Inference after separated hypotheses testing: an empirical investigation for linear models

Model selection problems arise while constructing unbiased or asymptotically unbiased estimators of measures known as discrepancies to find the best model. Most of the usual criteria are based on goodness-of-fit and parsimony. They aim to maximize a transformed version of likelihood. For linear regression models with normally distributed error, the situation is less clear when two models are equivalent: are they close to or far from the unknown true model? In this work, based on stochastic simulation and parametric simulation, we study the results of Vuong's test, Cox's test, Akaike's information criterion, Bayesian information criterion, Kullback information criterion and bias corrected Kullback information criterion and the ability of these tests to discriminate between non-nested linear models.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

An investigation of lag identification tools for vector nonlinear time series

Exploratory methods for determining appropriate lagged vsrlables in a vector nonlinear time series model are investigated. The first is a multivariate extension of the R statistic considered by Granger and Lin (1994), which is based on an estimate of the mutual information criterion. The second method uses Kendall's ρ and partial ρ statistics for lag determination. The methods provide nonlinear analogues of the autocorrelation and partial autocorrelation matrices for a vector time series. Simulation studies indicate that the R statistic reliabiy identifies appropriate lagged nonlinear moving average terms in a vector time series, while Kendall's ρ and partial ρ statistics have some power in identifying appropirate lagged nonlinear moving average and autoregressive terms, respectively, when the nonlinear relationship between lagged variables is monotonic. For illustration, the methods are applied to set of annual temperature and tree ring measurements at Campito Mountain In California.     

A comparison of robust versions of the AIC based on M-, S- and MM-estimators

Variable selection in the presence of outliers may be performed by using a robust version of Akaike's information criterion (AIC). In this paper, explicit expressions are obtained for such criteria when S- and MM-estimators are used. The performance of these criteria is compared with the existing AIC based on M-estimators and with the classical non-robust AIC. In a simulation study and in data examples, we observe that the proposed AIC with S and MM-estimators selects more appropriate models in case outliers are present.     

A weighted test of internal symmetry

This study examines extensions of McNemar's Test with multinomial responses, and proposes a linear weighting scheme, based on the distance of the response change, that is applied to one of these extensions (Bowker's test). This weighted version of Bowker's test is then appropriate for ordinal response variables. A Monte Carlo simulation was conducted to examine the Type I error rate of the weighted Bowker's test for a cross-classification table based on a five-category ordinal response scale. The weighted Bowker's test was also applied to a data set involving change in student attitudes towards mathematics. The results of the weighted Bowker's test were compared with the results of Bowker's test applied to the same set of data.     

A Discussion of Permutation Tests Conditional to Observed Responses in Unreplicated 2 M Full Factorial Designs

ABSTRACT

In this article we present a new solution to test for effects in unreplicated two-level factorial designs. The proposed test statistic, in case the error components are normally distributed, follows an F random variable, though our attention is on its nonparametric permutation version. The proposed procedure does not require any transformation of data such as residualization and it is exact for each effect and distribution-free. Our main aim is to discuss a permutation solution conditional to the original vector of responses. We give two versions of the same nonparametric testing procedure in order to control both the individual error rate and the experiment-wise error rate. A power comparison with Loughin and Noble's test is provided in the case of a unreplicated 2⁴ full factorial design.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

Detection of outliers in growth curve models

The growth curve model introduced by Potthoff and Roy (1964) is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. In this paper, we discuss procedures for detection of outliers in growth curve models for mean-slippage and dispersion-slippage outlier model. The distributions of the test statistics are discussed and the values of significant probabilities are given using Bonferronl's bounds. Some simulation results are also presented.     

The Finite-Sample Performance of White's Test for Heteroskedasticity Under Stochastic Regressors

We study the finite-sample properties of White's test for heteroskedasticity in stochastic regression models where explanatory variables are random and not given. We investigate by simulation the effect of non independence of explanatory variables and error term and heteroskedasticity on White's test. A standard bootstrap method in the computationally convenient form is found to work well with respect to the size and power.     

Criteria for Linear Model Selection Based on Kullback's Symmetric Divergence

Model selection criteria are frequently developed by constructing estimators of discrepancy measures that assess the disparity between the 'true' model and a fitted approximating model. The Akaike information criterion (AIC) and its variants result from utilizing Kullback's directed divergence as the targeted discrepancy. The directed divergence is an asymmetric measure of separation between two statistical models, meaning that an alternative directed divergence can be obtained by reversing the roles of the two models in the definition of the measure. The sum of the two directed divergences is Kullback's symmetric divergence. In the framework of linear models, a comparison of the two directed divergences reveals an important distinction between the measures. When used to evaluate fitted approximating models that are improperly specified, the directed divergence which serves as the basis for AIC is more sensitive towards detecting overfitted models, whereas its counterpart is more sensitive towards detecting underfitted models. Since the symmetric divergence combines the information in both measures, it functions as a gauge of model disparity which is arguably more balanced than either of its individual components. With this motivation, the paper proposes a new class of criteria for linear model selection based on targeting the symmetric divergence. The criteria can be regarded as analogues of AIC and two of its variants: 'corrected' AIC or AICc and 'modified' AIC or MAIC. The paper examines the selection tendencies of the new criteria in a simulation study and the results indicate that they perform favourably when compared to their AIC analogues.     

