- Email: [email protected]
Comparison of antitumor activities in tumor xenograft treatment
Contemporary Clinical Trials 28 (2007) 115 – 119 www.elsevier.com/locate/conclintrial
Comparison of antitumor activities in tumor xenograft treatment Hua Liang Department of Biostatistics and Computational Biology, University of Rochester Medical Center, 601 Elmwood Avenue, Box 630, Rochester, NY 14642, United States Received 6 December 2005; accepted 11 May 2006
Abstract To compare treatment effects with antitumor therapies, we proposed an intuitive approach to compare the antitumor effects of two different antitumor treatments by investigating tumor volumes which were measured in a given period of time. The approach is, in essence, a comparison of two unknown functions. The implementation of the approach is simple and straightforward. The approach is applied to analyze a real xenograft study of a new antitumor agent, irofulven, combined with irinotecan. © 2006 Elsevier Inc. All rights reserved. Keywords: Bootstrap; Comparison curve; Synergy; Tumor xenograft models
1. Introduction In the evaluation of antitumor therapies in cancer clinical research, tumor volume to treatment currently play an important role in antitumor therapies. For example, if tumor volume reduces, the response of tumors to the treatment is complete or partial response, and stable or progressive diseases otherwise. The difference of tumor volumes based on different antitumor treatments may be used to compare the antitumor activity of the treatments. Appropriate analysis of tumor activities and of comparison of treatment effects are, therefore, very important in cancer drug development. The combination of irofulven with agents that target DNA topoisomerase I has demonstrated in vitro and in vivo synergy [1]. Irofulven, in combination with irinotecan, activity was enhanced against colon carcinoma xenografts [2]. Irofulven and topotecan have demonstrated activity against xenografts derived from pediatric tumors, including brain tumors, and leukemia [3]. One of the aims of this study includes comparing the antitumor effects between two (or more) different treatments that may prove one treatment to be applicable to later large-scale studies. In an antitumor study conducted at St. Jude Children's Research Hospital [4], mice were grafted with human cancer cells to produce a xenograft model that was used to evaluate the efficacy of anticancer agents. In this study, the antitumor activity of irofulven and irinotecan were examined individually and in the brain tumor. The experiments were conducted in tumors within 30 serial passages of engraftment. Transplantations were from mouse to mouse. Each mouse bearing bilateral subcutaneous tumors received the chemotherapeutic agent(s) when tumors were ∼0.20–1 cm in diameter, as reported previously [5]. Briefly, two perpendicular tumor diameters were measured at seven day
E-mail address: [email protected]. 1551-7144/$ - see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.cct.2006.05.001
116
H. Liang / Contemporary Clinical Trials 28 (2007) 115–119
intervals. Assuming tumors to be spherical, volumes were calculated with the formula (π/6) × d3, where d represents the mean diameter. Tumor volumes were determined up to 12 weeks after starting treatment. The tumor volumes measured over 12 weeks of treatment with different agents are shown in Fig. 1, where some mice do not have observation over 12 weeks because they died prior to week 12. Six treatment agents are: A–Control; B–Irofulven 3.0 mg/kg; C–Irinotecan 1.25 mg/kg; D–Irofulven 3.0 mg/kg + Irinotecan 1.25 mg/kg; E–Irinotecan 0.61 mg/kg; and F–Irofulven 3.0 mg/kg + Irinotecan 0.61 mg/kg. Irofulven was administered daily for 5 days repeated every 21 days for 3 cycles. Irinotecan was given daily for 5 days on two consecutive weeks with cycles repeated every 21 days for 3 cycles. In the study of antitumor activity, a commonly used measurement is tumor failure time, which was defined as the time (in weeks) required by individual tumor to quadruple their volume from the initiation of therapy. Tumor failure times were termed as censored if a mouse died prior to week 12 and before a tumor grew to four times its initial volume. In xenograft studies where tumors were implanted in both lateral flanks, the tumor failure times from each mouse are clustered observations. Evidence of high correlation between failure times has been reported previously [5]. Because the individual mouse is the unit of the experiment, the time to failure was defined as the minimum of the failure times of the bilateral tumors. This approach accounts for the clustering effect attributable to the mouse, without explicitly specifying the correlation structure. For studies in which only one tumor was implanted in each mouse, tumor failure times were defined as above, but no manipulation of the data was needed to find the minimum times; individual data were used for those studies. For comparisons of time to tumor failure for different treatment regimens, survival distributions of each treatment group were compared to the survival distribution of the control group using the exact log rank test. The similar comparisons are conducted for treatment pairs B and D, C and D, and E and F. The comparison results are summarized
Fig. 1. Plots of tumor volume versus duration of treatment with different agents. A–Control; B–Irofulven 3.0 mg/kg; C–Irinotecan 1.25 mg/kg; D– Irofulven 3.0 mg/kg + Irinotecan 1.25 mg/kg; E–Irinotecan 0.61 mg/kg; and F–Irofulven 3.0 mg/kg + Irinotecan 0.61 mg/kg.
