- Proceedings
- Open Access
Joint analysis of case-parents trio and unrelated case-control designs in large scale association studies
- Jungnam Joo^{1}Email author,
- Xin Tian^{1},
- Gang Zheng^{1},
- Mario Stylianou^{1},
- Jing-Ping Lin^{1} and
- Nancy L Geller^{1}
https://doi.org/10.1186/1753-6561-1-S1-S28
© Joo et al; licensee BioMed Central Ltd. 2007
Published: 18 December 2007
Abstract
We present a new method for testing association when data from both case-parents trios and unrelated controls are available. Our method combines test statistics for case-parents trio and unrelated case-control studies by adjusting for the correlation that arises when the same set of cases is used for both tests. We further consider several analytical approaches for two-stage studies on a large number of markers, including methods based on the joint analysis. The performance of the proposed approaches is examined by analyzing the simulated data provided by the Genetic Analysis Workshop 15.
Keywords
Background
Genetic association studies are a popular method to detect genetic markers associated with a complex human disease. Two common designs in genetic association studies are family-based designs using case-parents trios and population-based designs using unrelated cases and controls. The transmission disequilibrium test (TDT) is frequently used to analyze the case-parents trio data [1]. The TDT tests for both linkage and association and is not sensitive to population admixture and stratification. Using a likelihood approach, Schaid and Sommer [2] proposed TDT-type statistics that are more powerful than the TDT for a specific genetic model (see also [3]). For the unrelated case-control design, a linear trend test [4], which is often more powerful than the TDT based on case-parents trios, can be considered specifically when obtaining a sufficient number of trios is difficult.
Data that contain both case-parents trios and unrelated cases and controls on the same set of markers are increasingly available. Nagelkerke et al. [5] provided a few situations where such a mixture of case-parents trios and unrelated cases and controls can occur: 1) a case-parents trio design was originally considered and then unrelated controls were added, 2) a case-control design was originally considered and then the parents of the cases were added to confirm the findings. Such designs are typically analyzed in two stages, and strategies for analyzing this type of data while fully utilizing the given information are important.
In this paper, we study several approaches for testing genome-wide association in such situations. Based on the design, either a TDT-type statistic or a linear trend test will be used in the first stage to select a proportion of markers that will be tested in the second stage. The other test will then be applied in the second stage while controlling the genome-wide false positive rates by adjusting for the correlation with the first stage. Following a recently proposed method by Skol et al. [6], we also study a joint analysis for the second stage.
Methods
Consider a marker with two alleles, M and N, where M itself is a risk allele or is in linkage disequilibrium with a risk allele with frequency p, and N is a normal allele with frequency q = 1 - p. Penetrances are defined as the probabilities of disease conditional on the genotypes, that is, f_{0} = Pr(disease|NN), f_{1} = Pr(disease|NM), and f_{2} = Pr(disease|MM). No association implies f_{0} = f_{1} = f_{2}, whereas f_{0} ≤ f_{1} ≤ f_{2} with at least one strict inequality implies there is an association between the marker and a disease. Using f_{0} as a baseline penetrance, the genotype relative risks are defined as ψ_{ i }= f_{ i }/f_{0} for i = 1, 2. A genetic model is recessive, additive, or dominant when f_{0} = f_{1} (or ψ_{1} = 1, ψ_{2} = ψ), f_{1} = (f_{0} + f_{2})/2 (or ψ_{1} = ψ, ψ_{2} = 2ψ-1), or f_{1} = f_{2} (or ψ_{1} = ψ_{2} = ψ).
