We present a metaanalysis of independent studies on the potential implication in the occurrence of coronary heart disease (CHD) of the singlenucleotide polymorphism (SNP) at the −308 position of the tumor necrosis factor alpha (TNFalpha) gene. We use Bayesian analysis to integrate independent data sets and to infer statistically robust measurements of correlation. Bayesian hypothesis testing indicates that there is no preference for the hypothesis that the −308 TNFalpha SNP is related to the occurrence of CHD, in the Caucasian or in the Asian population, over the null hypothesis. As a measure of correlation, we use the probability of occurrence of CHD conditional on the presence of the SNP, derived as the posterior probability of the Bayesian metaanalysis. The conditional probability indicates that CHD is not more likely to occur when the SNP is present, which suggests that the −308 TNFalpha SNP is not implicated in the occurrence of CHD.
Coronary heart disease (CHD) is now widely accepted to consist of a chronic inflammatory disease (
Among the genetic factors potentially implicated in the emergence of CHD, the tumor necrosis factor alpha (TNF
Among the several singlenucleotide polymorphisms (SNPs) that have been identified in the human TNF
In order to infer the risk of CHD derived from potential risk factors, it is important to develop a formalism that infers correlations among different intervening factors and combines independent data sets for a consistent inference of the correlations. In
The most exhaustive metaanalysis to date on this correlation is the frequentist analysis in
In this manuscript we propose a metaanalysis based on Bayesian analysis in an attempt to establish the potential implication of −308 TNF
Panel 1 of 3. Ellipses indicate the main actions. Rectangles indicate detailed actions. Rectangles with rounded corners indicate the main results.
Panel 2 of 3.
Panel 3 of 3.
This analysis is based on twenty data sets (indexed
Column 1: Studies selected for the metaanalysis. The index (A) indicates that a possible association was measured in the original publication; the index (NA) indicates that no association was measured in the original publication. Column 2: The phenotype of the patients in the studies grouped by ethnicity. Columns 3–6: Genotypic frequencies of TNF
Study  Phenotype  CHD patients  Controls  Bayes factor  

( 
( 
GG  GA/AA  GG  GA/AA 


Cauc CS  127  53  222  107  0.14 ± 0.05  
59  38  41  54  3.54 ± 1.12  0.049 ± 0.014  
613  236  222  92  0.08 ± 0.03  0.041 ± 0.016 

175  73  185  56  0.33 ± 0.11  
229  89  181  87  0.19 ± 0.07  0.048 ± 0.019 

231  110  159  48  1.33 ± 0.46  
Cauc MI  224  69  246  64  0.12 ± 0.04  
799  368  1,037  460  0.05 ± 0.02  
206  31  227  10  26.14 ± 8.56  0.026 ± 0.011  
325  120  376  158  0.11 ± 0.04  
117  79  97  79  0.19 ± 0.06  0.035 ± 0.015 

565  228  244  96  0.07 ± 0.03  0.030 ± 0.012 

120  28  114  34  0.17 ± 0.06  
365  182  337  168  0.07 ± 0.03  
242  64  177  69  0.60 ± 0.21  
Asian CS  29  11  21  9  0.27 ± 0.08  0.151 ± 0.057  
268  32  802  103  0.05 ± 0.02  
66  8  138  20  0.12 ± 0.04  0.114 ± 0.043 

