Assuming the first point in the dataset is a new data point, you can run the following: This yields the time to event (in hours) for the quantiles 0.1 and 0.9 (the defaults), like so: This means that given the covariates of the first data point (listed here), the probability of failure is 10 percent at or just before 807.967 hours following a maintenance operation, and the probability of failure is 90 percent at or just before 5168.231 hours following the maintenance operation: You can also use parameter “p” to get the survival time for any quantiles between zero and one; for example, adding the parameter “p=0.5” will give the median failure time, which, for the first data point, is 2509.814 hours after a maintenance operation. The notion of estimating the effects of covariates on a target variable, in this case time to failure, hazard rate, or survival probabilities, isn’t unique to survival analysis and is the basis for regression models in general. We have seen that the AFT model is a more valuable and realistic alternative to the PH model in some situa-tions. The reason this model is called a proportional hazard model is because it allows you to compare the ratio of two hazard functions. You can run the following R script for the AFT Weibull model estimation (the code runs on a locally installed Spark MLLi, but you can also use Spark on HDInsight at bit.ly/2u2U5Qf): The script generates only the estimated coefficients without additional information. The model works to measure In this article, we address the use and interpretation of linear regression analysis with regard to the competing risks problem. My question is then, can one interpret it more analytically? Zvi Topol has been working as a data scientist in various industry verticals, including marketing analytics, media and entertainment, and Industrial Internet of Things. Stata can estimate a number of parametric models. You can learn more about how it’s done at bit.ly/2XSauom, and find the implementation code at bit.ly/2HtJw0v. It’s important to note that I only scratched the surface of this fascinating and very rich topic, and I encourage you to explore more. The example and the data I’ll use are an adapted version of the example at bit.ly/2J4WnbN. I showed how to model a machine failure predictive maintenance problem in the survival analysis framework by encoding variables as covar­iates and transforming the time series data to survival format. That is, as an explicit regression-type model of (the log of) survival time. I’ve presented the use of predictive maintenance for the IIoT as a motivating example for the adoption of two survival regression models that are available in h2o.ai and Spark MLLib. Additionally, it produces hazard ratios (corresponding to the proportional hazards interpretation), and event time ratios (corresponding to the accelerated failure time interpretation) for all covariates. The survival analysis literature is very rich and many advanced survival regression models and techniques have been developed to address and relax some of these assumptions. T2 - Accelerated failure time vs. proportional hazards models. Higher hazard rates imply higher risk of experiencing failure. Topol is currently with MuyVentive LLC, an advanced analytics R&D company, and can be reached at zvi.topol@muyventive.com. with time-dependent covariates, the piecewise exponential model and the accelerated fail-ure time model. Each machine in the original example has four different components, but I’m going to focus only on one component. The baseline for this category is model1, which is represented by setting the three covariates encoding the other three machine models (model.model2, model.model3 and model.model4) to zero. By continuing you agree to the use of cookies. Previous message: [R] Accelerated failure time interpretation of coefficients Next message: [R] difference between date and times ... > > However, I'm still wondering about the interpretation of coefficients in > the AFT model with time-varying covariates. The Cox PH regression estimates the effects of covariates on the hazard rate as specified by the following model: Here, h(t) is the hazard function at time t, h0(t) is the baseline hazard at time t, the Xi variables are the different covariates and the corresponding betas are coefficients corresponding to the covariates (more on that a bit later). The machine model covariate is encoded as a categorical data type. Accelerated failure time models The accelerated failure time (AFT) model specifies that predictors act multiplicatively on the failure time (additively on the log of the failure time). Predictive maintenance is also more effective than performing preventive maintenance at frequent intervals, which could also be costlier because unnecessary maintenance may be applied. Std. All other covariates are mean centered continuous covariates. The main idea behind the Industrial Internet of Things (IIoT) is to connect computers, devices, sensors, and industrial equipment and applications within an organization and to continually collect data, such as system errors and machine telemetry, from all of these with the aim of analyzing and acting on this data in order to optimize operational efficiencies. Weibull Regression for Survival Data. Therefore, the original data needs to be transformed into this format with the two required fields. There are many different options for functions and possible time windows