Testing for homogeneity of gametic disequilibrium across strata
- Xiaolin Yin^{1},
- Wenqing Ma^{1},
- Manlai Tang^{2} and
- Jianhua Guo^{1}Email author
https://doi.org/10.1186/1471-2156-8-85
© Yin et al; licensee BioMed Central Ltd. 2007
Received: 25 April 2007
Accepted: 20 December 2007
Published: 20 December 2007
Abstract
Background
Assessing the non-random associations of alleles at different loci, or gametic disequilibrium, can provide clues about aspects of population histories and mating behavior and can be useful in locating disease genes. For gametic data which are available from several strata with different allele probabilities, it is necessary to verify that the strata are homogeneous in terms of gametic disequilibrium.
Results
Using the likelihood score theory generalized to nuisance parameters we derive a score test for homogeneity of gametic disequilibrium across several independent populations. Simulation results demonstrate that the empirical type I error rates of our score homogeneity test perform satisfactorily in the sense that they are close to the pre-chosen 0.05 nominal level. The associated power and sample size formulae are derived. We illustrate our test with a data set from a study of the cystic fibrosis transmembrane conductance regulator gene.
Conclusion
We propose a large-sample homogeneity test on gametic disequilibrium across several independent populations based on the likelihood score theory generalized to nuisance parameters. Our simulation results show that our test is more reliable than the traditional test based on the Fisher's test of homogeneity among correlation coefficients.
Background
Measuring gametic disequilibrium can provide important information about aspects of population histories and mating behavior [1] and can be useful in locating disease genes [2]. The term gametic disequilibrium is used in this article instead of the traditional term linkage disequilibrium to measure the extent of non-random association because such non-random association may be present between unlinked loci [3]. Various measures of gametic disequilibrium have been proposed [4–6], ranging from pairs of diallelic loci model to multiple multiallelic loci model. In this article, we consider the gametic disequilibrium which is defined as the difference between the gametic probability and its expected probability under the assumption of no statistical association of alleles, and the gametic disequilibrium calculations are based on two-allele, two-locus model [7].
where ${p}_{{A}_{i}}$ and ${p}_{{B}_{j}}$ denote the allele probabilities of A_{ i }and B_{ j }, ${p}_{{A}_{i}{B}_{j}}$ denotes the gamete probability of A_{ i }B_{ j }, i, j = 0, 1. Suppose that the gametic data are available from K strata and let p_{ ijk }denote the gametic probability of array of A_{ i }B_{ j }for the k-th stratum, i, j = 0, 1; k = 1,...,K, ∑_{i,j}p_{ ijk }= 1 for each k. According to the relationship between allelic probability and gametic probability, the allele probabilities of A_{0}, A_{1}, B_{0} and B_{1} are derived as p_{0+k}, p_{1+k}, p_{+0k}and p_{+1k}, respectively. Here "+" denote the summation over 0 and 1, for example, p_{0+k}= p_{00k}+ p_{01k}. For stratum k (k = 1,...,K), the gametic disequilibrium is calculated as
D_{ k }= p_{11k}- p_{1+k}p_{+1k}.
It is easy to show that D_{ k }is bounded by
D_{k,min}≤ D_{ k }≤ D_{k,max},
where D_{k,min}= -min{p_{1+k}p_{+1k}, p_{0+k}p_{+0k}}, D_{k,max}= min{p_{1+k}p_{+0k}, p_{0+k}p_{+1k}}. Testing for the homogeneity of gametic disequilibrium among strata can be informative in discriminating among the evolutionary agents generating them in natural population [8]. Detecting gametic disequilibrium can be informative in mapping gene and providing meaningful clues of population evolution. Combining the evidence of gametic disequilibrium across several strata may be more sufficient to support the clues, in contrast to analysis with each strata. In this case, it is crucial to test the homogeneity of gametic disequilibrium across strata before combining the data. For this purpose, it is interesting to consider the following hypothesis
H_{0} : D_{1} = ⋯ = D_{ K } versus H_{1} : D_{ i }≠ D_{ j }for at least a pair i ≠ j. (1)
Weir [9] recommended a homogeneity test on gametic disequilibrium, based on Fisher's test of homogeneity among correlation coefficients [10]. In his method, the gametic disequilibrium D_{ k }is first transformed to a correlation coefficient r_{ k }by r_{ k }= D_{ k }/$\sqrt{{p}_{1+k}{p}_{0+k}{p}_{+1k}{p}_{+0k}}$, r_{ k }is then transformed to a normal variable z_{ k }by Fisher's z transformation, and a weighted sum of squares of the z values which has χ^{2} distribution with K - 1 degrees of freedom is finally proposed for testing homogeneity of gametic disequilibrium. As pointed out by Zapata and Alvarez [8], this test is actually for homogeneity of r values instead of D values. They may not be equivalent when the allele probabilities are different across strata. Instead, Zapata and Alvarez [8] suggested the use of the normalized difference D' [11]. Specifically, ${{D}^{\prime}}_{k}$ is the ratio of D_{ k }to D_{k,max}when D_{ k }> 0, or the ratio of D_{ k }to -D_{k,min}when D_{ k }< 0. Zapata and Alvarez obtained the bias-corrected confidence interval for each D' value across strata via the bootstrap method. Hence, acceptance or rejection of homogeneity of D' values can be determined by evaluating the obtained confidence intervals. For the example considered in Zapata and Alvarez [8], there is no intersection for the confidence intervals obtained from all strata. Hence, one has evidence to reject the null hypothesis of homogeneity. Unfortunately, Zapata and Alvarez [8] did not discuss the decision rules for cases such as intersections exist but the extent are different. Hence, no rigorous rule based on this confidence interval approach was proposed and this makes their method less practicable. However, no rigorous rule based on this confidence interval approach was proposed and this makes their method less practicable. It should be noted that the homogeneity test of either r values or D' values is not equivalent to the homogeneity test of D values. In particular, transformation D' only guarantees that the range of D' is [-1, 1]. However, there remains difficulties in interpreting the value of D'. Lewontin [11] noted that values of D' at different loci and in different populations tend to vary with the values of the allele probabilities, so that the problem of cross-locus and cross-population comparisons is not fully overcome by the use of D'. In this article, without doing any transformation, we develop an asymptotic homogeneity test directly based on D values via score method.
