Plot of Laser Operating Current as a Function of Time Laser Test Data

Transcription

1 Chapter Repeated Measures Data and Random Parameter Models Repeated Measures Data and Random Parameter Models Chapter Objectives Understand applications of growth curve models to describe the results of statistical studies in which repeated measures are made on a sample of units from some population or process. Part of the Iowa State University NSF/ILI project Beyond Traditional Statistical Methods Copyright 999 D. Cook, W. M. Duckworth, M. S. Kaiser, W. Q. Meeker, and W. R. Stephenson. Developed as part of NSF/ILI grant DUE97. January 3, 7h min - Explore different kinds of models for repeated measure data. Empirical Mechanistic (e.g., from systems of differential equations) Understand how to fit nonlinear regression models. Understand the different sources of variability in repeated measures data. Become familiar with methods for data analysis and inference for repeated measure data. - Plot of Laser Operating Current as a Function of Time Laser Test Data Percent Increase in Operating Current 3 Hours Measured percentage increase in operating current over time for GaAs lasers tested at C. Fifteen () devices each measured every hours up to hours of operation. For these device and the corresponding application, a D f = % increase in current was the specified failure level. Engineers wanted to predict life of lasers operating at C Plasma Concentrations of Indomethicin Following Bolus Intravenous Injection Constant-Rate Growth Model For some types of growth, rate will be approximately constant: dy(t) = β With y() = β, this simple differential equation has the solution y(t) =β + β t where β is the intercept and β is the slope (growth rate). This linear model is useful describe certain growth patterns. Concentration (mg/l)..... Hours - -

2 Plasma Concentrations of Indomethicin Following Bolus Intravenous Injection Plasma Concentrations of Indomethicin Following Intravenous Injection Blood plasma concentrations of indomethicin following bolus intravenous injection. Data from Kwan et al. (97) Kinetics of indomethicin absorption, elimination, and enterohepatic circulation in man. Also, page of Davidian and Giltinan (99) Data collected on six human volunteers at time points from minutes to to hours. Pharmacokinetic researchers need to know the rate of passing to elimination and the time (or expected time) until a specified level of indomethicin is left. log(concentration) (log(mcg/ml)) Interest centers on the identification of demographic and physiological characteristics explaining the variability in drug response information to be used in choosing effective dosage regimes Time (hr) -7 - Compartmental Model for Indomethicin Absorption and Elimination Two-Compartment Model for Blood Plasma Concentrations of Indomethicin Following Bolus Intravenous Injection A two-compartment model provides useful model to describe the concentration in the blood over time. The following differential equations describe the two compartment model da (t) = k A (t), and da (t) = k A (t) k A (t). Injection Site Blood k A A k Elimination Then, using standard methods for solving systems of linear differential equations the concentration in the blood at time t is A (t) = θ exp( k t) θ exp( k t) θ = A ()k /(k k ) θ = θ + A () Splus has the biexponential function for this model Biexponential Function plot.function(biexp,:,a=.9, A=3., lrc=-.7, lrc=.) Serum Concentrations of Theophylline Following Oral Administration response Concentration (mg/l) time Time since drug administration (Hours) - -

3 Trellis Plot of Serum Concentrations of Theophylline Following Oral Administration Serum Concentrations of Theophylline Following Oral Administration 9 Data from Boeckmann et al. (99). Theophylline is an anti-asthmatic agent. Data collected on twelve subjects. Serum concentrations were measured at time points from minutes to to hours. Theophylline concentration (mg/l) 7 3 Dosage was varied in proportion to body weight. Time since drug administration (hr) Two-Compartment Model for Theophylline Absorption and Elimination Following Oral Administration Two-Compartment Model for Theophylline Absorption and Elimination Following Oral Administration Stomach k Blood A A k =V/Cl Elimination - The following two-compartment (gut and blood) model is used to describe the concentration in the blood as a function of time. da (t) = k A (t), and where da (t) = k A (t) Cl V A (t) A (t) is the amount in the stomach at time t. A (t) is the concentration in the blood at time t. The rate k has units /hour. Clearance Cl has units L/hr/kg. Volume of distribution V has units L/kg. The elimination rate k = Cl/V has units /hour. - Two-Compartment Model for Theophylline Absorption and Elimination With initial conditions A () = Dose and A () =, the concentration in the blood as a function of time is obtained by solving the system of differential equations to give: A (t) = Dose k V (k Cl/V ) [ exp ( Cl V t The amount left in the stomach at time t is A (t) =Dose exp( k t) ) ] exp( k t) The model fit to the blood concentration data was reparameterized as A (t)= Dose exp(θ ) exp(θ ) exp(θ 3 )[exp(θ ) exp(θ )] {exp[ exp(θ )t] exp[ exp(θ )t]} where θ =log(k ), θ =log(cl/v ), θ 3 = log(cl), are unrestricted in sign. response First Order Log Function plot.function(first.order.log, time=:, Dose=,lCl=-3.7,lka= -., lke=-.) Splus has function first.order.log function for this model. -7 time -

