On testing tumor onset times. (English) Zbl 0930.62102
Summary: In a typical assay toxicologists study a substance with respect to its cancerogenicity by comparing a control group of rats with a treatment group. One is interested in the distribution of the unobservable onset time \(T\) of a certain type of tumor. When a rat dies at a time point \(C\), say, one knows apart from \(C\) only whether \(T\leq C\) or \(T>C\) holds true. Data of this kind are called interval censored.
In the present paper by means of local asymptotic normality (LAN) combined with counting process theory asymptotically optimal nonparametric tests (in a specific sense) are derived for comparing the distributions of the tumor onset time of the control and of the treatment group. The resulting tests are under the null hypothesis of equal distributions of the onset times in both samples asymptotically distribution free even if the distributions of the times to death \(C\) are different in both samples.
In the present paper by means of local asymptotic normality (LAN) combined with counting process theory asymptotically optimal nonparametric tests (in a specific sense) are derived for comparing the distributions of the tumor onset time of the control and of the treatment group. The resulting tests are under the null hypothesis of equal distributions of the onset times in both samples asymptotically distribution free even if the distributions of the times to death \(C\) are different in both samples.
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G10 | Nonparametric hypothesis testing |
| 62M07 | Non-Markovian processes: hypothesis testing |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
interval censoring; two-sample tests; rank tests; optimality; LAN property; nonpredictable integrands; counting process; tumor onset time; distribution freeReferences:
| [1] | Andersen, P.K., 1993. A nonparametric test for comparing two samples where all observations are either left- or right-censored. Research Report 93/3, Statistical Research Unit, Univ. of Copenhagen.; Andersen, P.K., 1993. A nonparametric test for comparing two samples where all observations are either left- or right-censored. Research Report 93/3, Statistical Research Unit, Univ. of Copenhagen. · Zbl 0825.62458 |
| [2] | Andersen, P.K., Borgan, O., Gill, R.D., Keiding, N., 1993. Statistical Models Based on Counting Processes. Springer, Berlin.; Andersen, P.K., Borgan, O., Gill, R.D., Keiding, N., 1993. Statistical Models Based on Counting Processes. Springer, Berlin. · Zbl 0769.62061 |
| [3] | Behnen, K., Neuhaus, G., 1989. Rank Tests with Estimated Scores and their Application. Teubner, Stuttgart.; Behnen, K., Neuhaus, G., 1989. Rank Tests with Estimated Scores and their Application. Teubner, Stuttgart. · Zbl 0692.62041 |
| [4] | Fleming, Th.R., Harrington, D.P., 1991. Counting Processes and Survival Analysis. Wiley, New York.; Fleming, Th.R., Harrington, D.P., 1991. Counting Processes and Survival Analysis. Wiley, New York. · Zbl 0727.62096 |
| [5] | Götz, D., 1995. Asymptotische Testtheorie für das Zweistichprobenproblem mit Intervall-Zensierung. Diploma Thesis, Univ. Hamburg.; Götz, D., 1995. Asymptotische Testtheorie für das Zweistichprobenproblem mit Intervall-Zensierung. Diploma Thesis, Univ. Hamburg. |
| [6] | Groeneboom, P., Wellner, J.A., 1992. Information Bounds and Nonparametric Maximum Likelihood Estimation, DMV Seminar, vol.19. Birkhäuser, Basel.; Groeneboom, P., Wellner, J.A., 1992. Information Bounds and Nonparametric Maximum Likelihood Estimation, DMV Seminar, vol.19. Birkhäuser, Basel. · Zbl 0757.62017 |
| [7] | Heck-Boldebuck, D., Heimann, G., Neuhaus, G., 1997. Analyzing carcinogenicity assays without cause of death information. Drug Inform. J. 31 (2).; Heck-Boldebuck, D., Heimann, G., Neuhaus, G., 1997. Analyzing carcinogenicity assays without cause of death information. Drug Inform. J. 31 (2). |
| [8] | Heinemann, Th., 1995. Asymptotische Verteilungstheorie von stochastischen Integralen im Zweistichproben-problem mit Intervall-Zensierung. Diploma Thesis, Univ. Hamburg.; Heinemann, Th., 1995. Asymptotische Verteilungstheorie von stochastischen Integralen im Zweistichproben-problem mit Intervall-Zensierung. Diploma Thesis, Univ. Hamburg. |
| [9] | Huang, J., Efficient estimation for the proportional hazard model with interval censoring, Ann. Statist., 24, 540-568 (1996) · Zbl 0859.62032 |
| [10] | Klein, R. W., Specification tests for binary choice models based on index quantiles, J. Econometrics, 59, 343-375 (1993) · Zbl 0806.62098 |
| [11] | Klein, R. W.; Spady, R. H., An efficient semiparametric estimator for binary response models, Econometrica, 61, 387-421 (1993) · Zbl 0783.62100 |
| [12] | Kronmal, R.; Tarter, M., The estimation of probability densities and cumulatives by Fourier series methods, J. Am. Statist. Assoc., 63, 925-952 (1968) · Zbl 0169.21403 |
| [13] | Neuhaus, G., Zur Verteilungskonvergenz einiger Varianten der Cramér-von Mises-Statistik, Math. Operationsforsch. Statist., 4, 473-484 (1973) · Zbl 0274.62017 |
| [14] | Rosenblatt, M., Curve estimates, Ann. Math. Statist., 42, 1815-1842 (1971) · Zbl 0231.62100 |
| [15] | Schäbe, H., Estimating the tumor onset distribution using kernel estimators, Biomet. J., 33, 191-206 (1991) · Zbl 0738.62101 |
| [16] | Silverman, B. W., Weak and strong uniform consistency of the kernel estimator of a density and its derivatives, Ann. Statist., 6, 177-184 (1978) · Zbl 0376.62024 |
| [17] | Tang, M.; Tsai, W.; Marder, K.; Mayeux, R., Linear rank tests for doubly censored data, Statist. Med., 14, 2555-2563 (1995) |
| [18] | Witting, H., 1985. Mathematische Statistik I. Teubner, Stuttgart.; Witting, H., 1985. Mathematische Statistik I. Teubner, Stuttgart. · Zbl 0581.62001 |
| [19] | Witting, H., Müller-Funk, U., 1995. Mathematische Statistik II. Teubner, Stuttgart.; Witting, H., Müller-Funk, U., 1995. Mathematische Statistik II. Teubner, Stuttgart. · Zbl 0838.62001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
