Classical F-test Research Articles

A Random Projection Approach to Hypothesis Tests in High-Dimensional Single-Index Models

In this article, we consider the problem of hypothesis testing in high-dimensional single-index models. First, we study the feasibility of applying the classical F-test to a single-index model when the dimension of covariate vector and sample size are of the same order, and derive its asymptotic null distribution and asymptotic local power function. For the ultrahigh-dimensional single-index model, we construct F-statistics based on lower-dimensional random projections of the data, and establish the asymptotic null distribution and the asymptotic local power function of the proposed test statistics for the hypothesis testing of global and partial parameters. The new proposed test possesses the advantages of having a simple structure as well as being easy to compute. We compare the proposed test with other high-dimensional tests and provide sufficient conditions under which the proposed tests are more efficient. We conduct simulation studies to evaluate the finite-sample performances of the proposed tests and demonstrate that it has higher power than some existing methods in the models we consider. The application of real high-dimensional gene expression data is also provided to illustrate the effectiveness of the method. Supplementary materials for this article are available online.

Simultaneous test for linear model via projection

In this paper, we focus on the simultaneous test for the high-dimensional linear model coefficient. In high-dimensional setting, the traditional F-test is infeasible while the dimension is larger than the sample size because the sample covariance matrix is not invertible. Even when the dimension is smaller than the sample size, the efficiency of the classical F-test diminishes rapidly as the dimensionality increases. Several existing methods modified the classical F-test by removing the inverse term in the statistics and performed well. However, those modifications may lose efficiency due to the ignorance of correlation and those moment based procedures are sensitive to outlying observations and heavy tailed data. We propose a centered rank based method using projection which is robust in a broad range of distribution and alternative hypothesis. Under some mild conditions, we derive the best projection direction. The proposed procedure can retain type I error rate pretty well with the asymptotic distribution due to the data splitting strategy. Finite sample simulation studies are in accordance with the theoretical results.

Nonparametric modeling via two-way mixed effects design

The classical F-test for testing the hypothesis of no fixed main effects in a mixed effects design is valid under the assumption of normality, symmetry and variance homogeneity of the error terms assumption. We consider the two-way mixed effects design which does not require these three assumptions. A test procedure for the hypothesis of no main fixed effects is developed under this flexible model. The asymptotic distribution of the test statistic is studied for a large number of levels of the random effects.

Pengaruh Kualifikasi Pendidikan, Pengalaman Kerja dan Disiplin Kerja Terhadap Kinerja Guru Di SMA Frater Makassar

This research is a type of quantitative research that aims to determine and analyze how educational qualifications, work experience and work discipline influence the performance of teachers at Frater Makassar Senior High School. The independent variables in this study are educational qualifications, work experience and work discipline while the dependent variable is teacher performance. The population in this study were all teachers of SMA Frater Makassar consisting of ASN status teachers, permanent foundation teachers, contract teachers, and apprentice teachers as many as 41 teachers, while the sample used saturated sample techniques, namely the entire population was used as the research sample. Research data obtained by using a questionnaire, observation and documentation. The data analysis method used was validity test, reliability test and spearman rank correlation test. The data analysis techniques used were descriptive analysis (average score, standard deviation, variance and percentage), classical assumption test (normality test, linearity test, multicollinearity test, heteroscedasticity test) multiple regression test, t-test, f-test and determination coefficient delta test (ΔR2). The results showed that educational qualifications, work experience and work discipline simultaneously affect teacher performance, the amount of influence given is 24.2%, while the remaining 75.8% is explained or influenced by other variables not selected in this study.

The Effect of Leadership Style and Work Discipline on Employee Performance at PT Indonesia Nippon Seiki Cikande Serang

The role of HR management in carrying out aspects of HR must be managed properly so that policies and practices can be run in accordance with what the company wants. The purpose of this study was to determine the effect of partially and simultaneously leadership style and work discipline on employee performance at PT. Indonesia Nippon Seiki Cikande Serang. And to find out how much influence the leadership style and work discipline on employee performance at PT. Indonesia Nippon Seiki Cikande Serang. The research method used is quantitative research. The population in this study were all employees at PT. Indonesia Nippon Seiki Cikande Serang. The sample used in this study was 87 respondents. Data collection techniques by questionnaire, and data analysis techniques by instrument test, correlation coefficient, multiple linear regression, coefficient of determination, classical assumption test, t-test and F-test. The results showed that partially the leadership style had a positive and significant effect on employee performance, this could be proven from the tcount of 5.992> ttable 1.988 with a significant 0.000 <0.05 then H0 was rejected and Ha was accepted, meaning that there was a positive and significant influence of leadership style on performance the employee. Partially, work discipline has a positive and significant effect on employee performance, this can be proven from the value of tcount 7.088> ttable 1.988 with a significant 0.000 <0.05, then H0 is rejected and Ha is accepted, meaning there is a positive and significant effect of work discipline on employee performance. Simultaneously leadership style and work discipline have positive and significant effect on employee performance. This can be proven from the value of Fcount 62.884> Ftable 3.11 with a significance of 0.000 <0.05, thus Ho is rejected and Ha is accepted, meaning that there is a positive and significant influence of style leadership and work discipline simultaneously on employee performance. The magnitude of the effect of leadership style and work discipline simultaneously on employee performance by 60%, while the remaining 40% is influenced by other variables not examined in this study. And the value of R (Correlation) of 0.774 means that the level of relationship between leadership style (X1) and work discipline (X2) simultaneously on employee performance (Y) has a strong relationship level.