4 Device-B Power Drop Accelerated Degradation Test Results at C, 9 C,and37 C (Use conditions C) Device-B Power Drop Accelerated Test Results Power drop in db Degrees C 9 Degrees C 37 Degrees C RF amplifier IC device, used in a satellite communications system. System life was designed to be between and years. A six-month test at high temperature was to be used to predict life at C junction temperature. The physics/chemistry of the failure mechanism was wellunderstood. 3 Hours -9 - Device-B Power Drop Simple One-Step Chemical Reaction Leading to Failure Simple One-Step Chemical Reaction A (t) is the amount of harmful material available for reaction at time t A (t) is proportional to the amount of failure-causing compounds at time t. k A A Simple one-step chemical reaction: A k A Power drop proportional to A (t) The rate equations for this reaction are da = k A and da = k A - - Device-B Power Drop Simple One-Step Chemical Reaction Leading to Failure (contd.) Solution to system of differential equations: A (t) = A () exp( k t) A (t) = A () + A ()[ exp( k t)] where A () and A () are initial conditions. If A () =, then lim t A (t) =A () and the solution for A (t) (the function of primary interest) can be reexpressed as A (t) = A ()[ exp( k t)] A simple -step diffusion process has the same solution. Acceleration of Degradation The Arrhenius model describing the effect that temperature has on the rate of a simple one-step chemical reaction is ( ) E a R(temp) =γ exp k B (temp + 73.) where temp is temperature in C and k B =. is Boltzmann s constant in units of electron volts per C. The pre-exponential factor γ and the reaction activation energy E a are characteristics of the product or material. The Acceleration Factor between temp and temp U is AF(temp) =AF(temp, temp U,E R(temp) a)= R(temp U ) When temp > temp U, AF(temp, temp U,E a) >. -3 -

5 Simple Reversible Chemical Reaction Simple Reversible Chemical Reaction The rate equations for this reaction are da = k A + k A and da = k A k A k A A k In this model the failure causing reaction is reversible. With A () =, the solution of the system of differential equations for this model gives, for A k A (t) =A () { exp[ (k + k )t]}. k + k - - Orange Tree Circumference Growth Orange Tree Circumference Growth Data collected by measuring trunk circumference on five different trees grown in Riverside, Calfornia, between 99 and 973. Measurements taken at,,,, 3, 37, and days. Researchers are interested in characterizing the tree-to-tree variability and seeking explanations for why some trees grow faster than others. Trunk circumference (mm) Data from Draper and Smith (9), page. Time since December 3, 9 (days) -7 - Trellis Plot of Orange Tree Circumference Growth Logistic (Autocatalytic) Model For some types of growth, growth rate will increase to a point and then decline. Trunk circumference (mm) 3 The following differential equation has rate depending on has a basic rate, κ, current size f, and a limiting size α. dy(t) = Asym scale y(t) [Asym y(t)]. This differential equation has the general solution y(t) =Asym/{ + exp[ (t T )/scale]} where Asym is the asymptote, T is the time at which % growth has been achieved, and scale is a scale parameter describing the steepness of the growth curve. This S-shaped logistic curve is a useful curve to describe certain growth patterns. Time since December 3, 9 (days). -9 Splus has the logistic function for this model. -3

6 Orange Tree Circumference Growth Data and Fitted Logistic Function Logistic Function plot.function(logistic,:,asym=7,t=,scal=379) Orange Tree Circumference Growth response inches time Days -3-3 Fatigue Crack Size Observations for Alloy-A (Bogdanoff & Kozin 9) Bogdonoff-Kozin Fatigue Crack Growth Data Notched Compact Fatigue Test Specimen Inches Mcycles Alloy-A Fatigue Crack-Size Data Fatigue Crack Size Observations for Alloy-A (Bogdanoff & Kozin 9) Data from Hudak, Saxena, Bucci, and Malcolm (97) Bogdanoff and Kozin (9, page ), and Lu and Meeker (993). Bogdonoff-Kozin Fatigue Crack Growth Data Suppose investigators wanted to:... Estimate materials parameters related to crack growth. Inches Estimate time (measured in number of cycles) at which % of the cracks would reach. inches Assess adequacy of the paris model to describe crack growth rate Mcycles -3-3