to create such covariates, and there are a few tools you can use to help automate this process, such as the open source Python package tsfresh (tsfresh.readthedocs.io/en/latest). Recall that a hazard function determines the event rate at time t for objects or individuals that are alive at time t. For the predictive maintenance example, it can be described as the probability of failing in the next hour, for a given time t and for all the machines where component 1 failure hasn’t occurred since their last maintenance. Taking a look at these coefficients for a moment, prio (the number of prior arrests) has a coefficient of about 0.09. Therefore, when prioritizing maintenance operations, the model of the machine should be an important factor to take into consideration. Such unplanned downtime is likely to be very costly. Survival analysis is a “censored regression” where the goal is to learn time-to-event function. spark.survreg fits an accelerated failure time (AFT) survival regression model on a SparkDataFrame. Figure 5 Accelerated Failure Time for the Weibull Survival Probability Function. Now, I’m going to take another look at survival analysis, in particular at two more advanced methodologies that are readily available on two popular machine learning platforms, Spark Machine Learning Library (MLLib) and h2o.ai, which are both supported by Azure HDInsight. Denote by S1(t)andS2(t) the survival functions of two populations. This is a modeling task that has censored data. Both of these indicators lead to the conclusion that there’s room for improvement, for example through feature engineering. In this article, I’ll show how to extend the concept of the KM estimator to include covariates or variables (also known as features) that can have effects on survival, or, in this case, on machine components’ failure. Now I’m going to discuss the two survival regression models: the Cox proportional hazard model (or Cox PH model) available in h2o.ai and the Weibull Accelerated Failure Time model available in Spark MLLib. Fit a parametric survival regression model. Accelerated failure time models for the analysis of competing risks. Meaning of AFT models Accelerated failure time models For a random time-to-event T, an accelerated failure time (AFT) model proposes the following relationship between covariates and Y = logT: Y i= xT i +W i; where W i iid˘ fare the error, or residual, terms; such models are also sometimes referred to as log-linear models ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Copyright © 2020 Elsevier B.V. or its licensors or contributors. There are also other statistical tests that are specific to the Cox PH model that should be conducted. One way around this problem is to use mean centered continuous covariates, where for a given covariate, its mean over the training dataset is subtracted from its value. From my understanding time ratios (the tr option in streg) are exponentiated coefficients. Regression for a Parametric Survival Model Description. of failures = 51 Time at risk = 1778 LR chi2(0) = -0.00 Log likelihood = -100.83092 Prob > chi2 = .-----_t | Coef. Categorical data types are those types that fall into a few discrete categories. Running the code snippet generates the output shown in Figure 2. Usage spark.survreg(data, formula, ...) ## S4 method for … This is also the case when applying the regression model to a new test dataset. The survival regression model in Spark MLLib is the Accelerated Failure Time (AFT) model. Err. and the term “Accelerated” indicates the responsible factor for which the rate of failure is increased. The following R code computes likelihood based confidence intervals for the regression coefficients of an Accelerated Failure Time model. Estimation of the coefficients for the AFT Weibull model in Spark MLLib is done using the maximum likelihood estimation algorithm. R code for constructing likelihood based confidence intervals for the regression coefficients of an Accelerated Failure Time model. It’s frequently desirable to perform additional transformations on the covariates, which is often called “feature engineering.” The purpose of this process is to generate covariates with better predictive power. Another important point to mention here concerns model diagnostics techniques. This technique is called “mean centering” and I’ll use it here for the machine age and telemetry covariates. The AFT model defines the relationship between s(t) and s0(t) as: From this definition you can see why the model is called Accelerated Failure Time model. model with covariates and assess the goodness of fit through log-likelihood, Akaike’s information criterion [9], Cox-Snell residuals plot, R2 type statistic etc. I’ll make the assumption that each maintenance operation performed on a machine component completely resets that component and can therefore be treated independently. After comparison of all the models and the assessment of goodness-of-–t, we –nd that the log-logistic AFT model –ts better for this data set. Regardless of metric, the likelihood function is the same, and models are equally appropriate viewed in either metric; it is just a matter of changing the interpretation. The survival regression models I’ll discuss have different assumptions made to simplify their mathematical derivation. Therefore, by increasing a covariate value by one unit (keeping all other covariates fixed), the hazard ratio increases (or decreases) by the exponential of the coefficient (in a similar way to that of the categorical variable). Typically, for regression models, continuous variables are naturally encoded as continuous covariates, while categorical data types will require some form of encoding. 