Un critère de choix de variables en analyse en composantes principales fondé sur des modèles graphiques gaussiens particuliers

We focus on the problem of selection of a subset of the variables so as to preserve the multivariate data structure that a principal-components analysis of the initial variables would reveal. We propose a new method based on some adapted Gaussian graphical models. This method is then compared with those developed by Bonifas et al. (1984) and Krzanowski (1987a, b). It appears that the criteria for all methods consider the same correlation submatrices and often lead to similar results. The proposed approach offers some guidance as to the number of variables to be selected. In particular, Akaike's information criterion is used.     

Information criteria for inhomogeneous spatial point processes

The theoretical foundation for a number of model selection criteria is established in the context of inhomogeneous point processes and under various asymptotic settings: infill, increasing domain and combinations of these. For inhomogeneous Poisson processes we consider Akaike's information criterion and the Bayesian information criterion, and in particular we identify the point process analogue of ‘sample size’ needed for the Bayesian information criterion. Considering general inhomogeneous point processes we derive new composite likelihood and composite Bayesian information criteria for selecting a regression model for the intensity function. The proposed model selection criteria are evaluated using simulations of Poisson processes and cluster point processes.     

Testing the homogeneity of several two‐parameter populations

The authors extend Fisher's method of combining two independent test statistics to test homogeneity of several two‐parameter populations. They explore two procedures combining asymptotically independent test statistics: the first pools two likelihood ratio statistics and the other, score test statistics. They then give specific results to test homogeneity of several normal, negative binomial or beta‐binomial populations. Their simulations provide evidence that in this context, Fisher's method performs generally well, even when the statistics to be combined are only asymptotically independent. They are led to recommend Fisher's test based on score statistics, since the latter have simple forms, are easy to calculate, and have uniformly good level properties.     

A new revised version of McNemar's test for paired binary data

By considering separately B and C, the frequencies of individuals who consistently gave positive or negative answers in before and after responses, a new revised version of McNemar's test is derived. It improves upon Lu's revised formula, which considers B and C together. When both B and C are 0, the new revised version produces the same results as McNemar's test. When one of B and C is 0, the new revised test produces the same results as Lu's version. Compared to Lu's version, the new revised test is a more complete revision of McNemar's test.     

A complete solution to the problem of robustness of Grubbs's test

Srivastava (1980) showed that Grubbs's test for detecting a univariate outlier is robust against the effect of intraclass correlation structure. Young, Pavur, and Marco (1989) extended this result by proving that both the significance level and the power of Grubbs's test remain unchanged within a wider family of dispersion matrices, introduced by Baldessari (1966) in a different context. In this note, we derive a complete solution of the problem by establishing that the characteristics of Grubbs's test are invariant with respect to a given dispersion matrix if and only if it has Baldessari's structure.     

Mixed Graphical Model Selection Using Holm's Procedure

The problem of selecting a graphical model is considered as a performing simultaneously multiple tests. The control of the overall Type I error on the selected graph is done using the so famous Holm's procedure. We prove that when we use a consistent edge exclusion test the selected graph is asymptotically equal to the true graph with probability at least equal to a fixed level 1 ? α. This method is then used for the selection of mixed concentration graph models by performing the χ²-edge exclusion test. We also apply the method to two classical examples and to simulated data. We compare the overall error of the selected model with the one obtained using the stepwise method. We establish that the control is better when we use the Holm's procedure.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

A Cautionary Note on the Analysis of Extreme Data with Cox Regression

This article analyzes a small censored data set to demonstrate the potential dangers of using statistical computing packages without understanding the details of statistical methods. The data, consisting of censored response times with heavy ties in one time point, were analyzed with a Cox regression model utilizing SAS PHREG and BMDP2L procedures. The p values, reported from both SAS PHREG and BMDP2L procedures, for testing the equality of two treatments vary considerably. This article illustrates that (1) the Breslow likelihood used in both BMDP2L and SAS PHREG procedures is too conservative and can have a critical effect on an extreme data set, (2) Wald's test in the SAS PHREG procedure may yield absurd results from most likelihood models, and (3) BMDP2L needs to include more than just the Breslow likelihood in future development.     

Some robust anova procedures under heteroscedasticity and nonnormality

In this paper we consider the problem of comparing several means under heteroscedasticity and nonnormality. By combining Huber‘s M-estimators with the Brown-Forsythe test, several robust procedures were developed; these procedures were compared through computer simulation studies with the Tan-Tabatabai procedure which was developed by combining Tiku's MML estimators with the Brown-Forsythe test. The numerical results indicate clearly that the Tan-Tabatabai procedure is considerably more powerful than tests based on Huber's M-estimators over a wide range of nonnormal distributions.     

Data-driven smooth test for a location-scale family

A combination of a smooth test statistic and (an approximate) Schwarz's selection rule has been proposed by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. ((1997). Data-driven smooth tests for composite hypotheses. Ann. Statist. 25, 1222–1250) as a solution of a standard goodness-of-fit problem when nuisance parameters are present. In the present paper we modify the above solution in the sense that we propose another analogue of Schwarz's rule and rederive properties of it and the resulting test statistic. To avoid technicalities we restrict our attention to location-scale family and method of moments estimators of its parameters. In a parallel paper [Janic-Wróblewska, A. (2004). Data-driven smooth tests for the extreme value distribution. Statistics, in press] we illustrate an application of our solution and advantages of modification when testing of fit to extreme value distribution.