Methods
Homogeneity test
which asymptotically follows the chi-square distribution with K - 1 degrees of freedom under H_{0}.
Here, D* is analogous to the well-known Mantel-Haenszel estimator [13]. It is a consistent estimator to D. In general, it is not an efficient estimator to D. The proof of consistency and the conditions for achieving asymptotic efficiency for D* is presented in Appendix. We notice that the calculation of ${I}_{kD|{p}_{1+k}{p}_{+1k}}$ in (2) is quite tedious. Nonetheless, it is easy to show that ${I}_{kD|{p}_{1+k}{p}_{+1k}}$ is simply given by n_{ k }/w_{ k }(D, p_{1+k}, p_{+1k}) with ${w}_{k}(D,{p}_{1+k},{p}_{+1k})={p}_{11k}{p}_{00k}^{2}+{p}_{10k}{p}_{01k}^{2}+{p}_{01k}{p}_{10k}^{2}+{p}_{00k}{p}_{11k}^{2}-4{D}^{2}$ (see Appendix for the proof). It can be shown that X^{2*} has an asymptotic chi-square distribution with K - 1 degrees of freedom under H_{0}. Therefore, the homogeneity hypothesis H_{0} is rejected at level α when X^{2*} ≥ ${\chi}_{K-1,(1-\alpha )}^{2}$, where ${\chi}_{K-1,(1-\alpha )}^{2}$ is the 100 × (1 - α) percentile point of the chi-square distribution with K - 1 degrees of freedom. Finally, it is noteworthy that if the consistent estimators of D, p_{1+} and p_{+1} are the constrained MLEs under H_{0} then the second term of (2) vanishes, since ${\sum}_{k=1}^{K}{S}_{kD}({D}^{\ast},{p}_{1+k}^{\ast},{p}_{+1k}^{\ast})=0$, and (2) reduces to the likelihood score statistic.
Asymptotic power and sample size
where ${\chi}_{K-1,\beta}^{2}(\Delta )$ is the 100 × β percentile point of the non-central chi-square distribution with K - 1 degrees of freedom and non-centrality parameter Δ. The sample size n can be readily obtained by solving the above equation.
Availability and requirements
We have implemented the test procedures for computing our score statistic X^{2*} in a Matlab project. Project name: gametic disequilibrium homogeneity score test (GDHST); Project home page: http://math.nenu.edu.cn/jhguo/program.htm; Operating system: Windows XP; Programming language: Matlab 6.1; Licence: GNU GPL.
Results
Simulation results
where K is the total number of strata, n_{ k }is the total gamete number in stratum k, ${z}_{k}=\frac{1}{2}\mathrm{ln}\phantom{\rule{0.1em}{0ex}}(\frac{1+{r}_{k}}{1-{r}_{k}})$ is the Fisher's z transformation with ${r}_{k}=\frac{{n}_{k}{x}_{11k}-{x}_{1+k}{x}_{+1k}}{\sqrt{{x}_{0+k}{x}_{+0k}{x}_{1+k}{x}_{+1k}}}$ and (x_{00k}, x_{01k}, x_{10k}, x_{11k})' being the number of the gamete array in the k-th stratum, and $\overline{z}$ is the average of the z_{ k }values.
We investigate the performance of X^{2*} and T^{2} in terms of type I error rate and power. For type I error rates, we consider both equal and unequal allele probabilities varying from 0.1 to 0.5 across (K = 3 and 5) strata with equal sample sizes (n_{ k }= 50, 100 and 200) for k = 1,...,K and common disequilibrium (D = $\frac{1}{2}$D_{ min }, 0 and $\frac{1}{2}$D_{ max }), where D_{ min }= max{D_{1,min},...,D_{K,min}}, D_{ max }= min{D_{1,max},...,D_{K,max}}.
