee an appropriate significance level. In the field of human genetics, initial seminal research by Morton (1955) established an appropriate threshold for the LOD (logarithm of the odds favoring linkage) score statistic. Morton asserted that the null hypothesis of no linkage could be rejected when the LOD score exceeded a threshold value of 3.0. While this threshold applied to data from only a single marker and hypothesized trait locus, researchers have applied this threshold to data from genome-wide scans for many years.
In 1995 Lander and Kruglyak applied the work of Feingold and Siegmund (1993) to human genetics, and using the assumption of an infinitely dense marker map, showed that a MAXLOD threshold of 3.0 corresponds to a genome-wide significance level of only 0.09, and that a MAXLOD threshold of 3.3 corresponds to a genome-wide significance level of 5%. Here the term MAXLOD is defined as follows:
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) is the LOD score between the marker locus m and the trait locus at recombination fraction . This definition of MAXLOD is used throughout the remainder of this work. The solution presented by Lander and Kruglyak (1995) assumes an infinitely dense marker map (i.e., that the set G in the definition of MAXLOD is infinitely large and that the distance between any two markers in G is zero), and also is not restricted to humans. Lander and Kruglyak comment that their solution is relevant for genome scans in which fine mapping is performed. In the following analysis, we apply these formulas to our study of linkage analysis in dogs.
Dogs offer several key advantages over other animal systems for mapping genes relevant to human disease (Galibert et al. 1998; Ostrander and Giniger 1997; Ostrander et al. 2000; Patterson et al. 1982). First, intense selection for specific morphologic and behavioral features has resulted in more than 300 distinct pure breeding populations termed "breeds" (Ostrander and Giniger 1997; Ostrander and Kruglyak 2000; Wayne and Ostrander 1999). The strategies utilized to obtain such exceptional morphologic diversity between dog breeds, coupled with strong homogeneity within breeds in a relatively short period of time, has led to the increased expression of disease phenotypes in purebred dogs (Galibert et al. 1998; Ostrander and Giniger 1997; Patterson 2000). While this has had a direct and predictably negative impact on canine health (Ostrander et al. 1993), it presents a unique opportunity for investigators interested in understanding the genetics of human health to identify genes that have proven elusive through the analysis of human pedigrees.
Very recently, a collaborative effort on the part of multiple research groups has led to the development of an 1,800-marker integrated cytogenetic, meiotic linkage, and radiation hybrid (RH) dog map (Breen et al. 2001). The 1,500-marker RH map includes 1,078 microsatellites, 320 canine-specific genes, and 102 chromosome-specific markers. The 354-marker meiotic linkage map is composed largely of canine-specific microsatellites. Mapping of 266 chromosome-specific cosmid clones, each containing a microsatellite marker, to all 38 canine autosomes by fluorescence in situ hybridization (FISH) has facilitated both integration of the linkage and RH maps as well as positional orientation of the resulting linkage groups on individual canine chromosomes. The inclusion of 320 dog genes into the integrated map has enhanced existing comparative mapping data between human and dog by providing support for previous reciprocal paint studies (Breen et al. 1999; Yang et al. 1999). The current integrated map is estimated to cover more than 90% of the canine genome (Breen et al. 2001). While this level of coverage is much denser than that previously reported (Mellersh et al. 2000), it is unlikely that the canine will achieve the level of coverage currently featured in the human genome.
It is therefore the purpose of this work to compute MAXLOD statistics for linkage analysis in canine pedigrees using computer simulation, and to compare these results to those achieved by applying the analytic solution (equations 1 and 2). Furthermore, based on these results, recommendations for appropriate MAXLOD threshold statistics are provided for the dog.
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Materials and Methods |
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Pedigrees
The pedigree structures used for the simulations were those from linkage studies of actual canine diseases; two diseases were considered. The first, canine hereditary multifocal renal cystadenocarcinoma and nodular dermatofibrosis (RCND), is a naturally occurring inherited cancer syndrome observed in dogs whose mode of inheritance is autosomal fully penetrant dominant (Jonasdottir et al. 2000). The second, early retinal degeneration (ERD), is an early onset canine retinal disease phenotypically similar to human retinitis pigmentosa (RP), whose mode of inheritance is autosomal recessive (Acland et al. 1999).
For RCND, the pedigree structure was a single large pedigree of 82 dogs, of which 26 were affected. A pedigree may be found in the reference by Jonasdottir et al. (2000). We assumed that DNA was available for all dogs, as was the case in the actual study. This assumption allows us to simulate marker genotypes for all dogs at every marker. For ERD, there were eight families with a total of 125 dogs, of whom 72 are affected. Again, we assumed that DNA was available for all dogs. To address the question of whether our sample sizes are large enough to exceed the analytic MAXLOD thresholds, we note that, with the number of affected dogs in each pedigree, the power that MAXLOD exceeded 3.2 was greater than 0.90, given that the true recombination fraction between the disease locus and at least one marker locus in the genome was 0.05 or less (Ott 1999:88–89).
