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A class of goodness-of-fit tests based on the empirical characteristic function is studied. They can be applied to continuous as well as to discrete or mixed data with any arbitrary fixed dimension. The tests are consistent against any fixed alternative for suitable choices of the weight function involved in the definition of the test statistic. The bootstrap can be employed to estimate consistently the null distribution of the test statistic. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation.

This paper proposes a bootstrap goodness of fit test for the Generalized Pareto distribution (GPd) with shape parameter @c. The proposed test is an intersection-union test which tests separately the cases of @c>=0 and @c

Generalizing the Cramer-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L"p-norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L"p-norms should receive more attention. It is shown that, given a distribution function F"0 and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.

The Pearson's chi-squared statistic (X^2) does not in general follow a chi-square distribution when it is used for goodness-of-fit testing for a multinomial distribution based on sparse contingency table data. We explore properties of [Zelterman, D., 1987. Goodness-of-fit tests for large sparse multinomial distributions. J. Amer. Statist. Assoc. 82 (398), 624-629] D^2 statistic and compare them with those of X^2 and compare the power of goodness-of-fit test among the tests using D^2, X^2, and the statistic (L"r) which is proposed by [Maydeu-Olivares, A., Joe, H., 2005. Limited- and full-information estimation and goodness-of-fit testing in 2^n contingency tables: A unified framework. J. Amer. Statist. Assoc. 100 (471), 1009-1020] when the given contingency table is very sparse. We show that the variance of D^2 is not larger than the variance of X^2 under null hypotheses where all the cell probabilities are positive, that the distribution of D^2 becomes more skewed as the multinomial distribution becomes more asymmetric and sparse, and that, as for the L"r statistic, the power of the goodness-of-fit testing depends on the models which are selected for the testing. A simulation experiment strongly recommends to use both D^2 and L"r for goodness-of-fit testing with large sparse contingency table data.

In this paper we derive new tests for goodness of fit based on the fiducial empirical distribution function (EDF) after the probability integral transformation of the sample. Note that the fiducial EDF for a set of given sample observations is a randomized distribution function. By substituting the fiducial EDF for the classical EDF in the Kolmogorov-Smirnov, Cramer-von Mises statistics and so forth, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. It emerges from Monte Carlo simulations that in most cases there exist some of the new tests having better power properties than the corresponding tests based on the classical EDF and Pyke's modified EDF.

Semi-Markov processes have gained popularity as multistate models of disease progression, but have rarely been used in randomised clinical trials (RCTs) when many endpoints are collected over time. Reasons for this include the additional complexity created by censoring, the complexity of the model usually characterised by transition intensities, and difficulties in estimating clinically relevant quantities. In this paper, we show that an indirect approach can address these issues, while accommodating censored data and permitting accurate estimation of survivor functions, hazard ratio and net reductions in risk. The technique combines saddlepoint approximation and likelihood theory to fit a semi-Markov model, specified in terms of its transition probabilities and moment-generating functions (MGFs). A key feature of this approach is that the proportional hazard (PH) assumption commonly assumed for all transition time distributions is no longer needed. Inference for clinically relevant quantities is obtained through bootstrapping or randomisation tests; a simple goodness of fit procedure exploiting the link with the cumulative incidence function for competing risks is also introduced. As illustration, an illness-death model (with transition times between randomisation, non-fatal stroke, and death) for a cardiovascular trial (LIPID) is provided. Evidence of a cumulative benefit of continued treatment, not identified by standard analysis methods in earlier published work, is presented. The technique is flexible enough to be applied extensively in clinical trials.

The generalized estimating equations (GEE) approach has been widely used to analyze repeated measures data. However, in the absence of likelihood ratio tests, model diagnostic checking tools are not well established for the GEE approach, whereas they are for other likelihood-based approaches. Diagnostic checking tools are essential for determining a model's goodness of fit, especially for non-normal data. In this paper, we propose simple residual plots to investigate the goodness of fit of the model based on the GEE approach for discrete data. The proposed residual plots are based on the quantile-quantile (Q-Q) plots of a @g^2-distribution, and are particularly useful for comparing several models simultaneously.