H. Liang / Contemporary Clinical Trials 28 (2007) 115–119
117
Table 1 Comparison of the effect of individual and combination treatment by time to tumor failure Group
Time ± SD
P-value
A B C D E F D vs B F vs B D vs C F vs E
2.7 ± 0.2 9.5 ± 12 9.0 ± 3.0 >12 9.6 ± 0.8 >12
– 0.016 (vs A) 0.016 (vs A) 0.005 (vs A) 0.016 (vs A) 0.005 (vs A) 0.629 0.629 1.000 0.042
P-values obtained from asymptotic results.
in Table 1, and indicate that the treatment groups are consistently superior to the control group, and treatment F is more successful than treatment E. However, there was little difference in tumor failure times between combination and single agent treatments (P = 0.629 against irofulven alone and P = 1 against irinotecan 1.25 mg/kg alone). Observing the tumor volumes in treatments B, D, and F, one may see that the tumor volumes in groups D and F decline along the treatment times, whereas the tumor volumes in group B increase or keep flat status. The following scientific questions come to mind: (a) can we effectively show that two treatments are significantly different? and (b) can we use the results in clinical practice? These topics form the core of this note. In the next section we briefly introduce a test which was proposed to compare two functions. We will model tumor volume as a function of treatment time. 2. The test Assume that Y and Z are the population tumor volumes of groups 1 and 2 at treatment time X. We also assume that the tumor volumes Y and Z are the functions of X, such as f(X ) and g(X ). Comparison of the two treatment effects is equivalent to check if g(x) = f(x). Suppose we have observations (Xi, Zi, Yi) for i = 1, …, n derived from the model Yi ¼ f ðXi Þ þ ei ;
Zi ¼ gðXi Þ þ gi
where ε1, …, εn and η1, …, ηn are independent random errors, and E(ε12) = σ2 and E(η12) = τ2. The distributions of εi's and ηi's may be distinguished. How to handle this comparison has been studied over the past decade. Hall and Hart [6] used a bootstrap method to test the equality of two smooth curves. Young and Bowman [7] proposed an ANOVA-based non-parametric test to check the equality of two curves. The similar situations have been discussed in Refs. [8–11]. Liang [12] proposed a test to investigate f(x) = g(x). The test was defined as Z ⊤n ¼ Cn−2 Sn2 ðxÞdFn ; where Cn2 ¼
n−1 n 1X fYiþ1 −Ziþ1 −ðYi −Zi Þg2 1 X ; Sn ðxÞ ¼ pffiffiffi fðYi −Zi ÞIðXi VxÞg; n i¼1 2 n i¼1
and Fn is the empirical distribution based on (X1, …, Xn). One should reject the null hypothesis for a large value of ⊺n. The author also proposed a bootstrap [13] version of ⊺n for conveniently studying the level and power of ⊺n because the test statistic ⊺n may not be distribution-free. The bootstrap algorithms work as follows. Let {(Xi, Yic, Zic)} be the centered data, where Yic = Yi − ∑Yj / n and Zic = Zi − ∑Zj / n. Let {(X1⁎, Y1⁎, Z1⁎), …, (Xn⁎, Yn⁎, Zn⁎)} be the bootstrap sample from {(X1, Y1c, Z1c), …, (Xn, Ync, Znc)} directly where n X ⁎ ðxÞ ¼ p1ffiffiffi Sn;B fðYi⁎ −Zi⁎ ÞIðXi V xÞg: n i¼1
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H. Liang / Contemporary Clinical Trials 28 (2007) 115–119
The null hypothesis is rejected if the proportion of bootstrap statistics exceeding ⊺n is less than or equal to the appropriate nominal level. 