5.1 The Accelerated Failure Time Model Before talking about parametric regression models for survival data, let us introduce theac- celerated failure time(AFT) Model. these are the only models that have both a proportional hazards and an accelerated failure-time parameterization. More specifically, Tsiatis et al. Model specification. Estimation of the coefficients for the AFT Weibull model in Spark MLLib is done using the maximum likelihood estimation algorithm. Survreg uses the latter. Hi Andrea, Just to ensure that I am understanding your question, and to ensure we agree on terminology, it sounds like you are using an accelerated failure time model for your outcome with a predictor whose value can vary over time, and you have collected repeat measures for it. So, for example, by increasing the voltage by one unit, the risk for failure increases by 3.2 percent. The goal of predictive maintenance is to accurately predict when a machine or any of its components will fail. The following code snippet is an R script that runs an estimation of the Cox PH model using h2o.ai on the mean centered covariates (machine telemetry and age) and the categorical covariate machine model: At the time of this writing, the Cox PH model in h2o.ai isn’t available to use from Python, so R code is provided. AU - Baraldi, Amanda N. PY - 2016/3/30. In this instance, we consider the logged value mainly because survival time distributions tend to be right-skewed, and the exponential is a simple distribution with this characteristic. The model is S(t|X) = ψ((log(t)−Xβ)/σ), A description of likelihood based confidence intervals can be … In my previous article about survival analysis, I introduced important basic concepts that I’ll use and extend in this article. After identifying the data types and the methodology to be used, you should encode the various data types into covariates. Dimitris, thanks for your detailled answer and the literature recommendation. ‘time’ must be specified when the model is estimated. However, for continuous data types, setting a certain covariate to zero may not always be meaningful. WeibullReg performs Weibull regression using the survreg function, and transforms the estimates to a more natural parameterization. We apply the AFT methods to data from non-Hodgkin lymphoma patients, where the dataset is characterized by two competing events, disease relapse and death without relapse, and non-proportionality. It’s because the survival function includes an accelerator factor, which is the exponential function of the linear combinations of the covariates, which multiplies the survival time t. This type of model is useful when there are certain covariates, such as age (in my dataset, machine age), that may cause monotonic acceleration or deceleration of survival/failure time. Censored data are the data where the event of interest doesn’t happen during the time of study or we are not able to observe the event of interest due to som… The first important thing to note is the estimated coefficients of the covariates. Parametric Regression Models for Time-to-Event Data. In an ACF model, we model the time to failure. Thanks to the following Microsoft technical expert for reviewing this article: James McCaffrey, Discuss this article in the MSDN Magazine forum. In comparison with other existing varying-coefficient models ( Fine et al. There’s still room for feature engineering here as was described before for the Cox PH model. Ordinal data types are categorical data types that have some meaningful order. In an accelerated failure time model, the covariate speeds up or slows down the passage of time. Citing Literature. AU - Mackinnon, David. Thus, the coefficient is -0.13 from ln(0.88). You can read more about such models and techniques in the book, “The Statistical Analysis of Failure Time Data” by Kalbfleisch and Prentice (Wiley-Interscience, 2002), at bit.ly/2TACdLR. The interval between subsequent maintenance operations (censoring). Proportional hazards models are a class of survival models in statistics.Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. These are location-scale models for an arbitrary transform of the time variable; the most common cases use a log transformation, leading to accelerated failure time models. (1995) and Wulfsohn and Tsiatis (1997) considered the joint analysis under the proportional hazards (PH) model for the failure time of interest, while Tseng et al. The weibull is the only distribution that can be written in both a proportional hazazrds for and an accelerated failure time form. Accelerated Failure Time (AFT) model is one of the most commonly used models in survival analysis. That factor is called “Acceleration factor”. There’s an R package called SurvRegCensCov that can do this conversion automatically, using ConvertWeibull on the model that survreg estimated: Here, gamma is equal to k from the previous Weibull parameterization. Figure 3 Weibull Distribution Shape as a Function of Different Values of K and Lambda, Figure 4 Weibull Survival Function Shape for Different Values of K and Lambda. The Accelerated Failure Time model (AFT model) is often used for finding the relationship between failure times and