Tests for the Laplace distribution based on the sample skewness and kurtosis coefficients are shown to be related to components of smooth tests of goodness of fit and are compared with a number of tests including the Anderson-Darling test, a new data-driven smooth test, a new empirical characteristic function based test and a new maximum entropy test. This last would be our slight preference as the test of choice for testing for the Laplace distribution.

In areas such as biology, geology and meteorology data often occur as angles. This paper examines the use of Bayesian mixtures of triangular distributions for the semiparametric analysis of circular data. Applications to density estimation, goodness-of-fit testing and semiparametric regression are demonstrated.

Copulas are used to model multivariate data as they account for the dependence structure and provide a flexible representation of the multivariate distribution. A great number of copulas has been proposed with various dependence aspects. One important issue is the choice of an appropriate copula from a large set of candidate families to model the data at hand. A large number of copulas are compared via likelihood principle, showing that it is hard to recognize the true underlying copula from real data since copulas with similar dependence properties are very close together. A goodness of fit test based on Mahalanobis squared distance between original and simulated log-likelihoods through parametric bootstrap techniques is also proposed. The advantage of this approach is that it is applicable to all families of copulas.

This paper considers a class of distributions arising from the difference of two discrete random variables belonging to the Panjer family of distributions. Some distributional properties and computation of probabilities are discussed. Goodness of fit and tests of hypotheses involving the likelihood ratio, score and Wald tests have been considered. As an illustration, an application to paired count data is given.

Statistical models are often based on normal distributions and procedures for testing such distributional assumption are needed. Many goodness-of-fit tests are available. However, most of them are quite insensitive in detecting non-normality when the alternative distribution is symmetric. On the other hand all the procedures are quite powerful against skewed alternatives. A new test for normality based on a polynomial regression is presented. It is very effective in detecting non-normality when the alternative distribution is symmetric. A comparison between well known tests and this new procedure is performed by simulation study. Other properties are also investigated.

In this paper, we consider a family of generalized Birnbaum-Saunders distributions and present a lifetime analysis based mainly on the hazard function of this model. In addition, we carry out maximum likelihood estimation by using an iterative algorithm, which produces robust estimates. Asymptotic inference is also presented. Next, the quality of the estimation method is examined by means of Monte Carlo simulations. We then provide a practical example to illustrate the obtained results. From this example and based on goodness-of-fit methods, we show that the GBS distribution results in a more appropriate model for modeling fatigue data than other models commonly used to model this type of data. Finally, we estimate the hazard function and the critical point of this function.

In this article, statistical inference and prediction analyses for the Weibull process with incomplete observations via classical approach are studied. Specifically, observations in the early developmental phase of a testing program cannot be observed. We derive the closed-form expressions for the maximum likelihood estimates of the parameters in both the failure- and time-truncated Weibull processes. Confidence interval and hypothesis testing for the parameters of interest are considered. In addition, predictive inferences on future failures and the goodness-of-fit test of the model are developed. Two real examples from an engine system development study and a Boeing air-conditioning system development study are presented to illustrate the proposed methodologies.

When continuous predictors are present, classical Pearson and deviance goodness-of-fit tests to assess logistic model fit break down. The Hosmer-Lemeshow test can be used in these situations. While simple to perform and widely used, it does not have desirable power in many cases and provides no further information on the source of any detectable lack of fit. Tsiatis proposed a score statistic to test for covariate regional effects. While conceptually elegant, its lack of a general rule for how to partition the covariate space has, to a certain degree, limited its popularity. We propose a new method for goodness-of-fit testing that uses a very general partitioning strategy (clustering) in the covariate space and either a Pearson statistic or a score statistic. Properties of the proposed statistics are discussed, and a simulation study demonstrates increased power to detect model misspecification in a variety of settings. An application of these different methods on data from a clinical trial illustrates their use. Discussions on further improvement of the proposed tests and extending this new method to other data situations, such as ordinal response regression models are also included.

