摘要:

Additive isotonic models generalize linear models by replacing lines with isotonic (nondecreasing) transformations. Fitted transformations of several explanatory variables are added together and then transformed by a known function to yield fitted values of the response variable. The isotonic transformations are chosen to minimize an explicit criterion, such as the negative log-likelihood, by an algorithm that optimizes one transformation at a time while adjusting for the current fitted values of the others, cycling until the criterion converges. This approach can be used in various situations, notably for generalizing ordinary linear regression and linear logistic regression. At each step of the algorithm, the needed optimal isotonic transformation is found using a simple generalization of the standard pool-adjacent-violators algorithm (Ayer, Brunk, Ewing, Reid, and Silverman 1955). The fitted transformations are always made up of flat steps, so the technique is useful for finding optimal stratifications of the explanatory variables, but not for finding smooth transformations. The technique speeds the process of checking variables for possible addition to an existing model, because the possibility of finding a useful isotonic transformation can be ruled out if the best such transformation performs poorly. Isotonic regression is usually extended to multivariate settings by fitting a multivariate function that preserves a partial order on the values of all of the explanatory variables. Such models are more general than additive isotonic models, but they are less interpretable (because of their high dimension) and more prone to overfit the data. Transformations that preserve a partial order on a subset of the explanatory variables can be incorporated into additive isotonic models to model interactions within the subset. More general mixtures of techniques within the additive framework are also possible; smooth, isotonic, and parametric transformations can be applied to different explanatory variables within one additive model.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478768

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Logistic regression is a common technique for analyzing the effect of a covariate vector x on the number of successesyinmtrials whenyhas a binomial distribution. But at times either the logistic curve does not describe the probability of successp(x) adequately, ormis larger than 1 andyis more variable than the binomial distribution allows. Overdispersion relative to the binomial distribution is possible if themtrials in a set or “litter” are positively correlated, an important covariate is omitted, orxis measured with error. A simple way to accommodate departures from the logit link and overdispersion is to introduce a random intercept α and thus permit a random propensity toward success. When α varies between individual binary trials according to a discrete or multimodal distribution,p(x) has smooth steps andyhas a binomial(m, p(x)) distribution. When the random α is constant for a set ofmbinary trials and varies between sets ofmtrials according to a discrete or multimodal distribution,p(x) has smooth steps andyis overdispersed relative to the binomial distribution. In this article the distribution of α is left unspecified and estimated by nonparametric maximum likelihood. The estimated distribution of α is discrete, so the distribution ofyand all of its properties are easily estimated for anyx. Two examples are considered. In the first, the logit link is inadequate, butyappears to be binomial. Hence α is allowed to vary between binary trials. In the second,y's (withm> 1) from the same design point are more dispersed than the binomial distribution would predict, and there are outliers. Allowing α to vary randomly between sets ofmtrials accounts for the overdispersion and seems to temper the influence of outliers on the estimated probability of success.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478769

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

This study of statistical inference for repairable systems focuses on the development of estimation procedures for the life distributionFof a new system based on data on system lifetimes between consecutive repairs. The Brown—Proschan imperfect-repair model postulates that at failure the system is repaired to a condition as good as new with probabilityp, and is otherwise repaired to the condition just prior to failure. In treating issues of statistical inference for this model, the article first points out the lack of identifiability of the pair (p, F) as an index of the distribution of interfailure timesT1,T2, …. It is then shown that data pairs (Ti, Zi) (i= 1, 2, …) render the parameter pair (p, F) identifiable, whereZiis a Bernoulli variable that records the mode of repair (perfect or imperfect) following theith failure. Under the assumption that data of the form {(Ti, Zi)} are drawn via inverse sampling until the occurrence of themth perfect repair, the problem of estimating the parameter pair (p, F) of the Brown—Proschan model is studied. It is demonstrated that the nonparametric maximum likelihood estimator ofFexists only in special cases, but that a neighborhood maximum likelihood estimator [Fcirc] (using the language of Kiefer and Wolfowitz 1956) always exists and may be derived in closed form. Under mild assumptions, the strong uniform consistency of [Fcirc] is demonstrated, as is the weak convergence of an appropriately scaled version of [Fcirc] to a Gaussian process. It is noted that these results apply to other experimental designs, such as renewal testing, and that they can be extended to the age-dependent imperfect-repair model of Block, Borges, and Savits (1985).

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478770

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

In this article we consider experimental settings in which it is desired to optimally comparevtest treatments to a standard or control treatment and the experimental units are to be arranged inbblocks of sizek. This problem has received a good deal of attention in recent years. Majumdar and Notz (1983) developed some sufficient conditions that can be used to establish theA-optimality of balanced treatment block designs (BTBD's) in these situations, and several authors have since used these sufficient conditions to show theA-optimality of some specific BTBD's as well as characterize some infinite families ofA-optimal BTBD's. Here we develop some further sufficient conditions forA-optimality that generalize those given by Majumdar and Notz (1983) and can often be used to establish theA-optimality of BTBD's not covered by the results of Majumdar and Notz (1983), as well as theA-optimality of certain types of designs called group-divisible treatment designs. Several examples are given to illustrate how the sufficient conditions obtained here can be applied.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478771

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

We consider the problem of computing sums of squares in multifactor analysis of variance models with unequal but nonzero numbers per cell. Commonly used computing methods for obtaining main-effect and interaction hypothesis mean squares solve linear model equationsX′Xβ =X′Y. We present an algorithm based on Scheffé's method of contrasts that solves a smaller set of linear equations. It avoids subtraction of quadratic forms, which degrades floating-point accuracy and facilitates Scheffé's method of multiple comparisons. Selected efficiency calculations are presented.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478772

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

To test for treatment effects in a two-way model when the classical assumptions of normality of errors and constancy of variance cannot be verified, Hora and Conover (1984) proposed a rank test in which the entire data set is ranked, the ranks are scored, and then the classical analysis of varianceFstatistic is applied to the scored ranks. They showed that the limiting null distribution of this test statistic is a chi-squared distribution divided by its degrees of freedom. Simulation results suggest that this procedure, called the rank-transform procedure, has good power properties. This article determines the asymptotic relative efficiency of the rank-transform procedure relative to the classicalFstatistic. To do this, vectors of linear rank statistics are shown to have a limiting multivariate normal distribution under a sequence of Pitman alternatives. This work is based on the results of Hájek (1968). The rank-transform statistic is then expressed as a quadratic form in the vectors, divided by a consistent estimator of a variance component. It is then shown that the limiting distribution of this statistic is a noncentral chi-squared distribution divided by its degrees of freedom. The efficacy of the rank-transform procedure is the noncentrality parameter of this chi-squared distribution, which is shown to be close to the efficacy of the Kruskal—Wallis test. For the special case in which the data can be represented as a one-way layout, this efficacy coincides with the Kruskal—Wallis efficacy.

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478773

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

Statistical Analysis With Missing Data.; Roderick J. A. Little and Donald B. Rubin. New York: John Wiley, 1987. xiv + 278 pp. $34.95. reviewed by Maureen Lahiff

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478774

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478776

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor

摘要:

This article has no abstract

ISSN:0162-1459    

DOI:10.1080/01621459.1989.10478727

出版商:Taylor & Francis Group    

年代:1989

数据来源: Taylor