Empirical type I error rates for X^{2*} and T^{2} for equal allele probabilities across K = 3 strata under H_{0}
n | D | p_{1+} | p_{+1} | X ^{2*} | T ^{2} |
---|---|---|---|---|---|
50, 50, 50 | -0.125 | 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5 | 0.047 | 0.110 |
0.0 | 0.055 | 0.052 | |||
0.125 | 0.076 | 0.116 | |||
-0.075 | 0.5, 0.5, 0.5 | 0.3, 0.3, 0.3 | 0.051 | 0.061 | |
0.0 | 0.053 | 0.049 | |||
0.075 | 0.075 | 0.063 | |||
-0.045 | 0.3, 0.3, 0.3 | 0.3, 0.3, 0.3 | 0.042 | 0.021 | |
0.0 | 0.044 | 0.053 | |||
0.105 | 0.100 | 0.167 | |||
-0.025 | 0.5, 0.5, 0.5 | 0.1, 0.1, 0.1 | 0.061 | 0.036 | |
0.0 | 0.067 | 0.041 | |||
0.025 | 0.089 | 0.029 | |||
-0.015 | 0.3, 0.3, 0.3 | 0.1, 0.1, 0.1 | 0.024 | 0.017 | |
0.0 | 0.025 | 0.047 | |||
0.035 | 0.104 | 0.117 | |||
-0.005 | 0.1, 0.1, 0.1 | 0.1, 0.1, 0.1 | 0.031 | 0.024 | |
0.0 | 0.024 | 0.087 | |||
0.045 | 0.413 | 0.515 | |||
100, 100, 100 | -0.125 | 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5 | 0.049 | 0.106 |
0.0 | 0.052 | 0.051 | |||
0.125 | 0.065 | 0.112 | |||
-0.075 | 0.5, 0.5, 0.5 | 0.3, 0.3, 0.3 | 0.051 | 0.059 | |
0.0 | 0.049 | 0.051 | |||
0.075 | 0.055 | 0.063 | |||
-0.045 | 0.3, 0.3, 0.3 | 0.3, 0.3, 0.3 | 0.046 | 0.024 | |
0.0 | 0.048 | 0.051 | |||
0.105 | 0.048 | 0.156 | |||
-0.025 | 0.5, 0.5, 0.5 | 0.1, 0.1, 0.1 | 0.053 | 0.029 | |
0.0 | 0.048 | 0.047 | |||
0.025 | 0.089 | 0.030 | |||
-0.015 | 0.3, 0.3, 0.3 | 0.1, 0.1, 0.1 | 0.026 | 0.013 | |
0.0 | 0.029 | 0.048 | |||
0.035 | 0.075 | 0.109 | |||
-0.005 | 0.1, 0.1, 0.1 | 0.1, 0.1, 0.1 | 0.012 | 0.014 | |
0.0 | 0.013 | 0.057 | |||
0.045 | 0.278 | 0.474 | |||
200, 200, 200 | -0.125 | 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5 | 0.050 | 0.050 |
0.0 | 0.052 | 0.048 | |||
0.125 | 0.058 | 0.112 | |||
-0.075 | 0.5, 0.5, 0.5 | 0.3, 0.3, 0.3 | 0.050 | 0.057 | |
0.0 | 0.050 | 0.049 | |||
0.075 | 0.054 | 0.058 | |||
-0.045 | 0.3, 0.3, 0.3 | 0.3, 0.3, 0.3 | 0.049 | 0.025 | |
0.0 | 0.049 | 0.049 | |||
0.105 | 0.052 | 0.156 | |||
-0.025 | 0.5, 0.5, 0.5 | 0.1, 0.1, 0.1 | 0.053 | 0.030 | |
0.0 | 0.050 | 0.051 | |||
0.025 | 0.052 | 0.032 | |||
-0.015 | 0.3, 0.3, 0.3 | 0.1, 0.1, 0.1 | 0.044 | 0.014 | |
0.0 | 0.044 | 0.051 | |||
0.035 | 0.045 | 0.105 | |||
-0.005 | 0.1, 0.1, 0.1 | 0.1, 0.1, 0.1 | 0.020 | 0.010 | |
0.0 | 0.020 | 0.049 | |||
0.045 | 0.098 | 0.463 |
Empirical type I error rates for X^{2*} and T^{2} for unequal allele probabilities across K = 3 strata under H_{0}
n | D | p_{1+} | p_{+1} | X ^{2*} | T ^{2} |
---|---|---|---|---|---|
50, 50, 50 | -0.045 | 0.5, 0.4, 0.3 | 0.5, 0.4, 0.3 | 0.048 | 0.044 |
0.0 | 0.049 | 0.051 | |||
0.105 | 0.071 | 0.149 | |||
-0.015 | 0.5, 0.4, 0.3 | 0.5, 0.3, 0.1 | 0.046 | 0.037 | |
0.0 | 0.046 | 0.051 | |||
0.035 | 0.066 | 0.105 | |||
-0.005 | 0.5, 0.3, 0.1 | 0.5, 0.3, 0.1 | 0.063 | 0.036 | |
0.0 | 0.053 | 0.052 | |||
0.045 | 0.100 | 0.474 | |||
-0.015 | 0.5, 0.4, 0.3 | 0.3, 0.2, 0.1 | 0.040 | 0.032 | |
0.0 | 0.042 | 0.049 | |||
0.035 | 0.074 | 0.101 | |||
-0.005 | 0.5, 0.3, 0.1 | 0.3, 0.2, 0.1 | 0.056 | 0.033 | |
0.0 | 0.048 | 0.054 | |||
0.045 | 0.108 | 0.452 | |||
-0.005 | 0.3, 0.2, 0.1 | 0.3, 0.2, 0.1 | 0.048 | 0.031 | |
0.0 | 0.041 | 0.053 | |||
0.045 | 0.123 | 0.452 | |||
100, 100, 100 | -0.045 | 0.5, 0.4, 0.3 | 0.5, 0.4, 0.3 | 0.051 | 0.048 |
0.0 | 0.050 | 0.050 | |||
0.105 | 0.055 | 0.162 | |||
-0.015 | 0.5, 0.4, 0.3 | 0.5, 0.3, 0.1 | 0.047 | 0.043 | |
0.0 | 0.046 | 0.050 | |||
0.035 | 0.054 | 0.150 | |||
-0.005 | 0.5, 0.3, 0.1 | 0.5, 0.3, 0.1 | 0.064 | 0.037 | |
0.0 | 0.052 | 0.051 | |||
0.045 | 0.059 | 0.658 | |||
-0.015 | 0.5, 0.4, 0.3 | 0.3, 0.2, 0.1 | 0.044 | 0.035 | |
0.0 | 0.047 | 0.051 | |||
0.035 | 0.051 | 0.124 | |||
-0.005 | 0.5, 0.3, 0.1 | 0.3, 0.2, 0.1 | 0.055 | 0.033 | |
0.0 | 0.052 | 0.053 | |||
0.045 | 0.063 | 0.623 | |||
-0.005 | 0.3, 0.2, 0.1 | 0.3, 0.2, 0.1 | 0.055 | 0.033 | |
0.0 | 0.043 | 0.051 | |||
0.045 | 0.061 | 0.593 | |||
200, 200, 200 | -0.045 | 0.5, 0.4, 0.3 | 0.5, 0.4, 0.3 | 0.050 | 0.052 |
0.0 | 0.050 | 0.052 | |||
0.105 | 0.053 | 0.211 | |||
-0.015 | 0.5, 0.4, 0.3 | 0.5, 0.3, 0.1 | 0.049 | 0.049 | |
0.0 | 0.048 | 0.049 | |||
0.035 | 0.049 | 0.220 | |||
-0.005 | 0.5, 0.3, 0.1 | 0.5, 0.3, 0.1 | 0.058 | 0.037 | |
0.0 | 0.051 | 0.050 | |||
0.045 | 0.048 | 0.860 | |||
-0.015 | 0.5, 0.4, 0.3 | 0.3, 0.2, 0.1 | 0.048 | 0.043 | |
0.0 | 0.049 | 0.048 | |||
0.035 | 0.050 | 0.190 | |||
-0.005 | 0.5, 0.3, 0.1 | 0.3, 0.2, 0.1 | 0.056 | 0.035 | |
0.0 | 0.051 | 0.051 | |||
0.045 | 0.053 | 0.829 | |||
-0.005 | 0.3, 0.2, 0.1 | 0.3, 0.2, 0.1 | 0.051 | 0.034 | |
0.0 | 0.050 | 0.050 | |||