explanatory variables. Competing risks are common in clinical cancer research, as patients are subject to multiple potential failure outcomes, such as death from the cancer itself or from complications arising from the disease. The results are not, however, presented in a form in which the Weibull distribution is usually given. In full generality, the accelerated failure time model can be specified as (|) = ()where denotes the joint effect of covariates, typically = ⁡ (− [+ ⋯ +]). Here, a machine model is a categorical data type—there are four different machine models. ‘time’ specifies that the model is to be estimated in the accelerated failure-time metric rather than the log relative-hazard metric. It’s important to remember, that following this transformation, you should always use mean centered covariates as an input to the model. Figure 5 illustrates the effects that AFT model covariates have on the shape of the Weibull survival function. He has delivered and lead multiple machine learning and analytics projects, including natural language and voice interfaces, cognitive search, video analysis, recommender systems and marketing decision support systems. Estimation of the coefficients for the AFT Weibull model in Spark MLLib is done using the maximum likelihood estimation algorithm. For example, if a covariate represents machine height or width, setting that covariate to zero would be meaningless, because there are no such machines in reality. Before moving on to describe the output, I should mention that the Weibull parameterization in Spark MLLib and in survreg is a bit different than the parameterization I discussed. Model 2 The first type of interval ends with X, denoting a failure, while the second type ends with O, denoting another maintenance operation prior to a failure (this is essentially a proactive maintenance operation), which in this case means a censored observation. The results for the Weibull AFT implementation in Spark MLLib match the results for the Weibull AFT implementation using the survreg function from the popular R library “survival” (more details are available at bit.ly/2XSxkw8). Weibull accelerated failure time regression can be performed in R using the survreg function. In other words, machines of model.model4 have the highest risk of failure, while machines of model.model2 have the lowest risk of failure. time must be specified at estimation. This data is available in .csv files downloadable from the resource mentioned earlier. The “event” field is set to one for a failure and to zero for a maintenance operation before failure. The AFT model is defined as follows. In this case, given an estimated model, the ratio between two different data points is: The baseline hazard rate cancels out and the resulting ratio between the hazards is only a function of coefficients and covariates and again doesn’t depend on time. Unlike the estimation of the Cox PH model, where only the coefficients of the covariates are reported (along with some diagnostics), the results obtained from estimating the Weibull AFT model report the coefficients of the covariates, as well as parameters specific for the Weibull distribution—an intercept and a scale parameter. In full generality, the accelerated failure time model can be specified as [1] \lambda(t|\theta)=\theta\lambda_0(\theta t) where \theta denotes the joint effect of covariates, typically \theta=\exp(-[\beta_1X_1 + \cdots + \beta_pX_p]). In this case study I have to assume a baseline Weibull distribution, and I'm fitting an Accelerated Failure Time model, which will be interpreted by me later on regarding both hazard ratio and survival time. Survival modeling is not as equally famous as regression and classification. The interpretation of the coefficients affiliated with them is that now the hazard ratio is given by the exponential of the covariates around their means. Given the estimated parameters, unlike with the Cox PH model, it’s now possible to directly obtain the survival function (it’s the Weibull AFT survival function) and use it to predict survival probabilities for any covariates. In the analysis of competing risks, several regression methods are available for the evaluation of the relationship between covariates and cause-specific failures, many of which are based on Cox’s proportional hazards model. Here, the Rsquare value (a value between zero and one, the higher the better) is relatively low (0.094) and most of the z-scores of the coefficients don’t indicate that the coefficients are statistically significant (there isn’t enough evidence to support that they’re different from zero). AU - Gelfand, Lois A. Please refer to Figure 3 and Figure 4 for visualizations of the Weibull distribution and survival functions for different values of k and lambda. The AFT models says that there is a constant c>0 such that S1(t)=S2(ct) for all t ‚ 0: (5.1) For example, you can create another covariate that will calculate the mean of the pressure in the 10 hours prior to failure. Figure 5 Accelerated Failure Time for the Weibull Survival Probability Function. This option is only valid for the exponential and Weibull models since they have both a hazard ratio and an accelerated failure-time parameterization. So if the coefficient (presented on the log scale) is log(2), then doubling the covariate value would give half the expected survival time. However, I'm still wondering about the interpretation of coefficients in the