Alfio Marazzi

Fitting the distributions of length of stay by parametric models

OBJECTIVES The purpose of this study was to assess the adequacy of three widely used models--Lognormal, Weibull, and Gamma--for describing the distribution of length of stay. This is a fundamental step in the development of outliers resistant (robust) methods for the statistical analysis of this kind of data, where the main objective is to determine measures of average and total resource consumption of groups of patients. Current practice uses several types of trimming rules, many of which are based on the Lognormal model, although theoretical and experimental bases are still insufficient. METHODS The three models were adjusted using robust procedures based on M-estimators to approximately 5 million stays grouped by Diagnosis-Related Groups (DRGs). The resulting 3,279 samples were collected in five European countries during 3 years. RESULTS Most of the distributions observed could be fitted with one of these models. The descriptions provided by the Gamma and the Weibull models were similar, and the Gamma model could be omitted. The casemix description provided by the Log-normal-Weibull family was, for certain countries, significantly better than the one provided by the single Lognormal model. Often, for a given DRG and a given country, length of stay distributions could be described with the same model during several years. A given DRG, however, usually had to be described by means of different models for different countries. CONCLUSIONS Practical and conceptual consequences of the results are discussed. They can be extended to the analyses of other consumption variables used in health services. Statistical procedures for casemix description, including current rules of trimming, should be improved by means of more flexible families of models.

The truncated mean of an asymmetric distribution

This paper investigates a simple procedure to estimate robustly the mean of an asymmetric distribution. The procedure removes the observations which are larger or smaller than certain limits and takes the arithmetic mean of the remaining observations, the limits being determined with the help of a parametric model, e.g., the Gamma, the Weibull or the Lognormal distribution. The breakdown point, the influence function, the (asymptotic) variance, and the contamination bias of this estimator are explored and compared numerically with those of competing estimates.

Robust Box-Cox transformations based on minimum residual autocorrelation

Response transformations are a popular approach to adapt data to a linear regression model. The regression coefficients, as well as the parameter defining the transformation, are often estimated by maximum likelihood assuming homoscedastic normal errors. Unfortunately, consistency to the true parameters holds only if the assumptions of normality and homoscedasticity are satisfied. In addition, these estimates are nonrobust in the presence of outliers. New estimates are proposed, which are robust and consistent even if the assumptions of normality and homoscedasticity do not hold. These estimates are based on the minimization of a robust measure of residual autocorrelation.

Probabilistic algorithms for least median of squares regression

The basic probabilistic algorithm for Least Median of Squares regression (LMS) with p parameters is based on repeated drawings of random subsamples of p data points, followed by calculation of the median-squared residual with respect to the hyperplane fitted to these p points. The regression plane with the smallest median-squared residual is then treated as an approximate solution. Three improved variants of this algorithm are proposed and compared with regard to their capability to satisfy a necessary condition for LMS and to their empirical performances.

Robust accelerated failure time regression

Robust estimators for accelerated failure time models with asymmetric (or symmetric) error distribution and censored observations are proposed. It is assumed that the error model belongs to a log-location-scale family of distributions and that the mean response is the parameter of interest. Since scale is a main component of mean, scale is not treated as a nuisance parameter. A three steps procedure is proposed. In the first step, an initial high breakdown point S estimate is computed. In the second step, observations that are unlikely under the estimated model are rejected or down weighted. Finally, a weighted maximum likelihood estimate is computed. To define the estimates, functions of censored residuals are replaced by their estimated conditional expectation given that the response is larger than the observed censored value. The rejection rule in the second step is based on an adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Therefore, the final estimate attains full efficiency at the model, with respect to the maximum likelihood estimate, while maintaining the breakdown point of the initial estimator. Asymptotic results are provided. The new procedure is evaluated with the help of Monte Carlo simulations. Two examples with real data are discussed.

