Null Statistics Research Articles

Single-index relative error regression models

We consider the estimation and hypothesis test for single-index model by proposing a profile mean squares relative error estimation method, coupling with a leave-one-component-out method. The asymptotic normality for the PMSRE estimator is studied. For the hypothesis test of the single-index parameter, a restricted estimator under the null hypothesis and test statistics are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Lastly, we make some comparisons between the proposed estimator and the profile least squares estimator and the profile least product relative estimator through the simulation. Our simulation studies indicate that the profile mean squares relative error estimator provides a better alternative under some settings, particularly for multimodal or asymmetric distributions of the error term.

Smoothed Nested Testing on Directed Acyclic Graphs.

We consider the problem of multiple hypothesis testing when there is a logical nested structure to the hypotheses. When one hypothesis is nested inside another, the outer hypothesis must be false if the inner hypothesis is false. We model the nested structure as a directed acyclic graph, including chain and tree graphs as special cases. Each node in the graph is a hypothesis and rejecting a node requires also rejecting all of its ancestors. We propose a general framework for adjusting node-level test statistics using the known logical constraints. Within this framework, we study a smoothing procedure that combines each node with all of its descendants to form a more powerful statistic. We prove a broad class of smoothing strategies can be used with existing selection procedures to control the familywise error rate, false discovery exceedance rate, or false discovery rate, so long as the original test statistics are independent under the null. When the null statistics are not independent but are derived from positively-correlated normal observations, we prove control for all three error rates when the smoothing method is arithmetic averaging of the observations. Simulations and an application to a real biology dataset demonstrate that smoothing leads to substantial power gains.

Wavelet analysis in ecology and epidemiology: impact of statistical tests

Wavelet analysis is now frequently used to extract information from ecological and epidemiological time series. Statistical hypothesis tests are conducted on associated wavelet quantities to assess the likelihood that they are due to a random process. Such random processes represent null models and are generally based on synthetic data that share some statistical characteristics with the original time series. This allows the comparison of null statistics with those obtained from original time series. When creating synthetic datasets, different techniques of resampling result in different characteristics shared by the synthetic time series. Therefore, it becomes crucial to consider the impact of the resampling method on the results. We have addressed this point by comparing seven different statistical testing methods applied with different real and simulated data. Our results show that statistical assessment of periodic patterns is strongly affected by the choice of the resampling method, so two different resampling techniques could lead to two different conclusions about the same time series. Moreover, our results clearly show the inadequacy of resampling series generated by white noise and red noise that are nevertheless the methods currently used in the wide majority of wavelets applications. Our results highlight that the characteristics of a time series, namely its Fourier spectrum and autocorrelation, are important to consider when choosing the resampling technique. Results suggest that data-driven resampling methods should be used such as the hidden Markov model algorithm and the 'beta-surrogate' method.

An Expected Value Analysis of When to Avoid Type 1 and Type 2 Statistical Errors in Applied Research

For decades, applied researchers have relied heavily on either the null hypothesis significance testing procedure (NHSTP) or Bayesian statistics to help them determine the probability of the null hypothesis given their findings. While there may be some flaws in both methods, there are cases where the probability of the null hypothesis can be known, or at least estimated fairly accurately, thus making such analyses worthwhile. Assuming the worth of using such analyses, another issue in play is whether or not to avoid Type 1 or Type 2 statistical errors. While much has been said about the importance of avoiding both types of errors, we offer a new way of looking at the situation. We use an Expected Value analysis to help applied researchers determine when to avoid either type of error. We give examples and perform simulations to back up our mathematical arguments. Theoretical and applied issues are discussed.

Association of heat shock proteins with Parkinson’s disease

Association of heat shock proteins with Parkinson’s disease

Optimal significance analysis of microarray data in a class of tests whose null statistic can be constructed

Microarray data often consist of a large number of genes and a small number of replicates. We have examined testing the null hypothesis of equality of mean for detecting differentially expressed genes. The p-value for each gene is often estimated using permutation samples not only for the target gene but also for other genes. This method has been widely used and discussed. However, direct use of the permutation method for the p-value estimation may not work well, because two types of genes are mixed in the sample; some genes are differentially expressed, whereas others are not. To overcome this difficulty, various methods for appropriately generating null permutation samples have been proposed. In this paper, we consider two classes of test statistics that are naturally modified to null statistics. We then obtain the uniformly most powerful (UMP) unbiased tests among these classes. If the underlying distribution is symmetric, the UMP unbiased test statistic is similar to that proposed by Pan (Bioinformatics 19:1333–1340, 2003). Under another condition, the UMP unbiased test statistic has a different formula with one more degree of freedom and therefore is expected to give a more powerful test and a more accurate p-value estimation from a modified null statistic. In microarray data, because the number of replicates is often small, differences in the degree of freedom will produce large effects on the power of test and the variance of the p-value estimation. Some simulation studies and real data analyses are illustrated to investigate the performances of the methods.

Nonmonotonic power for tests of a mean shift in a time series§

The null hypothesis-based statistics CUSUM and QS are widely used for testing parameter stability. We provide examples, extensive simulation studies and theoretical results showing that these statistics fail to detect obvious shifts in the mean of a time series. Moreover, the detection probability can decrease when the magnitude of the shift in mean increases. Estimation of nuisance parameters under the null is identified as an important cause of this counterintuitive behavior of the power function. Results indicate that tests designed for the specific alternative of a shift in mean of a time series should be preferred. §This paper was presented at the 2002 Winter Meetings of the Econometric Society in Atlanta, GA.