234  52  142  34  0.10 ± 0.03  0.103 ± 0.037 

54  19  118  20  1.10 ± 0.34 
French cohort.
Irish cohort.
Excluding
Excluding
The data consist of frequencies of occurrence of the −308 TNF
The ratio of the fraction of SNP in the population of CHD patients to the fraction of SNP in the population of nonCHD patients,
(A) The ratio of the frequency of SNP in the CHD population to the frequency of SNP in the nonCHD population as a function of the sample size. (B) The Bayes factor for the two hypotheses discussed in the text as a function of the sample size.
In order to investigate the heterogeneity in the data sets, we compare the size of the effect (defined as a measure of the difference between CHD and nonCHD patients) in each study (
We plot this ratio of fractions for each study, grouped by ethnicity and CHD phenotype, in
In
First we test the hypothesis
The evidence of
The evidence of
We plot the Bayes factor for each study, grouped by ethnicity and CHD phenotype, in
In
The Bayes factor for each study, grouped by ethnicity and CHD phenotype. (A) Caucasians with coronary stenosis; (B) Caucasians with myocardial infarction; (C) Asians with coronary stenosis. The solid horizontal line marks the average Bayes factor of the data sets included in each panel. The dashed horizontal line marks the Bayes factor equal to one.
The Bayes factor for several realizations of CHD populations with the same
Comparing
To further explore how the ratio
We proceed to compute the probability for the occurrence of CHD, i.e., given the data on the presence of the SNP, we determine the probability that a patient has CHD. This is defined as the posterior probability
The prior probability
The evidence
Analogously we define the posterior probability
Finally, using the maximumlikelihood value of
In the case of
To each hypothesis there correspond several rows consisting of (A) the parameters
Hypothesis  Probabilities  Phenotype ( 


Cauc CS  Cauc MI  Asian CS  


0.299 ± 0.001  0.284 ± 0.001  0.141 ± 0.001 
(4.00 ± 1.31) ⋅ 10^{−3}  (1.00 ± 0.25) ⋅ 10^{−3}  (4.00 ± 0.91) ⋅ 10^{−3}  
(1.19 ± 0.39) ⋅ 10^{−3}  (0.28 ± 0.07) ⋅ 10^{−3}  (0.56 ± 0.13) ⋅ 10^{−3}  

0.298 ± 1.093  0.284 ± 1.752  0.141 ± 0.360  
0.299 ± 1.093  0.284 ± 1.572  0.141 ± 0.360  

(4.00 ± 14.65) ⋅ 10^{−3}  (1.00 ± 5.54) ⋅ 10^{−3}  (4.00 ± 10.22) ⋅ 10^{−3}  


0.295 ± 0.001  0.283 ± 0.001  0.158 ± 0.001 

0.305 ± 0.001  0.285 ± 0.001  0.132 ± 0.001  
(3.42 ± 7.94) ⋅ 10^{−3}  (0.98 ± 3.26) ⋅ 10^{−3}  (5.00 ± 7.02) ⋅ 10^{−3}  
(1.00 ± 2.34) ⋅ 10^{−3}  (0.28 ± 0.92) ⋅ 10^{−3}  (0.79 ± 1.11) ⋅ 10^{−3}  

0.304 ± 0.598  0.285 ± 0.926  0.131 ± 0.244  
0.305 ± 0.598  0.285 ± 0.926  0.132 ± 0.244  

(3.30 ± 10.02) ⋅ 10^{−3}  (0.98 ± 4.54) ⋅ 10^{−3}  (6.00 ± 13.84) ⋅ 10^{−3} 
We now proceed to compute the probability for the presence of the SNP, i.e., given the data, we determine the probability that a randomly selected patient (with or without CHD) has the SNP. This probability is defined as
For completion, using the Bayes theorem, we invert
In order to quantify the influence of CHD in the presence of the SNP, we compute the ratio of
In order to quantify the influence of the SNP in the occurrence of (CHD, we compute the ratio of
Excluded:
Hypothesis  Probabilities  Phenotype ( 


Cauc CS  Cauc MI  Asian CS  


0.288 ± 0.001  0.296 ± 0.001  0.136 ± 0.001 
(4.00 ± 1.26) ⋅ 10^{−3}  (1.00 ± 0.24) ⋅ 10^{−3}  (4.00 ± 0.89) ⋅ 10^{−3}  
(1.15 ± 0.36) ⋅ 10^{−3}  (0.30 ± 0.07) ⋅ 10^{−3}  (0.55 ± 0.12) ⋅ 10^{−3}  

0.287 ± 1.018  0.296 ± 1.605  0.136 ± 0.340  
0.289 ± 1.018  0.296 ± 1.605  0.136 ± 0.340  