3. Analysis of xenograft data In this section, we apply the test to the xenograft data introduced in Section 2. Raw data plots indicated that treatments D and F seem to be more successful than treatment B. We now statistically justify that the differences are significant using the test. First we calculate the averages of tumor volumes at each measured time for each group. These quantities are taken as Y or Z, and the time regarded as X. For the bootstrap procedure we generate 5000 bootstrapping samples. The proportions of bootstrap statistics exceeding ⊺n are both less than 0.001. The results indicate that treatments D and F are more effective than treatment B. So, biologically, treatments D and F should be pursued further in contrast to treatment B. A referee has asked a comparison of treatments D and F with treatment C. In the same way as above we had that the proportions of bootstrap statistics exceeding ⊺n are both less than 0.05. This indicated that the irofulven/irinotecan combination (treatment D and F) is more effective than irinotecan alone (treatment C). 4. Discussion To effectively compare two treatment effects of antitumor therapies we applied a new test to compare tumor volumes, which are modeled non-parametrically. The approach avoids non-parametric smoothing, which are required in most existing work. The test is easily implemented. A computer code conducted in R environment is available to conduct the implementation. The user only needs to specify a data file that provides group index and tumor volumes. This code, together with documentation and a sample dataset, is available from the author upon request. Although survival analysis is commonly used in study of antitumor activity, the analysis may not use fully information from data, and the associated conclusions may be rough. For example, (i) when two treatments fail at the same time, or (ii) two treatments are both successful based on the survival measurement, comparison of survival time can, by no means, give information. Our findings from the real data analysis indicate that the treatment with irofulven (3 mg/kg) and irinotecan (0.61 or 1.25 mg/kg) is more effective than the treatment with irinotecan (1.25 mg/kg) alone. This occurrence was not detected by comparing the survival time. For this typical study, one may use other modeling strategies to fit tumor volume [14], and comparison of antitumor effects is transferred to comparison of some parameters, which dominate the curve of tumor volume against time. However, the approach proposed in this paper is more flexible because we did not make any assumption on the model, and is more powerful than other approaches based on non-parametric smoothing techniques because the test can detect the root-n alternative hypothesis [12]. The approach is anticipated to be valuable in practice. One requirement of the test is that the design points of two groups are identical. However, our procedure can be generalized to models without common designs if the two designs are close. We omit the details but refer to Hall and Hart [6]. Acknowledgment This research was partially supported by the NIH/NAIAD grants AI62247 and AI059773. The author thanks two referees for their helpful comments. References [1] Britten CD, Hilsenbeck SG, Eckhardt SG, et al. Enhanced Antitumor Activity of 6-Hydroxymethylacylfulvene in Combination with Irinotecan and 5-Fluorouracil in the HT29 Human Colon Tumor Xenograft Model. Cancer Res 1999;59:1049–53. [2] McCreery H., Rowinsky E.K., Tolcher A., et al. Phase I trial of irofulven and CPT-11 in patients with advanced cancers. 2002 ASCO Annual Meeting. [3] Kelner MJ, McMorris TC, Estes L, et al. Anti-leukemic action of the novel agent MGI 114 (HMAF) and synergistic action with topotecan. Leukemia 2000;14:136–41. [4] Woo MH, Peterson JK, Billups C, et al. Enhanced antitumor activity of irofulven in combination with irinotecan in pediatric solid tumor xenograft models. Cancer Chemother Pharmacol 2005;55:411–9. [5] Houghton PJ, Cheshire PJ, Hallman JD, et al. Efficacy of topoisomerase I inhibitors, topotecan and irinotecan, administered at low dose levels in protracted schedules to mice bearing xenografts of human tumors. Cancer Chemother Pharmacol 1995;36:393–403.
