A generalization of Friedman's rank statistic

Abstract  In this paper a very natural generalization of the two-way analysis of variance rank statistic of F riedman is given. The general distribution-free test procedure based on this statistic for the effect of J treatments in a random block design can be applied in general two-way layouts without interactions and with different numbers of the continuous observations per cell provided the design scheme is connected. The asymptotic distribution under the null hypothesis of the test statistic is derived. A comparison with the method of m rankings of B enard and van E lteren is made. The disadvantage of B enard and van E lteren's test procedure is that the number of observations per block does influence the statistic twice, namely firstly by the number itself, as it should, and see ondly by the level of the ranks which will be different in different blocks if the numbers of observations per block are different. The proposed test statistic is not sensitive to differences in the levels of the ranks caused by the different numbers of observations per block. The test is derived from considerhg the K ruskal -W allis statistics per block.
Finally, the results of simulation experiments are given. The simulation is carried out for three designs and a number of normal location alternatives and gives some information about the power of the suggested test procedure. A comparison is made with B enard and van E lteren's test and with the classical analysis of variance technique. For some simple orthogonal designs the exact null distributions of B enard and van E lteren's test and the proposed test are compared.     

Exact linear rank tests for two‐sample equivalence problems with continuous data

The present paper introduces a methodology for the semiparametric or non‐parametric two‐sample equivalence problem when the effects are specified by statistical functionals. The mean relative risk functional of two populations is given by the average of the time‐dependent risk. This functional is a meaningful non‐parametric quantity, which is invariant under strictly monotone transformations of the data. In the case of proportional hazard models, the functional determines just the proportional hazard risk factor. It is shown that an equivalence test of the type of the two‐sample Savage rank test is appropriate for this functional. Under proportional hazards, this test can be carried out as an exact level α test. It also works quite well under other semiparametric models. Similar results are presented for a Wilcoxon rank‐sum test for equivalence based on the Mann–Whitney functional given by the relative treatment effect.     

An alternative model for the analysis of variance with random effects

The standard model for the analysis of variance with random effects implies, for the case of two independent variables, that single effects must be tested not against the error, but against the interaction mean squares. This causes, in comparison with the fixed effects AV, a considerable loss of test power, particularly for the 2 × 2 table. An alternative modelling of the interaction effect is proposed which completely avoids the loss of power.     

Type I error and test power of different tests for testing interaction effects in factorial experiments

A simulation study was conducted to investigate the effect of non normality and unequal variances on Type I error rates and test power of the classical factorial anova F‐test and different alternatives, namely rank transformation procedure (FR), winsorized mean (FW), modified mean (FM) and permutation test (FP) for testing interaction effects. Simulation results showed that as long as no significant deviation from normality and homogeneity of the variances exists, generally all of the tests displayed similar results. However, if there is significant deviation from the assumptions, the other tests are observed to be affected at considerably high levels except FR and FP tests. As a result, when the assumptions of factorial anova F‐test are not met or, in the case those assumptions are not tested whether met, it can be concluded that using FR and FP tests is more suitable than the classical factorial anova F‐test.     

Significance levels of linear rank tests using Edgeworth expansions

We study one aspect of applying Edgeworth expansions to linear rank statistics. Since the use of such expansions is often recommended already for moderate sample sizes we investigate for this case the gain of accuracy for the level of significance of some linear rank tests when their critical values are derived from an Edgeworth expansion instead of from a normal approximation. We verify Does' conditions (1983) for the validity of the expansions for four rank statistics of general interest and show by a numerical study that an Edge-worth expansion does not outperform the normal approximation in all situations. A considerable improvement shows up however for the Klotz test at the 5% level.     

A GENERALIZED SIGNED–RANK TEST FOR COMPARISON OF P TREATMENTS

Abstract. Several generalizations of the signed–rank test for testing of equality of p treatments in p related samples against general and trend alternative are discussed. A new generalized test is described and exemplified.     

Generalized reduced rank tests using the singular value decomposition

We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson [Annals of Mathematical Statistics (1951), 22, 327–351] sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald [Journal of the American Statistical Association (1996), 91, 1301–1309] and Gill and Lewbel [Journal of the American Statistical Association (1992), 87, 766–776] a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith [Econometric Theory (2000), 16, 151–175] usage of numerical optimization for the objective function statistic of Cragg and Donald [Journal of Econometrics (1997), 76, 223–250] and ignoring the non-negativity restriction on the singular values in Ratsimalahelo [2002, Rank test based on matrix perturbation theory. Unpublished working paper, U.F.R. Science Economique, University de Franche-Comté]. In the non-stationary cointegration case, the limiting distribution of the new rank statistic is identical to that of the Johansen trace statistic.     