A number of methods have been proposed to estimate the period of a variable star; e.g., a recent approach uses smoothing spline regression to fit tentative periodic functions (light curves) and selects the period minimizing a robust goodness-of-fit criterion. These methods assume that measurement errors vary independently over time. Empirical evidence, however, indicates substantial temporal dependence, possibly related to changes in observing conditions. Dependence complicates the period analysis in several respects: selection of a ''best'' period among several local optima, estimation of the light curve, and evaluation of uncertainty about period and light curve estimates. This article presents methods designed to accommodate dependent errors. An analysis of several data sets shows that the proposed approach can produce substantially different and arguably better results compared with other methods.

We develop a series of Bayesian statistical models for estimating survival of wild populations monitored using capture-recapture experiments and photoidentification data. The proposed methodology is based on Cormack-Jolly-Seber model [Cormack, R.M., 1964. Estimates of survival from the sighting of marked animals. Biometrika 51, 429-438; Jolly, G.M., 1965. Explicit estimates from capture-recapture data with both death and immigration-stochastic model. Biometrika 52, 225-247; Seber, G.A.F., 1965. A note on the multiple recapture census. Biometrika 52, 249-259]. Besides time effects in capture probabilities, the proposed models allow taking into account heterogeneity in capture probability caused by the existence of different groups of individuals in the population. For that purpose, the capture probabilities are fitted using a logistic model. Additionally, it is also possible to estimate group-specific survival rates. Goodness of fit is evaluated using Bayes factor methodology. The models are applied to an 11-year photoidentification capture-recapture experiment for bowhead whales, Balaena mysticetus. The best model provides an estimate close to the one obtained by Zeh et al. [2002. Survival of bowhead whales, Balaena mysticetus, estimated from 1981-1998 photoidentification data. Biometrics 58, 832-840] using the Jolly-Seber model, but accounting for heterogeneity in capture probabilities improves precision.

Consider the situation where the Structuration des Tableaux a Trois Indices de la Statistique (STATIS) methodology is applied to a series of studies, each study being represented by data and weight matrices. Relations between studies may be captured by the Hilbert-Schmidt product of these matrices. Specifically, the eigenvalues and eigenvectors of the Hilbert-Schmidt matrix S may be used to obtain a geometrical representation of the studies. The studies in a series may further be considered to have a common structure whenever their corresponding points lie along the first axis. The matrix S can be expressed as the sum of a rank 1 matrix @luu^T with an error matrix E. Therefore, the components of the vector @lu are sufficient to locate the points associated to the studies. Former models for S where vec(E) are mathematically tractable and yet do not take into account the symmetry of the matrix S. Thus a new symmetric model is proposed as well as the corresponding tests for a common structure. It is further shown how to assess the goodness of fit of such models. An application to the human immunodeficiency virus (HIV) infection is used for assessing the proposed model.

An extension of the Shapiro-Wilk test to verify the hypothesis of normality in the presence of nuisance regression and scale has been previously considered. Such a test is typically based on the pair of the maximum likelihood and BLUE estimators of the standard deviation in the linear regression model. It has been shown that the asymptotic null distribution of the test criterion, extended to the regression model, is equivalent to that of the original Shapiro-Wilk test for the location-scale model. A simulation study is performed in order to show that both criteria are close under the normality hypothesis for moderate as well for large data sets. The power of the test against various alternative distributions of the model errors is illustrated. Furthermore, it is shown that the probabilities of errors of both the first and second kinds do not depend on the design matrix or on the parameters of the linear model.

The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. Simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process are presented.

Methods of regression diagnostics for functional regression models are developed which relate a functional response to predictor variables that can be multivariate vectors or random functions. For this purpose, a residual process is defined by subtracting the predicted from the observed response functions. This residual process is expanded into functional principal components (FPC), and the corresponding FPC scores are used as natural proxies for the residuals in functional regression models. For the case of a univariate covariate, a randomization test is proposed based on these scores to examine if the residual process depends on the covariate. If this is the case, it indicates lack of fit of the model. Graphical methods based on the FPC scores of observed and fitted functions can be used to complement more formal tests. The methods are illustrated with data from a recent study of Drosophila fruit flies regarding life-cycle gene expression trajectories as well as functional data from a dose-response experiment for Mediterranean fruit flies (Ceratitis capitata).

