Multiplicative Intensity Model Research Articles

Course of treatment received by depressed patients

Using data from an observational study of affective disorders, we describe the rates of transition among levels of antidepressant treatment for subjects with Major Depressive Disorder (MDD), and relate these changes to changes in clinical status. We report on the treatment received during the first 10 years of follow-up in the Collaborative Depression Study by 555 patients with a diagnosis of MDD of at least one months duration. This work extends the initial examination of treatment received during the first eight weeks after entry into this study that showed depressed patients to be on low levels of treatment. Multiplicative intensity models which generalize survival analysis models were used to analyse these data. Description of the course of treatment of these depressed patients shows that low levels of treatment persist for these patients across subsequent episodes, and that these episodes, like the index one, are characterized by extended time in a symptomatic subcriterion state after acute symptoms have improved. These long-term descriptions of treatment support the initial hypothesis that these CDS patients were undertreated. The long-term tendency toward undertreatment seems to persist even as newer treatments become available and widely accepted in practice.

On the Problem of Identification in Multiplicative Intensity-Rate Models with Multiple Interactions

In this paper we examine a multiplicative intensity model in which a covariate interacts with two other covariates in the same model. We demonstrate, analytically, that in such situations a log-linear parameterization based on two pairs of baseline levels cannot be transformed, uniquely, to the, otherwise equivalent, multiplicative parameterization. We show that the problem lies in an oversight of the conditional independence between the two covariates interacting with a common third covariate. As a solution, therefore, we propose an approach that takes due account of such dependence. Our proposed approach uses a common baseline level for the three covariates involved in interaction while estimating the corresponding relative intensities. The issues addressed are illustrated with a demographic data set involving the estimation of rates of transition to parenthood.

Some central limit theorems for ℓ∞-valued semimartingales and their applications

This paper is devoted to the generalization of central limit theorems for empirical processes to several types of l∞(Ψ)-valued continuous-time stochastic processes t⇝X t n =(X t n ,ψ|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X t n ,ψ is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived.

Parametric Multiplicative Intensities Models Fitted to Bus Motor Failure Data

SUMMARY Classes of two- and three-parameter multiplicative intensities models are presented. These have the advantage that they can be simply fitted by standard software. The two-parameter models include the Weibull and extreme value models, whereas the three-parameter models can have bathtub-shaped forms. The bus motor failure data are reanalysed to show that all five failures of the 191 buses can be adequately described by the same two-parameter multiplicative intensity model based on the squared logarithm of distance travelled in service since previous repair. This is a new member of the multiplicative intensity family, based on the squared logarithm of distance, or of time for survival data.

Large sample properties of some survival estimators in heterogeneous samples

The Kaplan-Meier survival distribution estimator and Nelson hazard function estimator may not be appropriate descriptive devices for studies which involve nonhomogeneous populations. In this paper we investigate a mean survival estimator that incorporates important covariates associated with survival. The estimator is constructed based on Cox's regression model and a weighting according to any specified distribution of the baseline covariates. A mean hazard function estimator is also derived corresponding to the multiplicative intensity model which allows for recurrent outcome events. The weak convergence of each estimator is established using martingale and stochastic convergence results and its limiting variance is calculated. The strong and weak convergence of the Kaplan-Meier estimate and Nelson estimate are also explored in a nonhomogeneous population with an underlying proportional hazards model. It is shown that the Kaplan-Meier estimate remains a uniformly consistent estimate of S( t) = P( T > t) even in a heterogeneous sample if the censoring distribution is independent of covariates. However, when the censoring distribution depends on covariates, the Kaplan-Meier estimator can be quite biased even though the mean survival function estimator remains unbiased.

Nonparametric kernel regression when the regressor follows a counting process

The regression model Y i = m(τ i ) + e i where the the τ i can be thought of as times, is studied in a marked point process framework. This paper will show consistency and asymptotic normality for the Nadaraya-Watson kernel estimate of m(·) under Aalen's multiplicative intensity model for the τ i and provide some further results. The main contribution of the paper is the model formulation, which allows for quite general censoring schemes of the measurement times. The model appears useful for longitudinal data applications and is applied to repeated measurements of oestriol for 52 pregnant women, for a case where censoring is present.

Diagnostic Plots to Reveal Functional Form for Covariates in Multiplicative Intensity Models

We show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model. A smoothed plot of the martingale residues was suggested for this purpose by Therneau, Grambsch, and Fleming (1990, Biometrika 77, 147-160); however, its consistency required that the covariates be independent. They also noted that the plot could be biased for large covariate effects. We introduce two refinements which overcome these difficulties. The first is based on a ratio of scatter plot smooths, where the numerator is the smooth of the observed count plotted against the covariate, and the denominator is a smooth of the expected count. This is related to the Arjas goodness-of-fit plot (1988, Journal of the American Statistical Association 83, 204-212). The second technique smooths the martingale residuals divided by the expected count, using expected count as a weight. This latter approach is related to a GLM partial residual plot, as well as to the iterative methods of Hastie and Tibshirani (1990, Biometrics 46, 1005-1016) and Gentleman and Crowley (1991, Biometrics 47, 1283-1296). Applications to survival data sets are given.