Inference on the cointegration rank in fractionally integrated processes

For univariate time series we suggest a new variant of efficient score tests against fractional alternatives. This test has three important merits. First, by means of simulations we observe that it is superior in terms of size and power in some situations of practical interest. Second, it is easily understood and implemented as a slight modification of the Dickey–Fuller test, although our score test has a limiting normal distribution. Third and most important, our test generalizes to multivariate cointegration tests just as the Dickey–Fuller test does. Thus it allows to determine the cointegration rank of fractionally integrated time series. It does so by solving a generalized eigenvalue problem of the type proposed by Johansen (J. Econ. Dyn. Control 12 (1988) 231). However, the limiting distribution of the corresponding trace statistic is χ², where the degrees of freedom depend only on the cointegration rank under the null hypothesis. The usefulness of the asymptotic theory for finite samples is established in a Monte Carlo experiment.     

Stochastic labelling of biological images

Many hypotheses made by experimental researchers can be formulated as a stochastic labelling of a given image. Some stochastic labelling methods for random closed sets are proposed in this paper. Molchanov (I. Molchanov, 1984, Theor. Probability and Math. Statist. 29 , 113–119) provided the probabilistic background for this problem. However, there is a lack of specific labelling models. Ayala and Simó (G. Ayala and A. Simó, 1995, Advances in Applied Probability 27 , 293–305) proposed a method in which, given the whole set of connected components, every component is classified in a certain phase or category in a completely random way. Alternative methods are necessary in case the random labelling hypothesis is not reliable. A different kind of labelling method is proposed that considers the environment: the type of every connected component is a function of its location.
Two different biphase images are studied: a cross section of a nerve from a rat, and a cross section of an optic nerve from a lizard.     

Testing for the cointegration rank when some cointegrating directions are changing

We develop some tests for characterizing the cointegration space of a cointegrated vector autoregressive model when its long-run parameters are modified by a structural break at a known date. We first consider the case in which the break does not affect the loading factors and second the more general one in which all long-run parameters change. For each configuration, we design procedures to test for the cointegration rank as for the number of directions which are changing between the two regimes. For the simplest case, the cointegration rank test is also extended to the case of an unknown date of shift.     

Using orthogonal trend contrasts for testing ranked data with ordered alternatives

In the analysis of variance (Anova ) the use of orthogonal contrasts is quite common and is a traditional topic in many basic Anova courses. Similar ideas apply to rank tests. In this paper we present a simple and general method that allows an orthogonal contrast decomposition of rank test statistics such as the Kruskal‐Wallis, Friedman and Durbin statistics. The components of the test statistics are informative, particularly when ordered alternatives are of interest. The method can handle ties, and null distributions are readily available. Most of the methods are not new, but the way we present them is. Moreover, our formulation makes it easier to better understand and interpret the tests when the traditional location‐shift assumption does not hold. The methods are illustrated using several data sets.     

Control Function Approach for Partly Ordered Endogenous Treatments: Military Rank Premium in Wage

In treatment effect analysis, there are many cases where the treatment of interest is ordered (e.g. general‐education years or medicine doses) and the control treatment is not zero, but a different type of treatment (a vocational training or a surgery). We develop an approach to find effects of partly ordered treatments, while correcting for possible treatment endogeneity with nearly parametric control functions. We use this control function approach, along with its supplementary version, to estimate effects of military ranks (ordered treatments) on wage relative to non‐veteran status (control treatment) with the Wisconsin Longitudinal Study data. In our empirical analysis, the military rank effects differ much: officer has large positive effects, but enlisted ranks have small or no effects.     

Optimal and efficient repeated measurements designs for comparing test treatments with a control

Rearranging the sequence of test treatments and control treatment within each block of an A-optimal balanced treatment block design such that certain conditions are satisfied, the resulting design is an A-optimal repeated measurements designs when blocks are regarded as units or periods. The efficiencies of designs which are obtained from universally optimal repeated measurements designs with test treatments only by changing some treatment labels into control treatment are given.     

Identification Based on Difference‐in‐Differences Approaches with Multiple Treatments

This paper discusses identification based on difference‐in‐differences (DiD) approaches with multiple treatments. It shows that an appropriate adaptation of the common trend assumption underlying the DiD strategy for the comparison of two treatments restricts the possibility of effect heterogeneity for at least one of the treatments. The required assumption of effect homogeneity is likely to be violated because of non‐random assignment to treatment based on both observables and unobservables. However, this paper shows that, under certain conditions, the DiD estimate comparing two treatments identifies a lower bound in absolute values on the average treatment effect on the treated compared to the unobserved non‐treatment state, even if effect homogeneity is violated. This is possible if the treatments have ordered treatment effects, that is, in expectation, the effects of both treatments compared to no treatment have the same sign, and one treatment has a stronger effect than the other treatment on the respective recipients. Such assumptions are plausible if treatments are ordered or vary in intensity.     