Case-parents trio design
Conditional probabilities of genotype given parental mating types and offspring disease status
Parental mating type | Case genotype | Count | Probability of trio | Conditional probability |
---|---|---|---|---|
1) MM × MM | MM | n _{12} | p^{4}ψ_{2}/T | 1 |
2) MM × NM | MM | n _{22} | 2p^{3}q(ψ_{1} + ψ_{2})/T | ψ_{2}/(ψ_{1} + ψ_{2}) |
NM | n _{21} | ψ_{1}/(ψ_{1} + ψ_{2}) | ||
3) MM × NN | NM | n _{31} | 2p^{2}q^{2}ψ_{1}/T | 1 |
4) NM × NM | MM | n _{42} | 2p^{2}q^{2}(ψ_{2} + 2ψ_{1} + 1)/T | ψ_{2}/(ψ_{2} + 2ψ_{1} + 1) |
NM | n _{41} | 2ψ_{1}/(ψ_{2} + 2ψ_{1} + 1) | ||
NN | n _{40} | 1/(ψ_{2} + 2ψ_{1} + 1) | ||
5) NM × NN | NM | n _{51} | 2pq^{3}(ψ_{1} + 1)/T | ψ_{1}/(ψ_{1} + 1) |
NN | n _{50} | 1/(ψ_{1} + 1) | ||
6) NN × NN | NN | n _{60} | q^{4}1/T | 1 |
Schaid and Sommer [2] suggested an analysis conditional on parental mating types that provides unbiased estimates of genotype relative risks. Denote the likelihood function for a given model as L(ψ), then the score test for H_{0}: ψ = 1 can be obtained by ∂logL(ψ)/∂ψ/{-∂^{2}logL(ψ)/∂ψ^{2}}^{1/2}|_{ψ = 1}.
Unrelated case-control design
For the unrelated case-control design, denote the genotype counts of three genotypes NN, MN and MM as (r_{0}, r_{1}, r_{2}) in cases and (s_{0}, s_{1}, s_{2}) in controls that follow multinomial distributions mul(R: p_{0}, p_{1}, p_{2}) and mul(S: q_{0}, q_{1}, q_{2}). Then the null hypothesis of no association implies p_{ i }= q_{ i }for each i.
Sasieni [4] proposed a method that uses the marker genotype as a covariate in the logistic regression model where the genotype is coded by increasing scores, that is, 0, x, and 1 for NN, NM, and MM, where 0 ≤ x ≤ 1. The optimal scores for recessive, additive and dominant models are x = 0, 1/2, and 1 [4, 7] and the trend test [7] is given by ${Z}_{CC}=\frac{U(x)}{\sqrt{Var(U(x))}}$, where $U(x)={\displaystyle {\sum}_{i=0}^{2}{x}_{i}(1-R/N){r}_{i}-}{\displaystyle {\sum}_{i=0}^{2}{x}_{i}(R/N){s}_{i}}$, and $Var(U(x))={N}^{-1}RS\left\{{\displaystyle {\sum}_{i}{x}_{i}^{2}{p}_{i}-{({\displaystyle {\sum}_{i}{x}_{i}{p}_{i}})}^{2}}\right\}$ for (x_{0}, x_{1}, x_{2}) = (0, x, 1) and N = R+S. Under the null hypothesis, Z_{ CC }asymptotically follows the standard normal distribution.
Combined test of Z_{ TDT }and Z_{ CC }
Because the cases used in Z_{ TDT }and Z_{ CC }overlap, results from the two tests are correlated, and this correlation, ρ, must be considered when obtaining a combined test. By noting that both tests are functions of a multinomial random variable n with dimension 10 for the 10 n_{ ij }categories from Table 1, the correlation between Z_{ TDT }and Z_{ CC }can be obtained given a specific genetic model (Appendix). The probability of each category can be consistently estimated by the observed counts and ρ can be consistently estimated by the sample correlation between Z_{ TDT }and Z_{ CC }.
We propose the weighted average, ${Z}_{\text{joint}}=\frac{\sqrt{{w}_{1}}{Z}_{TDT}+\sqrt{{w}_{2}}{Z}_{CC}}{\sqrt{({w}_{1}+{w}_{2}+2\sqrt{{w}_{1}{w}_{2}}\rho )}}$, as a test statistic in a joint analysis. We consider a uniform weight, that is, w_{1} = w_{2} = 1 [8, 9] for simplicity. Other choices of weight, such as a weight proportional to the number of informative cases used in each test, can also be considered.