Logistic regression models are frequently used in epidemiological studies for estimating associations that demographic, behavioral, and risk factor variables have on a dichotomous outcome, such as disease being present versus absent. After the coefficients in a logistic regression model have been estimated, goodness-of-fit of the resulting model should be examined, particularly if the purpose of the model is to estimate probabilities of event occurrences. While various goodness-of-fit tests have been proposed, the properties of these tests have been studied under the assumption that observations selected were independent and identically distributed. Increasingly, epidemiologists are using large-scale sample survey data when fitting logistic regression models, such as the National Health Interview Survey or the National Health and Nutrition Examination Survey. Unfortunately, for such situations no goodness-of-fit testing procedures have been developed or implemented in available software. To address this problem, goodness-of-fit tests for logistic regression models when data are collected using complex sampling designs are proposed. Properties of the proposed tests were examined using extensive simulation studies and results were compared to traditional goodness-of-fit tests. A Stata ado function svylogitgof for estimating the F-adjusted mean residual test after svylogit fit is available at the author's website http://www.people.vcu.edu/~kjarcher/Research/Data.htm.

We consider testing for an exponential distribution with unspecified rate parameter when it is only possible to observe the counts in groups with boundaries specified before sighting the data. On the basis of a size and power study we recommend that tests of fit for the exponential distribution be based on the Anderson-Darling statistic and the SW2 statistic recommended by Gulati and Neus [(2001). Goodness-of-fit statistics for the exponential distribution when the data are grouped. In: Huber-Carol, C., Balakrishnan, N., Nikulin, M.S., Mesbah, M., (Eds.), Goodness-of-Fit Tests and Validity of Models. Birkhauser, Boston, pp. 113-123 (Chapter 9)]. We also suggest that inference based on one of these be complemented by examination of the components of the X^2 statistic. We illustrate how to use these components to give improved models.

An intensive simulation study to compare the spatio-temporal prediction performances among various space-time models is presented. The models having separable spatio-temporal covariance functions and nonseparable ones, under various scenarios, are also considered. The computational performance among the various selected models are compared. The issue of how to select an appropriate space-time model by accounting for the tradeoff between goodness-of-fit and model complexity is addressed. Performances of the two commonly used model-selection criteria, Akaike information criterion and Bayesian information criterion are examined. Furthermore, a practical application based on the statistical analysis of surface shortwave radiation budget (SSRB) data is presented.

A goodness of fit test for copulas based on Rosenblatt's transformation is investigated. This test performs well if the marginal distribution functions are known and are used in the test statistic. If the marginal distribution functions are unknown and are replaced by their empirical estimates, then the test's properties change significantly. This is shown in detail by simulation for special cases. A bootstrap version of the test is suggested and it is shown by simulation that it performs well. An empirical application of this test to daily returns of German assets reveals that a Gaussian copula is unsuitable to describe their dependence structure. A t"@n-copula with low degrees of freedom such as @n=4 or 5 fits the data in some cases.

This paper presents a two-step procedure using the method of moment or percentile to find initial values and then maximize the numerical log likelihood to fit the appropriate generalized lambda distribution to data. This paper demonstrates the use of this procedure to fit well-known statistical distributions as well as some empirical data. Overall, the use of numerical maximum log likelihood estimation is a valuable alternative among existing methods of fitting. It provides not only convincing results in terms of quantile plots and goodness of fit tests but also has the advantage of a lower variability in its parameter estimation compared to the existing starship (King and MacGillivray, 1999) and method of moment (Karian and Dudewicz, 2000) fitting schemes.

We develop a goodness-of-fit measure with desirable properties for use in the hierarchical logistic regression setting. The statistic is an unweighted sum of squares (USS) of the kernel smoothed model residuals. We develop expressions for the moments of this statistic and create a standardized statistic with hypothesized asymptotic standard normal distribution under the null hypothesis that the model is correctly specified. Extensive simulation studies demonstrate satisfactory adherence to Type I error rates of the Kernel smoothed USS statistic in a variety of likely data settings. Finally, we discuss issues of bandwidth selection for using our proposed statistic in practice and illustrate its use in an example.

