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Wood property relationships and survival models in reliability 2010
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Title | Wood property relationships and survival models in reliability |
Creator |
Cheng, Yan |
Publisher | University of British Columbia |
Date Created | 2010-08-30 |
Date Issued | 2010-08-30 |
Date | 2010 |
Description | It has been a topic of great interest in wood engineering to understand the relationships between the different strength properties of lumber and the relationships between the strength properties and covariates such as visual grading characteristics. In our mechanical wood strength tests, each piece fails (breaks) after surviving a continuously increasing load to a level. The response of the test is the wood strength property -- load-to-failure, which is in a very different context from the standard time-to-failure data in Biostatistics. This topic is also called reliability analysis in engineering. In order to describe the relationships among strength properties, we develop joint and conditional survival functions by both a parametric method and a nonparametric approach. However, each piece of lumber can only be tested to destruction with one method, which makes modeling these joint strengths distributions challenging. In the past, this kind of problem has been solved by subjectively matching pieces of lumber, but the quality of this approach is then an issue. We apply the methodologies in survival analysis to the wood strength data collected in the FPInnovations (FPI) laboratory. The objective of the analysis is to build a predictive model that relates the strength properties to the recorded characteristics (i.e. a survival model in reliability). Our conclusion is that a type of wood defect (knot), a lumber grade status (off-grade: Yes/No) and a lumber's module of elasticity (moe) have statistically significant effects on wood strength. These significant covariates can be used to match pieces of lumber. This paper also supports use of the accelerated failure time (AFT) model as an alternative to the Cox proportional hazard (Cox PH) model in the analysis of survival data. Moreover, we conclude that the Weibull AFT model provides a much better fit than the Cox PH model in our data set with a satisfying predictive accuracy. |
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Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2010-08-30 |
DOI | 10.14288/1.0071230 |
Degree |
Master of Science - MSc |
Program |
Statistics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2010-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/27953 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0071230/source |
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Wood Property Relationships and Survival Models in Reliability by Yan Cheng B.Econ., Hunan University, 2000 B.Sc., The University of British Columbia, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Statistics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 c© Yan Cheng 2010 Abstract It has been a topic of great interest in wood engineering to understand the relationships between the different strength properties of lumber and the relationships between the strength properties and covariates such as visual grading characteristics. In our mechanical wood strength tests, each piece fails (breaks) after surviving a continuously increasing load to a level. The response of the test is the wood strength property – load-to-failure[13], which is in a very different context from the standard time-to-failure[16] data in Biostatistics. This topic is also called reliability analysis[13] in engineering. In order to describe the relationships among strength properties, we de- velop joint and conditional survival functions by both a parametric method and a nonparametric approach. However, each piece of lumber can only be tested to destruction with one method, which makes modeling these joint strengths distributions challenging. In the past, this kind of problem has been solved by subjectively matching pieces of lumber, but the quality of this approach is then an issue. We apply the methodologies in survival analysis to the wood strength data collected in the FPInnovations (FPI) laboratory. The objective of the analysis is to build a predictive model that relates the strength properties to the recorded characteristics (i.e. a survival model in reliability). Our conclu- sion is that a type of wood defect (knot), a lumber grade status (off-grade: Yes/No) and a lumber’s module of elasticity (moe) have statistically signif- icant effects on wood strength. These significant covariates can be used to match pieces of lumber. This paper also supports use of the accelerated fail- ure time (AFT) model[12] as an alternative to the Cox proportional hazard (Cox PH) model[16] in the analysis of survival data. Moreover, we conclude that the Weibull AFT model provides a much better fit than the Cox PH model in our data set with a satisfying predictive accuracy. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Kaplan-Meier Method . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Cox Proportional Hazards Model . . . . . . . . . . . . . . . . 2 1.5 Accelerated Failure Time Model . . . . . . . . . . . . . . . . 4 2 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Exploratory Analysis and Preliminary Conclusions . . . . 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Graphical Presentation of Strength Properties Data . . . . . 13 3.2.1 Histogram of Strength Data . . . . . . . . . . . . . . 13 3.2.2 Exploring the Relationship Between the Strength Data and Covariates . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Univariate Approaches to Modeling the Distributions . . . . 17 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.2 Univariate Weibull Distribution . . . . . . . . . . . . 17 3.3.3 Univariate Kaplan–Meier Estimator . . . . . . . . . . 18 3.3.4 The 5th Percentile Estimators by the KM Approach . 22 3.4 Bivariate Approaches to Modeling the Distributions of (R,E) and (T,E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 iii Table of Contents 3.4.1 Bivariate Weibull Distribution . . . . . . . . . . . . . 23 3.4.2 Bivariate KM Estimator . . . . . . . . . . . . . . . . 23 3.5 Tests for the Difference of Distributions . . . . . . . . . . . . 26 3.5.1 Graphical Approach by the KM Estimator and Log- rank Test . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5.2 Test for the Difference Between Two MOE in the Two Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6 Exploring the Association between MSRC and FC . . . . . . 29 3.6.1 Two-way Contingency Table . . . . . . . . . . . . . . 29 3.6.2 Test of Independence (Chi-square and Related Tests) 30 3.6.3 Describing the Strength of Association . . . . . . . . 31 4 Semi-parametric Survival Model . . . . . . . . . . . . . . . . 33 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 AIC Procedure For Variable Selection . . . . . . . . . . . . . 33 4.3 Application to Variable Selection . . . . . . . . . . . . . . . . 34 4.3.1 Method I: step () to select the best model according to AIC statistic . . . . . . . . . . . . . . . . . . . . . 34 4.3.2 Method II: Single term deletions . . . . . . . . . . . . 34 4.3.3 Comparing Nested Models . . . . . . . . . . . . . . . 35 4.3.4 Checking for Interaction . . . . . . . . . . . . . . . . 36 4.4 Model Diagnostics for the Cox PH Model . . . . . . . . . . . 38 4.4.1 Checking for the Proportional Hazards Assumption . 38 4.4.2 Assessing Goodness-of-Fit . . . . . . . . . . . . . . . 42 4.4.3 Checking for Outliers . . . . . . . . . . . . . . . . . . 42 4.4.4 Influential Observations . . . . . . . . . . . . . . . . . 44 4.4.5 Dealing with the Violation of the Proportional Haz- ards Assumption . . . . . . . . . . . . . . . . . . . . . 44 5 Parametric Survival Models . . . . . . . . . . . . . . . . . . . 48 5.1 Exploring the Distribution of Load to Failure . . . . . . . . . 48 5.2 Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Q-Q Plot to Check the AFT Assumption . . . . . . . . . . . 52 5.4 Model Diagnostics for the AFT Model . . . . . . . . . . . . . 53 5.4.1 Overall Goodness-of-Fit . . . . . . . . . . . . . . . . . 53 5.4.2 Checking for Outliers . . . . . . . . . . . . . . . . . . 53 5.4.3 Influential Assessment . . . . . . . . . . . . . . . . . . 57 5.5 Interpretation of Results . . . . . . . . . . . . . . . . . . . . 57 5.6 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . 59 5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 59 iv Table of Contents 5.6.2 Description of Method . . . . . . . . . . . . . . . . . 59 5.6.3 Results for the Simulation . . . . . . . . . . . . . . . 60 5.7 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . 63 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 v List of Tables 2.1 Description of failures for dimension lumber. . . . . . . . . . 7 2.2 An example of coded single knot for three pieces of lumber . 8 2.3 An example of coded location of MSRC for four pieces of lumber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Description of variables in both samples. . . . . . . . . . . . . 9 2.5 Original bending data. . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Variables definition for the transformed bending data. . . . . 12 2.7 Transformed bending data. . . . . . . . . . . . . . . . . . . . 12 3.1 MLEs of univariate Weibull parameters for the bending data. 17 3.2 MLEs of univariate Weibull parameters for the tension data. 18 3.3 MLEs of ζ̂R0.05, ζ̂ T 0.05 and ρ̂. . . . . . . . . . . . . . . . . . . . . 18 3.4 KM estimation of survival function for MOR. . . . . . . . . . 20 3.5 KM estimation of survival function for MOT. . . . . . . . . . 20 3.6 KM estimators of percentiles in bending test. . . . . . . . . . 22 3.7 KM estimators of percentiles in tension test. . . . . . . . . . . 22 3.8 Ŝ(e, r) by Bivariate Weibull and by Bivariate KM. . . . . . . 24 3.9 Example of A Hypothetical Two-way Contingency Table. . . 29 3.10 Two-way Contingency Table of MSRC and FC. . . . . . . . . 30 4.1 Summary of the initial Cox PH model. . . . . . . . . . . . . . 34 4.2 Stepwise model path for the main effects model on the bend- ing data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 p-values of covariates in the model selected by step (). . . . . 35 4.4 Drop 1 model path for the main effects model on the bending data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 p-values of covariates in the final model. . . . . . . . . . . . . 37 5.1 Plots and associated distributions. . . . . . . . . . . . . . . . 48 5.2 Results from AFT models for the bending data. . . . . . . . . 51 5.3 AIC in the AFT models. . . . . . . . . . . . . . . . . . . . . . 52 5.4 Summary for the final Weibull AFT model. . . . . . . . . . . 57 vi List of Tables 5.5 True value α2 = −0.2. Expected value, standard deviation of parameter estimates. . . . . . . . . . . . . . . . . . . . . . . . 60 vii List of Figures 2.1 The bending test. . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The tension test. . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1 Distributions of the strength properties data. . . . . . . . . . 14 3.2 MOR against continuous variables, with a lowess smooth curve. 15 3.3 MOR against categorical variables. . . . . . . . . . . . . . . . 16 3.4 KM curves for MOR and MOE in the bending test. . . . . . . 19 3.5 KM curves for MOT and MOE in the tension test. . . . . . . 21 3.6 Comparison of Bivariate Weibull and KM Estimates. . . . . . 25 3.7 KM curves of MOR against categorical covariates. . . . . . . 27 3.8 KM curves of MOE in the two cases. . . . . . . . . . . . . . . 28 4.1 Scaled Schoenfeld residuals against load-to-failure. . . . . . . 40 4.2 Graphical check of the PH assumption. . . . . . . . . . . . . . 41 4.3 Cumulative hazard plot of the Cox-Snell residual for Cox PH model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Deviance residuals against the risk score,index and covariates. 45 4.5 Influence statistics. . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1 Exploring distribution of load to failure. . . . . . . . . . . . . 50 5.2 Q-Q plot for load to failure. . . . . . . . . . . . . . . . . . . . 