H. Liang / Contemporary Clinical Trials 28 (2007) 115–119 [6] [7] [8] [9] [10] [11] [12] [13] [14]
119
Hall P, Hart JD. Bootstrap test for difference between means in nonparametric regression. J Am Stat Assoc 1990;85:1039–49. Young SG, Bowman AW. Non-parametric analysis of covariance. Biometrics 1995;51:920–31. Härdle W, Mammen E. Comparing non-parametric versus parametric regression fits. Ann Stat 1993;21:1926–47. Härdle W, Marron S. Semiparametric comparison of regression curves. Ann Stat 1990;18:63–89. Cox DR, Koh E, Wahba G, Yandell BS. Testing the (parametric) null model hypothesis in (semi-parametric) partial and generalized spline model. Ann Stat 1988;16:113–9. Eubank RL, Hart JD. Commonality of Cusum, von Neumann and smoothing-based goodness-of-fit tests. Biometrika 1993;80:89–98. Liang H. Comparison of curves based on Cramér-von Mises statistic. Comput Stat Data Anal 2004;45:805–12. Efron B, Gong G. A leisurely look at the bootstrap, the jackknife and cross-validation. Am Stat 1983;37:36–48. Liang H, Sha NJ. Modeling antitumor activity by using a nonlinear mixed-effects models. Math Biosci 2004;189:61–73.
Comparison of antitumor activities in tumor xenograft treatment Hua Liang Department of Biostatistics and Computational Biology, University of Rochester Medical Center, 601 Elmwood Avenue, Box 630, Rochester, NY 14642, United States Received 6 December 2005; accepted 11 May 2006
Abstract To compare treatment effects with antitumor therapies, we proposed an intuitive approach to compare the antitumor effects of two different antitumor treatments by investigating tumor volumes which were measured in a given period of time. The approach is, in essence, a comparison of two unknown functions. The implementation of the approach is simple and straightforward. The approach is applied to analyze a real xenograft study of a new antitumor agent, irofulven, combined with irinotecan. © 2006 Elsevier Inc. All rights reserved. Keywords: Bootstrap; Comparison curve; Synergy; Tumor xenograft models
1. Introduction In the evaluation of antitumor therapies in cancer clinical research, tumor volume to treatment currently play an important role in antitumor therapies. For example, if tumor volume reduces, the response of tumors to the treatment is complete or partial response, and stable or progressive diseases otherwise. The difference of tumor volumes based on different antitumor treatments may be used to compare the antitumor activity of the treatments. Appropriate analysis of tumor activities and of comparison of treatment effects are, therefore, very important in cancer drug development. The combination of irofulven with agents that target DNA topoisomerase I has demonstrated in vitro and in vivo synergy [1]. Irofulven, in combination with irinotecan, activity was enhanced against colon carcinoma xenografts [2]. Irofulven and topotecan have demonstrated activity against xenografts derived from pediatric tumors, including brain tumors, and leukemia [3]. One of the aims of this study includes comparing the antitumor effects between two (or more) different treatments that may prove one treatment to be applicable to later large-scale studies. In an antitumor study conducted at St. Jude Children's Research Hospital [4], mice were grafted with human cancer cells to produce a xenograft model that was used to evaluate the efficacy of anticancer agents. In this study, the antitumor activity of irofulven and irinotecan were examined individually and in the brain tumor. The experiments were conducted in tumors within 30 serial passages of engraftment. Transplantations were from mouse to mouse. Each mouse bearing bilateral subcutaneous tumors received the chemotherapeutic agent(s) when tumors were ∼0.20–1 cm in diameter, as reported previously [5]. Briefly, two perpendicular tumor diameters were measured at seven day