The traditional regression analysis is usually applied to homogeneous observations. However, there are several real situations where the observations are not homogeneous. In these cases, by utilizing the traditional regression, we have a loss of performance in fitting terms. Then, for improving the goodness of fit, it is more suitable to apply the so-called clusterwise regression analysis. The aim of clusterwise linear regression analysis is to embed the techniques of clustering into regression analysis. In this way, the clustering methods are utilized for overcoming the heterogeneity problem in regression analysis. Furthermore, by integrating cluster analysis into the regression framework, the regression parameters (regression analysis) and membership degrees (cluster analysis) can be estimated simultaneously by optimizing one single objective function. In this paper the clusterwise linear regression has been analyzed in a fuzzy framework. In particular, a fuzzy clusterwise linear regression model (FCWLR model) with symmetrical fuzzy output and crisp input variables for performing fuzzy cluster analysis within a fuzzy linear regression framework is suggested. For measuring the goodness of fit of the suggested FCWLR model with fuzzy output, a fitting index is proposed. In order to illustrate the usefulness of FCWLR model in practice, several applications to artificial and real datasets are shown.

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.

Binary outcomes are very common in medical studies. Logistic regression is typically used to analyze independent binary outcomes while generalized estimating equations regression methods (GEE) are often used to analyze correlated binary data. Several goodness-of-fit (GoF) statistics for the GEE methods have been developed recently. The objective of this study is to compare the power and Type I error rates of existing GEE GoF statistics using simulated data under different conditions. The number of clusters was varied in each condition. Different tested models included discrete, continuous, observation-specific and/or cluster-specific covariates. Two or three observations per cluster were generated with various correlations between observations. No single GEE GoF statistic performed best across all conditions. Generally, the larger the number of clusters, the more powerful the GEE GoF statistics. The GEE GoF statistics with correctly specified working correlation matrices tended to be robust in terms of Type I error rates and more powerful. For data with two observations per cluster, both Evans and Pan's statistics [1998. Goodness of fit in two models for clustered binary data. Ph.D. Dissertation, University of Massachusetts; 2002a. Goodness-of-fit tests for GEE with correlated binary data. Scand. J. Stat. 29(1), 101-110.] and Barnhart-Williamson's statistics [1998. Goodness-of-fit tests for GEE modeling with binary data. Biometrics 54, 720-729.] performed well for detecting the effect of the omitted interaction between two binary covariates. Barnhart-Williamson's statistics were generally the most powerful for detecting other types of interactions in models with at least one continuous covariate. For data with three observations per cluster, Evans and Pan's statistics performed best. fit in two models for clustered binary data. Ph.D. Dissertation, University of Massachusetts; 2002a. Goodness-of-fit tests for GEE with correlated binary data. Scand. J. Stat. 29(1), 101-110.] statistics performed best.

Genes that are more closely spaced on the chromosome than expected by chance are said to be spatially clustered. Standard tests of clustering versus uniformity do not take into account two important features of genes-the high variability of gene length and the low probability that gene locations overlap (exclusion). We show by simulation that the standard null distributions which ignore length and exclusion do not appropriately approximate the true null distributions of standard tests such as the chi-squared test. We therefore recommend bootstrap sampling to estimate the null distributions. Simulations demonstrate that the chi-squared goodness-of-fit test is a more powerful test of clustering than two other commonly used tests-Kolmogorov and Cramer-von Mises-when the distribution of gene lengths and locations is modeled by a mixture of exponentials and there is a single cluster. The chi-squared test requires binning the gene locations-the number of genes in the bin can be compared to the expected maximum number under random distribution to determine the location of gene clusters and gene deserts. The bootstrap method to test clustering is illustrated using data from human chromosome 22.