0.045 | 0.049 | 0.791 |
Empirical type I error rates for X^{2*} and T^{2} for equal allele probabilities across K = 5 strata under H_{0}
n | D | p_{1+} | p_{+1} | X ^{2*} | T ^{2} |
---|---|---|---|---|---|
50, 50, 50, 50, 50 | -0.125 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.041 | 0.146 |
0.0 | 0.053 | 0.048 | |||
0.125 | 0.090 | 0.148 | |||
-0.075 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.049 | 0.059 | |
0.0 | 0.053 | 0.051 | |||
0.075 | 0.084 | 0.063 | |||
-0.045 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.036 | 0.018 | |
0.0 | 0.039 | 0.048 | |||
0.105 | 0.172 | 0.228 | |||
-0.025 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.075 | 0.029 | |
0.0 | 0.071 | 0.038 | |||
0.025 | 0.059 | 0.026 | |||
-0.015 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.015 | 0.012 | |
0.0 | 0.028 | 0.043 | |||
0.035 | 0.136 | 0.147 | |||
-0.005 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.025 | 0.024 | |
0.0 | 0.026 | 0.079 | |||
0.045 | 0.500 | 0.715 | |||
100, 100, 100, 100, 100 | -0.125 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.046 | 0.135 |
0.0 | 0.051 | 0.050 | |||
0.125 | 0.068 | 0.141 | |||
-0.075 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.050 | 0.059 | |
0.0 | 0.051 | 0.051 | |||
0.075 | 0.054 | 0.059 | |||
-0.045 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.046 | 0.020 | |
0.0 | 0.044 | 0.052 | |||
0.105 | 0.051 | 0.212 | |||
-0.025 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.063 | 0.030 | |
0.0 | 0.058 | 0.048 | |||
0.025 | 0.057 | 0.029 | |||
-0.015 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.026 | 0.009 | |
0.0 | 0.025 | 0.047 | |||
0.035 | 0.088 | 0.137 | |||
-0.005 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.016 | 0.007 | |
0.0 | 0.008 | 0.006 | |||
0.045 | 0.476 | 0.667 | |||
200, 200, 200, 200, 200 | -0.125 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.051 | 0.134 |
0.0 | 0.049 | 0.049 | |||
0.125 | 0.061 | 0.133 | |||
-0.075 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.051 | 0.055 | |
0.0 | 0.049 | 0.050 | |||
0.075 | 0.054 | 0.059 | |||
-0.045 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.053 | 0.022 | |
0.0 | 0.050 | 0.051 | |||
0.105 | 0.050 | 0.203 | |||
-0.025 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.054 | 0.027 | |
0.0 | 0.048 | 0.049 | |||
0.025 | 0.053 | 0.028 | |||
-0.015 | 0.3, 0.3, 0.3, 0.3, 0.3 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.037 | 0.008 | |
0.0 | 0.037 | 0.049 | |||
0.035 | 0.044 | 0.123 | |||
-0.005 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.1, 0.1, 0.1, 0.1, 0.1 | 0.017 | 0.007 | |
0.0 | 0.018 | 0.053 | |||
0.045 | 0.193 | 0.651 |
Empirical type I error rates for X^{2*} and T^{2} for unequal allele probabilities across K = 5 strata under H_{0}
n | D | p_{1+} | p_{+1} | X ^{2*} | T ^{2} |
---|---|---|---|---|---|
50, 50, 50, 50, 50 | -0.045 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.048 | 0.049 |
0.0 | 0.048 | 0.049 | |||
0.105 | 0.085 | 0.163 | |||
-0.025 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.047 | 0.041 | |
0.0 | 0.057 | 0.054 | |||
0.025 | 0.077 | 0.060 | |||
-0.005 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.048 | 0.035 | |
0.0 | 0.042 | 0.057 | |||
0.045 | 0.133 | 0.489 | |||
-0.015 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.045 | 0.037 | |
0.0 | 0.045 | 0.053 | |||
0.035 | 0.082 | 0.100 | |||
-0.015 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.032 | 0.023 | |
0.0 | 0.034 | 0.048 | |||
0.035 | 0.104 | 0.136 | |||
-0.005 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.045 | 0.034 | |
0.0 | 0.035 | 0.057 | |||
0.045 | 0.160 | 0.475 | |||
100, 100, 100, 100, 100 | -0.045 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.050 | 0.051 |
0.0 | 0.051 | 0.051 | |||
0.105 | 0.060 | 0.190 | |||
-0.025 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.050 | 0.050 | |
0.0 | 0.051 | 0.051 | |||
0.025 | 0.062 | 0.067 | |||
-0.005 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.050 | 0.037 | |
0.0 | 0.041 | 0.049 | |||
0.045 | 0.058 | 0.653 | |||
-0.015 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.046 | 0.040 | |
0.0 | 0.047 | 0.051 | |||
0.035 | 0.054 | 0.118 | |||
-0.015 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.037 | 0.027 | |
0.0 | 0.037 | 0.050 | |||
0.035 | 0.060 | 0.168 | |||
-0.005 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.047 | 0.035 | |
0.0 | 0.038 | 0.053 | |||
0.045 | 0.060 | 0.610 | |||
200, 200, 200, 200, 200 | -0.045 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.049 | 0.050 |