A class of linear nonparametric rank tests for dependent censoring

This paper studies linear nonparametric rank tests for dependent censoring in survival analysis. A method is proposed for testing the assumption of independent censoring, in which a random subset of lost-to-follow-up censored subjects is subsequently followed. Linear nonparametric rank tests and their asymptotic properties are derived by way of the multivariate counting process and multiplicative intensity model proposed by Gill (1980) and Aalen (1978). It is also shown that the proposed linear nonparametric rank tests are equivalent to a class of weighted score statistics for a Cox proportional hazards model. Some simulation results are given for the power of log rank and Wilcoxon-type tests under the propotional hazards alternative and nonpropotional hazards alternative, respectively.

Effects of spike discharge history on discharge probability and latency in frog basilar papilla units

Gaumond et al. [(1982) J. Neurophysiol. 48, 856–873] showed in the cat that a multiplicative-intensity model can generally account quite well for reduction of the probability of an auditory-nerve spike by another spike preceding it by 4 to 25 ms, and that for smaller separations there is also an increased latency of the following spike. Bosch [(1990) D. Sc. Dissertation, Washington University, St. Louis, MO] made important improvements in experimental design and estimation techniques for studying these effects, and confirmed their presence in the gerbil. However, direct application of these methods to the frog does not yield reliable estimates. A clearer separation of discharge probability and latency effects in frog basilar papilla units is provided by the paired-click paradigm used in this study, which is applicable to low-spontaneous-rate units that generally respond to click stimuli with zero or one spike within a short interval following the click. The results confirm the existence in the frog of both spike-probability and spike-latency effects that are qualitatively similar to those found in mammals, although the absolute refractory time is much longer in frog, and the relative refractory time usually shorter. The paired-click paradigm also reveals a stimulus-history effect at stimulus levels which are near threshold: when there is no response to the first click, responses to the second click occur with increased probability and reduced latency.

Relative Efficiency of the Log Rank Test Within a Multiplicative Intensity Model

Relative Efficiency of the Log Rank Test Within a Multiplicative Intensity Model

Relative efficiency of the log rank test within a multiplicative intensity model

SUMMARY For large-sample clinical trials with independent individuals randomly allocated to two treatment groups, in which survival times follow a log linear multiplicative intensity model with treatment group as one covariate, this paper calculates the asymptotic relative efficiency of the log rank test for treatment effect as compared with the optimal score test. The method is to exhibit the failure hazard intensity, not of proportional hazards form, obtained by ignoring all covariates other than treatment group. The efficiency formulae are illustrated in two examples, and estimation from data of the loss of efficiency is illustrated for two clinical trial datasets.

A penalty method for nonparametric estimation of the intensity function of a counting process

Nonparametric estimators are proposed for the logarithm of the intensity function of some univariate counting processes. An Aalen multiplicative intensity model is specified for our counting process and the estimators are derived by a penalized maximum likelihood method similar to the method introduced by Silverman for probability density estimation. Asymptotic properties of the estimators, such as uniform consistency and normality, are investigated and some illustrative examples from survival theory are analyzed.

A note of kernel smoothing of an estimator of a periodic function in the multiplicative intensity model

The estimator obtained by the kernel regularization of the sieve-based maximum likelihood estimator (MLE) of a periodic function in the multiplicative intensity model is considered. Under some regularity assumptions it is shown that this estimator is L 1 consistent and asymptotically normal. The paper generalizes the result of Leśkow (1988) obtained for a histogram estimator in this model.

Histogram maximum likelihood estimator in the multiplicative intensity model

In the paper the maximum likelihood estimator of a deterministic function of the stochastic intensity of a point process is derived by the method of sieves. Consistency and asymptotic normality of this estimator are proved.

Statistical modelling via partitioned counting processes

Partitioned counting processes arise from observation of the jumps of a counting process in disjoint stochastic time intervals. The multiplicative intensity model for a counting process is a statistical model for nonparametric interference about, e.g., the hazard function of a survival time distribution. In non-standard situations when the data are recorded only between two stopping times which vary from one individual to another, partitioned counting processes provide a simple approach to statistical modelling. Some recent extensions of Cox's regression model can be justified.

Maximum Likelihood Estimation in the Multiplicative Intensity Model: A Survey

Summary The nonparametric multiplicative intensity model for simple point processes on the line is considered. By introducing a suitable topology on the space of all point process paths, a topological extension of the model is found, accommodating point processes with multiple points. In the extended model the maximum likelihood estimator is determined and shown to be interpretable in terms of a multiple point process. Finally, the maximum likelihood estimator and the Aalen estimator are compared. The results are illustrated by examples involving a model for Poisson processes, a model for censored survival data and the Cox regression model.

Nonparametric Inference for a Class of Semi-Markov Processes with Censored Observations

A class of semi-Markov models, those which have proportional hazards and which are forward-going (if state $j$ can be reached from $i$, then $i$ cannot be reached from $j$), are shown to fit into the multiplicative intensity model of counting processes after suitable random time changes. Standard large-sample results for counting processes following this multiplicative model can therefore be used to make inferences on the above class of semi-Markov models, including the case where observations may be censored. Large-sample results for a four-state model used in clinical trials are presented.