Taking Parametric Assumptions Seriously: Arguments for the Use of Welch’s <i>F</i>-test instead of the Classical <i>F</i>-test in One-Way ANOVA

Student’s t-test and classical F-test ANOVA rely on the assumptions that two or more samples are independent, and that independent and identically distributed residuals are normal and have equal variances between groups. We focus on the assumptions of normality and equality of variances, and argue that these assumptions are often unrealistic in the field of psychology. We underline the current lack of attention to these assumptions through an analysis of researchers’ practices. Through Monte Carlo simulations, we illustrate the consequences of performing the classic parametric F-test for ANOVA when the test assumptions are not met on the Type I error rate and statistical power. Under realistic deviations from the assumption of equal variances, the classic F-test can yield severely biased results and lead to invalid statistical inferences. We examine two common alternatives to the F-test, namely the Welch’s ANOVA (W-test) and the Brown-Forsythe test (F*-test). Our simulations show that under a range of realistic scenarios, the W-test is a better alternative and we therefore recommend using the W-test by default when comparing means. We provide a detailed example explaining how to perform the W-test in SPSS and R. We summarize our conclusions in practical recommendations that researchers can use to improve their statistical practices.

A General Framework for Estimation and Inference From Clusters of Features

ABSTRACTApplied statistical problems often come with prespecified groupings to predictors. It is natural to test for the presence of simultaneous group-wide signal for groups in isolation, or for multiple groups together. Current tests for the presence of such signals include the classical F-test or a t-test on unsupervised group prototypes (either group centroids or first principal components). In this article, we propose test statistics that aim for power improvements over these classical approaches. In particular, we first create group prototypes, with reference to the response, and then test with likelihood ratio statistics incorporating only these prototypes. We propose a model, called the “prototype model,” which naturally models this two-step procedure. Furthermore, we introduce an inferential schema detailing the unique considerations for different combinations of prototype formation and univariate/multivariate testing models. The prototype model also suggests new applications to estimation and prediction. Prototype formation often relies on variable selection, which invalidates classical Gaussian test theory. We use recent advances in selective inference to account for selection in the prototyping step and retain test validity. Simulation experiments suggest that our testing procedure enjoys more power than do classical approaches. Supplementary materials for this article are available online.

Two Simple Tests for Heteroscedastic Two-Way ANOVA

Heteroscedastic two-way ANOVA models are frequently encountered in experimental sciences, e.g., biology, psychology and physics among others. In the literature, classical F-tests are often blindly employed although they are often biased even for moderate heteroscedasticity. To overcome this problem, several approximate tests, e.g., generalized P-values, bootstrap, and permutation-based tests among others, have been proposed in the literature. These tests, however, are computationally intensive or do not work well in terms of size controlling and power. In this paper, we study tests of linear hypotheses in heteroscedastic two-way ANOVA via proposing a modified Bartlett (MB) test and a parametric bootstrap (PB) test. The MB test is easy to implement via using the usual χ-distribution and it is shown to be invariant under affine transformations, different choices of the contrast matrix used to define the same hypothesis and different labelling schemes of the cell means. The PB test, on the other hand, is easy to implement but it is time-consuming. Simulations and real data applications show that the MB test and the PB test are comparable and they both outperform the classical F-test in terms of size controlling and power for various sample sizes and parameter configurations.

The relative effects of dimensionality and multiplicity of hypotheses on the $F$-test in linear regression

Recently, several authors have re-examined the power of the classical F-test in linear regression in a `large-p, large-n' framework (cf. Zhong and Chen (2011), Wang and Cui (2013)). They highlight the loss of power as the number of regressors p increases relative to sample size n. These papers essentially focus only on the overall test of the null hypothesis that all p slope coefficients are equal to zero. Here, we consider the general case of testing q linear hypotheses on the (p+1)-dimensional regression parameter vector that includes p slope coefficients and an intercept parameter. In the case of Gaussian design, we describe the dependence of the local asymptotic power function on both the relative number of parameters p and the number of hypotheses q being tested, showing that the negative effect of dimensionality is less severe if the number of hypotheses is small. Using the recent work of Srivastava and Vershynin (2013) on high dimensional sample covariance matrices we are also able to substantially generalize previous results for non-Gaussian regressors.

