Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials

This paper is concerned with the power behaviour of four goodness-of-fit test statistics in sparse multinomials with k cells. Most previous work has been concerned only with both Pearson's X² and the likelihood ratio test statistics. We consider in this study, two additional test statistics, namely, the Cressie-Read test statistic – I(2/3) and the modified Freeman-Tukey test (FT) statistic. Because k ≥ 10 in this study, a Monte Carlo procedure based on 1000 simulated samples is used to estimate the powers for the four test statistics. Alternatives on various line segments are employed. Results suggest that none of the test statistics completely dominate the other and that the choice of which test to use depends on the nature of the alternative hypothesis. These results are consistent with those obtained by West and Kempthorne (1972), although, the Pearson's χ² test statistic may be preferred because of its closer approximation to the χ² distribution in terms of the attained α levels.     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

Using Discrete Distributions to Compute Powers for Goodness-of-Fit Test Statistics in a One-Way Multinomial Setting

Both the Bionomial and Poisson distributions are employed in this study to compute approximate powers for goodness-of-fit test statistics. The procedure adopted involves simulating 1000 samples from each of these distributions. These samples are then employed to compute both the randomized nominal critical values and estimated powers. The type I error rates returned from the use of the randomized critical levels Cα fall within the acceptable regions. We illustrate the use of the procedure with the Pearson's X² test statistic and show that this can readily be extended to any of the other well known goodness-of-fit test statistics.     

Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

Comparisons of Some Chi-Squared Goodness-of-Fit Test Statistics in Sparse One-Way Multinomials

The behavior of Pearson's X² test, the likelihood ratio test Y² and the two of its derivatives, G² and G_k², the Freeman-Tukey test (FT) and the Cressie and Read test Statistic I(2/3) are examined in this study. Estimated attained α levels based on 1000 simulated samples when the approximating distribution is χ_k-1², are computed for these tests for the various values of k, n and seven null hypotheses. Results from estimated power computations indicate that none of the test statistics has a clear advantage over any others, and that the choice of which test to use must therefore rest mainly on the performances with regards to the attained α levels when the χ² approximation is invoked. In this respect, the log-normal approximation proposed by Lawal and Upton (1980) is strongly recommended. This is closely followed by the I(2/3).     

Diagnostic Measures of Influence for s2

We present three intuitive diagnostic measures for judging the influence that single or multiple observations exert on the estimate s², the mean square error. The three measures are shown to be equivalent statistics. We discuss distributional properties of these measures and conclude with an example.     

Double-check: validation of diagnostic statistics for PLS-DA models in metabolomics studies

Partial Least Squares-Discriminant Analysis (PLS-DA) is a PLS regression method with a special binary ‘dummy’ y-variable and it is commonly used for classification purposes and biomarker selection in metabolomics studies. Several statistical approaches are currently in use to validate outcomes of PLS-DA analyses e.g. double cross validation procedures or permutation testing. However, there is a great inconsistency in the optimization and the assessment of performance of PLS-DA models due to many different diagnostic statistics currently employed in metabolomics data analyses. In this paper, properties of four diagnostic statistics of PLS-DA, namely the number of misclassifications (NMC), the Area Under the Receiver Operating Characteristic (AUROC), Q ² and Discriminant Q ² (DQ ²) are discussed. All four diagnostic statistics are used in the optimization and the performance assessment of PLS-DA models of three different-size metabolomics data sets obtained with two different types of analytical platforms and with different levels of known differences between two groups: control and case groups. Statistical significance of obtained PLS-DA models was evaluated with permutation testing. PLS-DA models obtained with NMC and AUROC are more powerful in detecting very small differences between groups than models obtained with Q ² and Discriminant Q ² (DQ ²). Reproducibility of obtained PLS-DA models outcomes, models complexity and permutation test distributions are also investigated to explain this phenomenon. DQ ² and Q ² (in contrary to NMC and AUROC) prefer PLS-DA models with lower complexity and require higher number of permutation tests and submodels to accurately estimate statistical significance of the model performance. NMC and AUROC seem more efficient and more reliable diagnostic statistics and should be recommended in two group discrimination metabolomic studies.

