Computing power in case-control association studies through the use of quadratic approximations: application to meta-statistics

Power evaluation of asymptotic tests for comparing two binomial proportions to detect direct and indirect association in large-scale studies

Asymptotic tests are commonly used for comparing two binomial proportions when the sample size is sufficiently large. However, there is no consensus on the most powerful test. In this paper, we clarify this issue by comparing the power functions of three popular asymptotic tests: the Pearson's χ² test, the likelihood-ratio test and the odds-ratio based test. Considering Taylor decompositions under local alternatives, the comparisons lead to recommendations on which test to use in view of both the experimental design and the nature of the investigated signal. We show that when the design is balanced between the two binomials, the three tests are equivalent in terms of power. However, when the design is unbalanced, differences in power can be substantial and the choice of the most powerful test also depends on the value of the parameters of the two compared binomials. We further investigated situations where the two binomials are not compared directly but through tag binomials. In these cases of indirect association, we show that the differences in power between the three tests are enhanced with decreasing values of the parameters of the tag binomials. Our results are illustrated in the context of genetic epidemiology where the analysis of genome-wide association studies provides insights regarding the low power for detecting rare variants.

Statistical distributions of test statistics used for quantitative trait association mapping in structured populations

Background

Spurious associations between single nucleotide polymorphisms and phenotypes are a major issue in genome-wide association studies and have led to underestimation of type 1 error rate and overestimation of the number of quantitative trait loci found. Many authors have investigated the influence of population structure on the robustness of methods by simulation. This paper is aimed at developing further the algebraic formalization of power and type 1 error rate for some of the classical statistical methods used: simple regression, two approximate methods of mixed models involving the effect of a single nucleotide polymorphism (SNP) and a random polygenic effect (GRAMMAR and FASTA) and the transmission/disequilibrium test for quantitative traits and nuclear families. Analytical formulae were derived using matrix algebra for the first and second moments of the statistical tests, assuming a true mixed model with a polygenic effect and SNP effects.

Results

The expectation and variance of the test statistics and their marginal expectations and variances according to the distribution of genotypes and estimators of variance components are given as a function of the relationship matrix and of the heritability of the polygenic effect. These formulae were used to compute type 1 error rate and power for any kind of relationship matrix between phenotyped and genotyped individuals for any level of heritability. For the regression method, type 1 error rate increased with the variability of relationships and with heritability, but decreased with the GRAMMAR method and was not affected with the FASTA and quantitative transmission/disequilibrium test methods.

Conclusions

The formulae can be easily used to provide the correct threshold of type 1 error rate and to calculate the power when designing experiments or data collection protocols. The results concerning the efficacy of each method agree with simulation results in the literature but were generalized in this work. The power of the GRAMMAR method was equal to the power of the FASTA method at the same type 1 error rate. The power of the quantitative transmission/disequilibrium test was low. In conclusion, the FASTA method, which is very close to the full mixed model, is recommended in association mapping studies.

Independent replication and meta analysis of association studies establish TNFSF4 as a susceptibility gene preferentially associated with the subset of anticentromere-positive patients with systemic sclerosis

OBJECTIVE

Independent replication with large cohorts and metaanalysis of genetic associations are necessary to validate genetic susceptibility factors. The known tumor necrosis factor (ligand) superfamily, member 4 gene (TNFSF4) systemic lupus erythematosus (SLE) risk locus has been found to be associated with systemic sclerosis (SSc) in 2 studies, but with discrepancies between them for genotype-phenotype correlation. Our objective was to validate TNFSF4 association with SSc and determine the subset with the higher risk.

METHODS

Known SLE and SSc TNFSF4 susceptibility variants (rs2205960, rs1234317, rs12039904, rs10912580, and rs844648) were genotyped in 1031 patients with SSc and 1014 controls of French white ancestry. Genotype-phenotype association analysis and meta analysis of available data were performed, providing a population study of 4989 patients with SSc and 4661 controls, all of European white ancestry.