E-mail address: [email protected]. 1551-7144/$ - see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.cct.2006.05.001
116
H. Liang / Contemporary Clinical Trials 28 (2007) 115–119
intervals. Assuming tumors to be spherical, volumes were calculated with the formula (π/6) × d3, where d represents the mean diameter. Tumor volumes were determined up to 12 weeks after starting treatment. The tumor volumes measured over 12 weeks of treatment with different agents are shown in Fig. 1, where some mice do not have observation over 12 weeks because they died prior to week 12. Six treatment agents are: A–Control; B–Irofulven 3.0 mg/kg; C–Irinotecan 1.25 mg/kg; D–Irofulven 3.0 mg/kg + Irinotecan 1.25 mg/kg; E–Irinotecan 0.61 mg/kg; and F–Irofulven 3.0 mg/kg + Irinotecan 0.61 mg/kg. Irofulven was administered daily for 5 days repeated every 21 days for 3 cycles. Irinotecan was given daily for 5 days on two consecutive weeks with cycles repeated every 21 days for 3 cycles. In the study of antitumor activity, a commonly used measurement is tumor failure time, which was defined as the time (in weeks) required by individual tumor to quadruple their volume from the initiation of therapy. Tumor failure times were termed as censored if a mouse died prior to week 12 and before a tumor grew to four times its initial volume. In xenograft studies where tumors were implanted in both lateral flanks, the tumor failure times from each mouse are clustered observations. Evidence of high correlation between failure times has been reported previously [5]. Because the individual mouse is the unit of the experiment, the time to failure was defined as the minimum of the failure times of the bilateral tumors. This approach accounts for the clustering effect attributable to the mouse, without explicitly specifying the correlation structure. For studies in which only one tumor was implanted in each mouse, tumor failure times were defined as above, but no manipulation of the data was needed to find the minimum times; individual data were used for those studies. For comparisons of time to tumor failure for different treatment regimens, survival distributions of each treatment group were compared to the survival distribution of the control group using the exact log rank test. The similar comparisons are conducted for treatment pairs B and D, C and D, and E and F. The comparison results are summarized
Fig. 1. Plots of tumor volume versus duration of treatment with different agents. A–Control; B–Irofulven 3.0 mg/kg; C–Irinotecan 1.25 mg/kg; D– Irofulven 3.0 mg/kg + Irinotecan 1.25 mg/kg; E–Irinotecan 0.61 mg/kg; and F–Irofulven 3.0 mg/kg + Irinotecan 0.61 mg/kg.
H. Liang / Contemporary Clinical Trials 28 (2007) 115–119
117
Table 1 Comparison of the effect of individual and combination treatment by time to tumor failure Group
Time ± SD
P-value
A B C D E F D vs B F vs B D vs C F vs E
2.7 ± 0.2 9.5 ± 12 9.0 ± 3.0 >12 9.6 ± 0.8 >12
– 0.016 (vs A) 0.016 (vs A) 0.005 (vs A) 0.016 (vs A) 0.005 (vs A) 0.629 0.629 1.000 0.042
P-values obtained from asymptotic results.