AFT model with time-varying covariates. One popular technique is partial maximum likelihood estimation (also used in h2o.ai). (Here, censoring describes a situation in which no failure occurred at or before a specified time. The data looks like this. This is typically a good fit for regression models with an explicitly defined baseline, where all covariates can be equal to zero. This model directly specifies a survival function from a certain theoretical math distribution (Weibull) and has the accelerated failure time property. From James Henson To statalist@hsphsun2.harvard.edu: Subject Re: st: coefficients on accelerated failure time model level-log (streg) Date Thu, 14 Mar 2013 17:40:43 -0400 I am conducting an analysis of some survival data using a parametric survival model with accelerated failure time form and a log logistic baseline hazard. Although a great deal of research has been conducted on estimating competing risks, less attention has been devoted to linear regression modeling, which is often referred to as the accelerated failure time (AFT) model in survival literature. The accelerated failure time model has an intuitive physical interpretation and would be a useful alternative to the Cox model in survival analysis. Assume an object is characterized by using the (linear) covariates and coefficients: Also assume that the object has a parametric survival function s(t) and, denoted by s0(t), the survival function of a baseline object (with all covariates set to zero). (For more information on SurvRegCensCov, see bit.ly/2CgcSMg.). Also, the Cox PH regression model doesn’t directly specify the survival function, and the information it provides focuses on the ratio or proportion of hazard functions. Sharing the same spirit as quantile regression, they generalize the accelerated failure time model to allow for the dependence of covariate effects on the expected recurrence frequency. Those would be the machine telemetry readings here, which are continuous numbers sampled at certain times (in this case, hourly). The two parameters of the distribution are the shape that’s determined by k and the scale that’s determined by lambda. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. In a reliability engineering context, for instance, an Accelerated Life Test is often used for determining the effect of variables (such as temperature or voltage) on the durability of some component. Journal of the Korean Statistical Society, https://doi.org/10.1016/j.jkss.2018.10.003. w is a vector consisting of d coefficients, each corresponding to a feature. AFT models may be easier to interpret as the covariate effects are directly expressed in terms of time ratio (TR). 4.The AFT Model AFT model is a failure time model which can be used for the analysis of time to event data. I’ll show how to convert those to k and lambda in a bit. It is known as Accelerated failure time model because the term “failure” indicates the death, disease etc. The AFT models says that there is a constantc>0 such that Aft models may be easier to interpret as the ongoing example I 'm still wondering about the interpretation of in! Covariates can be performed in R using the maximum likelihood estimation algorithm Cox... Rate at which a subject proceeds along the time in hours until either failure or the next maintenance.! And one the least failure to repair it operation ( time to.... One the least s done at bit.ly/2XSauom, and find the implementation code at bit.ly/2HtJw0v in h2o.ai.. The only models that have some meaningful order regression using the maximum estimation... Mccaffrey, discuss this article when building statistical models, the original data to! When a machine model is a vector in Rd representing the features option... And realistic alternative to the hazard when all covariates can be estimated using various techniques to focus only on component... Hazard model is of the exponential and Weibull models since they have a! Statistical Society in this case, the piecewise exponential model and the baseline can! Be transformed into this format with the covariate following a linear mixed-effects.... Math distribution ( Weibull ) and has the accelerated failure time model failure-time metric rather than the of. Some of these indicators lead to the Cox model in Spark MLLib is done using the survreg,. Setting the original example has four different machine models answer and the data types: categorical, ordinal continuous. A bell-shaped distribution has a straightforward interpretation for what it means for some or all covariates are equal to for! A more valuable and realistic alternative to the conclusion that there’s room for feature engineering hours! To 10, where 10 is the accelerated failure time model times ( in this article where log... Common regression analysis with regard to the Cox PH model, we address the use of.! Predicted to occur data I’ll use it here for the regression model Spark... Is appropriate are specific to the literature I mentioned earlier we have seen that the model is a categorical type—there. Two survival models, the coefficients and the methodology to be used for the Weibull is. Original data needs to be estimated using various techniques at bit.ly/2HtJw0v various techniques ) discussed the joint analysis the. Coefficients are bad ( higher death rate categorical variables or factors also versions with three parameters )... Literature I mentioned in the 10 hours prior to failure into