     

Goodness of Fit Tests with Misclassified Data Based on φ‐Divergences

The well known χ² goodness of fit test for a multinomial distribution is generally biased when observations are subject to misclassification. In this paper, based on a double sampling scheme, the family of φ‐divergence test statistics is introduced for testing goodness of fit under misclassification of the data. The case of binomial data is discussed and an illustrative example is also given.     

A Modified Chi-squared Approach for Fitting WEIBULL Models to Synthetic Life Tables1

WEIBULL models are fitted to synthetic life table data by applying weighted least squares analysis to log log functions which are constructed from appropriate underlying contingency tables. As such, the resulting estimates and test statistics are based on the linearized minimum modified X²₁-criterion and thus have satisfactory properties in moderately large samples. The basic methodology is illustrated in terms of an example which is bivariate in the sense of involving two simultaneous, but non-competing, vital events. For this situation, the estimation of WEIBULL model parameters is described for both marginal as well as certain conditional distributions either individually or jointly.     

The effect of crowding on adult populations ofDelia (=Hylemya) antiqua (Meigen)

Using 5 density levels, the effect of increasing density was studied on several population statistics of adult Delia (=Hylemya) antiqua. Amongst the statistics studied were adult lifespan; female mating frequency, fecundity and egg hatchability. It was shown that females lived significantly longer than males and that increasing density significantly reduced lifespan. Density had no effect on mating frequency. Total fecundity/female was significantly reduced with increasing density but oviposition rate was density independent. Using these statistics, mean generation time T, net reproduction rate R₀ and capacity for increase r_c were calculated at the different density levels. Values of R₀ showed a one-tailed response but there was no clear effect of density on r_c. Multiple comparisons between variables revealed several important relationships.

1. Adult lifespan was the most important factor affecting egg production.
2. Number of mated females was more important in affecting total fecundity/cage than adult density.

It could also be calculated that to produce the effect of one single mated female on total fecundity/cage the overall density would have to be reduced by 0.0023 individuals/cm³. These results are discussed in the context of efficient laboratory rearing of the onion fly.     

Modelling night‐time ecosystem respiration by a constrained source optimization method

One of the main challenges to quantifying ecosystem carbon budgets is properly quantifying the magnitude of night‐time ecosystem respiration. Inverse Lagrangian dispersion analysis provides a promising approach to addressing such a problem when measured mean CO₂ concentration profiles and nocturnal velocity statistics are available. An inverse method, termed ‘Constrained Source Optimization’ or CSO, which couples a localized near‐field theory (LNF) of turbulent dispersion to respiratory sources, is developed to estimate seasonal and annual components of ecosystem respiration. A key advantage to the proposed method is that the effects of variable leaf area density on flow statistics are explicitly resolved via higher‐order closure principles. In CSO, the source distribution was computed after optimizing key physiological parameters to recover the measured mean concentration profile in a least‐square fashion. The proposed method was field‐tested using 1 year of 30‐min mean CO₂ concentration and CO₂ flux measurements collected within a 17‐year‐old (in 1999) even‐aged loblolly pine (Pinus taeda L.) stand in central North Carolina. Eddy‐covariance flux measurements conditioned on large friction velocity, leaf‐level porometry and forest‐floor respiration chamber measurements were used to assess the performance of the CSO model. The CSO approach produced reasonable estimates of ecosystem respiration, which permits estimation of ecosystem gross primary production when combined with daytime net ecosystem exchange (NEE) measurements. We employed the CSO approach in modelling annual respiration of above‐ground plant components (c. 214 g C m^?2 year^?1) and forest floor (c. 989 g C m^?2 year^?1) for estimating gross primary production (c. 1800 g C m^?2 year^?1) with a NEE of c. 605 g C m^?2 year^?1 for this pine forest ecosystem. We conclude that the CSO approach can utilise routine CO₂ concentration profile measurements to corroborate forest carbon balance estimates from eddy‐covariance NEE and chamber‐based component flux measurements.     