Two-stage method in large scale association studies
when the joint analysis is used or when the other test is used. Here, Z_{1i}and Z_{2i}denote the tests used in the first and the second stage for the i^{th} SNP (Z_{2i}is replaced by Z_{ jointi }when the joint analysis is used in the second stage). We need the subscript i because the correlations between two tests for different SNPs are generally not the same. Under HWE, however, we can show this correlation is a constant (Appendix), and these equations can then be simplified to P(|Z_{1}| > C_{1}, |Z_{joint}| > C_{joint}) = α/K and P(|Z_{1}| > C_{1}, |Z_{2}| > C_{2}, Z_{1}Z_{2} > 0) = α/K.
Data
The Genetic Analysis Workshop 15 provided simulated rheumatoid arthritis data that contain 1500 families with affected sib pairs and their parents, and 2000 unrelated controls on 9187 SNPs distributed throughout the genome. We used the first simulated data set and we randomly selected one from the affected sib pairs for data analysis. The minor allele frequencies of all 9187 SNPs were greater than 1%.
Results
Two-stage analysis: selected SNPs and their corresponding p-values in the second stage
Chromosome | SNP | Distance (Mb) | Z_{ TDT }then Z_{ CC } | Z_{ CC }then Z_{ TDT } | Z_{ TDT }or Z_{ CC }then Z_{joint} |
---|---|---|---|---|---|
6 | SNP6_128 | 7.13 | 1.02 × 10^{-8} | 1.61 × 10^{-7} | |
SNP6_129 | 7.10 | 4.01 × 10^{-13} | 1.45 × 10^{-8} | ||
SNP6_130 | 7.10 | 1.48 × 10^{-13} | 1.28 × 10^{-9} | ||
SNP6_134 | 6.41 | 5.76 × 10^{-10} | 1.33 × 10^{-}6 | ||
SNP6_138 ^{b} | 3.73 | 2.89 × 10 ^{-15} | 2.85 × 10 ^{-7} | 1.07 × 10 ^{-13} | |
SNP6_139 | 3.72 | 2.44 × 10 ^{-15} | 3.75 × 10 ^{-7} | 1.22 × 10 ^{-13} | |
SNP6_145 | 2.92 | 1.03 × 10 ^{-9} | 5.20 × 10 ^{-6} | 1.24 × 10 ^{-9} | |
SNP6_147 | 2.22 | 1.37 × 10 ^{-8} | 3.05 × 10 ^{-7} | ||
SNP6_150 | 1.39 | 5.99 × 10 ^{-7} | 6.25 × 10 ^{-7} | ||
SNP6_152 ^{c} | 0.04 | 2.38 × 10 ^{-221} | 1.55 × 10 ^{-94} | 4.31 × 10 ^{-193} | |
SNP6_153 | 0.01 | 0*^{a} | 7.33 × 10 ^{-206} | 0*^{a} | |
SNP6_154 | 0.04 | 0* | 1.23 × 10 ^{-182} | 0* | |
SNP6_155 | 0.29 | 4.49 × 10 ^{-87} | 1.91 × 10 ^{-49} | 7.36 × 10 ^{-86} | |
SNP6_160 | 0.65 | 8.62 × 19 ^{-10} | 8.20 × 10 ^{-9} | 1.16 × 10 ^{-11} | |
SNP6_162 | 0.13 | 9.15 × 10 ^{-24} | 1.78 × 10 ^{-15} | 9.36 × 10 ^{-26} | |
11 | SNP11_387 | 0.19 | 3.00 × 10 ^{-6} | ||
SNP11_389 | 0.03 | 2.78 × 10 ^{-28} | 3.33 × 10 ^{-15} | 2.41 × 10 ^{-27} | |
18 | SNP18_269 | 0.02 | 2.25 × 10 ^{-8} | 1.56 × 10 ^{-8} |
When we applied these three tests (Z_{ CC }, Z_{ TDT }, Z_{joint}) to a single-stage analysis, these tests found the same set of SNPs identified in a two-stage analysis with the corresponding test at the second stage. That is, Z_{ CC }, Z_{ TDT }, Z_{jointi} in a single-stage found 17, 10, and 18 SNPs in columns 4, 5, and 6 of Table 2. This implies that a two-stage analysis can maintain power with a substantially reduced genotyping cost while controlling the same genome-wide false-positive rate [6].