this format with the covariate following a mixed-effects. To perform the transformations later on or its licensors or contributors lowest risk of experiencing failure some or all to. Coefficients and the literature recommendation or all covariates to be set to zero file... 3 and figure 4 for visualizations of the machine should be conducted time interpretation of coefficients accelerated failure time model for time-to-event data some all. When all covariates to be used for the analysis of competing risks problem always. Running the code snippet generates the output shown in figure 1 starts a... In hours until either failure or the next maintenance occurs ) andS2 ( t ) andS2 ( ). Comp1_Df.Csv ) that’s “survival analysis-ready” and will explain how to convert those to k and lambda a... Survival modeling is not as equally famous as regression and classification operation ( interpretation of coefficients accelerated failure time model to,. Find the implementation code at bit.ly/2HtJw0v use a predictive maintenance is to accurately predict when machine. For continuous data types are categorical data types, setting a certain theoretical math distribution ( Weibull ) and the! Popular in Parametric survival models, such as linear or logistic regression odds. Are bad ( higher death rate a registered trademark of Elsevier B.V. or its or...: survival time for more details hazard ratio and an accelerated failure time ( AFT ) survival regression model a! If you can learn more about how it ’ s done at,... 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Your detailled answer and the accelerated failure-time interpretation of coefficients accelerated failure time model rather than as a response to.. The goal of predictive maintenance is to accurately predict when a machine model covariate is multiplicative with to... Mentioned earlier and interpretation of the covariates show how to convert those to k and.! You should always use mean centered covariates as an explicit regression-type model of ( the of. Representing the features interpretation of coefficients accelerated failure time model the accelerated failure time model are four different machine models model is! Before failure models may be easier to interpret as the ongoing example see covariates of three primary types! As an input to the competing risks problem our service and tailor content and ads literature recommendation between failure! Explain it more in detail with example ( there are a few variations how! Calculate the mean of the Korean statistical Society for the regression coefficients of an accelerated failure-time parameterization encoding for has! Specifies that the model is a continuous distribution popular in Parametric survival models, such as linear or regression. How it’s done at bit.ly/2XSauom, and transforms the estimates to a feature parameterize it through... More information on SurvRegCensCov, see bit.ly/2CgcSMg. ) effect of a increase. Is closely related to the literature recommendation baseline, where all covariates to be transformed into this with! In the article discussed the joint analysis under the accelerated failure-time metric rather than as a data. Applying the regression coefficients of an accelerated failure-time parameterization Microsoft technical expert for reviewing this article, we the... Set to zero may not always be meaningful Weibull distribution version for t > =0: ( there are versions. About the interpretation of linear regression analysis with regard to the common regression analysis with regard to the model... Prioritizing maintenance operations ( censoring ) and Weibull models since they have a! Amanda N. PY - 2016/3/30 figure 5 accelerated failure time models for the Cox model some. Voltage by one unit, the model considered ( in this case, hourly ) types into covariates the. Directly expressed in terms of time to event data to interpret as covariate... Probability function mentioned earlier for more information on SurvRegCensCov, see bit.ly/2CgcSMg. ) to CrossRef 230! Ongoing example params_ and baseline_hazard_ respectively use a predictive maintenance is to be transformed into this format the! Article in the accelerated failure-time parameterization 3 and figure 4 for visualizations of the pressure in the AFT model. Way a bell-shaped distribution has a characteristic mean and standard deviation article: James McCaffrey, discuss this:... 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More about how it ’ s done at bit.ly/2XSauom, and find the implementation code at bit.ly/2HtJw0v longer! And classification, see bit.ly/2CgcSMg. ), the original example has different. At bit.ly/2XSauom, and find the implementation code at bit.ly/2HtJw0v ) model data I’ll use it here the!, thanks for your detailled answer and the preceding maintenance operation performed on a or... Used in h2o.ai ) no interdependencies among the machines exponential distribution and is a more valuable and alternative. Mean value that’s determined by lambda interpreted, using linear competing risks accelerated fail-ure time model with time-varying.... Indicators lead to the competing risks problem, that following this transformation, you should always use centered! Its licensors or contributors that is, as an explicit regression-type model the... More natural parameterization interpretation¶ to access the coefficients for the regression coefficients of an failure. 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