54 5.3 Cumulative hazard plot of the Cox-Snell residual for theWeibull AFT model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4 Deviance residuals against the risk score, index and covariates. 56 5.5 Influence statistics. . . . . . . . . . . . . . . . . . . . . . . . . 58 5.6 Comparing observed and predicted survival curves. . . . . . . 62 viii Acknowledgements This thesis grew out of a research project provided by my supervisor Dr. Jim Zidek. I am deeply indebted to Dr. Zidek for his excellent guidance and immensely help without which the development of this thesis would not have been possible. I am sincerely grateful to my co-supervisor, Dr. Lang Wu, for his invaluable advice and constant patience. They have inspired me a lot in survival analysis with their expertise, support and encouragement, and broadened my knowledge beyond textbooks. They are undoubtedly two of the best professors that I have ever met throughtout my stay in UBC. Also, I thank Dr. John Petkau for his invaluable advice on my consulting projects, which will benefit me a lot in the future. I thank all the faulty, staff and graduate students in the Department of Statistics for making such a wonderful study environment. Last but not least, I am most thankful for the love, confidence and support from my parents. My huge thanks goes to my beloved husband and daughter, Yu Liu and Irene Qian Liu, for their love, tremendous support and encouragement that are always beside me. ix Chapter 1 Introduction 1.1 Background Survival analysis[16] is a collection of statistical techniques used to describe and quantify time to event data. The methodological developments with the most profound impact are the Kaplan-Meier method for estimating the sur- vival function, the log-rank test[16] for comparing the equality of two or more survival distributions, and the Cox proportional hazards (PH) model[16] for examining the covariate effects on the hazard function. The accelerated fail- ure time (AFT) model[16] was also proposed but less widely used. In this report, we present the basic concepts, parametric methods (univariate and bivariate Weibull distribution), nonparametric methods (the Kaplan-Meier method and the log-rank test), a semi-parametric model (the Cox PH model) and a parametric model (the AFT model) for analyzing survival data. 1.2 Weibull Distribution Results of mechanical tests on lumber, wood composites, and wood struc- tures are often summarized by a distribution function fit to data. The Weibull distribution (named after Waloddi Weibull, a Swedish physicist who used it in 1939 to describe the breaking strength of material) is playing an increasingly important role in this type of research and has become a part of several American Society of Testing and Materials standards. Due to one of the parameters - the shape parameter - which allows it to be like a vari- ety of other distributions, such as the normal, lognormal, and exponential distributions, it is very popular with researchers. Its flexibility to model experimental results makes the Weibull distribution a powerful tool in wood utilization research. The three-parameter Weibull distribution[10] is commonly used to char- acterize lumber strength. The density function of the Weibull is f(x;κ, λ, θ) = κ λ ( x− θ λ )κ−1 exp[−(x− θ λ )κ], (1.1) 1 1.3. Kaplan-Meier Method where x ≥ θ, κ > 0 is the shape, λ > 0 is the scale, and θ is the location. The distribution function of the Weibull is given by F (x;κ, λ, θ) = 1− exp[−(x− θ λ )κ]. (1.2) Methods must be available to fit the distribution to a data set and pro- vide statistically sound estimates of the parameters of the distribution. How- ever, the effect that different ways of estimating a parameter has on estimat- ing lower tail percentiles has not been widely researched. Fortunately, this limitation of using the Weibull distribution to estimate lumber properties does not affect our case since our data set is complete. 1.3 Kaplan-Meier Method The Kaplan-Meier[16] estimator of survival is a nonparametric method of inference concerning the survivor function S = Pr(Y > y). Let y(i) denote the ith distinct ordered observation and be the right endpoint of the interval Ii, i = 1, 2, ..., n. Also, let ni = # unbroken just before the level y(i), while di = # broken at the level y(i). The K–M estimator of the survivor function is then Ŝ(y) = k∏ i=1 ( ni − di ni ), where y(k) ≤ y < y(k+1). Compared to the parametric method, probability statements obtained from most nonparametric statistics are exact probabilities, regardless of the shape of the population distribution from which the random sample was drawn. However, the nonparametric method has several shortcomings such as low power and lack of software. Fortunately, there is a R function called by survfit which can calculate the K–M survival estimators. 1.4 Cox Proportional Hazards Model Let Y represent survival load and the survival function be S(y) = Pr(Y > y). One representation of the distribution of survival load is the hazard function, which represents the instantaneous risk of breaking at the load level y, conditional on survival to that time h(y) = lim 4y→0 Pr[(y ≤ Y < y +∆y)|Y ≥ y] ∆y . 2 1.4. Cox Proportional Hazards Model Models for survival data usually employs the hazard function or the log hazard. Survival analysis typically examines the relationship of the sur- vival distribution to covariates. Most commonly, this examination entails the specification of a linear-like model for the log hazard. For example, a parametric model based on the exponential distribution may be written as log hi(y) = α+ β1xi1 + β2xi2 + ...+ βkxik, or equivalently, hi(y) = exp(α+ β1xi1 + β2xi2 + ...+ βkxik), that is, as a linear model for the log-hazard or as a multiplicative model for the hazard. Here, i is a subscript for observation, and the x’s are the covariates. The constant α in this model represents a kind of log-baseline hazard, since log hi(y) = α (or hi(y) = eα) when all of the x’s are zero. The baseline hazard function α(y) = log h0(y) is unspecified, so the Cox PH model is log hi(y) = α(y) + β1xi1 + β2xi2 + ...+ βkxik, or again equivalently, hi(y) = h0(y) exp(β1xi1 + β2xi2 + ...+ βkxik). This model is semi-parametric because while the baseline hazard can take any form, the covariates enter the model linearly. Consider, now, two obser- vations i and j that differ in their x-values, with the corresponding linear predictors θi = β1xi1 + β2xi2 + ...+ βkxik and θj = β1xj1 + β2xj2 + ...+ βkxjk The hazard ratio for these two observations, hi(y) hj(y) = h0(y)eθi h0(y)eθj = eθi eθj is independent of the load y. This defines the “proportional hazards prop- erty”. The general rule is that if the hazard functions cross over load, the PH assumption is violated. We are not making assumptions about the form of h0(y) (the nonpara- metric part of model)– the shape of underlying hazard. Parameter estimates 3 1.5. Accelerated Failure Time Model are interpreted the same way as in parametric models, except that no shape parameter is estimated. Even though the baseline hazard is is not specified, we can still get a good estimate for regression coefficients β, hazard ratio, and adjusted hazard curves. The beauty of the Cox approach is that this vagueness creates no problems for such critical estimations. 1.5 Accelerated Failure Time Model The accelerated failure time model is an alternative to the Cox PH model for the survival time data. Under AFT models we measure the direct effect of the predictor variables on the survival time instead of the hazard as in the Cox PH model. This characteristic provides an easier interpretation of the results since the parameters measure the effect of the corresponding covariate on the mean survival time. As with the Cox PH model, the AFT model describes the relationship between survival probabilities and covariates. Given a set of covariates (X1, X2, ..., Xp), the model is S(y) = S0( yη(x)), where S0(y) is the baseline survival function and η(x) = exp(α1x1+α2x2+ ... + αpxp), an ‘acceleration factor’ that is a ratio of survival times corre- sponding to any fixed value of S(y). Under an accelerated failure time model, the covariate effects are as- sumed to be constant and multiplicative on the time scale, that is, the covariate impacts on survival by a constant factor (acceleration factor). Based on the relationship between the survival function and hazard func- tion, the hazard function for an individual with covariates X1, X2, ..., Xp is given by: h(y) = 1 η(x) h0( y η(x) ). The corresponding log-linear form of the AFT model with respect to load Y is given by: log Yi = µ+ α1X1i + α2X2i + ...+ αpXpi + σεi, where µ is the intercept, σ is the scale parameter and εi is a random variable assumed with a specified distribution. For each distribution of εi, there is a corresponding distribution for Y . The AFT models are named for the distribution of Y rather than the distribution of εi or log Y . 4 Chapter 2 Data Description The data come from tests conducted at a FPI/Forintek laboratory. We have two samples of lumber, each of size 98. We applied the bending (R) strength test to generate one sample and the tension (T) strength test to generate the other. In these two tests, as loads (bending or tension stress) increased, each piece will remain intact (“survive”) for a while until it breaks. The values of MOR and MOT are recorded (unit: psi 103) at the point where the stress is applied (usually at a random location near the center). The break occurs somewhere else along the board. Figure 2.1 and 2.2 show how a piece of lumber is broken in these two tests. Figure 2.1: The bending test. This is a transformed time-to-failure(load-to-failure) problem, and it is very typical in survival analysis. Stiffness or elasticity (E) is measured in both of the above two tests to give the values of MOE (unit: psi 106). As each piece of lumber can only be broken once, we only have MOE and MOR in the bending data, while in the tension data we only have MOE and 5 Chapter 2. Data Description Figure 2.2: The tension test. MOT. Interest lies in the relationships amongst MOR, MOT and MOE. As each piece of lumber is tested, the characteristic deemed most likely to cause the lumber’s failure during the test - maximum strength reduc- ing characteristic (MSRC) - is recorded in coded form. Examples of such characteristics are “knot”, “grain”, “shake” and “split”. The MSRC is the grader’s best guess before testing the board as to why it will fail. The failure code (FC) is the characteristic visually judged by the grader to have caused the piece to fail after testing. They could be the same if the failure occurs because of the MSRC. The association between them will be explored later. There are 10 different causes of failure recorded in the data set, including “knot combination”, “grain”, “shake” and “split”, while around 80% of defects in MSRC and FC are due to “knot” (including both a single knot and a combination of knots). We have the data set available in the form of an excel spreadsheet. The coding system of measurements[3, 14, 15] (e.g. MSRC) is quite complicated as shown in Table 2.1. 