First and foremost - we would be very interested in understanding the reliability of the device at a time of interest. I chose an arbitrary time point of t=40 to evaluate the reliability. I have all the code for this simulation for the defaults in the Appendix. Note: all models throughout the remainder of this post use the “better” priors (even though there is minimal difference in the model fits relative to brms default). This approach is not optimal however since it is generally only practical when all tested units pass the test and even then the sample size requirement are quite restricting. But we still don’t know why the highest density region of our posterior isn’t centered on the true value. Assume the service life requirement for the device is known and specified within the product’s requirements, Assume we can only test n=30 units in 1 test run and that testing is expensive and resource intensive, The n=30 failure/censor times will be subject to sampling variability and the model fit from the data will likely not be Weibull(3, 100), The variability in the parameter estimates is propagated to the reliability estimates - a distribution of reliability is generated for each potential service life requirement (in practice we would only have 1 requirement). This should give is confidence that we are treating the censored points appropriately and have specified them correctly in the brm() syntax. \( F(x) = 1 - e^{-(x^{\gamma})} \hspace{.3in} x \ge 0; \gamma > 0 \). The above gives a nice sense of the uncertainty in the reliability estimate as sample size increases, but you can’t actually simulate a confidence interval from those data because there aren’t enough data points at any one sample size. The function returns a tibble with estimates of shape and scale for that particular trial: Now that we have a function that takes a sample size n and returns fitted shape and scale values, we want to apply the function across many values of n. Let’s look at what happens to our point estimates of shape and scale as the sample size n increases from 10 to 1000 by 1. This delta can mean the difference between a successful and a failing product and should be considered as you move through project phase gates. Weibull’s Derivation n (1 ( )) (1 ) − = − = F x P e − ϕn n x ( ) ( ) 1 = − F x e −ϕx( ) x x o m u x x x F x e ( ) ( ) 1 − − = − A cdf can be transformed into the form This is convenient because Among simplest functions satisfying the condition is The function ϕ(x)must be positive, non … Calculate posterior via grid approximation:4. subset But on any given experimental run, the estimate might be off by quite a bit. 11 Each of the credible parameter values implies a possible Weibull distribution of time-to-failure data from which a reliability estimate can be inferred. \( Z(p) = (-\ln(p))^{1/\gamma} \hspace{.3in} 0 \le p < 1; \gamma > 0 \). Survival analysis is one of the less understood and highly applied algorithm by business analysts. All devices were tested until failure (no censored data). This is due to the default syntax of the survreg() function in the survival package that we intend to fit the model with:5. Once the parameters of the best fitting Weibull distribution of determined, they can be used to make useful inferences and predictions. In survival analysis we are waiting to observe the event of interest. Recall that each day on test represents 1 month in service. The operation looks like this:7. weights. This distribution gives much richer information than the MLE point estimate of reliability. In some cases, however, parametric methods can provide more accurate estimates. This looks a little nasty but it reads something like “the probability of a device surviving beyond time t conditional on parameters \(\beta\) and \(\eta\) is [some mathy function of t, \(\beta\) and \(\eta\)]. optional vector of case weights. The prior must be placed on the intercept when must be then propagated to the scale which further muddies things. For benchtop testing, we wait for fracture or some other failure. estimation for the Weibull distribution. Don’t fall for these tricks - just extract the desired information as follows: survival package defaults for parameterizing the Weibull distribution: Ok let’s see if the model can recover the parameters when we providing survreg() the tibble with n=30 data points (some censored): Extract and covert shape and scale with broom::tidy() and dplyr: What has happened here? Things look good visually and Rhat = 1 (also good). There’s a lot going on here so it’s worth it to pause for a minute. The syntax of the censoring column is brms (1 = censored). Flat priors are used here for simplicity - I’ll put more effort into the priors later on in this post. The likelihood is multiplied by the prior and converted to a probability for each set of candidate \(\beta\) and \(\eta\). De Weibull-verdeling wordt vaak gebruikt in plaats van de normale verdeling omwille van het feit dat een Weibull-verdeelde toevalsvariabele gegenereerd kan worden door inversie, terwijl