Markers and the Map
Marker genotypes were simulated for each dog in each of 44 syntenic groups, now known to comprise 38 chromosomes (Breen et al. 2001). Syntenic groups were initially established by Mellersh et al. (2000). It was assumed that the trait of interest is autosomal, so the X chromosome was excluded from these simulations. Recombination fractions were selected to approximate a 10 cM genome-wide map. Such map density is often used with genome scans, especially for human linkage analyses (e.g., Levy et al. 2000; Ohman et al. 2000). All recombination fractions were based on the linkage maps and not on the RH maps. In the case of syntenic groups whose recombination fractions were established strictly by RH mapping (and so reported in centiRays), we used the conversion 1 cR = 6.1 cM, as was reported by both Priat et al. (1998) and Mellersh et al. (2000) in the first and second versions of the canine RH map. The mapping function used was the Kosambi (1944) mapping function. A summary of the number of markers simulated for each of the 44 syntenic groups can be found in Table 1.
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Estimation of both the number of alleles per marker and corresponding allele frequencies was not practical, since alleles from each of the studies were labeled differently on different gels and in different families. Therefore we chose two different sets of allele frequencies for the simulations: either each marker had 10 equally frequent alleles (highHet) or 2 equally frequent alleles (lowHet). The corresponding heterozygosity for each marker is 0.9 and 0.5, respectively. The first set of allele frequencies (highHet) was chosen so that each replicate of each simulation would contain highly informative meioses for linkage and would most likely provide MAXLOD values close to those produced by a multipoint analysis. The second set of frequencies (lowHet) was chosen to create markers with lower average heterozygosities. Most dog breeds are highly inbred (Ostrander and Giniger 1997) and therefore the level of homogeneity is greater in purebred dogs than in mutts. This fact suggests to us that most founders for purebred lines should have a greater probability of being homozygous at a marker. Thus the lowHet simulations may be more realistic for purebred dogs. Finally, it should be noted that, when performing the linkage analyses, incorrect estimation of allele frequencies had no impact, as all dogs were assumed to be genotyped.
Simulations
For each of the diseases, genotypes for all markers on each dog were simulated using the SIMULATE program (Terwilliger and Ott 1994), which simulates genotype data using only recombination distances between marker loci and allele frequencies at each locus, that is, under the null hypothesis of no linkage of the trait to any location in the genome. Five thousand replicates were created for each disease (RCND and ERD) and within each disease for each heterozygosity of markers (highHet and lowHet). From this point forward, the term "simulation" shall refer to an ordered pair (disease, heterozygosity), where the value for the disease coordinate is either RCND or ERD, and the value for heterozygosity is either highHet or lowHet.
Test Statistics
Two statistics were computed in our analysis. The first is the MAXLOD statistic (defined in the introduction), which is the maximum over all markers m in the genome scan of the maximum LOD score at marker m. The second is the MAXHLOD statistic (Smith 1963), which equals
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,ß)) is called the heterogeneity LOD score (or HLOD score) at a marker m. As is indicated by the notation, this LOD score is a function of two parameters, the recombination fraction between the marker locus m and the trait locus for the "linked" families (those families for which < 0.5), and a second parameter, ß. This new parameter represents the proportion of families for which the true recombination fraction between marker and trait locus is less than 0.5. It should be noted that the HLOD (and therefore MAXHLOD) statistics do not differ from LOD and MAXLOD values when there is only one pedigree in the analysis. Hence MAXHLOD results were not reported for the RCND simulations. A description of how the MAXLOD score and the MAXHLOD score were computed for each replicate in each of simulation is reported in Materials and Methods.
Analytic Solution for Genome-Wide Significance Level
Applying the theory of Feingold and Siegmund (1993), Lander and Kruglyak (1995) showed that the number of regions in an entire genome where the MAXLOD score will achieve a pointwise significance of 
1 is distributed approximately as a Poisson random variable with mean
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1. Using equation 1, Lander and Kruglyak computed that the genome-wide significance level of MAXLOD is
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1 = 0.0001, and it follows that C = 13.8. Assuming the total length T of the human genome is 33 Morgans, it follows from equations 1 and 2 that µG = 0.09338 and thus = 0.089, or approximately 0.09, thus demonstrating Lander and Kruglyak's assertion that a MAXLOD score of 3.0 in humans corresponds to a genome-wide significance level of 0.09. For the analytic solution in dogs, we assumed that the length of the autosomes of the canine genome is T = 25.75 Morgans (Neff et al. 1999). In Table 2 we present a list of MAXLOD scores and the corresponding genome-wide significance levels for the canine genome.