6 C h ap ter 2. D ata D escrip tion Table 2.1: Description of failures for dimension lumber. Code Cause of Failure Code Cause of Failure 10 knot combination (pith present) nn % of cross-section displaced by knot (total) 20 knot combination (no pith) nn % of cross-section displaced by knot (total) 23 knot cluster (pith present) nn % of cross-section displaced 24 slope of grain (wide face) nn actual slope 25 grain deviation nn % of cross-section where deflection is greater than 1:4 26 cross grain (narrow face) nn actual slope 27 shake and checks 01 not through and less than 2’ long 02 not through and more than 2’ long 03 through and less than 2’ long 04 through and more than 2’ long 05 shake breaks less than 2/3 the edge 06 shake breaks more than 2/3 the edge 28 split nn average length of both sides 35 bark pocket 45 machine damage 01 saw cut through edge 02 all other saw cuts 03 mechanical damage at edge 04 all other mechanical damage 7 Chapter 2. Data Description For the single knot coding system, knots are allowed to be coded nu- merically with respect to size, orientation and location in the member of cross-section. All possible knot configurations have been incorporated into 10 ”knot classes”. For knot classes 1 through 9, the first digit designates the knot location on either the tension (0) or the compression (1) edge in bending tests. The second digit identifies the knot class (1-9). The next 4 to 8 digits are used for the required knot measurements. When the first two digits are 10, it indicates a knot class 10 and up to three sub-knots (starting from the largest) that can be individually coded with a 10 followed by the 10-digit knot code. As an example, in Table 2.2, for the 1st piece of lumber, a knot class 1 is considered to be MSRC. For the 2nd piece, a knot class 8 and a knot class 4 are considered to be MSRC 1 and MSRC 2. For the 3rd piece, a knot class 10 is considered to be the MSRC and up to three sub-knots (starting from the largest) are individually recorded as MSRC 1, MSRC 2 and MSRC 3. Here, the MSRC 1 is regarded as the most severe one. Table 2.2: An example of coded single knot for three pieces of lumber. Lumber MSRC1 MSRC2 MSRC3 1 0107001300 2 1810151104 1413002200 3 100810062710 100314152705 101314092920 In addition to the defects in MSRC and FC coded in the excel spread- sheet, we also have the corresponding location of MSRC coded. Location is a four-digit code describing the location of the defect or failure within the piece. The first digit indicates whether the defect or failure is located on the tension edge (0), compression edge (1), or both edges (2). The next three digits give the average location of the defect or failure along the length of the piece. As an example, look at Table 2.3. The random number location (RNL) is the number of inches from the centre of the test span to the worst MSRC (e.g. MSRC 1) - a random integer from 0 to 36. For most of our tests, the MSRC must be randomly located in the test span, and the test span is always less than the length of the lumber. In summary, we have two samples, MOR and MOT, each of size 98. For each specimen of these two samples, we have the following recorded variables as shown in Table 2.4. Table 2.5 presents the original layout of bending data in the spreadsheet. 8 Chapter 2. Data Description Table 2.3: An example of coded location of MSRC for four pieces of lumber. Lumber MSRC1 MSRC2 MSRC3 Loc1 Loc2 Loc3 1 0808131202 0025 2 1810151104 1413002200 1057 0058 3 101909142903 101315092722 100309122713 1042 1042 0042 4 2407 2705 2050 0052 Table 2.4: Description of variables in both samples. Variables Descriptions MOR/ MOT Module of bending or tension ( Load to break ) MOE Module of elasticity MSRC MSRC( 1-3 measures with 2 to 12 digits) MLoc locations of 3 MSRC ( 1-3 measures with 4 digits) FC Failure characteristic (1-3 measures with 2 to 12 digits) Floc locations of 3 FC ( 1-3 measures with 4 digits ) RNL Random number location ( 2 digits from 0 to 36) Off-grade Indicator of off-grade piece (1= yes, 0= no) Species 1 = Spruce, 2 = Pine, 3 = Fir Moisture Degree of moisture 9 C h ap ter 2. D ata D escrip tion Table 2.5: Original bending data. ] MSRC1 MSRC2 MSRC3 MLoc1 MLoc2 MLoc3 speci mois offg moe mor 1 0108131202 0025 2 14.8 0 1.65 6.04 2 1810151104 1413002200 1057 0058 2 13.7 0 1.44 6.59 3 101909142903 101315092722 100309122713 1042 1042 0042 2 15.5 0 1.43 7.46 4 2407 2705 2050 0052 2 14.4 0 1.58 8.95 5 101320172602 100904093015 1043 1028 2 13.6 0 1.36 3.09 6 1014 2111 2 15.7 0 1.46 8.74 7 101912103404 1068 2 15.4 0 1.83 9.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2. Data Description At this stage, to convert MSRC into meaningful covariates, I only look at MSRC1 as it is regarded as the most severe defect. Also, the first two digits of data strings in MSRC1 capture most of information of defect categories. Based on the “Forintek Knot and Failure Code” descriptions, it is reasonable to classify MSRC1 into 2 variables - knot and size of knot (ksize). To specify the categorical variable–“knot”, we take the first two digits of the MSRC1 data string as they capture most of relevant information on defects: 1. If the first two digits belong to (0, 9] ⋃ [11, 20), knot = 1(a single knot); 2. If the first two digits are equal to 10 ⋃ 20 ⋃ 23, knot = 2(a knot com- bination); 3. Otherwise, knot = 0(defects other than knot). To quantify the numerical variable – “ksize”: 1. The value of ksize for a single knot, class 20 or class 23 knot combina- tion is given by the 3rd and 4th digits of MSRC1 data string. 2. The value of ksize for a class 10 knot combination is mainly given by the 5th and 6th digits, or 3rd and 4th digits in some few cases. 3. The value of ksize for other defect is 0. Therefore, for bending data, we have variables defined as in Table 2.6, and the layout of bending data with transformed covariates is in Table 2.7. 11 Chapter 2. Data Description Table 2.6: Variables definition for the transformed bending data. Variables Descriptions knot 1=a single knot, 2=a knot combination, 0=other ksize the size of knot or 0 for non-knot defects rnl random number for location of MSRC Off-grade Indicator of off-grade piece (1= yes, 0= no) loc location of defect face edge of defect:0=tension, 1=compression, 2=both Species 1 = Spruce, 2 = Pine, 3 = Fir Moisture Degree of moisture moe module of elasticity mor module of rupture Table 2.7: Transformed bending data. Specimen knot ksize rnl offg loc face species moisture moe mor 1 1 12 7 0 44 0 2 14.8 1.65 6.0424 3 0 0 22 1 22 2 2 13.7 1.44 6.5902 5 1 101 9 0 29 1 2 15.5 1.43 7.4588 9 1 9 5 0 13 0 2 14.4 1.58 8.9549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 3 Exploratory Analysis and Preliminary Conclusions 3.1 Introduction Exploratory data analysis (EDA) is detective work. It comprises techniques to visualize patterns in data. 3.2 Graphical Presentation of Strength Properties Data 3.2.1 Histogram of Strength Data For bending and tension tests, let’s first explore the shape of distributions of the strength properties data: MOR, MOT and MOE in both tests. Based on their histograms and density curves in Figure 3.1, we see that all of the distributions are asymmetrical and in fact right-skewed, which is very typical for survival data. Moreover, the two density curves of MOE from the two tests seem to be identical, and the side-by-side boxplots of MOE in these two tests are almost overlapped. This indicates that there may be no significant difference between the two MOE’s in the two tests. 3.2.2 Exploring the Relationship Between the Strength Data and Covariates We next explore the relationships between the strength data and all other variables. With bending test data, we classify these variables into continuous and categorical. Then, we use scatterplots and side-by-side boxplots to visualize the relationships between MOR and these two types of variables respectively. Figure 3.2 displays MOR against continuous variables. The non-parametric curve using lowess shows the pattern of association between the MOR and 13 3.2. Graphical Presentation of Strength Properties Data Figure 3.1: Distributions of the strength properties data. mor D en si ty 0. 0 0. 1 0. 2 0. 3 0. 0 0. 1 0. 2 0. 3 mot D en si ty 0. 0 0. 1 0. 2 0. 3 0. 0 0. 1 0. 2 0. 3 1.2 1.4 1.6 1.8 2.0 0. 0 1. 0 2. 0 3. 0 0. 0 1. 0 2. 0 3. 0 moe D en si ty bending test tension test l bending test tension test 1. 2 1. 4 1. 6 1. 8 boxplot of moe in two tests m o e 14 3.2. Graphical Presentation of Strength Properties Data Figure 3.2: MOR against continuous variables, with a lowess smooth curve. l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 20 40 60 80 100 4 6 8 10 ksize m o r l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 5 10 15 20 25 30 4 6 8 10 rnl m o r l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0 10 20 30 40 50 4 6 8 10 loc m o r l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 13 14 15 16 17 4 6 8 10 moist m o r l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l ll l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 1.2 1.4 1.6 1.8 4 6 8 10 moe m o r other variables in pairs. We see that there is a positive association between MOR and MOE, but no specific patterns for MOR and other variables. Figure 3.3 shows the side-by-side boxplots of MOR against the categor- ical variables – “knot”,“offg”,“species” and “face”. It shows that a piece of lumber with a “single knot” as MSRC1, “off-grade”, “pine” species, or the defect is on the tension edge will produce a relatively lower MOR. The tension test data display the same patterns as the bending test data in terms of associations between the strength property MOT and other vari- ables. From the plots above, we can see that distributions of strength prop- erties are very typical for survival data. Thus, to model their distributions, we may consider both a parametric approach (e.g. Weibull distribution) and a non-parametric method (e.g. Kaplan-Meier estimator). 15 3.2. Graphical Presentation of Strength Properties Data Figure 3.3: MOR against categorical variables. l l l l l l other single comb 4 6 8 10 knot m o r l not offg offg 4 6 8 10 m o r l l Spruce Pine Fir 4 6 8 10 Species m o r l l l tension compression both 4 6 8 10 edge face of defect m o r 16 3.3. Univariate Approaches to Modeling the Distributions 3.3 Univariate Approaches to Modeling the Distributions 3.3.1 Introduction Interests lies in the relationships between the MOR, MOT and MOE. More- over, in lumber strength testing, people are usually interested in the weakest boards (e.g. the population strength 5th percentile– ζ0.05 or lower). Next, I will use both parametric and nonparametric approaches to estimate the ζ0.05 for each type of strength as well as their ratio. Using the population 5th percentiles for MOR and MOT as an example, the ratio is ρ = ζR0.05/ζ T 0.05. 3.3.2 Univariate Weibull Distribution Assuming Weibull population distributions and independent samples, the three parameters in (1.1) can be estimated using maximum likelihood via numerical optimization in R. Let (κi, λi, θi), i = 1, 2, be the true parameters for two independent 3-parameter Weibull distribution populations, and (κ̂i, λ̂i, θ̂i), i = 1, 2, be the corresponding maximum likelihood estimates from two samples, where κi > 0 is the shape, λi > 0 is the scale, and θi is the location. Table 3.1 displays the maximum likelihood estimates (MLEs) of parame- ters in the three parametric Weibull distribution for the MOR data. where Table 3.1: MLEs of univariate Weibull parameters for the bending data. Quantity Value Standard Error λ̂1 4.726 0.590 κ̂1 3.325 0.511 θ̂1 2.460 0.537 λ(psi× 103), κ(unitless) and θ(psi× 103): Similarly, Table 3.2 displays the maximum likelihood estimates (MLEs) of parameters in the three parametric Weibull distribution for the MOT data. where λ(psi× 103), κ(unitless) and θ(psi× 103): Since the distribution function of the Weibull is given by (1.1), F (x;κ, λ, θ) = P (X ≤ x) = 1− exp[−(x− θ λ )κ], 17 3.3. Univariate Approaches to Modeling the Distributions Table 3.2: MLEs of univariate Weibull parameters for the tension data. Quantity Value Standard Error λ̂2 3.610 0.362 κ̂2 2.556 0.335 θ̂2 0.901 0.297 the population 5th percentile ζ0.05 is then given by P (X ≤ ζ0.05) = 0.05 = 1− exp[−(ζ0.05 − θ λ )κ]. Solving this equation we get ζ0.05 = λ[− ln(0.95)] 1κ + θ. Thus, we can easily get the ratio given by ρ = ζR0.05 ζT0.05 = λ1[− ln(0.95)] 1 κ1 + θ1 λ2[− ln(0.95)] 1 κ2 + θ2 . By the invariance property of MLEs, we can obtain the corresponding MLEs of ζ̂R0.05, ζ̂ T 0.05 and ρ̂ can be calculated by substituting (κ̂i, λ̂i, θ̂i)i=1,2 in Table 3.3. Table 3.3: MLEs of ζ̂R0.05, ζ̂ T 0.05 and ρ̂. Quantity Value Standard Error 95% Confidence Interval ζ̂R0.05 4.394(psi× 103) 0.180(psi× 103) ( 4.041 , 4.747 ) (psi× 103) ζ̂T0.05 2.030(psi× 103) 0.137(psi× 103) ( 1.761 , 2.299 ) (psi× 103) ρ̂ 2.164 0.171 ( 1.829 , 2.499 ) 3.3.3 Univariate Kaplan–Meier Estimator To explore the distribution of MOR, MOE and MOT, we could also use the non-parametric Kaplan-Meier estimators of their survival functions S(y) = 18 3.3. Univariate Approaches to Modeling the Distributions Figure 3.4: KM curves for MOR and MOE in the bending test. 0 2 4 6 8 10 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 load to failure(psi x 10^3) pr op or tio n in re m ai ni ng 0.0 0.5 1.0 1.5 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 load to failure(psi x 10^6) 19 3.3. Univariate Approaches to Modeling the Distributions Pr(Y > y). Using the R function survfit, we plot the Kaplan-Meier curves of MOR (left) and MOE (right) with 95% error bands in Figure 3.4. As well as the KM estimators of Pr(MOR > mor) in Table 3.4, where ni= # at risk before mori, di= # that break at mori. Table 3.4: KM estimation of survival function for MOR. mori ni di Pr(MOR > mori) std.err lower 95% CI upper 95% CI 3.09 98 1 0.9898 0.0102 0.97010 1.0000 3.67 97 1 0.9796 0.0143 0.95199 1.0000 3.94 96 1 0.9694 0.0174 0.93587 1.0000 4.63 95 1 0.9592 0.0200 0.92080 0.9992 . . . . . . . . . . . . . . . . . . . . . Similarly, the KM curves for MOT (left) and MOE (right) with 95% error bands in the tension test are shown in Figure 3.5. as well as the KM estimators of Pr(MOT > mot) in Table 3.5, where ni= # at risk before moti, di= # that break at moti. Table 3.5: KM estimation of survival function for MOT. moti ni di Pr(MOT > moti) std.err lower 95% CI upper 95% CI 1.21 98 1 0.9898 0.0102 0.97010 1.0000 1.80 97 1 0.9796 0.0143 0.95199 1.0000 1.82 96 1 0.9694 0.0174 0.93587 1.0000 1.83 95 1 0.9592 0.0200 0.92080 0.9992 . . . . . . . . . . . . . . . . . . . . . 