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Much of the research on multiple comparison and simultaneous inference in the past 60 years or so has been for the comparisons of several population means. Spurrier seems to have been the first to investigate multiple comparisons of several simple linear regression lines using simultaneous confidence bands. In this paper, we extend the work of Liu et al. for finite comparisons of several univariate linear regression models using simultaneous confidence bands to finite comparisons of several multivariate linear regression models using simultaneous confidence tubes. We show how simultaneous confidence tubes can be constructed to allow more informative inferences for the comparison of several multivariate linear regression models than the current approach of hypotheses testing. The methods are illustrated with examples.

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RESUMEN

In many scientific problems the purpose of the comparison of two regression models, which describe the relationship between a same response variable and several same covariates for two different groups, is to demonstrate that one model is no higher than the other by a negligible amount, or to demonstrate that the models have only negligible differences and so they can be regarded as describing practically the same relationship between the response variable and the covariates. In this article, methods based on one-sided pointwise confidence bands are proposed for assessing the nonsuperiority of one model to the other and for assessing the equivalence of two regression models. Examples from QT/QTc study and from drug stability study are used to illustrate the methods.

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A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, it is proposed in Liu and Hayter (2007) to use the area of the confidence set that corresponds to a confidence band as an optimality criterion in comparison of confidence bands; the smaller is the area of the confidence set, the better is the corresponding confidence band. This minimum area confidence set (MACS) criterion can clearly be generalized to the minimum volume confidence set (MVCS) criterion in study of confidence bands for a multiple linear regression model. In this paper the hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and so the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and so the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band.

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This paper considers the problem of identifying which treatments are strictly inferior to the best treatment or treatments in a balanced one-way layout with three treatments, which has important applications in screening trials for new product development. A stepdown procedure is constructed that selects a subset of the treatments containing only treatments that are known to be strictly inferior to the best treatment or treatments. This stepdown procedure uses feedback from the first stage to the second stage that improves its operating characteristics. The advantages accruing from this feedback are demonstrated.

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Several C2-methylquinazoline-based antifolates have been prepared in which the C9,N10 bridge has been replaced by the reversed N9,C10 unit. This series was extensively studied by incorporating further substituents at N9 and C10 as well as by modifications to the p-aminobenzoate ring. The C2-methylquinazoline analogues 29, 30, and 31 containing the methyleneoxa, methylenethia, and thia bridge units were also synthesized. In general these isosteric replacements of the bridge unit in the parent C2-methyl-N10-propargylquinazoline antifolate 2 were much less potent as inhibitors of isolated thymidylate synthase (TS) but several were at least as potent as inhibitors of L1210 cell growth in culture. The fusion of the p-aminobenzoate ring into the bicyclic systems 75 and 76 also reduced activity against TS but again gave highly cytotoxic compounds. The cytotoxicities were largely prevented by thymidine, confirming that TS is the major locus.

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The synthesis is described of a series of C2-methyl-N10-alkylquinazoline-based antifolates in which the p-aminobenzoate ring is replaced by the heterocycles thiophene, thiazole, thiadiazole, pyridine, and pyrimidine. These were generally elaborated by the reaction of (bromomethyl)quinazoline 18 or its N3-[(pivaloyloxy)methyl]-protected derivative 36 with suitable heterocyclic amines although each heterocyclic system required its own particular synthetic approach. The compounds were tested as inhibitors of partially purified L1210 thymidylate synthase (TS). They were also examined for their inhibition of the growth of L1210 cells in culture. The thiophene system 7 and its related thiazole 8 gave analogues that were considerably more potent than the parent benzene series 2 as inhibitors of L1210 cell growth although in general these heterocycles were somewhat poorer inhibitors of the isolated TS enzyme. The enhanced cytotoxicities of the thiophene and thiazole analogues result, at least in part, from their efficient transport into the cells via the reduced folate carrier mechanism and very good substrate activity for folylpolyglutamate synthetase. The replacement of the C2-methyl group by C2-(fluoromethyl) and C2-(hydroxymethyl) substituents in the thiophene and thiazole series gave derivatives that were only slightly less potent inhibitors of the TS enzyme but which were considerably less cytotoxic.