(4.00 ± 14.16) ⋅ 10^{−3}  (1.00 ± 5.54) ⋅ 10^{−3}  (4.00 ± 10.00) ⋅ 10^{−3}  


0.290 ± 0.001  0.292 ± 0.001  0.151 ± 0.001 

0.287 ± 0.001  0.300 ± 0.001  0.128 ± 0.001  
(3.34 ± 7.57) ⋅ 10^{−3}  (0.99 ± 3.18) ⋅ 10^{−3}  (5.11 ± 6.96) ⋅ 10^{−3}  
(0.97 ± 2.19) ⋅ 10^{−3}  (0.29 ± 0.93) ⋅ 10^{−3}  (0.77 ± 1.05) ⋅ 10^{−3}  

0.286 ± 0.542  0.300 ± 0.947  0.128 ± 0.234  
0.287 ± 0.543  0.300 ± 0.947  0.129 ± 0.234  

(3.38 ± 9.96) ⋅ 10^{−3}  (0.96 ± 4.34) ⋅ 10^{−3}  (6.02 ± 13.67) ⋅ 10^{−3} 
Excluded:
Hypothesis  Probabilities  Phenotype ( 


Cauc CS  Cauc MI  Asian CS  


0.308 ± 0.001  0.271 ± 0.001  0.177 ± 0.001 
(4.00 ± 1.08) ⋅ 10^{−3}  (1.00 ± 0.20) ⋅ 10^{−3}  (4.00 ± 0.63) ⋅ 10^{−3}  
(1.12 ± 0.33) ⋅ 10^{−3}  (0.27 ± 0.05) ⋅ 10^{−3}  (0.71 ± 0.11) ⋅ 10^{−3}  

0.306 ± 0.923  0.270 ± 1.220  0.177 ± 0.314  
0.308 ± 0.923  0.271 ± 1.220  0.177 ± 0.314  

(4.00 ± 12.05) ⋅ 10^{−3}  (1.00 ± 4.51) ⋅ 10^{−3}  (4.00 ± 7.11) ⋅ 10^{−3}  


0.306 ± 0.001  0.270 ± 0.001  0.190 ± 0.001 

0.309 ± 0.001  0.271 ± 0.001  0.165 ± 0.001  
(3.93 ± 6.97) ⋅ 10^{−3}  (0.92 ± 2.58) ⋅ 10^{−3}  (3.90 ± 4.35) ⋅ 10^{−3}  
(1.20 ± 2.14) ⋅ 10^{−3}  (0.25 ± 0.70) ⋅ 10^{−3}  (0.74 ± 0.828) ⋅ 10^{−3}  

0.308 ± 0.535  0.271 ± 0.698  0.165 ± 0.187  
0.309 ± 0.535  0.271 ± 0.698  0.165 ± 0.187  

(3.90 ± 9.68) ⋅ 10^{−3}  (0.91 ± 3.48) ⋅ 10^{−3}  (4.48 ± 7.13) ⋅ 10^{−3} 
To test the robustness of this metaanalysis, we conceive two tests of the sensitivity of the results, namely to lowsignificance data sets, to data sets with extreme results and to extreme data sets.
To test the sensitivity of the results to lowsignificance data sets, we exclude the data sets with comparatively small sample sizes for the same CHD phenotype, namely the study by
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
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) lowsignificante data sets combined with data with extreme results, and to (b) extreme data sets, which renders this formalism significantly robust.
In this manuscript we investigated the correlation between the occurrence of CHD with the presence of the −308 TNF
Hypothesis testing on the combined data sets indicated that there is no evidence for a correlation between the occurrence of CHD and the presence of the SNP, either on Caucasians or on Asians. This result agrees with previous metaanalyses (
An interesting extension of this work for the sake of completion is the inclusion of studies referring to Africans and Indians which are currently too few to extract convincing results.
The author thanks E Vourvouhaki and G Tsiliki for useful discussions.
The author declares there are no competing interests.