20 3.3. Univariate Approaches to Modeling the Distributions Figure 3.5: KM curves for MOT and MOE in the tension test. 0 2 4 6 8 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 load to failure(psi x 10^3) pr op or tio n in re m ai ni ng 0.0 0.5 1.0 1.5 2.0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 load to failure(psi x 10^6) 21 3.3. Univariate Approaches to Modeling the Distributions 3.3.4 The 5th Percentile Estimators by the KM Approach It is very handy to use the KM method to get the 5th percentile estimates (ζ̂R0.05, ζ̂ T 0.05 and ζ̂ E 0.05) using a formula given by Mara and Jong (2004)[16]. Table 3.6 and 3.7 show the KM estimators of percentiles for bending and tension, respectively. Table 3.6: KM estimators of percentiles in bending test. Quantity Value Standard Error 95% Confidence Interval ζ̂R0.05 4.70(psi× 103) 0.503(psi× 103) ( 3.714 , 5.686 ) (psi× 103) ζ̂E0.05 1.30(psi× 106) 0.0.0395(psi× 106) ( 1.223 , 1.378 ) (psi× 106) Table 3.7: KM estimators of percentiles in tension test. Quantity Value Standard Error 95% Confidence Interval ζ̂T0.05 2.03(psi× 103) 0.285(psi× 103) ( 1.471 , 2.589 ) (psi× 103) ζ̂E0.05 1.30(psi× 106) 0.0.028(psi× 106) ( 1.245 , 1.355 ) (psi× 106) It has been shown that the KM estimators are pretty close to the esti- mators for the Weibull distribution approach in Table 3.3, but the standard errors of KM estimators are relatively larger than the ones by Weibull ap- proach. One reason is the nonparametric method is usually less precise than the parametric one. Also, the two estimated values of ζE0.05 for bending and tension are almost the same, which indicates that the lumbers in two different tests might be homogeneous in terms of elasticity. 22 3.4. Bivariate Approaches to Modeling the Distributions of (R,E) and (T,E) 3.4 Bivariate Approaches to Modeling the Distributions of (R,E) and (T,E) 3.4.1 Bivariate Weibull Distribution The density function of the bivariate Weibull[12] is: f(x, y;κ1, λ1, θ1, κ2, λ2, θ2, δ) = κ1 λ1 ( x− θ1 λ1 ) κ1 δ −1κ2 λ2 ( y − θ2 λ2 ) κ2 δ −1 ×{(x− θ1 λ1 ) κ1 δ + ( y − θ2 λ2 ) κ2 δ }δ−2{[(x− θ1 λ1 ) κ1 δ + ( y − θ2 λ2 ) κ2 δ ]δ + 1 δ − 1} × exp{−[(x− θ1 λ1 ) κ1 δ + ( y − θ2 λ2 ) κ2 δ ]δ} (3.1) For estimating of the bivariate Weibull parameters, a feasible method has been developed by Richard, James and David (1999)[2]. We first estimated the shape (κ), scale (λ) and location (θ) parameters from the two marginal distributions, using standard theory for the univariate Weibull. Given these parameter estimates (κ1, λ1, θ1, κ2, λ2, θ2), we can find the dependence pa- rameter estimate δ using maximum likelihood be numerical optimization in R. We can get the log of the likelihood logL for a random and uncen- sored sample, and the MLEs of parameters can be obtained by minimizing −2 logL. A three-parameter Weibull distribution has the survival function, F (x, y) = P [X > x, Y > y] = exp{−[(x− θ1 λ1 ) κ1 δ + ( y − θ2 λ2 ) κ2 δ ]δ}, 0 < δ ≤ 1 (3.2) Therefore, once the parameters κ1, λ1, θ1, κ2, λ2, θ2 and δ are estimated, we can easily estimate the survival probability for the bivariate data (x, y). 3.4.2 Bivariate KM Estimator A bivariate version of the KM estimator does exist. To describe it we let (Xi, Yi)(i = 1, ..., n) be n independent and identically distributed pairs of loads to failure with survival function F (x, y) = Pr(X ≥ x, Y ≥ y). Since Xi and Yi are the observed loads, it is natural to estimate Pr(X ≥ x, Y ≥ y) by the empirical survival function: Ŝ(x, y) = n−1 n∑ i=1 I(Xi ≥ x, Yi ≥ y) (3.3) 23 3.4. Bivariate Approaches to Modeling the Distributions of (R,E) and (T,E) And the asymptotic variance of this estimator is given by: V̂ ar(Ŝ(x, y)) = Ŝ(x, y)− [Ŝ(x, y)]2. Lin and Ying (1993)[9] provide evidence in favor of this approach. Also, our data are uncensored, which makes our problem much easier than the censored case. Then, as an example, for the MOR data, we can compare the estimates of the survival function S(e, r) computed with (3.2) and by (3.3) in the following Table 3.8: Table 3.8: Ŝ(e, r) by Bivariate Weibull and by Bivariate KM. (e, r)(psi× 106, psi× 103) Ŝ(e, r) by (4) Ŝ(e, r) by (5) (1.65, 6.042) 0.1327 0.1330 (1.65, 6.590) 0.1122 0.1146 (1.65, 7.459) 0.1020 0.0762 (1.36, 7.867) 0.2143 0.1960 (1.36, 4.791) 0.8061 0.7952 (1.36, 5.664) 0.7041 0.6840 (1.36, 5.363) 0.7347 0.7318 (1.36, 7.318) 0.3367 0.3112 (1.17, 7.459) 0.2857 0.2982 (1.17, 8.955) 0.0612 0.0558 (1.17, 3.095) 1.0000 0.9974 (1.17, 8.740) 0.0918 0.0757 (1.17, 9.939) 0.0204 0.0100 . . . . . . . . . It seems that these two estimates are pretty close to each other, which confirms that both parametric and nonparametric survival analysis approaches to lumber strength appears to work well. Besides, we could graph the 3- dimensional scatterplot for each method as shown in Figure , and it is ob- vious that these two estimates are almost the same. 24 3.4. Bivariate Approaches to Modeling the Distributions of (R,E) and (T,E) Figure 3.6: Comparison of Bivariate Weibull and KM Estimates. Bivairate Weibull estimator of P(E>e,R>r) 2 4 6 8 10 120. 00. 20. 40. 60. 81. 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 MOR M O E P( E> e,R >r ) Bivairate KM estimator of P(E>e,R>r) 2 4 6 8 10 120. 00. 20. 40. 60. 81. 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 MOR M O E P( E> e,R >t ) 25 3.5. Tests for the Difference of Distributions 3.5 Tests for the Difference of Distributions 3.5.1 Graphical Approach by the KM Estimator and Log-rank Test A central objective of the study described in this thesis is the relationship between strength and its covariates. For a categorical covariate, we may graph the the KM curves for strength data for different covariate categories, so that we can see if different categories make a difference in the distribution of strength. With bending data, Figure 3.7 displays the KM curves of “mor” against 4 categorical covariates – “knot”, “offg”, “species” and “face”, respectively. It seems that the KM curves are parallel for “offg” and “knot” (overall - there are slight cross-overs when MOR is either small or large). But they are decidedly nonparallel for “species” and “face”. That is, the differences between KM curves for “knot” and “offg” are relatively larger than the other two covariates. The KM curves give us an insight into the difference of survival func- tions in two or more groups, but whether this observed difference is sta- tistically significant requires a formal statistical test. One commonly used non-parametric tests for comparing two or more survival distributions is the log-rank test. The log-rank test compares the observed number of failures with the expected number of failures for each group. The null hypothesis asserts no difference between survival curves in two or more groups. That test yields p-values of 0.00623 (knot), 0.00215 (offg), 0.749 (species) and 0.312 (face). Therefore, the differences we observed above of MOR survival curves made by “knot” and “offg” are statistically significant, which indicates that “knot” and “off” may be the important predictors for MOR. 3.5.2 Test for the Difference Between Two MOE in the Two Tests Another topic of interest is that difference between the two MOE population distributions for bending and tension. The two KM curves are sketched in Figure 3.8 and we observe that they are almost identical. Also, by the log- rank test, their difference is not statistically significant with a very large p-value 0.995, a finding consistent with the previous conclusion suggested by Figure 3.1 – the two density curves of MOE in the two cases are almost identical. 26 3.5. Tests for the Difference of Distributions Figure 3.7: KM curves of MOR against categorical covariates. Notice that unlike the curves for “species” and “face”, those for “offg” and “knot” are quite parallel. 0 1 2 3 4 5 6 7 8 9 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 knot load to failure pr op or tio n of re m ai ni ng other single knot knot comb 0 2 4 6 8 10 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 off−grade load to failure not offgrade offgrade 0 1 2 3 4 5 6 7 8 9 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 species load to failure pr op or tio n of re m ai ni ng spruce pine fir 0 1 2 3 4 5 6 7 8 9 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 face load to failure tension compression both 27 3.5. Tests for the Difference of Distributions Figure 3.8: KM curves of MOE in the two cases. Notice that the curves for two tests are almost identical. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1. 0 load to failure(psi x 10^6) pr op or tio n of re m ai ni ng bending test tension test 28 3.6. Exploring the Association between MSRC and FC 3.6 Exploring the Association between MSRC and FC Recall that MSRC means the grader’s best guess before testing the board as to why it will fail, while FC is the characteristic visually judged by the grader to have caused the piece to fail after testing. They could be the same if the failure occurs because of MSRC. 3.6.1 Two-way Contingency Table If two variables are measured at categorical levels (eg. nominal or ordinal), we assess their relationship by crosstabulating the data in a two-way con- tingency table[1]. A two-way contingency table is a two-dimensional (rows × columns) table formed by ‘cross-classifying’ subjects or events on two categorical variables. One variable’s categories define the rows while the other variable’s categories define the columns. The intersection (crosstab- ulation) of each row and column forms a cell, which displays the count (frequency) of cases classified as being in the applicable category of both variables. Table 3.9 is a simple example of a hypothetical contingency ta- ble that crosstabulates student gender against answer on one question of an exam; a total of 100 students are described. So, we can set up the 2-way contingency table between MSRC and FC, as shown in Table 3.10, using the first two digits in the characteristic de- scriptions since they capture the most of the visual information on lumber defects. Note the total of observations is 195 (not 196), since we have one missing datum in the data set. Table 3.9: Example of A Hypothetical Two-way Contingency Table. Here we see “gender” being broken down by a subject’s answer to an examination question (1= “Yes”; 0= “No”). Answer Gender Yes No Total Male 38 12 50 Female 10 40 50 Total 48 52 100 29 3.6. Exploring the Association between MSRC and FC Table 3.10: Two-way Contingency Table of MSRC and FC. FC MSRC 01-09 10-19 20-60 Total 01-09 42 14 20 76 10-19 20 41 15 76 20-60 7 11 25 43 Total 69 66 60 195 3.6.2 Test of Independence (Chi-square and Related Tests) For ease of understanding, let’s take the data in Table 3.9 for example. If the characteristics Gender and Answer were not associated (the null hypothesis of independence), we can easily calculate the expected counts in each cell, i.e., the number of cases we would expect based on their total distribution in the sample. Given that the sample contains exactly 50% male and 50% female, were there no association between Gender and Answer, we would expect exactly half of those answering ‘Yes’ (48) to be male, i.e., 48÷2 = 24. The actual formula for computing the expected count (E) in any cell of a contingency table is: E = (row total× column total)÷ (grand total). Thus, for the “Male/Yes” cell, E = (50× 48)÷ 100 = 24. The larger the difference between the observed (O) and expected (E) cell counts, the less likely that the null hypothesis of independence holds true, i.e., the stronger the evidence that the two variables are related. In our example, the large difference between the observed (O = 38) and expected (E = 24) cell counts for the Male/Yes cell suggests that being male is associated with greater likelihood of answering ‘Yes’. To determine whether or not the row and column categories for the table as a whole are independent of each other, we compute Pearson’s chi-square statistic (X2): X2 = ∑ [ (O − E)2 E ] , where O = observed frequency and E = expected frequency. As indicated in the formula, one first computes the difference between the observed and expected frequencies in a cell, squares this difference, and then divides the squared difference by that cell’s expected frequency. These values are then summed (the ∑ symbol) over all the cells, yielding the value of X2. In our example, X2 = 31.41. 