An intuitively appealing lack-of-fit test to assess the adequacy of a regression model is introduced together with a graphical diagnostic tool. The graphical method itself includes a formal testing procedure, and, it is particularly useful to detect the location of lack-of-fit. The procedure is based on regional residuals, using subsets of the space of the independent variables. A simulation study shows that, the proposed procedures in simple linear regression have similar power as those of some popular classical lack-of-fit tests. In case of local departures from the hypothesized regression model, the new tests are shown to be more powerful. Therefore, when it becomes difficult to discriminate between systematic deviations and noise, regional residual plots are very helpful in formally locating areas of lack-of-fit in the predictor space. Data examples illustrate the ability of the new methods to detect and to locate lack-of-fit.

The transmission/disequilibrium test (TDT) and the mean test are the two major nonparametric methods used in linkage analysis. Using case-parent triad data the TDT method tests the equality of the probabilities of a marker allele transmitted or not transmitted with the disease gene of a heterozygous parent. On the other hand, the mean test tests the goodness-of-fit of the observed and expected distributions of the number of alleles shared identical by descent using affected sib-pair data. It has been clarified in the literature that the mean test has higher power than the TDT when linkage disequilibrium (LD) is weak but the TDT has higher power than the mean test when LD is strong. These results indicate that both tests are sensitive to linkage disequilibrium, though their responses are different. The development of a robust method against the interference of linkage disequilibrium is considered. The combination idea is employed to combine the TDT and the mean test to search for such a robust statistic using case-parent tetrad data (two parents and two affected children). Several available combining methods are taken into account in this study. Numerical comparisons of these methods with the published ones are presented.

The kurtosis coefficient is often regarded as a measure of the tail heaviness of a distribution relative to that of the normal distribution. However, it also measures the peakedness of a distribution, hence there is no agreement on what kurtosis really estimates. Another disadvantage of the kurtosis is that its interpretation and consequently its use is restricted to symmetric distributions. Moreover, the kurtosis coefficient is very sensitive to outliers in the data. To overcome these problems, several measures of left and right tail weight for univariate continuous distributions are proposed. They can be applied to symmetric as well as asymmetric distributions that do not need to have finite moments. Their interpretation is clear and they are robust against outlying values. The breakdown value and the influence functions of these measures and the resulting asymptotic variances are discussed and used to construct goodness-of-fit tests. Simulated as well as real data are employed for further comparison of the proposed measures.

A mixture model for preferences data, which adequately represents the composite nature of the elicitation mechanism in ranking processes, is proposed. Both probabilistic features of the mixture distribution and inferential and computational issues arising from the maximum likelihood parameters estimation are addressed. Moreover, empirical evidence from different data sets confirming the goodness of fit of the proposed model to many real preferences data is shown. © 2004 Elsevier B.V. All rights reserved.

There are many variety of tests for normality in the literature. Some are either too simple to be powerful or too difficult to apply. Most of them only suit for certain situations due to various kinds of biases. Powerful omnibus tests of normality based on the likelihood ratio are proposed. In terms of power for testing normality, they outperform the best tests in the literature, including the Shapiro-Wilk and Anderson-Darling tests.

One of the major problems for which satisfactory small-sample rank-based test procedures are only recently being developed is that of evaluating the possible presence of interaction in the standard two-way layout. This is especially true when there is only one observation on each treatment-block combination. In this paper we take a goodness-of-fit approach to this problem using the idea of grouping “similar” block rank permutations to obtain a workable number of collapsed categories. This grouping is based on the concepts of decomposable and indecomposable classes of rank vectors considered by Comtet (C.R. Acad. Sci. Paris 275 (1972) 569; in Advanced Combinatorics: the Art of Finite and Infinite Expansions, D. Reidel, Boston, 1974, p. 262). Alignment within treatment levels is utilized to adjust for the possible presence of treatment effects prior to the application of the goodness-of-fit procedure to test for interaction. We present the results of a substantial Monte Carlo simulation power study of the proposed procedure.