7 Fatigue Crack Size Observations for Alloy-A and Nonlinear Least Squares Paris Law Fit Bogdonoff-Kozin Fatigue Crack Growth Data Degradation Data in Reliability Analysis Sometimes possible to measure degradation directly over time Inches.... Continuously. At specific points in time. Degradation is natural response for some tests. Degradation data can provide considerably more reliability information than censored failure-time data (especially with few or no failures) Mcycles -37 Direct observation of the degradation process allows direct modeling of the failure-causing mechanism. -3 Paris Crack Growth Model Paris Model with no Variability The paris model is da(t) = C [ K(a)] m is a commonly used empirical model to describe the growth of fatigue cracks over some range of size. C>andm> are materials properties K(a) is the stress intensity function. Form of K(a) depends on applied stress, part dimensions, and geometry. To model a two-dimensional edge-crack in a plate with a crack that is small relative to the wih of the plate (say less than 3%), K(a) =Stress πa and the solution to the resulting differential equation is [ {a()} m/ +( m/) C (Stress π) m t ] m, m a(t) = a() exp [ C (Stress π) t ], m = Crack Size (mm)..... Cycles Paris Model with Unit-to-Unit Variability in Initial Crack Size but with Fixed Materials Parameters and Constant Stress. Models for Variation in Degradation and Failure Time If all manufactured units were identical, operated at exactly the same time, under exactly the same conditions, and in exactly the same environment, and if every unit failed as it reached a particular critical level of degradation, then all units would fail at exactly the same time. Crack Size (mm)... Need to identify and model important sources of variability in the degradation process. Quantities that might be modeled as random include: Initial conditions (flaw size, amount of material). Materials parameters (related to degradation rate). Level of degradation at which unit will fail.. Cycles Stochastic process variability (e.g., stress of other environmental variables changing over time). - -

8 Paris Model with Unit-to-Unit Variability in the Initial Crack Size and Materials Parameters but Constant Stress Paris Model with Unit-to-Unit Variability in the Initial Crack Size and Materials Parameters and Stochastic Stress.... Crack Size (mm). Crack Size (mm)..... Cycles -3 Cycles - General Response Path Model Estimation of Degradation Model Parameters D ij = D(t ij,β i,...,β ki ) is the degradation path for unit i at time t (measured in hours, cycles, etc.). Observed sample path of unit i at time t j is y ij = D ij + ɛ ij, i =,...,n, j =,...,m i Residuals ɛ ij NOR (,σ ɛ) describe a combination of measurement error and model error. For unit i, β i,...,β ki is a vector of k unknown parameters. Some of the β i,...,β ki are random from unit to unit. Model appropriate function of β i,...,β ki with multivariate normal distribution (MVN) with parameters µ β and Σ β. The likelihood for the random-parameter path model is L(µ β, Σ β,σ ɛ DATA) n mi =... φ nor(ζ ij) f β (β i,...,β ki ; µ β, Σ β )dβ i,...,dβ ki σ ɛ i= j= where ζ ij =[y ij D(t ij,β i,...,β ki )]/σ ɛ and f β (β i,...,β ki ; µ β, Σ β ) is the multivariate normal distribution density function. Each evaluation of L(µ β, Σ β,σ ɛ DATA) will, in general, require numerical approximation of n integrals of dimension k. Maximization of L(µ β, Σ β,σ ɛ DATA) computationally difficult. - - Alloy-A Fatigue Crack Size Observations and Fitted Paris-Rule Model Crack Length.... Estimates of Fatigue Data Model Parameters for Alloy-A Splus gives the following approximate ML estimates. ( ) ( ).7..9 ˆµ β =, ˆΣ 3.73 β =.9.9 and σ ɛ =.3. Here β = C, β = m Millions of Cycles -7 -

