Methods

Data pre-processing

Corresponding to strategy S1 (see Figure 3.3), statistical association testing was based on the fluorescence hybridization intensities at the autosomal non - polymor-phic copy number (CN) probe sets that are a measure of copy number variance.

For each individual and each probe set, raw intensity values were extracted from the individual ’.CEL’ files by use of the R-package ’affxparser’ (Bengtsson et al., 2008a).

Afterwards, the FBAT approach was genome-wide applied to the family-based obe-sity sample, that is to each of the 888 023 autosomal CN probe sets, in order to test the locus-specific CNV characteristics for an association with obesity.

3.5 Application of Strategy S1 to the Phenotype Obesity

Association testing

Since the offspring’s genotyping procedure was identical to those of the parents, inter-familial differences in hybridization intensity measurements should solely be derived from CNV inheritance or from de novo CNV events, but not from tech-nical artefacts. Applied to a binary trait, the FBAT is equivalent to a score test with a test statistic equaling the standardized sum of within-family components (see chapter 5.1.2.1 for a detailed description). Hence, a normalization of raw intensity values prior to the association testing is dispensable here. Consequently, the raw hy-bridization intensity measurements were directly tested without being transformed into raw copy number measurements (see chapter 6.1.1). The latter makes use of the fact that the FBAT approach is invariant under linear transformation (see chapter 5.1.2.1).

To account for multiple comparisons in testing multiple hypotheses (n = 888 023), the empirical Bayes method of local false discovery rates (lfdr) as proposed by Efron et al. (2001) was applied. The lfdr approach is motivated by the tail area false discovery rate (FDR), which was introduced by Benjamini and Hochberg (1995) in a frequentist framework. When a collection of hypotheses is tested simultaneously, the FDR equals the expected proportion of erroneously rejected null hypotheses among all rejected null hypotheses using a given rejection rule (Benjamini and Hochberg, 1995). Closely connected to a local version of the FDR, the lfdr is defined as the posteriori probability that a single null hypothesis is true given the observed value of the respective test statistic (Efron et al., 2001; Efron, 2004, 2007a,b).

In more detail, the lfdr method is based on a Bayesian two-class model that divides all test cases into two classes, ’null’ or ’non-null’, corresponding to whether or not they are generated according to the null hypothesis and with prior probabilities p0 and p1 = 1 − p0, and with associated test statistic densities f0 and f1. The test statistic density f can then be written as a mixture density f = p₀f₀ + p₁f₁. According to the Bayes theorem, the lfdr for an observed test statistic value z is given as the posteriori probability lfdr(z) = P(’null’|z) = p0f₀(z)/f (z).

Using the ’locfdr’ R package (Efron et al., 2011), lfdr estimates were obtained on the basis of empirical non-parametric estimates using central matching for the null distribution density ˆf₀, the mixture density ˆf and the factor ˆp₀. Thus, the applied lfdr methodology especially accounts for the fact that the null sub-density f₀ might differ from the theoretical null distribution. The natural choice for f₀ would be the standard N (0, 1) density in the underlying FBAT context with test statistics √

F BAT_k, k = 1, . . . , 888 023 (CNV FBAT z-values, cf. equation (5.4) in chapter 5.1.2.1). However, a deviation from the theoretical null distribution can

3.5 Application of Strategy S1 to the Phenotype Obesity

be caused by several reasons that are listed by Efron (2004, 2007a,b): (1) failed assumptions on the test statistic, (2) unobserved covariates, (3) correlation across probes and genes, (4) a large proportion of genuine but uninterestingly small ef-fects. As shown by Efron (2004, 2007a,b), in all these situations the application of the inappropriate theoretical null results in misclassified FDR and lfdr estimates.

Of note, Efron (2007a) emphasized that even if the theoretical null is singularly appropriate for each probe-wise test situation, correlation across probes can lead to an effectively deviated null distribution f₀ compared to the theoretical N (0, 1).

Moreover, Efron (2004) point out that the popular permutation methods, which provide a way of avoiding assumptions on an underlying correlation stucture and asymptotic approximations (like normality), do not automatically resolve the ques-tion of an appropriate null hypothesis f₀. As shown by Efron (2004), unobserved covariates such as personal characteristics of the analysed study patients (i.e. age, gender or geographical location) are likely to widen or narrow the empirical f0, and this effect is not detectable in permutationally derived null hypothesis. Efron (2004) point out that a permutation null distribution will not reveal correlation effects of hidden covariates, but will closely match the theoretical null distribution, irrespec-tive of whether or not there are unobserved covariates or other factors influencing the theoretical null distribution. Finally, results of each inference method, FDR, lfdr, Bonferroni, family-wise error rate (FWER), are doubtful if the null hypothesis is not chosen appropriately. Efron (2004) strongly argues to prefer the empirical null hypothesis in observational studies.

Evaluation of statistically significant results, CNV calling and follow-up analyses As stated by Ionita-Laza et al. (2008), it is challenging to evaluate whether sta-tistically significant association test results are caused by underlying CNV - trait associations or rather by hybridization intensity differences depending on probe-specificity and signal-to-noise properties of the platform, when the CNV FBAT methodology was applied genome-wide without an a priori selection of markers.

To address this concern, the HMM CNV detection algorithm implemented in the Affymetrix Genotyping Console (GTC) 3.0 was employed on the raw intensity data.

For each of the 1 272 individuals of the family-based obesity sample, CNV calls were estimated by comparing individual signal intensities against a reference sample. Due to computational constraints, the reference sample size was limited to 106 parental pairs of the obesity trio sample. In order to minimize the potential effect of the choice of the reference sample on the CNV calling results, two differently composed reference samples, each comprising 106 parental pairs, were used. One reference

3.5 Application of Strategy S1 to the Phenotype Obesity

sample (ref₁) was a random collection, whereas the other one (ref₂) was based on those parental pairs with the lowest mean BMI standard deviation scores out of all non-obese parental pairs. Phenotypical details on both reference samples can be found in Jarick et al. (2011) (Supplementary Table S1).

For probe sets with statistically significant CNV FBAT results, follow-up anal-yses were performed in the case-control sample. Significance of CNV FBATs was determined with respect to a lfdr level of 0.2, which was proposed to be a sensible threshold by Efron (2004). To address potential plate effects, quantile normaliza-tion (Bolstad et al., 2003) was applied to the raw intensity signals of the case-control sample. Subsequently, logistic regression with predictors normalized intensities, sex and age was used to test the CN probe sets for an association with obesity.