This paper extends the Bayes marginal model plot (BMMP) model assessment technique from a traditional logistic regression setting to a multilevel application in the area of criminal justice. Convicted felons in the United States receive either a prison sentence or a less severe jail or non-custodial sentence. Researchers have identified many determinants of sentencing variation across the country, some individual such as type of crime and race, and some based on geographical units such as county crime rate. Multilevel rather than conventional regression should be used to quantify any interplay between such individual- and county-level effects since the covariates have a hierarchical structure. Questions arise, however, as to whether a multilevel model provides an adequate fit to the data, and whether the computational burden of a multilevel model over a conventional model is justified. Residual plots, traditionally used to assess regression models, are difficult to interpret with a binary response variable and multilevel covariates, as in this case. BMMPs, an alternative graphical technique, can be used to visualize goodness of fit in such settings. The plots clearly demonstrate the need to use multilevel modeling when analyzing data such as these.

A copula based statistical method for fitting joint cumulative returns between a market index and a single stock to daily data is proposed. Modifying the method of inference functions for margins (IFM method), one performs two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-fit test. The copula fitting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory. The best overall fits are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel–Hougaard copulas. Finally, a significant application to covariance estimation for the linear Spearman copula is discussed.

New goodness-of-fit tests, based on bootstrap estimated expectations of probability integral transformed order statistics, are derived for the error distribution of autoregressive time-series models. The resulting test statistics are location and scale invariant, and are sensitive to discrepancies at the tails of the hypothesised distribution. It is shown how critical points for all sample sizes and significance levels can be obtained by using the parametric bootstrap. A simulation study shows that one of the new proposed tests is more powerful than the established Shapiro-Wilk test and the well-known score test (also known as the Lagrange multiplier test), for a wide range of alternative distributions and sample sizes.

Change point models are often used to model longitudinal data. To estimate the change point, Bayesian (Biometrika 62 (1975) 407; Appl. Statist. 41 (1992) 389; Biometrics 51 (1995) 236) or profile likelihood (Statist. Med. 19 (2000) 1555) methods may be used.We compare and contrast the two methods in analyzing longitudinal cognitive data from the Bronx Aging Study. The Bayesian method has advantages over the profile likelihood method in that it does not require all subjects to have the same change point. Caution must be taken regarding sensitivity to choice of prior distribution, identifiability, and goodness of fit. Analyses show that decline in memory precedes diagnosis of dementia by 7.5-8 years, and individual change points are not needed to model heterogeneity across subjects.

This contribution has two purposes. First, to review established methods for latent class model diagnostics and, second, to present and discuss less well-known methods. To the first group belong: (a) overall goodness-of-fit tests using, for large samples, the asymptotic chi-square approximation (Pearson, likelihood ratio, Read-Cressie power divergence statistic) and, for sparse data, resampling techniques such as parametric bootstrap; (b) residual analysis of the pattern frequencies for which also bootstrapping their reference distributions in the presence of smaller samples is recommended. Less well-known are (a) a chi-square goodness-of-fit test not being based on the full information provided by the pattern frequencies but rather using partial information available from lower-order marginals of the contingency table formed by the items, (b) a simple test for the odds ratios of pairs of items, (c) the Rudas-Clogg-Lindsay index of lack of fit of which (d) a new variant is considered, too. The methods are applied to both generated and empirical data to demonstrate their usefulness in latent class model diagnosis.

Suppose a random sample of individuals is drawn from a population with unknown number of disjoint classes. The population is said to be homogeneous if all the classes have the same proportions and heterogeneous otherwise. Although there is a vast literatutre towards estimation of the number of classes, and the performance of estimators is related to the homogeneity assumption, testing homogeneity has received little investigation. Tests and diagnostics for heterogeneity can be applied while the number of classes is eliminated as a nuisance parameter by conditioning. The goodness-of-fit test is discussed and a dispersion score test is presented. Two graphical diagnostics are used to detect the existence of heterogeneity. These tests and diagnostics are illustrated by two data sets.

In many Bradley-Terry models a more or less explicit assumption is that all decisions of the judges are independent. An assumption which might be questionable at least for the decisions of a given judge. In paired comparison studies, a judge chooses among objects several times, and in such cases, judgements made by the same judge are likely to be dependent. A log-linear representation for the Bradley-Terry model will be developed, which takes into account dependencies between judgements. The modelling of the dependencies is embedded in the analysis of multiple binomial responses, which has the advantage of interpretability in terms of conditional odds ratios. Furthermore, the modelling is done in the framework of generalized linear models, thus parameter estimation and the assessment of goodness of fit can be obtained in the standard way by using e.g. GLIM or another standard software.

