PSI - Issue 38

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514 Robin Hauteville, Xavier Hermite, Fabien Lefebvre / Structural Integrity Procedia 00 (2021) 000 – 000 With ̄ and  denoting the sample mean and standard deviation, respectively. Grubbs' test statistic is based on the identification of the largest difference between each experimental data and the sample mean, and the standard deviation of the sample. The assumption of no outliers is rejected if: > ( √ − 1) √ ( /(2 ), −2 2 − 2 + ( /(2 ), −2 2 (17) ( /(2 ), −2 2 indicating the critical value of the t-distribution with (N-2) degrees of freedom and a confidence level of  /(2N). 5.2. Bartlett test [11] Bartlett's test compares the variances of n dataset (n> 2), considering that they are normally distributed. The tests are carried out by setting a stress level and identifying the lifetimes at that level on different samples. The Bartlett test can also be performed on the variances of lifetimes by filtering the test results by stress level. To accept the assumption of a normal distribution over lifetime, they are transformed into logarithm. The Bartlett random variable is calculated and assumed to follow a chi² distribution, allowing estimates of the p-value. The p-value must be compared to the first-order risk, equal to 1 - the confidence level: if the p-value is inferior to the risks, the variances cannot be considered as equivalent. 5.3. Median curve The median curve is obtained using the maximum likelihood method as explained before. Linear regression is used to obtain initial values of parameters, and the sum of the likelihoods is then optimized through a Newtonian-type resolution algorithm. It was also indicated that for the Stromeyer and Bastenaire models, an assumption on the C-value is necessary to simplify the estimation of the other parameters. It is therefore proposed to estimate the parameters A, B (for Bastenaire) and E by an iteration of the parameter C (assumed as an integer) which will be between 1 and 3. Of course, if it turns out that none of the three curves is satisfying, it is possible to increase the maximum value of the parameter C or even to decrease the value of the iteration step. To obtain the initial values of each parameter, a linear regression is first performed on the simplified Bastenaire and Stromeyer models. log( ) + log( − ) = log + ( − ) (18) − 1 = ( − ) (19) A Newtonian algorithm then maximises the log-likelihood from dataset (failures and censored data), for each C value. The standard deviation on the stresses is also calculated. Then the curve with the most optimized log-likelihood is selected. Once the final parameters are obtained, fit tests of each model are then run. 5.4. Fischer-Snedecor test [12] The Fisher-Snedecor test compares the variances of 2 batches. It can also be performed to compare the intrinsic variance of a dataset with the residual variance of the Bastenaire/Stromeyer model to conclude on their goodness of fit. The intrinsic variance of the dataset (after a positive Bartlett test) and the residual variance of the model are based on the lifetime. The Bartlett p-value must be greater than the first-order risk to consider the homogeneity of variances and estimate the intrinsic variance. Robin Hauteville et al. / Procedia Structural Integrity 38 (2022) 507–518