2.5. Statistical Methods

In this study, the Bayesian statistical method was used to evaluate the confidence intervals (called in the Bayesian framework credibility intervals) and to compute the p values associated with our tests, as in [20]. Indeed, the Bayesian statistical methods are more appropriate in case of small samples and the “approximate value” of the variance of the logarithm of the Odds Ratio. In the Bayesian framework, the unknown parameter is considered as the realization of a random variable, which forces us to choose an a priori probability distribution for that random variable. We chose an “a priori” distribution for a pair of frequencies and decided that the two random variables are independent, each with the probability density equal to 1 on the interval (0.1). It is fair to say that our prior is uninformative: it is the uniform distribution on the interval (0.1). The “a posteriori” distribution of each parameter is easily shown from the Bayes formula to be a Beta distribution. More precisely, if while observing n realizations of 1 s and 0 s, we get n 1 s and n′ 0 s, then the a posteriori distribution of the proportion is the Beta (n + 1, n′ + 1) probability distribution. Since the two samples come from completely unrelated sources, it is fair to assume that the two proportions are conditionally independent, given the data. Consequently a posteriori distribution of the pair of parameters (corresponding to the proportions of the given allele among the patients and among the control individuals) is the product of two Beta distributions whose density is explicitly known. Our method involves simulating one million realizations of that pair of random variables, from which we deduce one million realizations of the Odds Ratio. The p value of the test is then obtained by looking at which proportion of those values is smaller (resp., larger) than 1; this gives us t. Finally we want to discuss the issue of multiple testing. We have used our Bayesian test method 6 times (Table 5) where the probability of an error among six successive tests is not controlled by the probability of an error in each single test. The p values were corrected for multiple comparisons using the Bonferroni correction in Table 2. PLINK 1.7 version software was used for analysis of association between haplotypes and disease, and Haploview 4.2 software was used for LD analysis. All haplotypes with frequencies less than 5% were ignored in analysis. Distributions of data were tested with the Shapiro-Wilk test. Chi² or Fisher test was used for categorical data; Yates correction was performed. For numerical data, comparisons of medians were performed using Mann–Whitney or Student's t-tests. For correlation study, statistical analyses were performed at the conventional two-tailed α level of 0.05 using R software version 3.0.2.10. Odds ratios (OR) with 95% confidence interval (CI) were also calculated. A p value less than 0.05 was considered statistically significant.

TMEM187-IRAK1 genotypes for samples of RA and healthy controls in Tunisian and French female populations.

^∗ p values compared genotypes according to the dominant/recessive model.