in Table 1, and indicate that the treatment groups are consistently superior to the control group, and treatment F is more successful than treatment E. However, there was little difference in tumor failure times between combination and single agent treatments (P = 0.629 against irofulven alone and P = 1 against irinotecan 1.25 mg/kg alone). Observing the tumor volumes in treatments B, D, and F, one may see that the tumor volumes in groups D and F decline along the treatment times, whereas the tumor volumes in group B increase or keep flat status. The following scientific questions come to mind: (a) can we effectively show that two treatments are significantly different? and (b) can we use the results in clinical practice? These topics form the core of this note. In the next section we briefly introduce a test which was proposed to compare two functions. We will model tumor volume as a function of treatment time. 2. The test Assume that Y and Z are the population tumor volumes of groups 1 and 2 at treatment time X. We also assume that the tumor volumes Y and Z are the functions of X, such as f(X ) and g(X ). Comparison of the two treatment effects is equivalent to check if g(x) = f(x). Suppose we have observations (Xi, Zi, Yi) for i = 1, …, n derived from the model Yi ¼ f ðXi Þ þ ei ;
Zi ¼ gðXi Þ þ gi
where ε1, …, εn and η1, …, ηn are independent random errors, and E(ε12) = σ2 and E(η12) = τ2. The distributions of εi's and ηi's may be distinguished. How to handle this comparison has been studied over the past decade. Hall and Hart [6] used a bootstrap method to test the equality of two smooth curves. Young and Bowman [7] proposed an ANOVA-based non-parametric test to check the equality of two curves. The similar situations have been discussed in Refs. [8–11]. Liang [12] proposed a test to investigate f(x) = g(x). The test was defined as Z ⊤n ¼ Cn−2 Sn2 ðxÞdFn ; where Cn2 ¼
n−1 n 1X fYiþ1 −Ziþ1 −ðYi −Zi Þg2 1 X ; Sn ðxÞ ¼ pffiffiffi fðYi −Zi ÞIðXi VxÞg; n i¼1 2 n i¼1
and Fn is the empirical distribution based on (X1, …, Xn). One should reject the null hypothesis for a large value of ⊺n. The author also proposed a bootstrap [13] version of ⊺n for conveniently studying the level and power of ⊺n because the test statistic ⊺n may not be distribution-free. The bootstrap algorithms work as follows. Let {(Xi, Yic, Zic)} be the centered data, where Yic = Yi − ∑Yj / n and Zic = Zi − ∑Zj / n. Let {(X1⁎, Y1⁎, Z1⁎), …, (Xn⁎, Yn⁎, Zn⁎)} be the bootstrap sample from {(X1, Y1c, Z1c), …, (Xn, Ync, Znc)} directly where n X ⁎ ðxÞ ¼ p1ffiffiffi Sn;B fðYi⁎ −Zi⁎ ÞIðXi V xÞg: n i¼1
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H. Liang / Contemporary Clinical Trials 28 (2007) 115–119
The null hypothesis is rejected if the proportion of bootstrap statistics exceeding ⊺n is less than or equal to the appropriate nominal level. 3. Analysis of xenograft data In this section, we apply the test to the xenograft data introduced in Section 2. Raw data plots indicated that treatments D and F seem to be more successful than treatment B. We now statistically justify that the differences are significant using the test. First we calculate the averages of tumor volumes at each measured time for each group. These quantities are taken as Y or Z, and the time regarded as X. For the bootstrap procedure we generate 5000 bootstrapping samples. The proportions of bootstrap statistics exceeding ⊺n are both less than 0.001. The results indicate that treatments D and F are more effective than treatment B. So, biologically, treatments D and F should be pursued further in contrast to treatment B. A referee has asked a comparison of treatments D and F with treatment C. In the same way as above we had that the proportions of bootstrap statistics exceeding ⊺n are both less than 0.05. This indicated that the irofulven/irinotecan combination (treatment D and F) is more effective than irinotecan alone (treatment C). 