30 3.6. Exploring the Association between MSRC and FC The value of X2 is then compared to a critical value that is based on the number of rows and columns (df = degrees of freedom = (number of rows− 1)× (number of columns− 1)) and obtained from a chi-square distribution table. If the value of X2 is less than this critical value, then we cannot reject the null hypothesis and we conclude that the data do not provide evidence of an association. If the value of X2 exceeds the critical value, then we re- ject the null hypothesis and conclude that the variable categories are indeed associated. In our example, df = 1 and the chi-square critical value for a significance level of α = 0.05 is 3.84. Since our calculated X2 is 31.41 which clearly exceeds this critical value, we may conclude that gender is associated with answer in the exam. If the minimum expected count for any cell in a contingency table is less than 5, then the chi-square approximation to the distribution of the X2 statistic may not be accurate. In this case, an alternative is Fisher’s Exact Test. If one or more of the expected counts in the cells of a contingency table are less than 5 or when the row or column totals are very uneven, Fisher’s exact test is more desirable. In our real 2-way contingency table,where H0: there is no association between MSRD and FC H1: there is association between MSRC and FC our calculatedX2 is 43.9383, and the corresponding p-value is approximately 0, which indicates that we should reject the null hypothesis and in favor of the hypothesis that independence doesn’t hold here, there is association between MSRD and FC. The Fisher’s exact test also produces a p-value close to 0, which confirms the conclusion of the Chi-square test. 3.6.3 Describing the Strength of Association If there is an association, it may be desirable to then describe the strength of the association. We use correlation-like measures such as the Phi coefficient and Cramer′s V to describe the strength of relationship between nominal variables, since MSRC and FC are measured at nominal level. These co- efficients range from 0 to 1 since you cannot have a ‘negative’ relationship between nominal variables. The Phi coefficient (φ) is a measure of nominal association applicable only to 2× 2 tables. It is calculated as: φ = √ X2 N 31 3.6. Exploring the Association between MSRC and FC whereX2 = the value of Pearson′s chi−square, andN = the sample size. In our example, the Phi coefficient = √ 31.41 100 = 0.56, suggesting a moder- ately strong association. For contingency tables that are larger than 2× 2, Cramer′s V [1] is the choice of nominal association measure. The formula for Cramer′s V is given by: V = √ X2 N(k − 1) where N is the sample size and k is the lesser of the number of rows or columns. Since in 2×2 tables k = 2, Cramer′s V equals the Phi coefficient for 2× 2 tables. Therefore, since our calculated X2 is 43.9383, the strength of association between MSRC and FC is √ 43.9383 195(3−1) = 0.34, suggesting a relatively weak association. However, making a low V level is inevitable with such a small data set. If we also include information on MSRC2 and MSRC3 to construct the two-way contingency table, a larger V should be produced in no doubt. 32 Chapter 4 Semi-parametric Survival Model 4.1 Introduction Let’s first fit a semi-parametric survival regression model - CoxPH model[6, 7, 17]. Since in the CoxPH model, the baseline hazard function h0(t) is nonparametric and no distributional assumption is needed for the survival data, it is easier to start with it. As an example, for bending data, we may fit a CoxPH model for MOR with covariates: knot, ksize, random number location(rnl), off-grade indica- tor(offg), location of defect(loc), face of defect(0=on the tension edge, 1=on the compression edge, 2=on the both edges) , species, moisture and MOE. 4.2 AIC Procedure For Variable Selection Comparisons between a number of possible models, which need not neces- sarily be nested nor have the same error distribution, can be made on the basis of the statistic AIC = −2× log(maximumlikelihood) + k × p, where p is the number of parameters in each model under consideration and k is a predetermined constant. This statistic is called Akaike’s (1974) information criterion (AIC); the smaller the value of this statistic, the better the model. This statistic trades off goodness of fit (measured by the maximized log likelihood) against model complexity (measured by p). Here we shall take k as 2. So, we can rewrite the AIC in the context of the Cox PH model: AIC = −2× log(maximumlikelihood) + 2× b, where b is the number of β coefficients in each model under consideration. The maximum likelihood is replaced by the maximum partial likelihood. The smaller the AIC value the better is the model. 33 4.3. Application to Variable Selection 4.3 Application to Variable Selection First, we fit the initial Cox PH model for the bending data using all possible covariates: coxph.fit1 < −coxph(Surv(mor) ∼ factor(knot) + ksize+ rnl + factor(offg) + loc+ factor(face) + factor(species) +moist+moe) Table 4.1 presents summary(coxph.fit1) as below: Table 4.1: Summary of the initial Cox PH model. coef exp(coef) se(coef) z p factor(knot)1 1.81 6.13 0.42 4.27 0.00 ∗ ∗ ∗ factor(knot)2 0.92 2.50 0.39 2.32 0.02 ∗ ksize 0.00 0.99 0.00 -0.20 0.84 rnl 0.02 1.01 0.01 1.36 0.17 offg 1.70 5.49 0.53 3.16 0.00 ∗∗ loc 0.00 1.00 0.00 0.14 0.88 factor(face)1 -0.48 0.62 0.26 -1.81 0.07 factor(face)2 0.46 1.57 0.35 1.28 0.20 factor(species)2 0.47 1.59 0.43 1.07 0.28 factor(species)3 1.25 3.47 1.18 1.04 0.29 moist 0.22 1.24 0.13 1.56 0.12 moe -5.82 0.00 1.06 -5.46 0.00 ∗ ∗ ∗ Thus, we can see the covariates “ knot”, “off-grade” and “moe” are significant at level of 0.05. 4.3.1 Method I: step () to select the best model according to AIC statistic Table 4.2 shows p-values corresponding to variables selected by step(coxph.fit1). From Table 4.3, we may see that the stepwise method chooses 3 covari- ates: knot, off-grade and moe. 4.3.2 Method II: Single term deletions Table 4.4 displays the result of single term deletions method drop 1 (coxph.fit1, test = “Chi”): 34 4.3. Application to Variable Selection Table 4.2: Stepwise model path for the main effects model on the bending data. Step Df AIC 667.16 - moist 1 667.89 - factor(face) 2 668.45 - factor(offg) 1 672.74 - factor(knot) 2 682.11 - moe 1 698.23 Table 4.3: p-values of covariates in the model selected by step (). coef exp(coef) se(coef) z p factor(knot)1 1.440 4.22207 0.363 3.97 7.3e-05 ∗ ∗ ∗ factor(knot)2 0.730 2.07554 0.367 1.99 4.7e-02 ∗ factor(offg)1 1.692 5.42779 0.528 3.20 1.4e-03 ∗∗ factor(face)1 -0.407 0.66565 0.241 -1.69 9.1e-02 factor(face)2 0.374 1.45321 0.350 1.07 2.9e-01 moist 0.214 1.23845 0.130 1.65 9.9e-02 moe -5.931 0.00266 1.069 -5.55 2.9e-08 ∗ ∗ ∗ So, we see that deletion of knot, off-grade and moe will lead to a significant increase in AIC values, which indicates that these 3 variables are likely to be the most important covariates. 4.3.3 Comparing Nested Models So far, we obtain the same reduced model by Method I and Method II. Next, we will compare this reduced model to the initial full model. Nested models can be compared using the likelihood ratio test (LRT). Symbolically we may describe a model as follows: full model : coxph.fit1 < −coxph(Surv(mor) ∼ factor(knot) + ksize + rnl+ factor(offg) + loc+ factor(face) + factor(species) +moist+ moe) 35 4.3. Application to Variable Selection Table 4.4: Drop 1 model path for the main effects model on the bending data. Df AIC LRT Pr(Chi) 673.69 factor(knot) 2 690.73 21.042 2.697e-05 ∗ ∗ ∗ ksize 1 671.73 0.044 0.834587 rnl 1 673.56 1.866 0.171930 factor(offg) 1 679.24 7.554 0.005987 ∗∗ loc 1 671.71 0.021 0.883448 factor(face) 2 676.11 6.417 0.050423 factor(species) 2 671.32 1.635 0.441493 moist 1 674.17 2.483 0.115061 moe 1 704.00 32.315 1.311e-08 ∗ ∗ ∗ reduced model by method I and II : cox1 < −coxph(Surv(mor) ∼ factor(knot) + factor(offg) +moe) anova(cox1, coxph.fit1) gives: loglik Chisq Df p 1 -330.38 2 -324.84 11.063 8 0.20 Conclusion: the LRT test shows no evidence against the reduced model (p− value= 0.20), which indicates the difference between these two models is not significant, and we prefer the smaller reduced model cox1. 4.3.4 Checking for Interaction step(cox1,∼ .2) Conclusion: Adding the interaction term makes the AIC values increase and we may conclude that there is no need to add interactions, so our final model is cox1. cox1 < −coxph(Surv(mor) ∼ factor(knot) + offg) Table 4.5 presents the results of summary(cox1): Based on the above summary output of cox1, we may make the following comments: 36 4.3. Application to Variable Selection Step Df AIC 668.75 + factor(offg):moe 1 669.66 + factor(knot):moe 2 671.33 + factor(knot):factor(offg) 2 672.65 - factor(offg) 1 673.06 - factor(knot) 2 677.90 - moe 1 696.64 Table 4.5: p-values of covariates in the final model. coef exp(coef) se(coef) z p factor(knot)1 1.049262 2.855543 0.316801 3.312 0.000926 ∗ ∗ ∗ factor(knot)2 0.691564 1.996836 0.361133 1.915 0.055495 factor(offg)1 1.452420 4.273444 0.491035 2.958 0.003098 ∗∗ moe -4.811266 0.008138 0.936015 -5.140 2.75e-07 ∗ ∗ ∗ 1. The estimated coefficient for the single knot as MSRC is 1.049 with very small p-value. Hence, fixing other covariates, the hazard ratio between the lumber with a single knot as MSRC and the one with knot combination as MSRC is exp(1.049)/exp(0.692) = 2.85554/1.99684 = 1.43, which means that the prior ones are 1.43 times more likely than the later ones to fail( having shorter survival). Similarly, the hazard ratio between the lumber with a single knot as MSRC and the ones with other defects than knot is 2.856, which means that the prior ones are 2.856 times more likely than the later ones to fail( having shorter survival). This is consistent with the side-by-side boxplots of “mor” against “knot” in the exploratory data analysis (EDA), which shows lumbers with a single knot as MSRC posses the lowest “mor” than ones with other two categories of “knot” as MSRC. 2. The estimated coefficient for the offgrade pieces of lumber is 1.452, and exp(1.452) = 4.273, which means the offgrade pieces of lumbers are 4.273 times weaker than standarded ones. This is also consistent with the conclusion in EDA. 3. Fixing other covariates, lumbers with higher moe have a decreased 37 4.4. Model Diagnostics for the Cox PH Model hazard than the ones with lower moe. This is quite reasonable in common sense as the higher elasticity a piece of lumber the less likely that the failure will occur. 