RESULTS

Allelic and genotypic associations were observed for the 5 single-nucleotide polymorphisms (SNP) with the subset of patients with SSc who are positive for anticentromere antibodies (ACA) and only a trend for association with SSc and limited cutaneous SSc. Rs2205960 exhibited the strongest allelic association in ACA+ patients with SSc [p = 0.0015; OR 1.37 (1.12-1.66)], with significant intra-cohort association when compared to patients with SSc positive for ACA. Metaanalysis confirmed overall association with SSc but also raised preferential association with the ACA+ subset and strongest effect with rs2205960 [T allele p = 0.00013; OR 1.33 (1.15-1.54) and TT genotype p = 0.00046; OR 2.02 (1.36-2.98)].

CONCLUSION

We confirm TNFSF4 as an SSc susceptibility gene and rs2205960 as a putative causal variant with preferential association in the ACA+ SSc subphenotype.

On the power of the Cochran–Armitage test for trend in the presence of misclassification

The Cochran–Armitage (CA) test is commonly used in both epidemiology and genetics to test for linear trend in two-way tables with a binary outcome. There has been increasing interest in the power and size of the test and in determination of sample size, especially when there is potential misclassification in the ‘exposure’ category. This article provides a unified approach to determination of the power function over different sampling strategies (fixed overall sample size or fixed marginal sample sizes) and allowing for misclassification in one or both variables. The misclassification may be either differential or non-differential. In addition to the standard CA test, results are also given which provide some insight into the performance of the modified CA test, which utilizes a standard error obtained without invoking the null hypothesis. Even without misclassification, some new expressions are also obtained for determining power with a fixed overall sample size. Numerical illustrations are presented with an emphasis on the more commonly occurring problem of misclassification in the exposure category.

Association study of ITGAM, ITGAX, and CD58 autoimmune risk loci in systemic sclerosis: results from 2 large European Caucasian cohorts

OBJECTIVE

Accumulating evidence shows that shared autoimmunity is critical for the pathogenesis of many autoimmune diseases. Systemic sclerosis (SSc) belongs to the connective tissue disorders, and recent data have highlighted strong associations with autoimmunity genes shared with other autoimmune diseases. To determine whether novel risk loci associated with systemic lupus erythematosus or multiple sclerosis may confer susceptibility to SSc, we tested single-nucleotide polymorphisms (SNP) from ITGAM, ITGAX, and CD58 for associations.

METHODS

SNP harboring associations with autoimmune diseases, ITGAM rs9937837, ITGAX rs11574637, and CD58 rs12044852, were genotyped in 2 independent cohorts of European Caucasian ancestry: 1031 SSc patients and 1014 controls from France and 1038 SSc patients and 691 controls from the USA, providing a combined study population of 3774 individuals. ITGAM rs1143679 was additionally genotyped in the French cohort.

RESULTS

The 4 polymorphisms were in Hardy-Weinberg equilibrium in the 2 control populations, and allelic frequencies were similar to those expected in European Caucasian populations. Allelic and genotypic frequencies for these 3 SNP were found to be statistically similar in SSc patients and controls. Subphenotype analyses for subgroups having diffuse cutaneous subtype disease, specific autoantibodies, or fibrosing alveolitis did not reveal any difference between SSc patients and controls.

CONCLUSION

These results obtained through 2 large cohorts of SSc patients of European Caucasian ancestry do not support the implication of ITGAM, ITGAX, and CD58 genes in the genetic susceptibility of SSc, although they were recently identified as autoimmune disease risk genes.

What's the best statistic for a simple test of genetic association in a case-control study?