The present paper deals with main-effects models for the analysis of two-way two-mode frequency data that are obtained by summing replicated binary associations over replications. Main-effects generalized linear models (GLM) which explain the transformed expected frequencies as an additive function of row- and column parameters, include many models that only differ in the particular transformation—called link function—that is applied. In this paper, besides the standard logit link, we also consider a log-, a complementary log-, and an identity link as models with these link functions are equivalent to one-feature probability matrix decomposition models (PMD). Reparameterizing main-effects GLMs into one-feature PMD models yields two advantages: First, it offers a process interpretation for the way the observed data are generated; in particular, in PMD models it is assumed that covert events with binary outcomes take place on both the row- and the column side and that these outcomes are subsequently mapped into an observed data point according to a prespecified mapping rule which may reflect a communality or a dominance relation. Second, it may facilitate parameter estimation of these models as for the original parameterization maximum likelihood estimates in the interior of the parameter space may not exist. Methods to evaluate the goodness of fit of the models under study are discussed and the models are applied to data on psychiatric diagnosis and on educational measurement.

Graphical displays of multivariate data often clearly exhibit features of the expectations even though the data themselves are poorly fitted by the displays. Thus, it often occurs that ordinations and biplots that poorly fit the sample data still reveal salient characteristics such as clusters of similar individuals and patterns of correlation. This paper provides an explanation of this seemingly paradoxical phenomenon and shows that when many variables are analyzed, the common measure of goodness of fit of a lower rank approximation often seriously underestimates the closeness of the fit to underlying patterns. The paper also provides some guidelines on better estimates of the latter goodness of fit.

Goodness-of-fit test statistics based on the empirical distribution function (EDF) are considered for the generalized logarithmic series distribution. The α levels of the tests for small or moderate sample sizes are close to the chosen nominal 5% and 10% significance levels. For small or moderate sample sizes, the tests are compared with respect to their simulated power of detecting some alternative hypotheses against a null hypothesis of generalized logarithmic series distribution. The discrete version of the Cramer–von Mises and Anderson–Darling tests are found to be the most powerful pair among the EDF tests.

The main statistical packages for analysing linear models in categorical data either permit a wide application of the weighted least-squares methodology or confine the application of the maximum-likelihood approach to specific forms of these models. In this work, the likelihood equations for the general linear model are derived in a neat form suitable to their iterative solving and computational implementation.

Based on the notion of multivariate quantiles in terms of minimum volume ellipsoids as introduced in Einmahl and Mason (1992), a generalized chi-square quantile plot is constructed from which tests for multivariate normality are derived. The generalized quantile plot also offers some extra possibilities for data analysis, e.g. in identifying and describing non-normality. An algorithm is proposed to approximate the generalized quantiles, extending existing methods to approximate median quantiles with only a minor extra computational cost and preserving the effectivity of the method. A power analysis is included and the method is illustrated with some case-studies.

Because of the low power of the Kolmogorov-Smirnov goodness-of-fit test in case of multimodal alternatives this paper investigates a family of tests indexed by an integer-valued parameter K≥2. The sensitivity of these tests to multimodal alternatives increases as K increases. The Kolmogorov-Smirnov test is obtained for K=2. The other elements of this family measure local deviations from uniformity in the same way as the Kolmogorov-Smirnov test measures the global deviation. Hence they may be regarded as generalized Kolmogorov-Smirnov tests. Tables of critical values are given and applications to the problem of testing for white noise are discussed. In addition, a further family of related tests is proposed. A Monte Carlo power study compares all the tests proposed in this note with Neyman's smooth test and different versions of the length test introduced by Reschenhofer and Bomze (Biometrika, 78 (1991) 207–216; Biometrika, 79 (1992) 859). A secondary result of this simulation study is that the possibly most widely used version of the length test can break down when a peak is located near the boundary.