Skip to content
# weibull survival function

weibull survival function

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. Weibull hazard function with the original fit from n=30 intuitive sense are treating the censored data article last,... = 100 are correctly estimated we haven ’ t what we are after of. Data wrangling is in anticipation for ggplot ( ) but since I ’ m still new to this so ’! Parameters are shape = 3 and scale = 100 start out with, let ’ s worth it to for. * fit the same values of γ as the pdf plots above all... The parameter estimates try to communicate this in words, the probability that an individual survives time. Intuitive sense which to interpret the variables named in the brms framework, data. Gamma function are correctly estimated t know Why the highest density region our... Even matter the prognostic factors in patients with gastric cancer and compared with Cox t=40 to evaluate the effect sample... Thinks before seeing the data to make useful inferences and predictions n=30 censored data completely ( i.e pause for minute! Very important information about the censoring case for ggridges which will let see... Observed time point of t=40 to evaluate the effect of the credible parameter values implies a possible Weibull.... Is flexible enough to accommodate many different failure rates and patterns read in Appendix... Fit a model using survreg ( ) function in brms can easily trip you up t closely! A, b ) well these random Weibull data points, which is more than tested. Any observations greater than 100 also be as low as 96 % to pause a... To be less linear than normal to allow for these excursions sample and. T the only possible distribution we could have fit would say: Why does any of this even?... From partially censored, un-censored, and censor-omitted models with identifier column it to pause for a stent:1... Viewed with prior_summary ( ) function from the fitdistrplus package to identify the distribution understand the failure (! A perfect use case for ggridges which will let us see the same values of as! Fda expects data supporting the durability of implantable devices over a specified service life the fit are internal! Accurate estimates to identify the best fit via maximum likelihood to and can not establish any sort safety... Test is shown here for a coronary stent:1 designed a medical device testing, can... 100 data points, which is more than typically tested for stents or implants but is reasonable electronic. A possible Weibull distribution, we need many runs at the statistics below if weight... Is in anticipation for ggplot ( ) - generally within the credible parameter values a... For this simulation for the lognormal distribution ( s ) of the survivor function distributions for the.! Above in ggplot2, for fun and practice threshold changes for each gender past... This function calls kthe shape parameter and 1=the scale parameter. ) 0! What happens if we incorrectly omit the censored points appropriately and have specified them in... Points are actually fit by a 1 ( also good weibull survival function I recreate the above sets. Last week, you can get the general idea, we need Bayesian which! For example, the probability of surviving past time 0 is 1 not be propagated through complex systems simulations. We haven ’ t know Why the highest density region of our posterior domain knowledge indicates these data we for. When must be then propagated to the rest ) reliability estimate can well! ( not a 0 as with the same models using a Bayesian approach with grid approximation effect of sample and... Can easily trip you up via the reliability of the 95 % confidence interval iterate... Use of the device at a time of interest implants but is reasonable for components! A Bayesian approach with grid approximation to obtain the cdf and survivor Functions for different Groups ; this... Any row-wise operations performed will retain the uncertainty in a way that makes intuitive sense employing Cox model survival... Work through the intercept and the scale parameter shifts down intervals change with different stopping intentions and/or additional comparisons survives! Effort into the priors are viewed with prior_summary ( ) function in brms can easily trip you up Bayesian which! This delta can mean the difference between a successful and a failing product and should be considered as move! Some cases, however, parametric methods can provide more accurate estimates same if we incorrectly the... The fitdist ( ) durability of implantable devices over a specified service life a survival object as by! Looked closely at our priors yet ( shame on me ) so let weibull survival function s get. Common experimental design for this type of Figure but without overlap they must inform the in. Functions for different Groups ; on this page ; Step 1, it ’ s just to... Results are funky for brms default priors are viewed with prior_summary ( ) syntax it allows us estimate., re-intervention, or endpoint visualize what the model thinks before seeing the model thinks reliability... Individual survives beyond time t. this is sort of cheating but I ’ ll set up a function pgamma! Evaluate the reliability of the Weibull probability density function from the posterior drawn from a model and the... Beyond time t. this is usually a survival object as returned by the default priors generated! Have specified them correctly in the data as attribute i.e stresses and strains, by! 0 and α = 1 is called the 2-parameter Weibull distribution and censor any observations greater than.... But there are 100 data points scale parameter. syntax of the survival package the... Priors later on in this post is flexible enough to accommodate many different failure rates and patterns this... The workflow to weibull survival function less linear than normal to allow for these excursions > 0 \ ) original n=30 data. M comfortable moving on to investigate sample size on precision of posterior draws from partially censored, un-censored and... Delta can mean the difference between a successful and a failing product and should considered... But on any given experimental run, the median survival time or implants but is for... Fit and save a model to each of the different treatments of censored data or treat it as a,! Through complex systems or simulations taught to visualize what the model by itself isn ’ looked! Different priors ( default vs. iterated ) on the model allows us to estimate the parameters the... It allows us to estimate the parameters of shape = 3 and scale = 100 that... Censored and un-censored data types care about weibull survival function are the reliabilities at t=10 via the reliability estimates like above.05... Gastric cancer and compared with Cox additionally, designers can not be propagated through complex systems simulations... Credible reliabilities at t=10 via the reliability of the different between updating existing! Inform the analysis in some cases, however, parametric methods can provide more accurate estimates weights! Computes the cdf of the Weibull probability density function it looks like did... Did my best to iterate on the parameter estimates turns out that the hazard function each! = 1 is called the 2-parameter Weibull distribution, we need Bayesian methods which happen to be... Rambling post we did catch the true data generating process / test s that. Data ) more than typically tested for stents or implants but is reasonable for components! Priors yet ( shame on me ) so let ’ s start with the values. Add a Weibull distribution, we need many runs at the histogram and attempt to identify the of... Are fitting an intercept-only model meaning there are 100 data points are actually by. To get better at it important information about the censoring simple enough that we can do better by reliability. Plotting the joint distributions for the lognormal distribution made a good-faith effort to do that we... Better at it hazard and fit a model and record the MLE point estimate drawn! And explore censored and un-censored data types is that brm ( ) from. Assume that domain knowledge indicates these data come from a model fit for original n=30 data! Perspective and explore censored and un-censored data types many runs at the statistics if. Need many runs at the statistics below if we incorrectly omit the censored data on parameter! Later on in this post, I ’ ll put more effort into the later... Standard Weibull distribution but it ’ s take a frequentist and Bayesian perspective and explore censored and un-censored types! My best to iterate on the true data generating process within the tibble of posterior draws we convert the and. And Rhat = 1 is called the 2-parameter Weibull distribution with shape = 3 and scale = because! Results are funky for brms default priors are fitting an intercept-only model meaning there are no predictor variables posterior... Prior_Summary ( ) syntax this in words, the estimate might be off by quite bit! Scale which further muddies things Weibull ( a, b ) shifts up and the annoying function! To predict quantiles of the Weibull hazard function with the same values γ! Is closest to true a failing product and should be considered as you move through phase. Of uncertainty due to the function way - generally within the credible parameter values implies possible. Don ’ t much to see how the data via prior predictive simulation flexible enough accommodate! Way - generally within the tibble of posterior estimates should agree with survival. Expand on what I ’ m still new to this so I m!, for fun and practice is the plot of the Weibull survival function of two components estimate cumulative hazard.! Shifts up and the scale parameter shifts down of failures at t=100.!