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Data Analysis
For each disease, the data were analyzed assuming the correct mode of inheritance for the disease. The disease allele frequency for each disease was set at 0.001. The MLINK module from the FASTLINK suite of programs (Cottingham et al. 1993; Lathrop et al. 1984; Schaffer et al. 1994) was used to compute LOD scores for each marker. The LOD score for each marker in each replicate was evaluated using
settings ranging from 0.0 to 0.5, in increments of 0.05. For each replicate of a given disease and heterozygosity level, the MAXLOD score was recorded as follows: let (Zm(S)(t)) represent the LOD score of the mth marker in the Sth syntenic group for the setting = t. We recorded maxS (maxm (maxt(Zm(S)(t)))), where t ranges from 0.0 to 0.5 in increments of 0.05, m ranges from 1 to the maximum number of markers for syntenic group S, and S ranges from 1 to 44. Similarly, for MAXHLOD, we recorded for each replicate the value maxS (maxm (maxt,ß (Zm(S)(t,ß)))), where ß ranges from 0.0 to 1.0 in increments of 0.02. The HLOD scores were computed using the HOMOG program (Ott 1999). |
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Analytic Solution
Table 2 provides a list of MAXLOD scores and the corresponding genome-wide significance levels for the canine genome, based on our assumed parameter values. The MAXLOD score that most closely corresponds to a genome-wide significance level of 5% (respectively, 1%) is 3.2 (respectively, 3.95). It should be noted that the MAXLOD threshold corresponding to a 5 % significance level in dogs (3.2) is quite close to the MAXLOD threshold of 3.3 for humans.
Simulations
Table 3 provides the MAXLOD and MAXHLOD scores corresponding to empirical 5% and 1% significance levels for each of the disease families (RCND and ERD) and each type of pedigree genotype data (lowHet or highHet). With each value (5%, 1%) the empirical significance levels in Table 3 were computed as follows: the set of all 5,000 (one for each replicate) MAXLOD statistics (respectively, MAXHLOD statistics) of a given simulation were listed from largest to smallest. This list shall hereafter be referred to as the ordered list. The 5,000 x yth value in the ordered list (y = 0.05 or 0.01, corresponding to 5% and 1% significance levels) is the number reported in Table 3 for the empirical (100 x y)% significance level. As noted previously, no HLOD scores are computed for the RCND pedigrees, since there is only one pedigree per replicate and the MAXHLOD score equals the MAXLOD score in that case.
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In addition to the LOD and HLOD scores, Table 3 also reports in parentheses the 95% confidence intervals (CI) computed for the MAXLOD and MAXHLOD statistics corresponding to the 5% and 1% significance levels. These confidence intervals are computed using the formula employed in the BINOM program (Ott 1999). For example, the 95% CI corresponding to the proportion p = 250/5,000 is (0.0441, 0.0564). The MAXLOD scores that make up the lower and upper bounds of the 95% CI are then the 5,000 x 0.0441 = 221st and 5,000 x 0.0564 = 282nd values, respectively, on the ordered list of MAXLOD scores.
As Table 3 demonstrates, for each simulation the MAXLOD threshold corresponding to a particular significance level (5% or 1%) is lower than the value computed via the analytic solution. The highest MAXLOD corresponding to a 5% significance level is 2.69, for the ERD lowHet case. This value is 0.51 LOD units lower than that predicted by the analytic solution. The lowest MAXLOD corresponding to a 5% significance level is 2.38 for the RCND lowHet case, which is 0.82 LOD units lower than that predicted by the analytic solution. The highest MAXLOD corresponding to a 1% significance level is 3.376, for the ERD lowHet case. This value is 0.59 LOD units lower than that predicted by the analytic solution. Finally, the lowest MAXLOD corresponding to a 1% significance level is 3.04, for the RCND lowHet case. This value is 0.91 LOD units lower than that predicted by the analytic solution. These results are not surprising, given the density of the map for our simulations (10 cM). Other authors (Lander and Kruglyak 1995; Sawcer et al. 1997) also observed a decrease in MAXLOD thresholds when performing simulations with genetic maps of lower density.
We note also that for the MAXHLOD thresholds there is a smaller range of values for the lowHet and highHet scores as compared to the ranges observed for the MAXLOD scores. In fact, for the 5% significance level, the difference between the lowHet and highHet MAXHLOD scores is 0.024. The difference is 0.0 for the 1% significance level. Also, at least at the 1% level of significance, the MAXHLOD threshold is approximately 0.3 units (rounded to the nearest tenth) larger than the corresponding MAXLOD threshold for RCND. Such a difference was also observed in human genetics simulations with MAXLOD and MAXHLOD (Hodge et al. 1997).