Clonal heterogeneity and chromosomal instability at disease presentation in high hyperdiploid acute lymphoblastic leukemia

Although aneuploidy has many possible causes, it often results from underlying chromosomal instability (CIN) leading to an unstable karyotype with cell-to-cell variation and multiple subclones. To test for the presence of CIN in high hyperdiploid acute lymphoblastic leukemia (HeH ALL) at diagnosis, we investigated 20 patients (10 HeH ALL and 10 non-HeH ALL), using automated four-color interphase fluorescence in situ hybridization (I-FISH) with centromeric probes for chromosomes 4, 6, 10, and 17. In HeH ALL, the proportion of abnormal cells ranged from 36.3% to 92.4%, and a variety of aneuploid populations were identified. Compared with conventional cytogenetics, I-FISH revealed numerous additional clones, some of them very small. To investigate the nature and origin of this clonal heterogeneity, we determined average numerical CIN values for all four chromosomes together and for each chromosome and patient group. The CIN values in HeH ALL were relatively high (range, 22.2-44.7%), compared with those in non-HeH ALL (3.2-6.4%), thus accounting for the presence of numerical CIN in HeH ALL at diagnosis. We conclude that numerical CIN may be at the origin of the high level of clonal heterogeneity revealed by I-FISH in HeH ALL at presentation, which would corroborate the potential role of CIN in tumor pathogenesis.

New approaches to reimbursement schemes based on patient classification systems and their comparison.

We propose reimbursement schemes based on patient classification systems (PCSs) that include adjustments for length of stay (LOS) and exceptional costs and are designed to minimize undesirable effects of economic incentives. In addition, a statistical approach to compare the schemes and the underlying PCSs is proposed, where costs and LOSs for two successive years are used. The first year data provides estimates of the class cost means and the next years reimbursements which are compared with the second years costs. This method focuses on the predictive power of a PCS and differs from the usual retrospective analyses based on the proportion of explained variance for single year data. The approach is applied to discharge data of Swiss hospitals where stays are grouped according to five PCSs: All Patient Diagnosis-Related Groups (AP-DRGs), All Patient Refined Diagnosis-Related Groups (APR-DRGs), International Refined Diagnosis-Related Groups (IR-DRGs), Australian Refined Diagnosis-Related Groups (AR-DRGs), and SQLape. When adjusting for LOS and outliers, these systems do not differ substantially in their ability to predict cost of stay. Therefore, increasing the number of classes does not necessarily improve cost predictions. However, the payment of a fixed amount per diem (not exceeding the marginal cost) and correcting the reimbursements for exceptional costs substantially reduces the average discrepancy between costs and reimbursements.

Robust Estimators of the Generalized Log-Gamma Distribution

We propose robust estimators of the generalized log-gamma distribution and, more generally, of location-shape-scale families of distributions. A (weighted) Qτ estimator minimizes a τ scale of the differences between empirical and theoretical quantiles. It is n1/2 consistent; unfortunately, it is not asymptotically normal and, therefore, inconvenient for inference. However, it is a convenient starting point for a one-step weighted likelihood estimator, where the weights are based on a disparity measure between the model density and a kernel density estimate. The one-step weighted likelihood estimator is asymptotically normal and fully efficient under the model. It is also highly robust under outlier contamination. Supplementary materials are available online.

Robust parametric indirect estimates of the expected cost of a hospital stay with covariates and censored data.

We consider the problem of estimating the mean hospital cost of stays of a class of patients (e.g., a diagnosis-related group) as a function of patient characteristics. The statistical analysis is complicated by the asymmetry of the cost distribution, the possibility of censoring on the cost variable, and the occurrence of outliers. These problems have often been treated separately in the literature, and a method offering a joint solution to all of them is still missing. Indirect procedures have been proposed, combining an estimate of the duration distribution with an estimate of the conditional cost for a given duration. We propose a parametric version of this approach, allowing for asymmetry and censoring in the cost distribution and providing a mean cost estimator that is robust in the presence of extreme values. In addition, the new method takes covariate information into account.

Optimal robust estimates using the Hellinger distance

Optimal robust M-estimates of a multidimensional parameter are described using Hampel’s infinitesimal approach. The optimal estimates are derived by minimizing a measure of efficiency under the model, subject to a bounded measure of infinitesimal robustness. To this purpose we define measures of efficiency and infinitesimal sensitivity based on the Hellinger distance. We show that these two measures coincide with similar ones defined by Yohai using the Kullback–Leibler divergence, and therefore the corresponding optimal estimates coincide too. We also give an example where we fit a negative binomial distribution to a real dataset of “days of stay in hospital” using the optimal robust estimates.