0.0 | 0.048 | 0.049 | |||
0.105 | 0.053 | 0.230 | |||
-0.025 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.049 | 0.062 | |
0.0 | 0.051 | 0.050 | |||
0.025 | 0.053 | 0.081 | |||
-0.005 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.051 | 0.039 | |
0.0 | 0.043 | 0.048 | |||
0.045 | 0.052 | 0.865 | |||
-0.015 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.048 | 0.044 | |
0.0 | 0.049 | 0.049 | |||
0.035 | 0.048 | 0.169 | |||
-0.015 | 0.5, 0.4, 0.3, 0.2, 0.1 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.042 | 0.029 | |
0.0 | 0.045 | 0.053 | |||
0.035 | 0.045 | 0.229 | |||
-0.005 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.3, 0.2, 0.1, 0.2, 0.3 | 0.047 | 0.035 | |
0.0 | 0.040 | 0.053 | |||
0.045 | 0.051 | 0.815 |
1. When D is large (i.e., $\frac{1}{2}$D_{ max }), both tests generally appear to be quite liberal (e.g., empirical size being 10 times of the nominal level), especially for small sample size (e.g., n_{ k }= 50) and small allele probability (e.g., p_{1+} = p_{+1} = (0.1, 0.1, 0.1)'). Such liberty in empirical size is more severe in T^{2} than in our asymptotic homogeneity test X^{2*} and is significantly alleviated in X^{2*} when sample size increases. However, sample size increase does not alleviate the liberty of T^{2} much. In fact, even for n_{ k }= 3200 for k = 1, 2, 3, T^{2} is still very liberal for D = 0.045 with empirical type I errors rate being 0.456 (data are not shown).
2. For other settings, both tests perform quite satisfactorily in the sense that their empirical sizes are well controlled around the pre-chosen nominal level. In general, the larger the sample size, the closer the empirical type I error rate to the pre-chosen nominal level.
Table 2 reports the empirical size performance of X^{2*} and T^{2} for unequal allele probabilities across K = 3 strata. We observe similar phenomena above. However, our asymptotic homogeneity test X^{2*} performs quite well in all settings under consideration for moderate to large sample sizes (i.e., n_{ k }= 100 and 200) while it is not the case for T^{2}. For T^{2}, the resultant empirical type I error rate can be extremely inflated even for large sample design (e.g., more than 17 times of the nominal level when n_{ k }= 200 (for k = 1, 2, 3), p_{1+} = p_{+1} = (0.5, 0.3, 0.1)', and D = 0.045).
Table 3 and 4 shows the empirical type I error rate performance of X^{2*} and T^{2} for K = 5. The parameter settings are similar to Table 1 and 2. According to the simulation results, liberty issue becomes more serious and larger sample sizes are required to attain similar performance when K increases from 3 to 5 under similar parameter settings.
where m = 5000 and W represents the empirical type I error rate of X^{2*} or T^{2}. Here, the t-test is almost identical to the z-test for the sample size is very large. Those empirical type I error rates which are significantly different from the nominal level of 0.05 are underlined in Tables 1 to 4. In Table 1, the total number of significant difference from the nominal level of 0.05 for X^{2*} and T^{2} is 28 and 38, respectively. The pair (28, 38) can be further decomposed to (14, 14), (8, 13) and (6, 11) according to n = 50, 100 and 200. The decreasing rate of the number of empirical type I error rates which is significant different from the nominal level of 0.05 for X^{2*} is 14/18-6/18 = 44.4% as n increases from 50 to 200. While the corresponding decreasing rate for T^{2} is 14/18-11/18 = 16.7%. It is easy to see that our X^{2*} is less liberal than T^{2} as sample size increases.
For Table 2, the total number of significant difference from the nominal level of 0.05 for X^{2*} and T^{2} is 17 and 33, respectively. The pair (17, 33) can again be decomposed to (14, 14), (8, 13) and (6, 11) according to n = 50, 100 and 200. The decreasing rate of the number of empirical type I error rates which is significant different from the nominal level of 0.05 for X^{2*} is 10/18-1/18 = 50.0% as n increases from 50 to 200. While the decreasing rate for T^{2} is 12/18-10/18 = 11.1%. The decreasing rate of our X^{2*} is again more significant than that of T^{2}.
In Table 3 to 4, the strata increases from 3 to 5. However, the decreasing rates of the number of empirical type I error rates which is significant different from the nominal level of 0.05 for Tables 3 and 4 is very close to that of Tables 1 and 2, respectively. Therefore, we have reason to believe that this decreasing rate is not greatly affected by the number of strata.