4.4 Model Diagnostics for the Cox PH Model As in the case of a linear or generalized linear model, it is desirable to determine whether a fitted Cox regression model adequately describes the data. The model checking procedures below are based on residuals. In linear regression methods, residuals are defined as the difference between the observed and predicted values of the dependent variable. However, when the partial likelihood function is used in the Cox PH model, the usual concept of residual is not applicable. We will discuss three major residuals that have been proposed for use in connection with the Cox PH model: the Scaled Schoenfeld residuals[5], the Deviance residuals[18] and the Cox-Snell residuals[8]. Then we will talk about influence assessment and strategies for analysis of nonpro- portional data. 4.4.1 Checking for the Proportional Hazards Assumption The main assumption of the Cox PH models is proportional hazards[16]. Proportional hazard means that the hazard function of one individual is proportional to the hazard function of the other individual, i.e., the hazard ratio is constant over time. There are several methods for verifying that a model satisfies the assumption of proportionality. The kth Schoenfeld residual (Schoenfeld, 1982) defined for the kth sub- ject on the jth explanatory variable xj is given by rsjk = δkx j k − ajk, where δk is the kth subject’s censoring indicator, x j k is the value of the jth explanatory variable on the kth individual in the study, ajk = ∑ m∈R(yk) exp(x ′ mβ̂)x j m∑ m∈R(yk) exp(x ′ mβ̂) , and R(yk) is the risk set at time yk. The MLE β̂ is obtained from maxi- mizing Cox’s partial likelihood function. The Shoenfeld residuals for each predictor xj must sum to zero. We define the scaled Schoenfeld residuals 38 4.4. Model Diagnostics for the Cox PH Model by the product of the inverse the estimated variance-covariance matrix of the kth Schoenfeld residual and the kth Schoenfeld residual, so that the kth Schoenfeld residual has an easily computable variance-covariance matrix. Tests and graphical diagnostics for proportional hazards may be based on the scaled Schoenfeld residuals. Conveniently, the cox.zph function calculates tests of the pro- portional hazards assumption for each covariate, by correlating the corre- sponding set of scaled Schoenfeld residuals with a suitable transformation of load (the default is based on the Kaplan−Meier estimate of the survival function, i.e., Ŝ(r) for the bending data). If the PH assumption holds for a particular covariate then the scaled Schoenfeld residual for that covariate will not be related to survival time. Using the cox.zph function, rho is the Pearson product-moment correlation between the scaled Schoenfeld residu- als and survival time. The null hypothesis is that the correlation between the scaled Schoenfeld residuals and the ranked survival time is zero. Rejection of the null hypothesis concludes that the PH assumption is violated. As mentioned, cox.zph computes a test for each covariate, along with a global test for the model as a whole: cox.zph(cox1) gives: factor(knot)1 -0.0875 0.694 0.4048 factor(knot)2 0.1215 1.491 0.2220 offg 0.0439 0.190 0.6633 moe -0.0524 0.319 0.5722 GLOBAL NA 9.179 0.0568 Therefore, there is no statistically significant evidence of non-proportional hazards for any of the covariates, and the global test is also not quite statis- tically significant. These tests are sensitive to linear trends in the hazard. Moreover, we may plot the scaled Schoenfeld residuals against load-to-failure for each covariate in Figure 4.1: Interpretation of these graphs is greatly facilitated by smoothing, for which purpose cox.zph uses a smoothing spline, shown on each graph by a solid line; the broken lines represent ±2−standard−error envelopes around the fit. Systematic departures from a horizontal line are indicative of non-proportional hazards. The assumption of proportional hazards appears to be supported for the covariate offg (which is, recall, a dummy variable, accounting for the two bands in the graph) and moe. However, there appears to be a trend in the plot for knot, with the knot effect 39 4.4. Model Diagnostics for the Cox PH Model Figure 4.1: Scaled Schoenfeld residuals against load-to-failure. 5 5.9 6.8 8 − 8 − 4 0 2 4 6 l l l l l llll l l ll lll l l l l l l llllll l l l ll ll l l l l l lll l l l l l l l ll l l l lll l l l ll ll l l l l l l l l l l l l l knot load to failure be ta fo r kn ot 5 5.9 6.8 8 − 5 0 5 10 15 20 l l l l l llll l ll l lll l l l l l l lllllll l l l ll ll l l l l l lll l l l l l l l ll l l l llll l l ll ll ll l l ll l l ll offgrad load to failure be ta fo r o ffg 5 5.9 6.8 8 − 20 − 10 0 10 20 l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l lll l l l l l l l l l l l l l l l l ll l l l l l l l moe load to failure be at a fo r m o e 40 4.4. Model Diagnostics for the Cox PH Model Figure 4.2: Graphical check of the PH assumption. 1.2 1.6 2.0 − 4 − 3 − 2 − 1 0 1 knot Log Load to failure Lo g cu m u la tiv e h az ar d knot0 knot1 knot2 1.2 1.6 2.0 − 4 − 3 − 2 − 1 0 1 off−grade Log Load to failure offg0 offg1 increasing with load. That is, the variability band for knot (a categorical variable with 3 levels, accounting for the 3 bands in the graph) displays a positive slope over load, suggesting non-proportionality of hazard and conflicting with the finding of the cox.zph test. An alternative (and less sensitive) means of testing the proportional hazards assumption is to plot log[− logS(r)] vs log(r) in Figure 4.2. We conclude that the log[− logS(r)] vs load plots are parallel for offg while nonparallel for knot, implying that the proportional hazards assump- tion has been violated for knot, which is supported by the Schoenfeld resid- ual plots. Therefore, it gives us some concern about whether the Cox PH model is appropriate. 41 4.4. Model Diagnostics for the Cox PH Model 4.4.2 Assessing Goodness-of-Fit The ith Cox-Snell residual is defined as rCi = Ĥ0(ti)× exp(x′iβ̂) = Ĥi(ti) = − log Ŝi(ti), where Ĥ0(ti) and β̂ are the MLE’s of the baseline cumulative hazard function and coefficient vector, respectively. rCi = − log Ŝi(ti) will have a unit exponential distribution with fR(r) = exp(−r). Let SR(r) denote the survival function for the Cox-Snell residual rCi. Then, SR(r) = ∫ ∞ r exp(−x)dx = exp(−r), and HR(r) = − logSR(r) = − log(exp(−r)) = r. Therefore, we plot the cumulative hazard function HR(rCi) versus Cox- Snell residual rCi to check the fit of the model. This gives a straight line with unit slope and zero intercept if the fitted model is correct. Note the Cox-Snell residuals will not be symmetrically distributed about zero and cannot be negative. Then, we assess the goodness of fit for this Cox PH model by residual plots. A plot of the Cox-Snell residuals against the cumulative hazard of Cox-Snell residuals is presented in Figure 4.3. There is some obvious evi- dence of a systematic deviation from the straight line with an intercept zero and a slope one, which gives us some concern about the adequacy of the fitted model. 4.4.3 Checking for Outliers The ith deviance residual is defined by rDi = sign(rmi) √ −2{rmi + δi log(δi − rmi)}, where the function sign() is the sign function which takes the value 1 if rmi is positive and -1 if rmi is negative; rmi = δi−rCi is the martingale residual; and δi = 1 for uncensored observation, δi = 0 for censored observation. In a fitted Cox PH model, the hazard of failure for the ith individual at any time depends on the value of exp(β′xi) that is called the risk score. A plot of deviance residuals versus the risk score is a helpful diagnostic to assess a given individual on the model. Potential outliers will have deviance 42 4.4. Model Diagnostics for the Cox PH Model Figure 4.3: Cumulative hazard plot of the Cox-Snell residual for Cox PH model. llll lll lll lll lll lll lll l ll ll ll ll lll lll lll ll ll ll ll ll l l l l l l l l l l l l l l 0 1 2 3 4 0 1 2 3 4 5 Cox−Snell residual Cu m u la tiv e h az ar d of re sid ua l 43 4.4. Model Diagnostics for the Cox PH Model residuals whose absolute values are very large. This plot will give infor- mation about characteristics of observations that are not well fitted by the model. A plot of deviance residuals against the covariates can also be obtained. Any unusual patterns may suggest features of the data that have not been adequately fitted for the model. Very large or very small values suggest that the observation may be an outlier in need of special attention. The plots of deviance residuals against the risk score, index and covari- ates are given in Figure 4.4. They show only one possible outlier, but none of them seems to be systematically distributed about zero. Therefore, overall, we have some concern about the adequacy of the fitted Cox PH model. 4.4.4 Influential Observations Figure 4.5 shows the change in each regression coefficient when each obser- vation is removed from the data (influence statistics). The changes plotted are scaled in units of standard errors and changes of less than 0.1 are of little concern. These plots give us an idea of the influence individual observations have on the estimated regression coefficients for each covariate. Most of the changes in the regression coefficients are less than 0.1 s.e.’s of the coeffi- cients and all others are less than 0.2 s.e.’s. Therefore, data sets where the influence plot is tightly clustered around zero indicate an absence of influential observations. 4.4.5 Dealing with the Violation of the Proportional Hazards Assumption From the analyses conducted so far, we conclude that the proportional haz- ards assumption has been violated for the variable “knot”. One method of dealing with this is to stratify the model by “knot”. This means that we produce a separate baseline hazard function for each level of “knot”. How- ever, by stratifying, we cannot obtain a hazard ratio for “knot” since the ‘knot effect’ is absorbed into the baseline hazard. The two models are given as below: cox1 < −coxph(Surv(mor) ∼ factor(knot) + offg +moe,method = “breslow”) cox2 < −coxph(Surv(mor) ∼ strata(factor(knot)) + offg +moe,method = “breslow”) 44 4.4. Model Diagnostics for the Cox PH Model Figure 4.4: Deviance residuals against the risk score,index and covariates. l l l l l l l l ll l l l l l l l l ll l l l l l ll l l l l ll l l l l l l l ll l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l ll l l −2 −1 0 1 2 − 2 0 1 2 risk score D ev ia nc e re si du al s l l l l l l l l ll l l l l l l l l ll l l l l l lll l l llll l l l l l lll l l l l l l l l l l l l ll l l l l l l ll l l l l l l l l l l l l l l l l ll l l l l ll l ll l l l ll l l 0 20 40 60 80 100 − 2 0 1 2 Index l l l l l l l l ll l l l l l l ll l l l l l l l ll l l l l l l l l l l l l l ll l ll l 0.0 0.5 1.0 1.5 2.0 − 2 0 1 2 knot D ev ia nc e re si du al l l l l l l l l ll l l l l l l l l l l l ll l l l l l l l l l l 0.0 0.2 0.4 0.6 0.8 1.0 − 2 0 1 2 offg l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l ll ll l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l l l ll l l l l l l ll l l 1.2 1.3 1.4 1.5 1.6 1.7 1.8 − 2 0 1 2 moe D ev ia nc e re si du al 45 4.4. Model Diagnostics for the Cox PH Model Figure 4.5: Influence statistics. 0 20 40 60 80 100 − 0. 3 − 0. 1 0. 1 0. 3 Observation Ch an ge in c oe ffi cie nt knot 0 20 40 60 80 100 − 0. 3 − 0. 1 0. 1 0. 3 Observation offgrade 0 20 40 60 80 100 − 0. 3 − 0. 1 0. 1 0. 3 Observation Ch an ge in c oe ffi cie nt moe 46 4.4. Model Diagnostics for the Cox PH Model We may compare these two models using the AIC criterion. Since the stratified model cox2 provides a smaller AIC value than the previous model cox1, we may conclude that the stratified model gives a better fit for this data. However, if the covariate “knot” is of primary interest, this method is not recommended. Therefore, we may try other appropriative alternatives, such as the accelerated failure time model that will be discussed in the sequel. 47 Chapter 5 Parametric Survival Models The accelerated failure time (AFT) model[4, 7, 11] is another alternative of the Cox PH model when the PH assumption is violated. The AFT model can be used to express the magnitude of effect in a more accessible way in terms of the difference between groups in survival strength. Under AFT models we measure the direct effect of the explanatory variables on the survival strength instead of hazard, as we do in the PH model. This characteristic allows for an easier interpretation of the results because the parameters measure the effect of the corresponding covariate on the mean survival strength. 