normale toevalsvariabelen typisch gegenereerd worden met de complexere Box-Müller-transformatie, die twee uniform verdeelde toevalsvariabelen vereist. To further throw us off the trail, the survreg() function returns “scale”" and “intercept”" that must be converted to recover the shape and scale parameters that align with the rweibull() function used to create the data. * Fit the same models using a Bayesian approach with grid approximation. These data are just like those used before - a set of n=30 generated from a Weibull with shape = 3 and scale = 100. Now the function above is used to create simulated data sets for different sample sizes (all have shape 3, scale = 100). \( f(x) = \frac{\gamma} {\alpha} (\frac{x-\mu} Researchers in the medical sciences prefer employing Cox model for survival analysis. They represent months to failure as determined by accelerated testing. given for the standard form of the function. A lot of the weight is at zero but there are long tails for the defaults. This plot looks really cool, but the marginal distributions are bit cluttered. Since Weibull regression model allows for simultaneous description of treatment effect in terms of HR and relative change in survival time, ConvertWeibull() function is used to convert output from survreg() to more clinically relevant parameterization. a formula expression as for other regression models. Let’s fit a model to the same data set, but we’ll just treat the last time point as if the device failed there (i.e. Not too useful. Again, it’s tough because we have to work through the Intercept and the annoying gamma function. Our boss asks us to set up an experiment to verify with 95% confidence that 95% of our product will meet the 24 month service requirement without failing. Given the low model sensitivity across the range of priors I tried, I’m comfortable moving on to investigate sample size. The Weibull distribution is named for Professor Waloddi Weibull whose papers led to the wide use of the First – a bit of background. Weibull probability plot: We generated 100 Weibull random variables using \(T\) = 1000, \(\gamma\) = 1.5 and \(\alpha\) = 5000. \( G(p) = (-\ln(1 - p))^{1/\gamma} \hspace{.3in} 0 \le p < 1; \gamma > 0 \). In the following section I work with test data representing the number of days a set of devices were on test before failure.2 Each day on test represents 1 month in service. And the implied prior predictive reliability at t=15: This still isn’t great - now I’ve stacked most of the weight at 0 and 1 always fail or never fail. In an example given above, the proportion of men dying each year was constant at 10%, meaning that the hazard rate was constant. In a clinical study, we might be waiting for death, re-intervention, or endpoint. This article describes the characteristics of a popular distribution within life data analysis (LDA) – the Weibull distribution. To obtain the CDF of the Weibull distribution, we use weibull(a,b). In this post, I’ll explore reliability modeling techniques that are applicable to Class III medical device testing. Plot survivor functions. Fit and save a model to each of the above data sets. If you have a sample of n independent Weibull survival times, with parameters , and , then the likelihood function in terms of and is as follows: If you link the covariates to with , where is the vector of covariates corresponding to the i th observation and is a vector of regression coefficients, the log-likelihood function … That is a dangerous combination! Survival analysis is used for modeling and analyzing survival rate (likely to survive) and hazard rate (likely to die). We plot the survivor function that corresponds to our Weibull(5,3). Just like with the survival package, the default parameterization in brms can easily trip you up. Thank you for reading! Since the priors are flat, the posterior estimates should agree with the maximum likelihood point estimate. remove any units that don’t fail from the data set completely and fit a model to the rest). Here’s the TLDR of this whole section: Suppose the service life requirement for our device is 24 months (2 years). Intervals are 95% HDI. This is Bayesian updating. However, if we are willing to test a bit longer then the above figure indicates we can run the test to failure with only n=30 parts instead of n=59. In both cases, it moves farther away from true. \( H(x) = x^{\gamma} \hspace{.3in} x \ge 0; \gamma > 0 \). * Explored fitting censored data using the survival package. The precision increases with sample size as expected but the variation is still relevant even at large n. Based on this simulation we can conclude that our initial point estimate of 2.5, 94.3 fit from n=30 is within the range of what is to be expected and not a software bug or coding error. The default priors are viewed with prior_summary(). Now another model where we just omit the censored data completely (i.e. The case where μ = 0 is called the To start, I’ll read in the data and take a look at it. 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