Statistical perturbations in personal exposure meters caused by the human body in dynamic outdoor environments

Personal exposure meters (PEM) are routinely used for the exposure assessment to radio frequency electric or magnetic fields. However, their readings are subject to errors associated with perturbations of the fields caused by the presence of the human body. This paper presents a novel analysis method for the characterization of this effect. Using ray‐tracing techniques, PEM measurements have been emulated, with and without an approximation of this shadowing effect. In particular, the Global System for Mobile Communication mobile phone frequency band was chosen for its ubiquity and, specifically, we considered the case where the subject is walking outdoors in a relatively open area. These simulations have been contrasted with real PEM measurements in a 35‐min walk. Results show a good agreement in terms of root mean square error and E‐field cumulative distribution function (CDF), with a significant improvement when the shadowing effect is taken into account. In particular, the Kolmogorov–Smirnov (KS) test provides a P‐value of 0.05 when considering the shadowing effect, versus a P‐value of 10⁻¹⁴ when this effect is ignored. In addition, although the E‐field levels in the absence of a human body have been found to follow a Nakagami distribution, a lognormal distribution fits the statistics of the PEM values better than the Nakagami distribution. As a conclusion, although the mean could be adjusted by using correction factors, there are also other changes in the CDF that require particular attention due to the shadowing effect because they might lead to a systematic error. Bioelectromagnetics 32:209–217, 2011. © 2010 Wiley‐Liss, Inc.     

Genetic diversity-fitness correlation revealed by microsatellite analyses in European alpine marmots (Marmota marmota)

The relationship between individual genetic diversity and fitness-related traits are poorly understood in the wild. The availability of highly polymorphic molecular markers, such as microsatellites, has made research on this subject more feasible. We used three microsatellite-based measures of genetic diversity, individual heterozygosity H, mean d ² and mean d ² _outbreeding to test for a relationship between individual genetic diversity and important fitness trait, juvenile survival, in a population of alpine marmots (Marmota marmota), after controlling for the effects of ecological, social and physiological parameters that potentially influence juvenile survival in marmots. Analyses were conducted on 158 juveniles, and revealed a positive association between juvenile survival and genetic diversity measured by mean H. No association was found with mean d ² and with mean d ² _outbreeding. This suggests a fitness disadvantage to less heterozygous juveniles. The genetic diversity-fitness correlation (GDFC) was somewhat stronger during years with poor environmental conditions (i.e. wet summers). The stressful environmental conditions of this high mountain population might enhance inbreeding depression and make this association between genetic diversity and fitness detectable. Moreover the mating system, allowing extra pair copulation by occasional immigrants, as well as close inbreeding, favours a wide range of individual genetic diversity (mean H ranges from 0.125 to 1), which also may have facilitated the detection of the GDFC. The results further suggest that the observed GDFC is likely to be explained by the “local effect” hypothesis rather than by the “general effect” hypothesis.     

Cluster Randomization Trials: A Simulation Study

A simulation study is conducted to compare several methods that test the common log odds ratio in multiple 2 × 2 tables when the data are correlated within clusters. Allowing cluster size to vary within each table, we evaluate the unadjusted Mantel‐Haenszel chi‐square statistic (χ²_MH), the adjusted Mantel‐Haenszel chi‐square statistics of Rao and Scott using both an unpooled design effect (χ²_RSN) and a pooled design effect (χ²_RSP), the adjusted Mantel‐Haenszel chi‐square statistic of Donald and Donner (χ²_DD), the chi‐square statistic using the GEE approach (χ²_GEE), the adjusted Mantel‐Haenszel chi‐square statistic of Begg (χ²_B), the Wald (χ²_W), the robust Wald (χ²_RW), the score (χ²_S), the robust score (χ²_RS), and the adjusted Mantel‐Haenszel chi‐square statistics of Zhang and Boos (χ²_ZBP and χ²_ZBN). The test statistics above are compared in terms of empirical significance levels and empirical power levels. The robust score statistic χ²_RS and the adjusted Mantel‐Haenszel chi‐square statistics of Zhang and Boos (χ²_ZBP and χ²_ZBN) generally have empirical significance levels closer to the nominal value than the other statistics. These three statistics have similar empirical power levels when the intracluster correlation is zero or the cluster sizes are balanced. χ²_RS performs better in terms of empirical power levels when a positive intracluster correlation exists in the imbalance setting.     