9 Estimates β i Versus β i, i =,..., and Contours for the Fitted Bivariate Normal Distribution 7 Models Relating Degradation and Failure Beta Soft failures are defined to occur at a specified degradation level. In some products there is a gradual loss of performance (e.g., decreasing light output from a fluorescent light bulb). Use fixed D f to denote the critical level defining failure for a degradation path. 3 Beta -9 - Alloy-A Fatigue Crack Size Observations and Fitted Paris-Rule Model Evaluation of F (t) Distribution of Time to First Crossing Crack Length Millions of Cycles Direct evaluation of F (t): Closed forms available for simple problems (e.g., a single random variable and other special cases). Numerical integration: Useful for a small number of random variables (e.g., or 3). FORM (first order) approximation: Rapid computation, but uncertain approximation. Used frequently in engineering problems with large number of random variables. Monte Carlo simulation: General method. Needs much computer time to evaluate small probabilities. Can use importance sampling. Estimate failure probabilities by evaluating at ML estimates. - - Evaluation of F (t) by Numerical Integration Bootstrap Estimates of F (t) =Pr(T t) The failure (crossing probability) can be expressed as Pr(T t) =F (t) =F (t; θ β )=Pr[D(t, β,...,β k ) > D f ]. If (β,β ) follows a bivariate normal distribution with parameters µ β,µ β σ β,σ β,ρ β,β,thenp (T t) = Φnor g(d f,t,β ) µ β β σ β β φ nor σ β ( ) β µ β dβ σ β where g(d f,t,β )isthevalueofβ for given β, that gives D(t) =D f. Cumulative Probability.... Method generalizes to multivariate normal, but requires correspondingly higher-order integration Millions of Cycles -3 -

10 Degradation Estimate of F (t) with Pointwise Two-Sided 9% and % Bootstrap Bias-Corrected Percentile Confidence Intervals, Based on the Crack-Size Data Censored at t c =.. The Nonparametric Estimate of F (t) Indicated by Dots Probability <- Censoring time Millions of Cycles cdf Estimate % Confidence Intervals 9% Confidence Intervals - > Indometh.lis <- nlslist(data=indometh, + conc ~ biexp(time, A, A,lrc,lrc), cluster= ~ Subject) A A lrc lrc A A lrc lrc A A lrc lrc A A lrc lrc A A lrc lrc A A lrc lrc > Indometh.nlme <- nlme(indometh.lis) > plot(indometh.nlme) - Fitted Values of Plasma Concentrations of Indomethicin Following Intravenous Injection Plasma Concentrations of Indomethicin Estimates of Random Parameters A -... Concentration (mcg/ml).... Time since drug administration (Hours) A lrc lrc > anova(indometh.nlme) Value Std.Error z ratio A A lrc lrc > > summary(indometh.nlme) Call: Model: conc ~ biexp(time, A, A, lrc, lrc) Fixed: list(a ~., A ~., lrc ~., lrc ~.) Random: list(a ~., A ~., lrc ~., lrc ~.) Cluster: ~ Subject Data: Indometh Estimation Method: ML Convergence at iteration: Approximate Loglikelihood:.3 AIC: -9.7 BIC: Variance/Covariance Components Estimate(s): Structure: matrixlog Standard Deviation(s) of Random Effect(s) A A lrc lrc Correlation of Random Effects A A lrc A.993 lrc.9.33 lrc Cluster Residual Variance:.9 Fixed Effects Estimate(s): Value Approx. Std.Error z ratio(c) A A lrc lrc

11 Conditional Correlation(s) of Fixed Effects Estimates A A lrc A.39 lrc lrc Random Effects (Conditional Modes): A A lrc lrc Standardized Population-Average Residuals: Min Q Med Q3 Max Number of Observations: Number of Clusters: - > predict(indometh.nlme,data=indometh,cluster= ~ Subject) cluster fit.cluster fit.population Fitted Values of Serum Concentrations of Theophylline Following Oral Administration Fitted Values of Orange Tree Circumference Growth Concentration (mg/l) Time since drug administration (Hours) Trunk circumference (mm) Time since December 3, 9 (days) -3 - Trellis Plot of the Results of an Experiment to Compare Two Genotypes of Soybeans Summary Repeated measure data occur in many different areas of application from biological to engineering F 99 P For many purposes, it is better to have a mechanistic (e.g., one derived from kinetics) but empirical models can also be useful Leaf weight/plant (g) 99 F 99 P 3 Modeling the unit-to-unit variability in model parameters is useful for answering many important questions about variability in the population or process being studied. 3 9 F 9 P Modern software (e.g., the Splus function nlme) make it easy to fit nonlinear regression models with random parameters to reflect unit-to-unit variability. Days after planting - It is also possible to fit models with autocorrelated residual terms. -