Integrated probability of coronary heart disease subject to the −308 tumor necrosis factor-alpha SNP: a Bayesian meta-analysis

Data selection

This analysis is based on twenty data sets (indexed i) on two CHD phenotypes (indexed j) selected from the studies compiled in Zhang et al. (2011), following a well-documented study identification, data acquisition and selection strategy, including also statistical tests (Hardy–Weinberg equilibrium, heterogeneity, publication bias). The selected data sets are the studies that report the genotypes of both CHD patients and non-CHD (control) patients for the two CHD phenotypes separately. In particular, there were included: fifteen data sets from studies on Caucasians, where six studies are on the CHD phenotype coronary stenosis (CS) (Allen et al., 2001; Elahi et al., 2008; Georges et al., 2003; Sbarsi et al., 2007; Szalai et al., 2002; Vendrell et al., 2003) and nine studies are on the CHD phenotype myocardial infarction (MI) (Antonicelli et al., 2005; Bennet et al., 2006; Dedoussis et al., 2005; Herrmann et al., 1998; Koch et al., 2001; Padovani et al., 2000; Tobin et al., 2004; Tulyakova et al., 2004); and five data sets from studies on Asians on the CHD phenotype coronary stenosis (Hou et al., 2009; Liu et al., 2009; Shun et al., 2009; Li et al., 2003; Chen et al., 2001). The rejected data sets are: three studies on Caucasians (for not reporting data on non-CHD patients); four studies on Asians (three for not reporting data on non-CHD patients and one for not separating the CHD phenotypes); the study on Indians and the study on Africans (both for not separating the CHD phenotypes).

The data consist of frequencies of occurrence of the −308 TNF-α SNP in randomly selected CHD patients and non-CHD patients, respectively n_SNP,CHD and $n_{\text{SNP},\overline{\text{CHD}}}$. The data are summarized in Table 1 (columns 3–6). The errors indicated were computed from error propagation. Assuming that the methods for measuring the presence of the SNP have a success rate of r_suc = 0.88 (Sharifian, 2010), and furthermore that the error of a counting result is given by the Poisson approximation $\sqrt{n}$, then the error of a counting result n on the presence of the SNP is given by $\left(1-r_{\mathrm{suc}}\right)\sqrt{n}/2.$

Hypotheses testing

First we test the hypothesis H₁ that the presence of TNF-α SNP is related to the occurrence of CHD against the null hypothesis H₀ that the presence of the SNP is unrelated to the occurrence of CHD. By the Bayes theorem, the probability of a hypothesis H_i given the data D_SNP is the posterior probability of the corresponding hypothesis (1)${\displaystyle P\left(H_i|D_{\text{SNP}}\right)=\frac{P\left(D_{\text{SNP}}|H_i\right)P\left(H_i\right)}{P\left(D_{\text{SNP}}\right)},}$ where P(D_SNP|H_i) is the evidence, P(H_i) is the prior probability of H_i and $P\left(D_{\text{SNP}}\right)=\sum _{n}P\left(D_{\text{SNP}}|H_i\right)P\left(H_i\right).$ The subscript in D_SNP reminds us that the random variable is the occurrence of the SNP. In order to infer which hypothesis is more likely in view of the data, we compare the evidence computed for the two hypotheses. The evidence is the integral of the likelihood over the j-dimensional parameter space p_i,j of the hypothesis H_i(2)${\displaystyle P\left(D_{\text{SNP}}|H_i\right)=\int d^j{p}_{i,j}P\left(D_{\text{SNP}}|p_{i,j},H_i\right)P\left(p_{k,n}|H_i\right).}$ Assuming equal prior probabilities for the two hypotheses, then from Eq. (1) it follows that (3)${\displaystyle \frac{P\left(H_1|D_{\text{SNP}}\right)}{P\left(H_0|D_{\text{SNP}}\right)}=\frac{P\left(D_{\text{SNP}}|H_1\right)}{P\left(D_{\text{SNP}}|H_0\right)}.}$ We compute the evidence of the two hypotheses, for each data set separately and for the combined data sets grouped by CHD phenotype. We follow the procedure detailed in Vourvouhaki & Carvalho (2011), which we here summarize for one data set and then generalize for the combined data sets. In all cases, we choose a uniform distribution for the prior of the parameters, which is justified by the absence of an a priori bias on the values of the parameters (MacKay, 2003).