Empirical powers for X^{2*} and T^{2}
n | D | p_{1+} | p_{+1} | X ^{2*} | T ^{2} |
---|---|---|---|---|---|
50, 50, 50 | -0.03, 0.0, 0.03 | 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5 | 0.205 | 0.210 |
100, 100, 100 | 0.311 | 0.306 | |||
200, 200, 200 | 0.560 | 0.558 | |||
50, 50, 50 | -0.03, 0.0, 0.03 | 0.5, 0.4, 0.3 | 0.5, 0.4, 0.3 | 0.196 | 0.203 |
100, 100, 100 | 0.360 | 0.368 | |||
200, 200, 200 | 0.630 | 0.641 | |||
50, 50, 50 | -0.03, 0.0, 0.03 | 0.5, 0.5, 0.5 | 0.5, 0.4, 0.3 | 0.197 | 0.187 |
100, 100, 100 | 0.354 | 0.340 | |||
200, 200, 200 | 0.612 | 0.612 | |||
50, 50, 50 | -0.05, 0.0, 0.05 | 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5 | 0.430 | 0.421 |
100, 100, 100 | 0.724 | 0.720 | |||
200, 200, 200 | 0.958 | 0.958 | |||
50, 50, 50 | -0.05, 0.0, 0.05 | 0.5, 0.4, 0.3 | 0.5, 0.4, 0.3 | 0.474 | 0.483 |
100, 100, 100 | 0.797 | 0.806 | |||
200, 200, 200 | 0.980 | 0.981 | |||
50, 50, 50 | -0.05, 0.0, 0.05 | 0.5, 0.5, 0.5 | 0.5, 0.4, 0.3 | 0.457 | 0.446 |
100, 100, 100 | 0.763 | 0.760 | |||
200, 200, 200 | 0.973 | 0.973 | |||
50, 50, 50, 50, 50 | -0.06, -0.03, 0.0, 0.03, 0.06 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.523 | 0.512 |
100, 100, 100, 100, 100 | 0.846 | 0.841 | |||
200, 200, 200, 200, 200 | 0.993 | 0.993 | |||
50, 50, 50, 50, 50 | -0.06, -0.03, 0.0, 0.03, 0.06 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.526 | 0.526 |
100, 100, 100, 100, 100 | 0.854 | 0.853 | |||
200, 200, 200, 200, 200 | 0.995 | 0.995 | |||
50, 50, 50, 50, 50 | -0.06, -0.03, 0.0, 0.03, 0.06 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.535 | 0.522 |
100, 100, 100, 100, 100 | 0.855 | 0.850 | |||
200, 200, 200, 200, 200 | 0.994 | 0.994 | |||
50, 50, 50, 50, 50 | -0.1, -0.05, 0.0, 0.05, 0.1 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.957 | 0.953 |
100, 100, 100, 100, 100 | 1.000 | 1.000 | |||
200, 200, 200, 200, 200 | 1.000 | 1.000 | |||
50, 50, 50, 50, 50 | -0.1, -0.05, 0.0, 0.05, 0.1 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.960 | 0.960 |
100, 100, 100, 100, 100 | 1.000 | 1.000 | |||
200, 200, 200, 200, 200 | 1.000 | 1.000 | |||
50, 50, 50, 50, 50 | -0.1, -0.05, 0.0, 0.05, 0.1 | 0.5, 0.5, 0.5, 0.5, 0.5 | 0.5, 0.4, 0.3, 0.4, 0.5 | 0.957 | 0.954 |
100, 100, 100, 100, 100 | 1.000 | 1.000 | |||
200, 200, 200, 200, 200 | 1.000 | 1.000 |
In view of the above results, we prefer the proposed homogeneity test X^{2*} to the traditional T^{2} which is based on the Fisher's test of homogeneity among correlation coefficient.
Real and hypothetical examples
It is reported that mutations at the cystic fibrosis transmembrane conductance regulator gene (CFTR) cause cystic fibrosis, the most prevalent severe genetic disorder in individuals of European descent. Mateu [15] conducted a worldwide genetic analysis of the CFTR region and analyzed normal allele and haplotype variation at two single-nucleotide polymorphisms (SNPs), namely the T854/Ava II (2694 T/G) and TUB20/PVU II (4006-200 G/A). The T854 and TUB20 markers can be used to define the core haplotypes since they are diallelic, have presumably much lower mutation rates than the other polymorphisms and the ancestral state can be inferred for them.
T854-TUB20 haplotype counts by 18 populations and some related statistics
Gametic counts | Allele frequencies | Disequilibrium | Estimation | ||||||
---|---|---|---|---|---|---|---|---|---|
Population | 1 - 1 | 1 - 2 | 2 - 1 | 2 - 2 | 1 | 1 | r | D' | D |
Africa: | |||||||||
Biaka | 5 | 16 | 12 | 29 | 0.339 | 0.274 | -0.058 | -0.132 | -0.012 |
Mbuti | 0 | 14 | 5 | 14 | 0.424 | 0.152 | -0.363 | -1.000 | -0.064 |
Tanzanian | 0 | 13 | 3 | 20 | 0.361 | 0.083 | -0.227 | -1.000 | -0.030 |
North Africa: | |||||||||
Saharawi | 5 | 22 | 12 | 16 | 0.491 | 0.309 | -0.263 | -0.401 | -0.061 |
Middle East: | |||||||||
Yemenites | 2 | 29 | 4 | 5 | 0.775 | 0.150 | -0.444 | -0.570 | -0.066 |
Druze | 2 | 47 | 10 | 4 | 0.778 | 0.191 | -0.713 | -0.786 | -0.116 |
Europe: | |||||||||
Adygei | 1 | 34 | 9 | 5 | 0.714 | 0.204 | -0.689 | -0.860 | -0.125 |
Russians | 0 | 17 | 10 | 5 | 0.531 | 0.313 | -0.718 | -1.000 | -0.166 |
Finns | 0 | 23 | 6 | 4 | 0.697 | 0.182 | -0.715 | -1.000 | -0.127 |
Catalans | 3 | 53 | 18 | 9 | 0.675 | 0.253 | -0.661 | -0.788 | -0.135 |
Basques | 4 | 72 | 15 | 17 | 0.704 | 0.176 | -0.500 | -0.701 | -0.087 |
Asia: | |||||||||
Kazakhs | 1 | 18 | 2 | 12 | 0.576 | 0.091 | -0.155 | -0.421 | -0.022 |
Chinese | 0 | 22 | 1 | 20 | 0.512 | 0.023 | -0.158 | -1.000 | -0.012 |
Japanese | 0 | 32 | 0 | 12 | 0.727 | 0 | NaN | NaN | 0 |
Yakut | 0 | 18 | 1 | 4 | 0.783 | 0.044 | -0.405 | -1.000 | -0.034 |
Pacific: | |||||||||
Nasioi | 1 | 20 | 0 | 22 | 0.488 | 0.023 | 0.158 | 1.000 | 0.012 |
America: | |||||||||
Maya | 2 | 15 | 0 | 31 | 0.354 | 0.042 | 0.282 | 1.000 | 0.027 |
Surui | 0 | 7 | 0 | 35 | 0.167 | 0 | NaN | NaN | 0 |
It is noticed that the gametic counts for the populations of Japanese (14th) and Surui (18th) are (0, 32, 0, 12)' and (0, 7, 0, 35)', respectively and their estimated gametic disequilibrium D_{ k }, D_{k,min}and D_{k,max}are all equal to zero. Therefore, we will exclude these two populations for subsequent homogeneity testings. We consider the following scenarios.