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Discussion |
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The availability of more than 300 pure breeding populations of dogs, or breeds, offers researchers a unique opportunity to simplify the locus heterogeneity problems typically associated with mapping of complex human diseases like cancer, epilepsy, deafness, and heart disease, simply by studying variant forms of disease in unrelated breeds of dog (Ostrander and Kruglyak 2000). Indeed, more than 300 medical genetic disorders have been described in domestic dogs so far, constituting the largest known number of naturally occurring genetic disorders in any nonhuman species (Patterson 2000, 2001). More than 70% of these are believed to be autosomal recessive, X-linked, or genetically complex, with nearly 60% believed to be homologues of human disorders. Thus the domestic dog is an ideal species in which to undertake genetic studies of human inherited disorders. Integral to the success of such studies, however, is a clear understanding of the test statistics, particularly MAXLOD.
Based on our analysis, the simulation-based MAXLOD threshold is at least 0.5 units lower than the analytic threshold. Although other authors (Lander and Kruglyak 1995; Sawcer et al. 1997) observed a decrease in MAXLOD thresholds when using genetic maps of lower density, we make a distinction between humans and dogs. For humans, it appears likely that there will be an infinitely dense human marker map sometime in the near future. This is unlikely to ever be the case for the canine genome, however. Nonetheless, researchers performing fine mapping in canines may create significantly dense maps for regions of interest, and therefore we recommend that researchers performing fine mapping in canine linkage studies use a MAXLOD threshold of 3.2 for a 5% genome-wide significance level, and those not performing fine mapping perform a simulation study to empirically determine the MAXLOD threshold corresponding to a 5% genome-wide significance level.
In canine linkage analyses performed to date, exceeding the MAXLOD threshold of 3.2 has not been a problem (Acland et al. 1998; Jonasdottir et al. 2000; Lingaas et al. 1998; Yuzbasiyan-Gurkan et al. 1997), but previous studies have all benefited from "purpose-bred" pedigrees that were carefully designed to include dogs from different genetic backgrounds. It is likely that single-breed pedigrees, for which not all dogs might be available, will be more inbred and thus present a greater challenge. In addition, it is always possible, simply by chance, to observe a data set in which a MAXLOD threshold of 3.2 is not exceeded. To prevent the possibility of a type II error (the nonrejection of a false null hypothesis) and corresponding loss of power, we suggest that researchers performing linkage analyses in dogs perform similar simulations to establish appropriate genome-wide significance levels.
It is important to note that we chose canine pedigrees that are large enough so that large MAXLOD and MAXHLOD scores could be achieved in simulations, that is, we observed MAXLOD and MAXHLOD scores in the tails of distribution. It is true that considerably smaller pedigrees with smaller numbers of individuals will have theoretical upper bounds on their MAXLOD and MAXHLOD scores that are less than 2.7. It is not true, however, that as the number of pedigrees approaches infinity, the MAXLOD score under the null hypothesis approaches infinity. Rather the MAXLOD value corresponding to a 5% significance level, assuming an infinitely dense map for the canine genome and a sample size that approaches infinity, will approach the theoretical result of 3.2 (Table 2) (Lander and Kruglyak 1995).
We observe that there appears to be a range of MAXLOD thresholds, depending on the mode of inheritance of the disease (dominant versus recessive) and the heterozygosity of the markers (highHet versus lowHet). The simulations with a dominant mode of inheritance (RCND) always had a lower MAXLOD threshold than the simulations with a recessive mode of inheritance, at both the 5% and 1% levels of significance. One explanation for this fact is that the recessive disease pedigree had more affected offspring and therefore a larger number of informative meioses. Because of this fact, the MAXLOD scores from the recessive study are most probably closer to the true MAXLOD threshold for a 10 cM map.
Two potentially interesting findings of this work are that MAXHLOD is robust to the heterogeneity of the markers used in the analysis and that there appears to be an interaction between heterogeneity and mode of inheritance for MAXLOD. We recognize that the current simulation is not powerful enough to resolve the issue. Also, the number of pedigrees and mode of inheritance are confounded in our study design. Despite these limitations, we expect the information provided here will prove useful to investigators seeking to map canine disease genes and in interpreting their data.
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Acknowledgments |
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M. B. Corwin was supported by the Hebrew Technical Institute as part of the Rockefeller University Summer Outreach Program. C. S. Mellersh was funded by a postdoctoral fellowship from Ralston Purina. E. A. Ostrander also gratefully acknowledges the support of the American Kennel Club Canine Health Foundation and a Burroughs Wellcome Award for Excellence in Functional Genomics. We thank Gregory Acland and Gustavo Aguirre for use of the ERD data and Thora Jonasdottir, Lars Moe, and Frode Lingaas for use of the RCND data set. We also thank anonymous reviewers for helpful comments. This paper was delivered at the Advances in Canine and Feline Genomics symposium, St. Louis, MO, May 16–19, 2002.
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Footnotes |
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Corresponding Editor: William Murphy
Received July 15, 2002
Accepted October 9, 2002
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