New PDF release: Analytical Chemistry Vol 1

By F. P. Treadwell

Excerpt from Analytical Chemistry, Vol. 1: Qualitative Analysis

The translator needs to precise his honest because of Professor Treadwell for his kindness and solid needs in regards to the instruction of this translation, to Professor A. A. Noyes, Professor Henry Fay, and Mr. H. S. Walker of the Massachusetts Institute of know-how, for his or her suggestion and' feedback of the facts, and particularly to Dr. W. O. Holway of Newton Centre for his many necessary suggestions.

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Tarone and Ware (1977) first considered the general class. Harrington and Fleming (1982) considered a family of weight functions indexed by a parameter ρ that included the Wilcoxon and log-rank tests as special cases. They derive the asymptotic null distribution of the maximum weighted log-rank statistic in the class. 2 28 INTRODUCTION Fleming and Harrington (1991, Chap. 7) give an extensive discussion of log-rank and weighted log-rank procedures and have collected numerous references. References for counting processes and associated asymptotics are collected in Chapter 5.

In modeling adult human mortality, a more rapidly increasing hazard function than that represented by, say, the Weibull distribution is necessary. In fact, a relationship in which the hazard function is an exponential function of follow-up time (age at death) has been found to be descriptive in many investigations, at least for ages greater than 35. Such a relation leads to the Gompertz (1825) hazard A(i) = λ exp(7i). Sometimes the exponential term is generalized to a polynomial function of t. The Makeham (1860) generalization adds a constant to the hazard to give λ(ί) = Q + λ exp(7f).

Using subscripts 1 and 2 for the respective models and changing the sign of the accelerated failure time regression parameter, we require that λο,(ί) exp(Z'/3,) = Ami* exp(Z'ft)] cxp(Z%) for all (r,Z). The value Ζ = 0 gives λοι (•) = λο2(·) = λ (·)> say, while ζ = ( - l o g r / 0 2 \ , 0 , . . , 0) gives, at that t, 0 =λο(1)Γ', \ (ήΐ " » β β 0 where β\\ and β \ are the first components of β and β , respectively. It follows that for all t, 2 { 2 Ao(f) = A ( A i ) - , 1 1 7 where 7 = β β^ and λ = [ λ ( 1 ) / 7 ] ' .

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Analytical Chemistry Vol 1 by F. P. Treadwell

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