(i) Homogeneity of gametic disequilibrium among the 16 populations (i.e., excluding Japanese and Surui). The statistic value of our proposed X^{2*} is 121.35 with p-value being less than 0.0001 while that of T^{2} yields 99.64 with p-value being less than 0.0001. In this case, both tests reject the homogeneity hypothesis at the 0.05 nominal level.
(ii) Homogeneity of gametic disequilibrium among those populations with the same numbers of participants for both markers T854 and TUB20 (i.e., Mbuti, Yemenites, Druze, Adygei, Catalans, Basques, Chinese, and Nasioi).
Our proposed statistic X^{2*} yields 50.56 with p-value being less than 0.0001 while T^{2} gives 39.72 with p-value being less than 0.0001. Again, both tests suggest rejection of the homogeneity hypothesis at the 0.05 nominal level. Suppose that another research team wants to reconduct the same genetic analysis. In this regard, it is sensible to ask, "How large is the sample size for each population in order to achieve, say, 90% power at the 0.05 nominal level". Based on the present study, we have $\overline{D}$ = (-0.064, -0.066, -0.116, -0.125, -0.135, -0.087, -0.012, 0.012)', ${\overline{p}}_{1+}$ = (0.576, 0.225, 0.222, 0.286, 0.325, 0.296, 0.488, 0.512)' and ${\overline{p}}_{+1}$ = (0.849 0.850, 0.810, 0.796, 0.747, 0.824, 0.977, 0.977)'. By solving equation (3), n = 157 subjects are required for each of the eight populations under the balanced design.
(iii) Homogeneity of gametic disequilibrium among those populations in Europe.
Our statistic X^{2*} yields 7.48 with p-value being 0.11 and T^{2} yields 7.26 with p-value being 0.12. Both tests do not reject the homogeneity hypothesis at the 0.05 nominal level. In this case, we have evidence to believe that populations in Europe reach their gametic equilibrium.
Hypothetical example of gametic disequilibrium between two loci (A, B) with twoalleles (A_{0}, A_{1} and B_{0}, B_{1}, respectively) across ten populations
Gamete counts | Allele frequencies | Disequilibrium | Estimation | ||||||
---|---|---|---|---|---|---|---|---|---|
Population | A _{0} B _{0} | A _{0} B _{1} | A _{1} B _{0} | A _{1} B _{1} | A _{0} | B _{0} | r | D' | D |
1 | 495 | 405 | 5 | 95 | 0.90 | 0.50 | 0.300 | 0.900 | 0.045 |
2 | 540 | 360 | 1 | 90 | 0.90 | 0.55 | 0.300 | 0.814 | 0.045 |
3 | 479 | 371 | 21 | 129 | 0.85 | 0.50 | 0.300 | 0.714 | 0.054 |
4 | 460 | 340 | 40 | 160 | 0.80 | 0.50 | 0.300 | 0.600 | 0.060 |
5 | 671 | 229 | 29 | 71 | 0.90 | 0.70 | 0.300 | 0.589 | 0.041 |
6 | 539 | 261 | 61 | 139 | 0.80 | 0.60 | 0.300 | 0.490 | 0.059 |
7 | 615 | 185 | 85 | 115 | 0.80 | 0.70 | 0.300 | 0.393 | 0.055 |
8 | 373 | 227 | 127 | 273 | 0.60 | 0.50 | 0.300 | 0.367 | 0.073 |
9 | 403 | 197 | 147 | 253 | 0.60 | 0.55 | 0.300 | 0.332 | 0.073 |
10 | 325 | 175 | 175 | 325 | 0.50 | 0.50 | 0.300 | 0.300 | 0.075 |
Discussion
Verification of the homogeneity assumption of gametic disequilibrium across several populations is crucial in gametic disequilibrium analysis. We note that traditional homogeneity test on gametic disequilibrium is based on the Fisher's test of homogeneity among correlation coefficients. However, our simulations demonstrate that this traditional test may not perform satisfactorily. Specifically, it can be very conservative or liberal, for almost all the cases in which the common true gametic disequilibrium D is bounded away from zero. Most importantly, these kinds of conservativeness and liberty can not effectively alleviated with increased sample sizes.
Our proposed large-sample homogeneity score test on gametic disequilibrium across several independent populations requires the count of haplotypes as input. In practice, only genotype data can be obtained in most situations. To employ our method, one can use some haplotyping software, such as PHASE, HAPLOTYPER, to resolve the genotype data as haplotype data. In this way, it separates haplotype phasing and gametic disequilibrium homogeneity test. Naturally, it is more promising to extend our method which can directly handle the genotype data. In this sense, model assumptions are based on genotype data. However, the haplotype phase uncertainty for the double heterozygotes makes the definition of gametic disequilibrium can not be directly expressed by the genotype data even assuming Hardy-Weinberg equilibrium holds. It may severely affect the further derivation of the corresponding score test. Thus, extending our method to handle genotype data is an avenue we intend to explore future.