The evidence of H₀, P(D_SNP|H₀), is computed assuming that the presence of the SNP is described by a binomial distribution with one parameter only, namely the probability p₀ that the SNP occurs in a given population. For n_SNP occurrences of the SNP and $n_{\overline{\text{SNP}}}$ non-occurrences of the SNP in a sample of size $n=n_{\text{SNP}}+n_{\overline{\text{SNP}}}$, the likelihood P(D_SNP|p₀, H₀) is given by (4)${\displaystyle P\left(D_{\text{SNP}}|p_0,H_0\right)=p_0^{n_{\text{SNP}}}{\left(1-p_0\right)}^{n_{\overline{\text{SNP}}}}.}$ Moreover, assuming a uniform prior distribution for p₀, P(p₀) = 1, we find that (5)${\displaystyle P\left(D_{\text{SNP}}|H_0\right)={\int }_0^{1}d{p}_{0}P\left(D_{\text{SNP}}|p_0,H_0\right)P\left(p_0|H_0\right)=\frac{n_{\text{SNP}}!n_{\overline{\text{SNP}}}!}{\left(n_{\text{SNP}}+n_{\overline{\text{SNP}}}+1\right)!},}$ where n! stands for the factorial of n.

The evidence of H₁, P(D_SNP|H₁), is computed assuming that the presence of the SNP is described by a binomial distribution with two parameters, namely the probability p_1,CHD that the SNP occurs in the subset of CHD patients and the probability $p_{1,\overline{\text{CHD}}}$ that the SNP occurs in the subset of non-CHD patients, (6)${\displaystyle P\left(D_{\text{SNP}}|H_1\right)={\int }_0^{1}d{p}_{1,\mathrm{CHD}}{\int }_0^{1}d{p}_{1,\overline{\text{CHD}}}P\left(D_{\text{SNP}}|p_{1,\mathrm{CHD}},p_{1,\overline{\text{CHD}}},H_1\right)P\left(p_{1,\mathrm{CHD}},p_{1,\overline{\text{CHD}}}|H_1\right).}$ For n_SNP,CHD occurrences of the SNP and $n_{\overline{\text{SNP}},\text{CHD}}$ non-occurrences of the SNP in a subset of CHD patients $n_{\text{CHD}}=n_{\text{SNP,CHD}}+n_{\overline{\text{SNP}},\text{CHD}}$, and also for $n_{\text{SNP},\overline{\text{CHD}}}$ occurrences of the SNP and $n_{\overline{\text{SNP}},\overline{\text{CHD}}}$ non-occurrences of the SNP in a subset of non-CHD patients $n_{\overline{\text{CHD}}}=n_{\mathrm{SNP},\overline{\text{CHD}}}+n_{\overline{\text{SNP}},\overline{\text{CHD}}},$ the likelihood $P\left(D_{\text{SNP}}|p_{1,\mathrm{CHD}},p_{1,\overline{\text{CHD}}},H_1\right)$ is separable, i.e., it can be decomposed into the product of the likelihoods P(D_SNP|p_1,CHD, H₁) and $P\left(D_{\text{SNP}}|p_{1,\overline{\text{CHD}}},H_1\right)$, as follows (7)${\displaystyle P\left(D_{\text{SNP}}|p_{1,\text{CHD}},p_{1,\overline{\text{CHD}}},H_1\right)=p_{1,\text{CHD}}^{n_{\text{SNP, CHD}}}{\left(1-p_{1,\text{CHD}}\right)}^{n_{\overline{\text{SNP}},\text{CHD}}}{p}_{1,\overline{\text{CHD}}}^{n_{\text{SNP},\overline{\text{CHD}}}}{\left(1-p_{1,\overline{\text{CHD}}}\right)}^{n_{\overline{\text{SNP}},\overline{\text{CHD}}}}\equiv P\left(D_{\text{SNP}}|p_{1,\text{CHD}},H_1\right)P\left(D_{\text{SNP}}|p_{1,\overline{\text{CHD}}},H_1\right).}$ Assuming a uniform probability for p_1,CHD and $p_{1,\overline{\text{CHD}}},$ $P\left(p_{1,\text{CHD}},p_{1,\overline{\text{CHD}}}|H_1\right)=$1 and moreover that the priors on p_1,CHD and $p_{1,\overline{\text{CHD}}}$ are separable, the posterior distribution will also be separable and given by (8)${\displaystyle P\left(D_{\text{SNP}}|H_1\right)={\int }_0^{1}d{p}_{1,\text{CHD}}P\left(D_{\text{SNP}}|p_{1,\text{CHD}},H_1\right)P\left(p_{1,\text{CHD}}|H_1\right){\int }_0^{1}d{p}_{1,\overline{\text{CHD}}}P\left(D_{\text{SNP}}|p_{1,\overline{\text{CHD}}},H_1\right)P\left(p_{1,\overline{\text{CHD}}}|H_1\right)=\frac{n_{\text{SNP,CHD}}!n_{\overline{\text{SNP}},\text{CHD}}!}{\left(n_{\text{SNP,CHD}}+n_{\overline{\text{SNP}},\text{CHD}}+1\right)!}\frac{n_{\text{SNP},\overline{\text{CHD}}}!n_{\overline{\text{SNP}},\overline{\text{CHD}}}!}{\left(n_{\text{SNP},\overline{\text{CHD}}}+n_{\overline{\text{SNP}},\overline{\text{CHD}}}+1\right)!}.}$ In order to compare the hypotheses, we take the ratio of the corresponding evidences, B₁₀ = P(H₁|D)/P(H₀|D), which we present in Table 1 (columns 7–8). This quantity is known as the Bayes factor and gives empirical levels of significance for the strength of the evidence of the test hypothesis over that of the null hypothesis. It also encapsulates the Occam’s factor, which measures the adequacy of a hypothesis to the data over the parameter space of the hypothesis (MacKay, 2003). The levels of significance ascribed to the Bayes factor are calibrated by the Jeffrey’s scale (Kass & Raftery, 1995). According to this scale, a Bayes factor larger than one indicates that H₁ is favoured over H₀. Otherwise, H₀ is favoured over H₁. For the data sets taken separately, the results from this hypothesis test mostly agree with the corresponding results presented in the meta-analysis by Chu et al. (2013, see Fig. 1).