4. Discussion To effectively compare two treatment effects of antitumor therapies we applied a new test to compare tumor volumes, which are modeled non-parametrically. The approach avoids non-parametric smoothing, which are required in most existing work. The test is easily implemented. A computer code conducted in R environment is available to conduct the implementation. The user only needs to specify a data file that provides group index and tumor volumes. This code, together with documentation and a sample dataset, is available from the author upon request. Although survival analysis is commonly used in study of antitumor activity, the analysis may not use fully information from data, and the associated conclusions may be rough. For example, (i) when two treatments fail at the same time, or (ii) two treatments are both successful based on the survival measurement, comparison of survival time can, by no means, give information. Our findings from the real data analysis indicate that the treatment with irofulven (3 mg/kg) and irinotecan (0.61 or 1.25 mg/kg) is more effective than the treatment with irinotecan (1.25 mg/kg) alone. This occurrence was not detected by comparing the survival time. For this typical study, one may use other modeling strategies to fit tumor volume [14], and comparison of antitumor effects is transferred to comparison of some parameters, which dominate the curve of tumor volume against time. However, the approach proposed in this paper is more flexible because we did not make any assumption on the model, and is more powerful than other approaches based on non-parametric smoothing techniques because the test can detect the root-n alternative hypothesis [12]. The approach is anticipated to be valuable in practice. One requirement of the test is that the design points of two groups are identical. However, our procedure can be generalized to models without common designs if the two designs are close. We omit the details but refer to Hall and Hart [6]. Acknowledgment This research was partially supported by the NIH/NAIAD grants AI62247 and AI059773. The author thanks two referees for their helpful comments. References [1] Britten CD, Hilsenbeck SG, Eckhardt SG, et al. Enhanced Antitumor Activity of 6-Hydroxymethylacylfulvene in Combination with Irinotecan and 5-Fluorouracil in the HT29 Human Colon Tumor Xenograft Model. Cancer Res 1999;59:1049–53. [2] McCreery H., Rowinsky E.K., Tolcher A., et al. Phase I trial of irofulven and CPT-11 in patients with advanced cancers. 2002 ASCO Annual Meeting. [3] Kelner MJ, McMorris TC, Estes L, et al. Anti-leukemic action of the novel agent MGI 114 (HMAF) and synergistic action with topotecan. Leukemia 2000;14:136–41. [4] Woo MH, Peterson JK, Billups C, et al. Enhanced antitumor activity of irofulven in combination with irinotecan in pediatric solid tumor xenograft models. Cancer Chemother Pharmacol 2005;55:411–9. [5] Houghton PJ, Cheshire PJ, Hallman JD, et al. Efficacy of topoisomerase I inhibitors, topotecan and irinotecan, administered at low dose levels in protracted schedules to mice bearing xenografts of human tumors. Cancer Chemother Pharmacol 1995;36:393–403.
H. Liang / Contemporary Clinical Trials 28 (2007) 115–119 [6] [7] [8] [9] [10] [11] [12] [13] [14]
119
Hall P, Hart JD. Bootstrap test for difference between means in nonparametric regression. J Am Stat Assoc 1990;85:1039–49. Young SG, Bowman AW. Non-parametric analysis of covariance. Biometrics 1995;51:920–31. Härdle W, Mammen E. Comparing non-parametric versus parametric regression fits. Ann Stat 1993;21:1926–47. Härdle W, Marron S. Semiparametric comparison of regression curves. Ann Stat 1990;18:63–89. Cox DR, Koh E, Wahba G, Yandell BS. Testing the (parametric) null model hypothesis in (semi-parametric) partial and generalized spline model. Ann Stat 1988;16:113–9. Eubank RL, Hart JD. Commonality of Cusum, von Neumann and smoothing-based goodness-of-fit tests. Biometrika 1993;80:89–98. Liang H. Comparison of curves based on Cramér-von Mises statistic. Comput Stat Data Anal 2004;45:805–12. Efron B, Gong G. A leisurely look at the bootstrap, the jackknife and cross-validation. Am Stat 1983;37:36–48. Liang H, Sha NJ. Modeling antitumor activity by using a nonlinear mixed-effects models. Math Biosci 2004;189:61–73.