SIMULTANEOUS CONFIDENCE BANDS FOR THE PTH PERCENTILE AND THE MEAN LIFETIME IN EXPONENTIAL AND WEIBULL REGRESSION MODELS. Ping Sa and S.J.

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1 SIMULTANEOUS CONFIDENCE BANDS FOR THE PTH PERCENTILE AND THE MEAN LIFETIME IN EXPONENTIAL AND WEIBULL REGRESSION MODELS " # Ping Sa and S.J. Lee " Dept. of Mathematics and Statistics, U. of North Florida, Jacksonville, FL.34 # Dept. of Mathematics and Statistics, U. of West Florida, Pensacola, FL.3514 ABSTRACT Simultaneous confidence bands are developed for the : th percentile and the expected lifetime of the Weibull regression model, using the confidence region about the parameters and Lagrange multiplier procedure. An example of real data is analyzed to illustrate the procedure. Key Words and Phrases: simultaneous confidence band; confidence region; maximum likelihood estimator; Lagrange multiplier procedure. 1. INTRODUCTION The most commonly used Weibull regression model has the survival function for lifetime X, given the vector of regressor variables B, of the form / WÐ>ß BÑ œ /B: Š Š >Î) B ß >! (1) here / ž! is a shape parameter hich does not depend upon B, and ) is a function of B that involves unknon parameters ". The most frequently used ) B is ) B œ /B:Ð " Ñ, here B is the transpose of the vector B. Note that the exponential regression model is a special case of (1) as / œ". In reliability practice, one is often interested in obtaining simultaneous confidence intervals for mean or median response under various values of regressor variables hen the model is Weibull regression ith unknon parameters. In this paper, e construct confidence bands for the : th percentile and the mean life of the Weibull regression ith 5 regressor variables involved. That is, ) B œ /B:( B "), here B œ Ð"ß B ß ÞÞÞß B Ñ, B Vß 3 œ 5 œ " ß " ß ÞÞÞß " B " 5 3 1,,..., and " Š. Scheffe' (1959, Appendix III) originated the idea! " 5 B 1

2 to project the ellipsoidal confidence region of " onto a linear combinations of " s in linear regression. Similar to Scheffe's idea, the construction of the confidence bands in this paper is based on an existing confidence region about the parameters along ith Lagrange multiplier procedure hich give a concise derivation and more general results than those using maximization and minimization procedure by Brand, Pinnock and Jackson (1973) and Khorasani and Milliken (198). Khorasani and Milliken (198) used the maximization and minimization approach to construct simultaneous confidence bands for the mean of the Michaelis-Menten Kinetic model ith i.i.d. normal error terms and for the Logistic response curve. Brand, Pinnock and Jackson (1973) also constructed large sample confidence bands for the Logistic response curve and its inverse by the same technique. Both papers considered only one regressor variable in the models. The folloing section provides detailed derivation of the confidence bands using Lagrange multiplier procedure. A numerical example ith three regressor variables is presented in Section 3 to illustrate the method. The conclusion of this procedure is given in Section 4.. THE PROCEDURE Consider model (1). Let shon to be UÐ:ß BÑ be the : th percentile of the model and can be UÐ:ß B Ñ œ )BŠ 6og Ð" :Ñ ß! : " / "Î/ () here 6og / is the natural logarithm. Note that UÐ:ß BÑ is the median lifetime of the model hen : œ "Î#. Let.ÐBÑbe the mean lifetime of the model and can be derived to have the form, ÐBÑ œ B > Ð" "Î/ Ñ. ) (3) here > (.) is the Gamma function. To find the simultaneous confidence bands of () or (3) for all B3 V, 3 œ 1,,..., 5, one needs to find confidence band of )B œ /B:( B " ), for all B3 V, 3 œ 1,,..., 5. This implies that e need to find confidence band for B ", for all B V, 3 œ 1,,..., 5, then apply the monotonicity property of the exponential function to it. Note that model (1) ith )B œ /B:ÐB " Ñ can be ritten as 3