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The synthesis of 16 new N10-propargylquinazoline antifolates with methylamino, ethylamino, (2-aminoethyl)amino, [2-(dimethylamino)ethyl]amino, (2-hydroxyethyl)amino, (carboxymethyl)amino, dimethylamino, imidazol-1-yl, methoxy, ethoxy, phenoxy, 2-methoxyethoxy, 2-hydroxyethoxy, mercapto, methylthio, and chloro substituents at C2 is described. In general, the synthetic route involved the coupling of diethyl N-[4-(prop-2-ynylamino)benzoyl]-L-glutamate (5a) with 6-(bromomethyl)-2-chloro-3,4-dihydro-4-oxoquinazoline in N,N-dimethylformamide with calcium carbonate as the base, displacement of the C2-chloro substituent with nitrogen and sulfur nucleophiles, and deprotection using mild alkali. The C2-ether analogues were most conveniently prepared by coupling 5a with 6-(bromomethyl)-2,4-diakoxy(or diphenoxy)quinazolines. In this series the final deprotection step with aqueous alkali gave simultaneous selective hydrolysis of the C4-alkoxy or C4-phenoxy substituent. The compounds were tested as inhibitors of partially purified L1210 thymidylate synthase (TS). As a measure of cytotoxicity, they were examined for their inhibition of the growth of L1210 cells in culture. The C2-methoxy analogue 11a was equivalent to the previously described tight binding TS inhibitor N10-propargyl-5,8-dideazafolic acid (CB3717, ICI 155387, 1a) against the TS enzyme and exhibited enhanced potency in culture. The C2-methoxy substituent also gave a 110-fold enhancement in aqueous solubility relative to the C2-amine. These results suggest that 11a will be an interesting compound for further study as a potential antitumor agent in vivo. A further series of 2-methoxyquinazoline antifolates with modified alkyl substituents at N10 is also described. None of these analogues equalled the activity of 11a. Thus the propargyl group appears to be the optimum N10 substituent in both 2-amino- and 2-methoxyquinazoline antifolates.

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The synthesis of 12 new 5,8-dideazafolates with isopropyl, cyclopropylmethyl, 2-fluoroethyl, carbamoylmethyl, phenacyl, 3-fluorobenzyl, 5-uracilylmethyl, carboxymethyl, 2-carboxyethyl, 3-cyanopropyl, 3-hydroxypropyl, and cyanomethyl substituents at N10 is described. In general, the synthetic route involved monoalkylation of diethyl N-(4-amino-benzoyl)-L-glutamate, coupling of the resulting secondary amine with 2-amino-6-(bromomethyl)-4-hydroxyquinazoline hydrobromide in N,N-dimethylacetamide with calcium carbonate as the base, and deprotection using mild alkali. The cyanomethyl derivatives was found to be unexpectedly base labile and was therefore prepared by mild acid deprotection of a di-tert-butyl ester. The compounds were tested as inhibitors of purified L1210 thymidylate synthase (TS). Four members of the series were more potent that the N10-hydrogen compound, but none was superior to the previously described N10-propargyl-5,8-dideazafolic acid. Selected compounds were examined as inhibitors of purified L1210 dihydrofolate reductase (DHFR). As desired, N10 substitution in general reduced DHFR inhibitory activity; these results are discussed. As a measure of cytotoxicity, the compounds were examined for their inhibition of the growth of L1210 cells in culture. None of the new substituents conferred enhanced potency relative to N10-propargyl-5,8-dideazafolic acid (ID50 = 5 microM), which, as the best TS inhibitor and a relatively poor DHFR inhibitor, continues to lead this series.

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