Genome-wide genetic association studies typically start with univariate statistical tests of each marker. In principle, this single-SNP scanning is statistically straightforward--the testing is done with standard methods (e.g. chi(2) tests, regression) that have been well studied for decades. However, a number of different tests and testing procedures can be used. In a case-control study, one can use a 1 df allele-based test, a 1 or 2 df genotype-based test, or a compound procedure that combines two or more of these statistics. Additionally, most of the tests can be performed with or without covariates included in the model. While there are a number of statistical papers that make power comparisons among subsets of these methods, none has comprehensively tackled the question of which of the methods in common use is best suited to univariate scanning in a genome-wide association study. In this paper, we consider a wide variety of realistic test procedures, and first compare the power of the different procedures to detect a single locus under different genetic models. We then address the question of whether or when it is a good idea to include covariates in the analysis. We conclude that the most commonly used approach to handle covariates--modeling covariate main effects but not interactions--is almost never a good idea. Finally, we consider the performance of the statistics in a genome scan context.

Robust tests for single-marker analysis in case-control genetic association studies

Choosing an appropriate single-marker association test is critical to the success of case-control genetic association studies.An ideal single-marker analysis should have robust performance across a wide range of potential disease risk models. MAX was designed specifically to achieve such robustness. In this work, we derived the power calculation formula for MAX and conducted a comprehensive power comparison between MAX and two other commonly used single-marker tests,the one-degree-of-freedom (1-df) Cochran-Armitage trend test and the 2-df Pearson chi2 test. We used a single-marker disease risk model and a two-marker haplotype risk model to explore the performances of the above three tests. We found that each test has its own "sweet" spots. Among the three tests considered, MAX appears to have the most robust performance.

Association between the IRF5 rs2004640 functional polymorphism and systemic sclerosis: a new perspective for pulmonary fibrosis

OBJECTIVE

There is now growing evidence that connective tissue diseases, including systemic sclerosis (SSc), share a common genetic background. Microarray studies support a pivotal role of type I interferon (IFN) in the pathophysiology of connective tissue diseases. Interferon regulatory factors coordinate the expression of type I IFNs, and the IRF5 gene has been identified as a susceptibility gene of systemic lupus and Sjögren's syndrome. The aim of this study was to determine whether the IRF5 rs2004640 single-nucleotide polymorphism is associated with SSc.

METHODS

The IRF5 rs2004640 (GT) functional polymorphism was genotyped in 1,641 subjects of French European Caucasian origin: a discovery set comprising 427 patients with SSc and 380 control subjects and a replication set comprising 454 patients with SSc and 380 control subjects.

RESULTS

In both the discovery set and the replication set, the TT genotype was significantly more common in patients with SSc than in control subjects, with an odds ratio (OR) for the combined populations of 1.58 (95% confidence interval [95% CI] 1.18-2.11 [P for trend 0.002]). Analyses of the whole SSc population showed a significant association between homozygosity for the T allele and the presence of antinuclear antibodies (corrected P [Pcorr]=0.04, OR 1.59, 95% CI 1.16-2.17) and fibrosing alveolitis (Pcorr=0.001, OR 2.07, 95% CI 1.38-3.11). In a multivariate analysis model including the diffuse cutaneous subtype of SSc and positivity for anti-topoisomerase I antibodies, the IRF5 rs2004640 TT genotype remained associated with fibrosing alveolitis (P=0.029, OR 1.92, 95% CI 1.07-3.44).

CONCLUSION

The IRF5 rs2004640 GT substitution is associated with susceptibility to SSc. These data provide new insight into the pathogenesis of SSc, including clues to the mechanisms leading to fibrosing alveolitis.

[A review of power and sample size estimation in genomewide association studies]

Power and sample size estimation is one of the crucially important steps in planning a genetic association study to achieve the ultimate goal, identifying candidate genes for disease susceptibility, by designing the study in such a way as to maximize the success possibility and minimize the cost. Here we review the optimal two-stage genotyping designs for genomewide association studies recently investigated by Wang et al(2006). We review two mathematical frameworks most commonly used to compute power in genetic association studies prior to the main study: Monte-Carlo and non-central chi-square estimates. Statistical powers are computed by these two approaches for case-control genotypic tests under one-stage direct association study design. Then we discuss how the linkage disequilibrium strength affects power and sample size, and how to use empirically-derived distributions of important parameters for power calculations. We provide useful information on publicly available software developed to compute power and sample size for various study designs.