Discussion
In this paper, we presented a new method for testing association when both case-parents trios and unrelated controls are available. Because parents are selected for having an affected child, we consider the characteristics of non-affected parents to be different from those of unrelated controls in case-control studies. Thus, the genotype information of parents was used only for Z_{ TDT }and not for Z_{ CC }. By adjusting for the correlation between the two test statistics (Z_{ TDT }and Z_{ CC }), we proposed a combined test statistic for analyzing such data.
For data with a large number of markers in a two-stage analysis, we considered several analytical approaches following the method by Skol et al. [6]. Even with a slightly larger threshold required, more SNPs near the major genes were found using the joint analysis in the second stage. Also, we noticed the choice of test for the first stage was important when two separate tests were used in the two stages, but when the joint analysis was used, the impact of which test was used first seemed to be less important. The added benefit of the joint analysis was rather minor compared to what was studied by Skol et al. [6] because the two tests for the first and the second stages were highly correlated even without using the joint analysis. Nevertheless, the joint analysis found slightly more significant SNPs and is robust against the choice of the first stage test. These properties suggest that the joint analysis would be desirable.
Our method can be generalized to data with missing genotypes by either imputing the missing genotypes based on partially available data [5, 10], or by omitting cases without complete parental information from Z_{ TDT }. In this situation, the correlation between Z_{ TDT }and Z_{ CC }will decrease, and therefore, the advantage of the joint analysis could be accentuated. Complete justification, however, requires further study.
Conclusion
We presented a new method for testing association when data from both case-parents trios and unrelated controls are available. By deriving the correlation of test statistics for these two designs, we proposed a combined test as a joint analysis. In a two-stage analysis for testing a large number of markers, we found that the joint analysis detects more SNPs near the major genes than other methods that do not use the combined test in the second stage. This approach is also robust against the choice of the first stage test.
Appendix
When the conditional likelihood is used for Z_{ TDT }, n_{1} = n_{12}, n_{2} = (n_{21}, n_{22}), n_{3} = n_{31}, n_{4} = (n_{40}, n_{41}, n_{42}), n_{5} = (n_{50}, n_{51}), and n_{6} = n_{60} are independent random variables conditional on parental mating types (m) where n_{2} and n_{5} follow a binomial distribution and n_{4} follows a trinomial distribution with probabilities given in column 5 of Table 1[2]. The score test for H_{0}: ψ = 1 is then written as ${Z}_{TDT}=\frac{{U}_{T}(n)-E({U}_{T}(n)|m)}{\sqrt{{\text{Var(U}}_{\text{T}}\text{(n)|}m\text{)}}}$, where U_{ T }(n) = n_{22}+n_{42}, n_{22}+n_{42}+0.5(n_{21}+n_{41}+n_{51}) and n_{42}+n_{41}+n_{51} for the recessive, additive, and dominant models. By applying the variance decomposition formula, we obtain the correlation between Z_{ TDT }and Z_{ CC }as $(1-R/N)\frac{E(\sqrt{{\text{Var(U}}_{\text{T}}\text{(n)|}m\text{)}})}{\sqrt{\text{Var(U(x))}}}$. An additional distributional assumption needs to be made for parental genotypes. We considered six parental mating types as a six dimensional multinomial distribution, and the corresponding probabilities were consistently estimated by the observed counts.
Under HWE, we can show that the correlation for three models can be simplified to $\sqrt{1-R/N}$ when all cases have parental genotypes available. When only a proportion of cases overlaps between case-parents and case-control designs, we can introduce an additional parameter η < 1 such that ${\sum}_{ij}{n}_{ij}}=\eta R$, and the correlation between Z_{ TDT }and Z_{ CC }is reduced to $\eta \sqrt{1-R/N}$.
Declarations
Acknowledgements
This article has been published as part of BMC Proceedings Volume 1 Supplement 1, 2007: Genetic Analysis Workshop 15: Gene Expression Analysis and Approaches to Detecting Multiple Functional Loci. The full contents of the supplement are available online at http://www.biomedcentral.com/1753-6561/1?issue=S1.
Authors’ Affiliations
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