5.1 Exploring the Distribution of Load to Failure The most commonly used AFT models include the exponential AFT model, Weibull AFT model, log-logistic AFT model, and log-normal AFT model. The AFT models are named for the distribution of survival data. Since each parametric distribution is defined by a different hazard func- tion, we can check the consistency of survival data with a specific distribution by investigating the corresponding underlying linearity. Four different plots can be obtained and the corresponding distributions indicated if these plots form a straight line pattern. The plots and their associated distributions are given in Table 5.1, where Z(p) means the pth quantile from the standard normal distribution. Table 5.1: Plots and associated distributions. Plot Distribution indicated by a straight line pattern -log[S(t)] vs. t Exponential, through the origin log[−log(S(t))] vs. log(t) Weibull log[(1− S(t))/S(t)] vs. log(t) Log-logistic Z[1-S(t)] vs. log(t) Log-normal 48 5.2. Variable Selection We present these four different plots based on the bending data in Fig- ure 5.1. By comparing the straightness of these lines, we may see that the distribution of bending data is more likely to be one of Weibull, log-normal, or log-logistic. Also, it seems that the bending data should not be from the exponential distribution because the line is far away from the straight line through the origin. Note it is the left hand tail that accounts in applications, and the left hand tail observations seem more likely to be from the Weibull distribution as they present a slightly better straight line through the origin. 5.2 Variable Selection We fit the bending data using exponential, Weibull, log-logistic, and log- normal AFT models. In both univariate and multivariate AFT models, “knot” , “offg” and “moe” are statistically significantly associated with load to failure MOR. No interactions are statistically significant in multivariate AFT models. There is no big difference for the estimated Weibull, log- logistic and log-normal models, but the estimated exponential model is quite different. This indicates the distribution of MOR may be far away from the exponential distribution. The results from the different AFT models applied to the bending data are presented in Table 5.2, where η is the estimated acceleration factor. 49 5.2. Variable Selection Figure 5.1: Exploring distribution of load to failure. l l l lllll llllll llll lll lll lll lll ll lll ll lll ll l l ll ll l l l l l 4 6 8 10 0 1 2 3 4 Checking for exponential distn t − lo g[S (t) ] l l l l ll ll ll lll ll lll lll lll lll lll llll llll llll lll lll lll ll l 1.2 1.4 1.6 1.8 2.0 2.2 − 4 − 3 − 2 − 1 0 1 Checking for Weibull distn log(t) lo g [−l og (S (t) )] l l l l ll lll ll llll lll lll lll lll llll lll lll lll lll ll lll ll l l l l 1.2 1.4 1.6 1.8 2.0 2.2 − 4 − 2 0 2 4 Checking for Log−logistic distn log(t) lo g [(1 −S (t) )/S (t) ] l l l l ll lll ll lll lll lll lll lll lll lll lll lll lll lll ll lll ll ll l l l 1.2 1.4 1.6 1.8 2.0 2.2 − 2 − 1 0 1 2 Checking for Log−normal distn log(t) Z[1 −S (t) ] 50 5.2. V ariab le S election Table 5.2: Results from AFT models for the bending data. Coef Exponential Weibull Log-logistic Log-normal α sd η p α sd η p α sd η p α sd η p µ 1.41 0.45 1.44 0.00 1.41 0.00 1.42 0.00 knot1 -0.21 0.36 0.81 0.55 -0.23 0.04 0.79 0.00 -0.23 0.06 0.79 0.00 -0.21 0.05 0.81 0.00 knot2 -0.10 0.38 0.90 0.78 -0.12 0.04 0.88 0.02 -0.10 0.06 0.90 0.10 -0.10 0.06 0.90 0.11 ksize -0.00 0.01 1.00 0.98 0.00 0.00 1.00 0.97 0.00 0.00 1.00 0.97 -0.00 0.00 1.00 0.88 rnl -0.00 0.01 1.00 0.85 0.00 0.00 1.00 0.15 -0.00 0.00 1.00 0.32 -0.00 0.00 1.00 0.24 offg -0.30 0.50 0.74 0.54 -0.23 0.06 0.79 0.00 -0.25 0.10 0.77 0.02 -0.33 0.08 0.71 0.00 loc 0.00 0.01 1.00 0.95 0.00 0.00 1.00 0.99 0.00 0.00 1.00 0.77 0.00 0.00 1.00 0.61 face1 0.07 0.24 1.07 0.76 0.07 0.03 1.07 0.03 0.07 0.03 1.07 0.07 0.07 0.03 1.07 0.06 face2 -0.00 0.34 1.00 0.99 -0.06 0.04 0.94 0.19 -0.02 0.05 0.98 0.77 0.01 0.05 1.01 0.83 spec2 -0.04 0.41 0.96 0.91 -0.06 0.05 0.94 0.29 -0.05 0.06 0.95 0.43 -0.04 0.06 0.96 0.55 spec3 -0.11 1.13 0.89 0.92 -0.18 0.15 0.83 0.23 -0.13 0.15 0.87 0.38 -0.09 0.18 0.91 0.62 mois -0.02 0.12 0.98 0.86 -0.03 0.01 0.97 0.16 -0.02 0.01 0.98 0.30 -0.02 0.01 0.98 0.29 moe 0.66 0.83 1.93 0.42 0.77 0.11 2.15 0.00 0.64 0.12 1.89 0.00 0.64 0.13 1.89 0.00 51 5.3. Q-Q Plot to Check the AFT Assumption For the parametric models we discuss here, the AIC is given by AIC = −2× log(maximumlikelihood) + 2× (a+ b), where a is the number of parameters in the specific model and b the num- ber of one-dimensional covariates. For example, a = 1 for the exponential model, a = 2 for the Weibull, log-logistic, and log-normal models. In Ta- ble 5.3, we compared all these AFT models using statistical criteria–AIC. Note the smaller AIC is the better. The Weibull AFT model appears to be an appropriate AFT model according to AIC compared to other AFT models. However, the exponential model provides the worst fit, which is consistent with the conclusion we drawn from Figure 5.1. Table 5.3: AIC in the AFT models. Model Log-likelihood a b AIC Exponential -283.6 1 12 593.1685 Weibull -139 2 12 306.0937 Log-logistic -144.4 2 12 316.8026 Log-normal -144.5 2 12 317.0219 5.3 Q-Q Plot to Check the AFT Assumption An initial method for assessing the potential for an AFT model is to produce a quantile-quantile plot. For any value p in the interval (0,100), the pth percentile is t(p) = S−1( 100− p 100 ). Let t0(p) and t1(p) be the pth percentiles estimated from the survival functions of the two groups of survival data. The percentiles for the two groups may be expressed as t0(p) = S−10 ( 100− p 100 ), t1(p) = S−11 ( 100− p 100 ), 52 5.4. Model Diagnostics for the AFT Model where S0(t) and S1(t) are the survival functions for the two groups. So we can get S1[t1(p)] = S0[t0(p)]. Under the AFT model, the assumption is S1(t) = S0[t/η], and so S1[t1(p)] = S0[t1(p)/η]. Therefore, we get t0(p) = η−1t1(p). The percentiles of the survival distributions for the two groups can be estimated by the KM estimates of the respective survival functions. If the accelerated failure time model is appropriate, a plot of percentiles of the KM estimated survival function from one group against another should be given an approximate straight line through the origin. The slop of this line will be an estimate of the acceleration factor η−1. For the 3-level categorical covariate “knot”, we have 3 possible pairwise combinations. The Q-Q plot in Figure 5.2 approximates well a straight line from the origin indicating that the AFT model may be appropriate. 5.4 Model Diagnostics for the AFT Model 5.4.1 Overall Goodness-of-Fit We check the goodness of fit of the model using residual plots. The cumula- tive hazard plot of the Cox-Snell residuals in the Weibull model is presented in Figure 5.3. The plotted points mostly lie on a line that has a unit slope and zero intercept. So there is no reason to doubt the suitability of this fitted Weibull model. Comparing Figure 4.3 with Figure 5.3, we may see that the Weibull AFT model provides a much better fit than the Cox PH model. We conclude that the Weibull produces the best fitting AFT model based on AIC criteria and residuals plot. 5.4.2 Checking for Outliers Similarly, the plots of deviance residuals against the risk score, index and covariates are given in Figure 5.4. They display only one possible outlier, but none of them seem to be systematically distributed about zero. Therefore, overall, we have little concern about the adequacy of the fitted log-normal AFT model. 53 5.4. Model Diagnostics for the AFT Model Figure 5.2: Q-Q plot for load to failure. l l l l 0.0 0.2 0.4 0.6 0.8 1.0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Load to failure for non−knot group Lo ad to fa ilu re fo r si ng le k no t g ro up l l l l 0.0 0.2 0.4 0.6 0.8 1.0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Load to failure for non−knot group Lo ad to fa ilu re fo r kn ot c om b gr ou p l l l l 0.0 0.2 0.4 0.6 0.8 1.0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Load to failure for knot comb group Lo ad to fa ilu re fo r si ng le k no t g ro up l l l l 0.0 0.2 0.4 0.6 0.8 1.0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Load to failure for not offgrade group Lo ad to fa ilu re fo r o ffg ra de g ro up 54 5.4. Model Diagnostics for the AFT Model Figure 5.3: Cumulative hazard plot of the Cox-Snell residual for the Weibull AFT model. lll llll llll lll lll lll lll ll llll ll ll lll lll lll ll ll ll ll ll l l l l l l l l l l l l l l l 0 1 2 3 4 0 1 2 3 4 5 Cox−Snell residual Cu m u la tiv e h az ar d of re sid ua l 55 5.4. Model Diagnostics for the AFT Model Figure 5.4: Deviance residuals against the risk score, index and covariates. l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l ll ll l l l l l l l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l l l l l l ll l l l l ll l l l l ll l l l 1.6 1.7 1.8 1.9 2.0 2.1 2.2 − 2 0 1 2 risk score D ev ia nc e re si du al s l l l l l ll l l l l l l l l l l l l l l l l l l lll l l llll l l l l l l l l l l l l l l l l l l l lll l l l l lll ll l l l l l l l l l l l l l l l ll l l l l ll l ll l l ll l l l 0 20 40 60 80 100 − 2 0 1 2 Index l l l l l l l l l l l l l l ll l l l ll ll l l l l l l l l l l l l ll l ll l l l l l l l l 0.0 0.5 1.0 1.5 2.0 − 2 0 1 2 knot D ev ia nc e re si du al l l l l l l l l l l l l ll l l l l ll l l l l l l l l l l l 0.0 0.2 0.4 0.6 0.8 1.0 − 2 0 1 2 offg l l l l l l l l l l l l l l l l l l l l l l l l l l ll l l ll ll l l l l l l l l l l l l l l l l l l l l l l l l l ll ll l l l l l l l l l l l l l l ll l l l l ll l l l l l ll l l l 1.2 1.3 1.4 1.5 1.6 1.7 1.8 − 2 0 1 2 moe D ev ia nc e re si du al 56 5.5. Interpretation of Results 5.4.3 Influential Assessment Figure 5.5 shows the change in each regression coefficient when each obser- vation is removed from the data (influence statistics). The changes plotted are scaled in units of standard errors. Changes of less than 0.04 are of little concern. These plots give us an idea of the influence individual observations have on the estimated regression coefficients for each covariate. Most of the changes in the regression coefficients are less than 0.02 s.e.’s of the coef- ficients and all others are less than 0.03 s.e.’s. Therefore, data sets where the influence plot is tightly clustered around zero indicate an absence of influential observations. 5.5 Interpretation of Results Finally, we may fit the Weibull AFT model with only statistically significant covariates – “knot”, “offg” and “moe”: wei < −survreg(Surv(mor) ∼ factor(knot)+offg+moe, dist = “weibull”) The summary for this model is given in Table 5.4: Table 5.4: Summary for the final Weibull AFT model. coef se(coef) η(coef) p (Intercept) 1.0174 0.1796 1.47e-08 factor(knot)1 -0.1428 0.0412 0.8669274 5.31e-04 ∗ ∗ ∗ factor(knot)2 -0.0978 0.0471 0.9068303 3.79e-02 ∗ offg -0.2118 0.0661 0.8091265 1.35e-03 ∗∗ moe 0.7057 0.1119 2.0252639 2.86e-10 ∗ ∗ ∗ Log(scale) -1.9688 0.0786 - 2.00e-138 Loglik(model)= -145 Conclusion: the acceleration factor (η) for “offg” is 0.81 (less than 1), which indicates that the smaller survival load is more likely for off-grade lumber. The η’s for “knot” is also less than 1 imply that this variable yields a lower load to failure, and the “single knot” group is more likely to break than the “knot combination” group since it has a even smaller acceleration factor. The acceleration factor (η) for “moe” is 2.03 (more than 1), which indicates that the larger survival load is more likely for the piece of lumber with higher MOE. These conclusions are consistent with the ones drawn from application of the Cox PH model. 57 5.5. Interpretation of Results Figure 5.5: Influence statistics. 0 20 40 60 80 100 − 0. 05 0. 00 0. 05 Observation Ch an ge in c oe ffi cie nt knot 0 20 40 60 80 100 − 0. 05 0. 00 0. 05 Observation offgrade 0 20 40 60 80 100 − 0. 05 0. 00 0. 