An Improved Nonparametric Approach for Detecting Differentially Expressed Genes with Replicated Microarray Data

Previous nonparametric statistical methods on constructing the test and null statistics require having at least 4 arrays under each condition. In this paper, we provide an improved method of constructing the test and null statistics which only requires 2 arrays under one condition if the number of arrays under the other condition is at least 3. The conventional testing method defines the rejection region by controlling the probability of Type I error. In this paper, we propose to determine the critical values (or the cut-off points) of the rejection region by directly controlling the false discovery rate. Simulations were carried out to compare the performance of our proposed method with several existing methods. Finally, our proposed method is applied to the rat data of Pan et al. (2003). It is seen from both simulations and the rat data that our method has lower false discovery rates than those from the significance analysis of microarray (SAM) method of Tusher et al. (2001) and the mixture model method (MMM) of Pan et al. (2003).

Construction of null statistics in permutation-based multiple testing for multi-factorial microarray experiments

The parametric F-test has been widely used in the analysis of factorial microarray experiments to assess treatment effects. However, the normality assumption is often untenable for microarray experiments with small replications. Therefore, permutation-based methods are called for help to assess the statistical significance. The distribution of the F-statistics across all the genes on the array can be regarded as a mixture distribution with a proportion of statistics generated from the null distribution of no differential gene expression whereas the other proportion of statistics generated from the alternative distribution of genes differentially expressed. This results in the fact that the permutation distribution of the F-statistics may not approximate well to the true null distribution of the F-statistics. Therefore, the construction of a proper null statistic to better approximate the null distribution of F-statistic is of great importance to the permutation-based multiple testing in microarray data analysis. In this paper, we extend the ideas of constructing null statistics based on pairwise differences to neglect the treatment effects from the two-sample comparison problem to the multifactorial balanced or unbalanced microarray experiments. A null statistic based on a subpartition method is proposed and its distribution is employed to approximate the null distribution of the F-statistic. The proposed null statistic is able to accommodate unbalance in the design and is also corrected for the undue correlation between its numerator and denominator. In the simulation studies and real biological data analysis, the number of true positives and the false discovery rate (FDR) of the proposed null statistic are compared with those of the permutated version of the F-statistic. It has been shown that our proposed method has a better control of the FDRs and a higher power than the standard permutation method to detect differentially expressed genes because of the better approximated tail probabilities.

On Some Tests of the Covariance Matrix Under General Conditions

We consider the problem of testing the hypothesis about the covariance matrix of random vectors under the assumptions that the underlying distributions are nonnormal and the sample size is moderate. The asymptotic expansions of the null distributions are obtained up to n−1/2. It is found that in most cases the null statistics are distributed as a mixture of independent chi-square random variables with degree of freedom one (up to n−1/2) and the coefficients of the mixtures are functions of the fourth cumulants of the original random variables. We also provide a general method to approximate such distributions based on a normalization transformation.

TESTING FOR NONLINEARITY OF THE CONTRACTION SEGMENTS IN UTERINE ELECTROMYOGRAPHY

In this report, we test for possible nonlinearity of the contraction segments interspersed in a uterine electromyography (EMG), recorded externally with abdominal electrodes. There have been several reports in which the uterine contractility had been assumed to be an auto-regressive process and others have hypothesized it as a nonlinear process and possibly chaotic. The surrogate data testing was used successfully to detect nonlinear behavior of physiological systems. However, there have been case studies, which discuss spurious identification of nonrandom structures. The proper choice of the null hypothesis and discriminant statistics plays a crucial role in the surrogate data testing. We have chosen the approximate entropy as the discriminant statistic for our tests. The null hypothesis addressed here is that the uterine contraction is a linearly correlated noise transformed by a nonlinear function. We applied the Amplitude Adjusted Fourier Transform (AAFT) and the Iterated Amplitude Adjusted Fourier Transform (IAAFT) tests to the uterine contraction data. The Kolmogorov Smirnov (D) statistics identified the discriminant values of the surrogates to be from a Gaussian distribution. Parametric testing showed a very low significance value, (~2σ), which indicated the absence of nonrandom structure in the contraction segment.

Generalized Fock spaces, new forms of quantum statistics and their algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ``doubly-infinite'' statistics, many representations of orthostatistics, Hubbard statistics and its variations.

An algebra for ordered particles

We construct an algebra of creation and destruction operators for a system whose multiparticle states have the property that the permutation of coordinates is forbidden. Although this “null statistics” can be related to a singularity of quantum-group-related structures, it exists as an independent new statistics.

Modeling Total Particulate Measurement in Cotton

The in situ native standard method is presented to measure total particulate content (nonlint trash plus dust) in cotton. The method is based on first rendering a sample free of particulate by repetitive mechanical cleaning and then spiking the cleaned lint with the recoverable particulate. A calibration curve is generated with property constant λi. The same property is then measured in the uncleaned cotton. An apparent particulate burden, τi, is determined from the calibration curve. By observing different properties and using null hypothesis statistics it is possible to calculate a mean value for the accepted total particulate content, τ. Earlier modeling predicted the null hypothesis error in τi is a function of percentage particulate recovered for spiking and the ratio λi/θ i where θ is the nonrecoverable particulate ith property constant. The model is refined to minimize error contributions from additional sources in the measurement system. A positive bias is expected if particulate size distribution in the spikes is different from that in the stock. A positive or negative error may occur if indigenous fiber material isolated from the cleaned lint is excluded in spike prepara tion. Experimental data is presented. to verify model predictions.