Modified GEE and Goodness of the Marginal Fit (GOMF) Test with Correlated Binary Responses for Contingency Tables

This paper addresses testing the goodness of fit of models for marginal probabilities estimated by generalized estimating equations. We develop a modified version of generalized estimating equation and a goodness‐of‐fit test based on the fitted marginal means. The test statistic is easy to compute and has a simple reference distribution. Its performance is evaluated asymptotically and in small samples. It is also compared to the deviance and Pearson X² statistics. Example applications are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conspecific Sharing of Breeding Sites by Anopheline Female Mosquitoes (Diptera: Culicidae) Inferred from Microsatellite Markers

The number of Anopheles gambiae and Anopheles arabiensis females that used each of the 33 sampled breeding sites in west Kenya was estimated by microsatellite markers and related statistics to test the hypothesis that conspecific females share aquatic sites. Totally, 166 An. gambiae and 168 An. arabiensis larvae were identified and were genotyped. The mean number of larvae per breeding site was 8.3 for An. gambiae and 8.4 for An. arabiensis. The likelihood method estimated that, for An. gambiae, the mean number of females that would have laid eggs per breeding site was 5.2 and ranged from 2 to 9, and for An. arabiensis, the mean was 5.0 with a range of 2–10. The clustering method estimated that the mean number of females laying eggs per breeding site was 6.8 for An. gambiae. The results provide molecular evidence that females of one or both species share breeding sites.     

Comparisons of Some Chi-squared Tests for the Test of Independence in Sparse Two-Way Contingency Tables

We consider in this paper, the behaviour of a class of the CRESSIE READ (1984) power divergence test statistics indexed by parameter λ - I (λ), with the modified X₂ test statistics (LU) proposed by LAWAL and UPTON (1984), for sparse contingency tables ranging from the 3×3 to the 10×10. We present a sample of our results here. The results indicate that the LU test out-performs either the Cressie-Read suggested test I(2/3) or the Pearson's test - I(1). Our results further show that the modification to the likelihood ratio test [Y₂ = I'(0)] proposed by WILLIAMS (1976) performs like the parent Y₂ test, very poorly compared with either the I(2/3), X₂ or the LU test statistics. Power results also indicate that the powers of the LU test are in all cases considered in this study slightly higher than those of X₂ and I(2/3) tests. The LU test is therefore strongly recommended for use with sparse two-way contingency tables because in all of the cases considered, none of the other test statistics consistently out-performs the LU test with respect to attained α level or power.     

Test Statistics for Simple MARKOV Chains. A Monte Carlo Study

The statistical method for the analysis of frequency tables which has been published by GRIZZLE, STARMER, and KOCH (1969) is outlined for the case of linear functions of relative frequencies. Its use for the analysis of aggregate time series data under the model of simple MARKOV chains is suggested. Weighty objections against the resulting test statistics were grounds for the planning and performance of a Monte Carlo study. The results of the study demonstrated that the test statistic SS ( Cb =0) which is used for the comparison of different MARKOV chains is in fact X²-distributed. The test statistic SS ( F (p) = Xb ) for evaluation of the lack of fit of the model also follows a X²-distribution when a simple MARKOV chain ～ is used instead of the observed time series ～ can be determined by simply iterating the original GSK algorithm.     

Simple diagnostic statistical tests of models for DNA substitution

The accuracy of models for DNA substitution used in phylogenetic analyses is becoming more important with the increasing availability and analysis of molecular sequence data. It is natural to look for ways of improving these models, and to do this in a planned manner it is useful to be able to identify features of sequences that may not be described adequately. In this paper, I describe three statistics which may give useful diagnostic information on departures from models' predictions. The statistical distributions of these statistics are discussed and simple significance tests are derived. These tests are based on the (estimated) phylogeny of the sequences and so have the advantage of using the information contained in this tree. Examples are given of the application of the new tests to Markov chain models describing the evolution of primate pseudogene sequences and small-subunit RNA sequences.Abbreviations b(N,p) binomial distribution of N trials, each with probability p of success - m(N,p ₁,p ₂, ..., p _r ) multinomial distribution of N trials, with r possible outcomes having probabilities p ₁, p ₂, ..., p_r, respectively - N(, ²) Normal distribution with mean and variance ² - p() Poisson distribution with mean - bp base pairs - cdf cumulative distribution function - i.i.d. independent, identical distribution