05 Observation Ch an ge in c oe ffi cie nt moe 58 5.6. Simulation Study 5.6 Simulation Study 5.6.1 Introduction In practice the model relating the strength of a piece of lumber to its co- variates cannot be known and we explore through simulations studies the inferential effect of mis-specifying that model. However, to constrain the scope of our study to a practical limit, we will assume that the structural link between the response and the covariates is correct based on our belief that diagnostic assessments of data would suggest a reasonable choice for that link. Thus we restrict our studies to the effect of mis-specifying the random error component of an AFT model for the strength. More precisely, we looked at the estimates for the coefficients in that link when the standard Normal distribution, the Cauchy t1 distribution, the Student t2 distribution and the standard Gumbel distribution are assumed for the error distribution when the true distribution is none of these. The details follow below. A simulation study was conducted to compare the estimates for the AFT models with Weibull, exponential, log-normal and log-logistic distri- bution assumptions. Also, one of our interests is to investigate predic- tive accuracy. One commonly used measure of predictive accuracy is the expected squared error of the estimate. This quantity is defined as the ex- pected squared difference between predicted and observed values, that is, the average squared difference between predicted and observed values if the experiment were repeated finitely often and new estimates were made at each replication. 5.6.2 Description of Method Our final log-linear form of the AFT model with respect to load Y is given by: log Yi = µ+ α1 knoti + α2 offgi + α3moei + σ εi, where µ = 1.0, α11 = −0.1,α12 = −0.1,α2 = −0.2, α2 = 0.7 and σ = 0.1 are fixed. The significant X variables knot, offg andmoe values from the origi- nal sample are also fixed with respect to replication of the study. The errors εi was generated parametrically from a standard Normal distribution, from a Cauchy t1 distribution, from a Student t2 distribution and from a standard Gumbel distribution. The response values Yi, however, are randomly gener- ated by the AFT model, because of the error component of the model. We would then regress the response values Yi on the fixed X matrix (knot, offg and moe) to obtain the regression coefficients estimates at each replication. 59 5.7. Cross-Validation We may also obtain the average squared difference between predicted and observed values 198 ∑ (Yi − Ŷi)2 at each replication. Since there are 4 settings of the errors term distributions and 4 settings of AFT models with different distribution assumptions ( Weibull, exponential, log-normal and log-logistic), there were total 4 × 4 (16) different settings of simulation conducted. Each simulation involved 1000 replications with a sample size 98. 5.6.3 Results for the Simulation For each simulation, the estimates were computed using the Weibull AFT model, the exponential AFT model, the log-normal AFT model and the log-logistic model. Let’s take the coefficient α2 for the covariate “offg” for example. Table 5.5 shows the average values of the parameter estimates for α2 and their standard deviations over the 1000 replications with 4 different error terms. Table 5.5: True value α2 = −0.2. Expected value, standard deviation of parameter estimates. Setting Weibull Exponential Log-normal Log-logistic Normal -0.211(0.08) -0.214(0.07) -0.212(0.07) -0.211(0.07) Cauchy t1 -0.224(0.15) -0.261(0.32) -0.240(0.35) -0.253(0.34) Student t2 -0.233(0.46) -0.276(0.57) -0.265(0.47) -0.269(0.57) Gumbel -0.245(2.10) -0.292(6.12) -0.283(5.10) -0.288(5.12) Overall, based on this simulation study, the Weibull AFT model shows better estimations on this coefficient, which are closer to the true values. The patterns of other coefficients are the same in most cases. Moreover, the mean of predictive accuracy 198 ∑ (Y − Ŷ )2 over 1000 replications shows that the Weibull AFT model with a standard normal error performed better than other models since it gave the smallest mean predictive accuracy. This confirms our choice of the Weibull AFT model once again. 5.7 Cross-Validation A stringent test of a model is an external validation - the application of the ‘frozen’ model to a new population. It is often the case that the failure 60 5.7. Cross-Validation of a model to validate externally could have been predicted from an hon- est (unbiased) ‘internal’ validation. One well-known method for obtaining nearly unbiased internal assessments of accuracy is cross-validation. To un- cover problems that may make prediction models misleading or invalid, the predictive accuracy has to be unbiasedly validated using cross-validation. Each time, we drop one record from the sample and the remaining data are used as a training (model development) sample. That model is ‘frozen’ and applied to the dropped out sample for computing predictive survival probability. For example, we drop record 98, then fit a model on records 1 to 97 and use this model to predict the 98th record, so on so forth. The following plot Figure 5.6 gives us an idea of how well the predicted survival curve from the final Weibull AFT model tracks observed Kaplan- Meier estimates. The predicted survival is slightly larger than the observed in the lower tail and smaller than the observed in the upper tail. However, we see that predicted survival curve mainly falls within the 95% error bounds of the observed survival curve. Therefore, it does not produce large deviations from the true values. 61 5.7. Cross-Validation Figure 5.6: Comparing observed and predicted survival curves. 0 2 4 6 8 10 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 load to failure(psi*10^3) pr op or tio n in re m ai ni ng observed predicted 62 Chapter 6 Conclusion and Discussion This study is based on the wood strength data collected in a FPInnova- tions (FPI) laboratory. We employed survival analysis methods in this very different context - load to failure problem. A finding of the present study shows that a type of wood defect (knot), a lumber grade status (off-grade: Yes/No) and a lumber’s modulus of elasticity (moe) have statistically sig- nificant effects on wood strength properties including bending strength and tension strength. Forms of non-parametric and parametric bivariate-strength survival func- tions (Biv-KM and Biv-Weibull) have been explored to obtain the joint strength distributions. Association between MSRC and FC was also ex- amined by the Cramer’s V statistic and found to be just 0.3, indicating the strength of association is not that strong. However, this measure of strength highly depends on how the covariate values are aggregated into sub cate- gories and in our case, these lumber categories were fairly fine, making a low V level inevitable with such a small dataset. The Cox PH model is routinely applied to the analysis of survival data, but the proportional hazards (PH) assumption does not hold for ‘knot’ in this analysis. We also use four different accelerated failure time (AFT) models to fit the data. We found that the Weibull AFT model was the best fit for this dataset. The study considered here provides an example of a situation where Cox PH model is inappropriate and where the Weibull AFT model provides a better description of the data. We see that the Weibull AFT model is a more valuable and realistic alternative to the Cox PH model in some situation. Moreover, the AFT model has a more realistic interpretation in terms of an effect on expected load to failure and provides more informative results. To this content the AFT model has explanatory advantage in that covariates have a direct effect on load to failure rather on hazard functions as in the Cox PH model. Therefore, we suggest that using the Cox PH model may not be the optimum approach. The AFT model may provide an alternative method to fit some survival data. This final Weibull AFT model can be used to make the current lumber grading system (currently highly relays on graders’ experience) more powerful and 63 Chapter 6. Conclusion and Discussion reliable. Both of the Cox PH model and the Weibull AFT model yield exactly the same significant covariates - ‘knot’, ‘off-grade’ and ‘moe’, indicating these three are the most important predictors in our reliability modeling. In our study, a piece of lumber with a ‘knot’ defect is more likely to break than one with other defects; in particular, a piece of lumber with a ‘single knot’ defect is even more likely to break than one with a ‘knot combination’ defect. Not surprising, off-grade lumber is more likely to have lower survival loads than the standard ones. Also, the piece of lumber with a higher ‘moe’ is more likely to have a higher survival load. As mentioned above, after applying these survival analysis methods to wood strength properties, we obtained the same significant covariates - ‘knot’, ‘offg’ and ‘moe’ in both bending strength data and tension strength data. These significant covariates can be used to match pieces of lumber in describing the relationships among strength properties. Obtaining matched pairs in this way helps solve the challenging problem that a single piece of lumber cannot be broken twice by two different strength tests. This is a major potential application of reliability modeling analysis conducted here, and this could even be used in other situations for example in analyzing the duration of load for lumber. We should also recognize the fact that a small sample size makes it difficult to find significant predictors and that in future work, a larger sample should be collected to find more others. In practice the model relating the strength of a piece of lumber to its covariates cannot be known and we explore through simulations studies the inferential effect of misspecifying that model. These studies were conducted to compare the coefficients estimates from the AFT models with Weibull, ex- ponential, log-normal and log-logistic distribution assumptions. TheWeibull AFT model leads to somewhat better estimates of coefficients than the other incorrectly specified models. As well, it provides the best mean predictive accuracy. This confirms our choice of the Weibull AFT model once again. Finally, to uncover problems that may make prediction models mislead- ing or invalid, predictive accuracy has been unbiasedly assessed using cross- validation. We observe that predicted survival curve from the final Weibull AFT model tracks the observed Kaplan-Meier estimates very well. This study has shown the power of employing survival analysis methods in re- liability in this very different context from that which originally led to its development. 64 Bibliography [1] Agresti A. An Introduction to Categorical Data Analysis. Wiley- Interscience, 2008. [2] Richard A.J., James W.E., and David W.G. Some Bivariate Distribu- tions for Modeling the Strength Properties of Lumber. Forest Products Laboratory Research Paper, 1999. [3] National Lumber Grades Authority. Canadian Lumber Grading Man- ual. National Lumber Grades Authority, 8th edition, 2001. [4] Collett D. Modelling Survival Data in Medical Research. Chapman and Hall, London, 2003. [5] Schoenfeld D. Partial residuals for the proportional hazards regression model. Biometrika, 69:239–241, 1982. [6] Cox D.R. Regression Models and Life Tables(with Discussion). Journal of the Royal Statistical Society, 34:187–220, 1972. [7] Cox D.R. and Oakes D. Analysis of Survival Data. Chapman and Hall, 1984. [8] Cox D.R. and Snell E.J. A general definition of residuals with discus- sion. Journal of the Royal Statistical Society, 30:248–275, 1968. [9] Lin D.Y. and Ying Z. A simple Nonparametric Estimator of the Bivari- ate Survival Function Under Univariate Censoring. Biometrika, 80:573– 581, 1993. [10] P. Hougaard. A class of multivariate failure time distributions. 73:671– 678, 1986. [11] Lawless J.F. Statistical Models and Methods for Lifetime Data Analysis. Wiley, New York, 1982. [12] Peter J.S. Analysis of Failure and Survival Data. Chapman and Hall/CRC, 2002. 65 Bibliography [13] Crowder M.J., Kimber A.C., Smith R.L., and Sweeting T.J. Statistical Analysis of Reliability Data. Chapman and Hall, 1991. [14] Abbott R. Commentary on the Maximum Strength Reducing De- fects(MSRD) and Failure Coding System. 2002. [15] Abbott R. Forintek Knot and Failure Code. 2002. [16] Mara T. and Jong S.K. Suvival Analysis Using S: Analysis of Time-to- Event Data. Chapman and Hall/CRC, 2004. 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