We plot the Bayes factor for each study, grouped by ethnicity and CHD phenotype, in Fig. 6. For the data sets taken separately, we observe that the Bayes factor is asymmetrically distributed about the Bayes factor equal to one, with most Bayes factors being smaller than one. The exceptions are Elahi et al. (2008), Vendrell et al. (2003) and Dedoussis et al. (2005) for the Caucasian population, and Shun et al. (2009) for the Asian population. This asymmetry indicates heterogeneity in the results. For the combined data sets included in each panel, the Bayes factor takes values 0.03–0.05 for the Caucasian population and 0.15 for the Asian population, which indicates that there is no evidence for H₁ over H₀. We also observe that, for the Caucasian population, the Bayes factor of the combined data sets is outside the range of variability of the Bayes factor of the data sets considered separately. This suggests that the combination of the Caucasian data sets causes a new data pattern to emerge. Conversely the combination of the Asian data sets leads to an approximately average data pattern. Hence we conclude that the data favour H₀ over H₁. Since H₀ yields trivial results, in the subsequent subsections we present the results also for H₁ to illustrate the application of the formalism to a more general setup. It is also instructive to compare the subsequent results using both hypotheses.

In Fig. 5B, we plot the Bayes factor as a function of the sample size. We observe that smaller data sets are distributed across a wide range of values of the Bayes factor, whereas larger data sets are distributed across values smaller than one. This implies that smaller data sets favour either H₀ or H₁, whereas larger data sets favour H₀.

Inference of conditional probabilities

Sensitivity of the results

To test the robustness of this meta-analysis, we conceive two tests of the sensitivity of the results, namely to low-significance data sets, to data sets with extreme results and to extreme data sets.

To test the sensitivity of the results to low-significance data sets, we exclude the data sets with comparatively small sample sizes for the same CHD phenotype, namely the study by Elahi et al. (2008) and the study by Chen et al. (2001), from the combination. We also exclude the studies with extreme results (i.e., the studies with the largest Bayes factor), namely the study in Dedoussis et al. (2005). We recompute both the Bayes factors (Table 1) and the probabilities of CHD (Table 3). We observe that the Bayes factor in the new combination changes by 18%, −38% and 24%, respectively for the CS Caucasian, the MI Caucasian and the CS Asian population. The inferred parameters and probabilities vary by −6 to 6%, −5 to 2%, and −1 to 4%, respectively for the CS Caucasian, the MI Caucasian and the CS Asian population. The largest difference is observed for the CS Caucasian population due to the exclusion of the study by Elahi et al. (2008). The exclusion of the study by Dedoussis et al. (2005) from the MI Caucasian population causes predominantly negative differences.

To test the sensitivity of the results to extreme data sets, we exclude the data sets with comparatively large samples sizes for the same CHD phenotype, namely the study by Georges et al. (2003), the study by Bennet et al. (2006) and the study by Hou et al. (2009), from the combination. These are also the studies with the smallest Bayes factor for each CHD phenotype. We recompute both the Bayes factors (Table 1) and the probabilities of CHD (Table 4). We observe that the Bayes factor in the new combination changes by 3%, −19% and 32%, respectively for the CS Caucasian, the MI Caucasian and the CS Asian population. The inferred parameters and probabilities vary by −20 to −1%, 5 to 11%, and −26 to 25%, respectively for the CS Caucasian, the MI Caucasian and the CS Asian population. The largest difference is observed for the CS Asian population due to the exclusion of the study by Hou et al. (2009). The exclusion of the study by Georges et al. (2003) from the CS Caucasian population causes predominantly negative differences.

In both tests, the differences in the Bayes factor leave the result of the hypothesis testings unchanged, while the differences in the inferred parameters and probabilities also leave the conclusions unchanged. We thus infer that this formalism is largely insensitive to (a) low-significante data sets combined with data with extreme results, and to (b) extreme data sets, which renders this formalism significantly robust.