Conclusion
In this article, we propose a large-sample homogeneity test on gametic disequilibrium across several independent populations based on the likelihood score theory generalized to nuisance parameters. Our simulation results show that our test is more reliable than the traditional test based on the Fisher's test of homogeneity among correlation coefficients. Although our test may also demonstrate conservativeness and liberty in some cases, unlike the traditional test these issues can be effectively resolved by increasing sample sizes. For design purpose, sample size formula that controls power is derived.
Appendix
Consistency and the condition to attain asymptotic efficiency for D*
By Cauchy-Schwarz inequality ${({\displaystyle {\sum}_{k=1}^{K}{v}_{k}})}^{2}\le ({\displaystyle {\sum}_{k=1}^{K}{b}_{k}/{w}_{k}})({\displaystyle {\sum}_{k=1}^{K}{w}_{k}{v}_{k}^{2}/{b}_{k})}$, we have AsyVar($\overline{D}$) = AsyVar(D*). To this end, we obtain the sufficient and necessary condition for the asymptotic efficiency of D*, that is, w_{ k }v_{ k }= c, k = 1, 2,...,K, where c is a constant independent of all parameters. When D = 0, the condition is satisfied. From this, we know that D* is inefficient for general cases.
A simple expression for ${I}_{kD|{p}_{1+k}{p}_{+1k}}$
where ${\widehat{D}}_{k},{\widehat{p}}_{1+k}={x}_{1+k}/{n}_{k}$ and ${\widehat{p}}_{+1k}={x}_{+1k}/{n}_{k}$ are the MLEs of D_{ k }, p_{1+k}and p_{+1k}, respectively. Hence, the asymptotic variance of $\sqrt{{n}_{k}}{\widehat{D}}_{k}$ is ${n}_{k}/{I}_{kD|{p}_{1+k}{p}_{+1k}}({D}_{k},{p}_{1+k},{p}_{+1k})$. On the contrary, by the Central Limit Theorem, $\sqrt{{n}_{k}}({y}_{k}-{g}_{k})$ follows an asymptotic normal distribution N(0, Σ_{ k }). By δ method, we immediately get that $\sqrt{{n}_{k}}({\widehat{D}}_{k}-{D}_{k})$ follows an asymptotic normal distribution N(0, w_{ k }(D_{ k }, p_{1+k}, p_{+1k})). Therefore, we can obtain the exact expression ${I}_{kD|{p}_{1+k}{p}_{+1k}}$(D_{ k }, p_{1+k}, p_{+1k}) = n_{ k }/w_{ k }(D_{ k }, p_{1+k}, p_{+1k}). Naturally, the expression of ${I}_{kD|{p}_{1+k}{p}_{+1k}}$(D, p_{1+k}, p_{+1k}) is just ${I}_{kD|{p}_{1+k}{p}_{+1k}}$(D_{ k }, p_{1+k}, p_{+1k}) by substituting D for D_{ k }.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant Numbers 10431010 and 10701022), National 973 Key Project of China (2007CB311002), NCET-04-0310, EYTP, the Jilin Distinguished Young Scholars Program (Grant Number 20030113) and the Program Innovative Research Team (PCSIRT) in University (#IRT0519). The work of ML Tang was fully supported by a grant from the Research Grant Council of the Hong Kong Special Administration (Project no. HKBU261007).
Authors’ Affiliations
References
- Lewontin RC: The genetic basis of evolutionary change. 1974, New York: Columbia University PressGoogle Scholar
- Jorde LB: Linkage disequilibrium as a gene mapping tool. Am J Hum Genet. 1995, 56: 11-14.PubMed CentralPubMedGoogle Scholar
- Hedrick PW, Jain S, Holden L: Multilocus systems in evolution. Evol Biol. 1978, 11: 101-182.View ArticleGoogle Scholar
- Weir BS: Inferences about linkage disequilibrium. Biometrics. 1979, 35: 235-254. 10.2307/2529947.View ArticlePubMedGoogle Scholar
- Hedrick PW: Gametic disequilibrium measures: proceed with caution. Genetics. 1987, 117: 331-341.PubMed CentralPubMedGoogle Scholar
- Mueller JC: Linkage disequilibrium for different scales and applications. Brief Bioinform. 2004, 5: 355-364. 10.1093/bib/5.4.355.View ArticlePubMedGoogle Scholar
- Lewontin RC, Kojima K: The evolutionary dynamics of complex polymorphisms. Evolution. 1960, 14: 458-472. 10.2307/2405995.View ArticleGoogle Scholar
- Zapata C, Alvarez G: Testing for homogeneity of gametic disequilibrium among populations. Evolution. 1997, 51: 606-607. 10.2307/2411132.View ArticleGoogle Scholar
- Weir BS: Genetic Data Analysis II. 1996, Sunderland, Massachusetts: Sinauer AssociatesGoogle Scholar
- Fisher RA: Statistical methods for research workers. 1925, New York: Oliver and BoydGoogle Scholar
- Lewontin RC: The interaction of selection and linkage. I. General considerations; heterotic models. Genetics. 1964, 49: 49-67.PubMed CentralPubMedGoogle Scholar
- Tarone RE: Homogeneity score tests with nuisance parameters. Commun Stat-Theor M. 1988, 17: 1549-1556. 10.1080/03610928808829697.View ArticleGoogle Scholar
- Mantel N, Haenszel W: Statistical aspects of the analysis of data from retrospective studies of disease. J Natl Cancer Inst. 1959, 22 (4): 719-748.PubMedGoogle Scholar
- Guo JH, Ma YP, Shi NZ, Lau TS: Testing for homogeneity of relative difference under inverse sampling. Comput Stat Data An. 2004, 44: 613-624. 10.1016/S0167-9473(02)00262-1.View ArticleGoogle Scholar
- Mateu E, Calafell F, Lao O, Batsheva BT, Kidd JR, Pakstis A, Kidd KK, Bertranpetit J: Worldwide genetic analysis of the CFTR region. Am J Hum Genet. 2001, 68: 103-117. 10.1086/316940.PubMed CentralView ArticlePubMedGoogle Scholar
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