3 (4) ]œ B " 5^ here ]œ691x /, 5 œ"î/ and ^ has the standard Gumbel distribution ith pdf /B:ÐD /B:ÐDÑÑ, _ D _. That is, the p.d.f. of ] œ 691/ X, given B, is C B " C B " 0ÐClB Ñ œ /B: /B: Š Î5, _ C _. 5 5 The estimates of parameters can be obtained in different ays. Equation (4) is essentially a linear regression for hich exact methods are available (see Laless, 198) but are not usually feasible because of the intensive computations. The least squares method can be used, though it can be inefficient. For more details about the efficiency of least squares estimators, see Prentice and Shillington (1975) and Williams (1978). The more efficient methods are the likelihood ratio methods and the normality approximation of the maximum likelihood estimators in large sample. We can obtain the maximum likelihood estimates (m.l.e.) of " and 5 as follos. Suppose a random sample of n observations is available, ( C", B" ), ( C#, B# ),..., ( C8, B8), here C4 can be either a log lifetime or a log censoring time and B 4 œ ÐB!4 ß B"4ß ÞÞÞß B54 Ñ, B!4 œ 1, for 4 œ 1,..., n. The likelihood function is then 1 L( ", 5) œ # C 4 B " C 4 B " C 4 B " /B: /B: Š #/B: /B: Š, E here E and F denote the sets for hich C 4 is a log lifetime and a log censoring time, respectively. The maximum likelihood equations for " and 5 are, respectively, 4 F œ! BÎ! D4 B /Î œ0, for 6œ0, 1,,..., 5, `" ` 691P 4 E 8 4œ" ` 691P and œ <Î! D Î! D / Î, `5 5 5 D E 4œ" C 4 B" 4 here D 4 œ, for 4œ 1,..., n, and < is the number of lifetimes in set E. The m.l.e. 5 of " and 5 can be obtained by solving the above equations using the Neton-Raphson method or some other iteration procedures (see Laless, 198). To find the explicit forms of maximum and minimum, hich denoted by - Y ÐBÑ and - P ÐBÑ, respectively, of B ", e assume that sample size n is large and use the asymptotic 8 3

4 distribution of the maximum likelihood estimators or other asymptotically equivalent estimators of " œ Š " ß " ß ÞÞÞß ". For a sufficiently large n, the distribution of! " 5 Ð s Ñ " Ð s Ñ " " D " " is approximately a chi-square distribution ith Ð5 1 Ñ degrees of freedom, here s œ Ð s ß s ß ÞÞÞß s Ñ " " " ", is ( 5 " ) by ( 5 1) matrix and is obtained from the observed " D! " 5 I! information matrix belo: D " deleted, here and œ Î Ð Ï `" 6 `" = `" 6 `5 ` 5 `" 6 `" = `" 6 `5 `5 `" = `5 8! Ñ Ó Ò Ð" s ßsÑ 5 D œ B B / Î 4 5, 6ß = œ 0, 1,..., 5; 4œ" 64 =4 8 8, ith the last ro and last column œ!! D4! D , 0, 1,..., B Î B / Î B D / Î 6 œ E 4œ" 4œ" 8 8 5! 5! D4 5! D E 4= " 4œ"! " œ <Î D Î D / Î D / Î. The approximate 100 Ð" Ñ% confidence region for then satisfies, " " D " " Ð s Ñ " Ð s # Ñ Ÿ Ð5 "Ñ, here ; # Ð5 "Ñ is the upper! percentage point of a chi-square distribution ith Ð5 "Ñ! degrees of freedom. No, in order to construct confidence bands for B ", e need to determine the extreme values of over the boundary of the confidence region B " ;! " " D " " Ð s Ñ " Ð s # Ñ œ Ð5 "ÑÞ ;! (5) " " That is, optimize a linear function 0Ð Ñ œ B subject to the constraint (5). Let 1Ð ÑœÐs Ñ " Ðs Ñand applying Lagrange multiplier technique, the solution is " " " D " " found by solving That is, hich gives f" 0 œ -f" 1. B œ -D " Ð " s " Ñ, " œ " s " D #- BÞ 4

5 Substituting into the constraint equation Substituting into the expression for (5) gives "Î- œ # È ; # Ð5 "ÑÎB D B "! above, e obtain the solutions " œ " Ð5 "ÑÎ D D s È; Š # B B B!. Þ So, the extreme values of 0 are È ;! Ð5 "Ñ " D B s # B B. (6) It can be verified that the maximization and minimization procedure suggested in Brand, Pinnock and Jackson (1973) and Khorasani and Milliken (198) produces the exact same extreme values as those of (6) after simplification hen 5œ", but it involves more steps and is almost untractable to find extreme points hen there are more than one regressor variable in the model. Let -PÐBÑ œ B s" È ;! # Ð5 "Ñ B DB È ;! Ð5 "Ñ " D and -Y ÐBÑ œ B s # B B. Applying the monotonicity of the exponential function, the probability statement becomes, T Ð/B:Ð-PÐBÑÑ /B:ÐB" Ñ /B:Ð- YÐBÑÑß a B3 Vß 3 œ 1,,..., 5Ñ "! Þ Therefore, the large sample simultaneous confidence bands for B and B are U Ñ Ð. ß. Ñ Y and P Y hich satisfy, respectively, 1,..., TÐUP UÐ:ßBÑ UYß ab3 Vß 3 œ ß 5Ñ "! ß here U œ /B:Ð Ð ÑÑ 691 Ð" :Ñ, and P P / "Î s - B Š / "Î s UY œ /B:Ð- Y ÐBÑÑŠ 691/ Ð" :Ñ / s/ / 1,,..., ith the m.l.e. of ; TÐ.. ÐBÑ. ßaB3 Vß 3œ 5Ñ "! ß P Y here. œ /B:Ð- ÐBÑÑ> Ð" "Îs/ Ñ and. œ /B:Ð- ÐBÑÑ> Ð" "Îs/ Ñ. P P Y Y UÐ:ß Ñ. Ð Ñ ÐUPß 3. NUMERICAL EXAMPLE 5