Nonparametric methods for paired samples

The small sample and asymptotic properties of nonparametric tests for paired sampled are examined. Linear rank statistics are compared with the paired t-test and the Wilcoxon-signed-rank test in simulation studies. From a minimax point of view the linear rank statistics turn out to be the best. Moreover, it is illustrated that the Wilcoxon-signed-rank test should not be used if it is not clear that the differences of the pairs have a symmetric distribution.     

PERFORMANCE OF MURPHY'S TEST FOR TWO OUTLIERS IN A LINEAR MODEL

Abstract. In JOSHI and LALITHA (1986) a test for two outliers in the same direction in a linear model is discussed. Here the performance of this statistic is studied. For this, the exact non–null density function of the random variables involved in defining the statistic is obtained. Then a measure of performance is defined and it is applied to the case of a random sample from a normal distribution, as in this case the above said statistic reduces to the well known Murphy's test statistic. These values are then compared with the power values obtained by HAWKINS (1978).     

Mildly explosive autoregression under weak and strong dependence

A limit theory is developed for mildly explosive autoregression under both weakly and strongly dependent innovation errors. The asymptotic behaviour of the sample moments is affected by the memory of the innovation process both in the form of the limiting distribution and, in the case of long range dependence, in the rate of convergence. However, this effect is not present in least squares regression theory as it is cancelled out by the interaction between the sample moments. As a result, the Cauchy regression theory of Phillips and Magdalinos (2007a) is invariant to the dependence structure of the innovation sequence.     

Measure for Measure: Exact F Tests and the Mixed Models Controversy

We consider exact F tests for the hypothesis of null random factor effect in the presence of interaction under the two factor mixed models involved in the mixed models controversy. We show that under the constrained parameter ( CP ) model, even in unbalanced data situations, MSB/MSE (in the usual ANOVA notation) follows an exact F distribution when the null hypothesis holds. We also obtain an exact F test for what is generally (and erroneously) assumed to be an equivalent hypothesis under the unconstrained parameter ( UP ) model. For unbalanced data, such a corresponding test statistic does not coincide with MSB/MSAB (the test statistic advocated for balanced data cases). We compute the power of the exact test under different imbalance patterns and show that although the loss of power increases with the degree of imbalance, it still remains reasonable from a practical point of view.     

Exact distribution of Durbin's distribution-free test statistic for Balanced Incomplete Block Designs, and comparison with the Chi-square and F approximation

Durbin's distribution-free rank test can be used to test the null hypothesis that there are no differences among the treatments in a Balanced Incomplete Block Design. Until now only the chi-square and F approximations for the test statistic were known. In this paper the exact distribution has been given for 15 designs and comparison is made with the chi-square and F approximation.     

On Tests for a Normal Mean with Known Coefficient of Variation

Hinkley (1977) derived two tests for testing the mean of a normal distribution with known coefficient of variation (c.v.) for right alternatives. They are the locally most powerful (LMP) and the conditional tests based on the ancillary statistic for μ. In this paper, the likelihood ratio (LR) and Wald tests are derived for the one‐ and two‐sided alternatives, as well as the two‐sided version of the LMP test. The performances of these tests are compared with those of the classical t, sign and Wilcoxon signed rank tests. The latter three tests do not use the information on c.v. Normal approximation is used to approximate the null distribution of the test statistics except for the t test. Simulation results indicate that all the tests maintain the type‐I error rates, that is, the attained level is close to the nominal level of significance of the tests. The power functions of the tests are estimated through simulation. The power comparison indicates that for one‐sided alternatives the LMP test is the best test whereas for the two‐sided alternatives the LR or the Wald test is the best test. The t, sign and Wilcoxon signed rank tests have lower power than the LMP, LR and Wald tests at various alternative values of μ. The power difference is quite large in several simulation configurations. Further, it is observed that the t, sign and Wilcoxon signed rank tests have considerably lower power even for the alternatives which are far away from the null hypothesis when the c.v. is large. To study the sensitivity of the tests for the violation of the normality assumption, the type I error rates are estimated on the observations of lognormal, gamma and uniform distributions. The newly derived tests maintain the type I error rates for moderate values of c.v.