6 The numerical illustration of the confidence band computations is the lung cancer data set studied by Prentice (1973). It is of interest that the confidence bands of mean survival time and median survival time (ie. : œ 1Î in UÐ:ß BÑ) of patients are constructed under various values of some regressors studied in this subject. Weibull regression model ith shape parameter / and ) " " " ", is considered for the data set, here B " B œ /B:Ð! " B " B 3B3Ñ is the performance status (a general medical status measure on a scale of 0 to 100: 10, 0, and 30 mean that the patient is completely hospitalized; 40, 50 and 60, partially confined to hospital; and 70, 80 and 90 able to care for himself), B is patient's age; and B 3 denotes months from diagnosis to entry into the study. Data set produced the maximum likelihood estimates of " s and / as " s œ Î Ð Ï Ñ Ó Ò and /s œ , respectively. The estimated variance-covariance matrix of D œ 10 3 x Î Ð Ï " s is: Ñ Ó Ò. The above information along ith the formulation established in Section give the large sample simultaneous confidence bands for mean and : th percentile survival times. Since three regressors are involved in this example, the confidence bands of mean and median survival times are tabulated for selected values of regressors belo to avoid complexity: Table 1. 95% Confidence Bands for Mean and Median Survival Times Regressor values 95% confidence band for.( B) 95% confidence band for U(1/, B) B" B B3 loer band upper band loer band upper band Note that Table 1 gives relatively loer bands for median lifetime than those for mean lifetime. This reflects the right skeness of a Weibull distribution (the median is smaller than the mean under a right skeed distribution). For different age groups, B œ 0 6

7 and 50, the bands of the mean and median survival times under the same conditions of performance status ( B" œ 30) and months from diagnosis to entry into the study ( B3 œ 5, 10, 15, and 0) are significantly different. Note that the idths of the bands for age group 0 are ider than those of age group 50. This is due to that more uncertainty (or variations) is involved in estimating percentile and mean lifetimes corresponding to age levels outside the range of the data set ( B, patient's age, is beteen 36 and 70 in the data set studied). 4. CONCLUSION In this article, e have presented a procedure to construct simultaneous confidence bands for either the : th percentile or the mean lifetime hen the regressor variables are unconstrained in Weibull regression. This confidence band depends strongly upon the confidence region of the parameters. Without an appropriate region, the confidence band ould be meaningless. It is noted that the result applies to more than one regressor variable in the model. In some applications, the user may be interested in UÐ:ß BÑ or. ÐBÑ hen realistic constraint information of B is incorporated in the analysis. For example, the patients aged from 40 to 50 may be of interest only in lung cancer study. Tighter confidence bands are expected. The amount of improvement of confidence bands under constrained information of regressors is of interest and the authors are currently pursuing these avenues. REFERENCES Brand, R.J., Pinnock, D.E. and Jackson, K.L. (1973), Large Sample Confidence Bands for the Logistic Response Curve and its Inverse, American Statistician, 7, Khorasani, F. and Milliken, G.A. (198), Simultaneous Confidence Bands for Nonlinear Regression Models, Comm. in Stat. - Theory and Method, 11(11), Laless, J.F. (198), Statistical Models and Methods for Lifetime Data Sons, Inc., Ne York). (John Wiley & Prentice, R.L. (1973), Exponential Survival ith Censoring and Explanatory Variables Biometrika, 60, pp Prentice, R.L. and Shillington, R. (1975), Regression Analysis of Weibull Data and the Analysis of Clinical Trials, Utilitas Math. 8, Scheffe', H. (1959), The Analysis of Variance, (John Wiley & Sons, Inc., Ne York). 7

8 Williams, J.S. (1978), Efficient Analysis of Weibull Survival Data from Experiments on Heterogeneous Patient Populations, Biometrics, 34,

PART I. Multiple choice. 1. Find the slope of the line shown here. 2. Find the slope of the line with equation $ÐB CÑœ(B &.

PART I. Multiple choice. 1. Find the slope of the line shown here. 2. Find the slope of the line with equation $ÐB CÑœ(B &. Math 1301 - College Algebra Final Exam Review Sheet Version X This review, while fairly comprehensive, should not be the only material used to study for the final exam. It should not be considered a preview