Special Issue on ``Estimation, Testing and Forecasting in Econometrics''

One of us (King) joined Monash University as a young lecturer in econometrics in 1979. At about that time, there was a noticeable leap in the availability of computer power that allowed us to contemplate a likelihood based approach to econometrics. For example, the Faculty of Economics and Politics had taken delivery of its own VAX11/780 mini computer which allowed serious statistical and econometric computations to be made in double precision if required. Econometricians no longer had their CPU time rationed. We finally had the computational power and ability to run maximum likelihood estimation for most models and consequently conduct likelihood ratio (LR) tests or if more convenient, Lagrange multiplier (LM) or Wald tests. Did these advances mean that the theory of econometrics was “sorted”? It didn’t take long for cracks to appear. One testing problem that generated a disproportionately large literature in econometrics is that of testing for first-order autoregressive disturbances in the linear regression model, see King [5] for a survey. Here the LR test was found to be a very poor performer [1]. Unit root testing is another area where the classical tests (LR, LM and Wald) were found to perform poorly. In general, problems were caused by the small sample sizes and difficulties in dealing with nuisance parameters that can cause biases in estimated parameters that can also affect the performance of tests and forecasts. The last 35 years has seen all sorts of challenges overcome in an unending search for improvement in model based estimation, testing and forecasting. Many of these developments have become feasible because of the advances in computer hardware and software during that time. Also the ready availability of simulation methods have helped assess the small sample performance of proposed methods, both absolutely and comparatively. This special issue makes a contribution to this literature with the aim of bringing some of these advances to the notice of statisticians and others interested in model based statistical inference. The main ideas behind point optimal testing plus a strategy for the general application of such tests were outlined by King [6]. This issue opens with a paper that surveys the post 1987 literature on point optimal testing with particular emphasis on dealing with nuisance parameters. That period has witnessed the development of asymptotic point optimal testing led by the pioneering work of Elliott, Rothenberg and Stock [2] and, more recently, Muller [10] with a range of applications involving unit root testing, testing for cointegration and breaks or variation in regression coefficients. The review closes with an outline of a new class of point optimal tests for multi-dimensional testing, an area with few applications to date. Jahar Bhowmik contributes a paper that demonstrates the t-test of a classical regression coefficient is uniformly most powerful for a wider class of invariant tests. It is widely known that the F-test is uniformlymost powerfulwithin the class of tests that are invariant to four separate sets of transformations of the data. Because the t-test statistic is the square root of the classical F-test statistic, research papers and standard texts (see for example [3,8,13]) typically state that the t-test is uniformly most powerful for the same four sets of transformations as the F-test. Bhowmik

Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity

It is often of interest to undertake a general linear hypothesis testing (GLHT) problem in the one-way ANOVA without assuming the equality of the group variances. When the equality of the group variances is valid, it is well known that the GLHT problem can be solved by the classical F-test. The classical Ftest, however, may lead to misleading conclusions when the variance homogeneity assumption is seriously violated since it does not take the group variance heteroscedasticity into account. To our knowledge, little work has been done for this heteroscedastic GLHT problem except for some special cases. In this paper, we propose a simple approximate Hotelling T 2 (AHT) test. We show that the AHT test is invariant under a‐ne-transformations, difierent choices of the coe‐cient matrix used to deflne the same hypothesis, and difierent labeling schemes of the group means. Simulations and real data applications indicate that the AHT test is comparable with or outperforms some well-known approximate solutions proposed for the k-sample Behrens-Fisher problem which is a special case of the heteroscedastic GLHT problem.

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.

On parallelism of regression lines

ABSTRACT Testing of parallelism of regression lines is discussed and the classical F-test is compared by means of simulations with a rank test computed by means of results of Sen’s 1969 AMS paper. A rank based multiple comparisons rule for detecting regression lines with different slopes is constructed and asymptotic distribution of the underlying statistic is derived. This rule is compared by means of simulations with the rule derived from the assumption that the random fluctuations are Gaussian.

An approximate degrees of freedom test for heteroscedastic two-way ANOVA

Heteroscedastic two-way ANOVA are frequently encountered in real data analysis. In the literature, classical F-tests are often blindly employed although they are often biased even for moderate heteroscedasticity. To overcome this problem, several approximate tests have been proposed in the literature. These tests, however, are either too complicated to implement or do not work well in terms of size controlling. In this paper, we propose a simple and accurate approximate degrees of freedom (ADF) test. The ADF test is shown to be invariant under affine-transformations, different choices of contrast matrix for the same null hypothesis, or different labeling schemes of cell means. Moreover, it can be conducted easily using the usual F-distribution with one unknown degree of freedom estimated from the data. Simulations demonstrate that the ADF test works well in various cell sizes and parameter configurations but the classical F-tests work badly when the cell variance homogeneity assumption is violated. A real data example illustrates the methodologies.

A Perturbative Approach to Novelty Detection in Autoregressive Models

We propose a new method to perform novelty detection in dynamical systems governed by linear autoregressive models. The method is based on a perturbative expansion to a statistical test whose leading term is the classical F-test, and whose O(1/n) correction can be approximated as a function of the number of training points and the model order alone. The method can be justified as an approximation to an information theoretic test. We demonstrate on several synthetic examples that the first correction to the F-test can dramatically improve the control over the false positive rate of the system. We also test the approach on some real time series data, demonstrating that the method still retains a good accuracy in detecting novelties.

Evidence for ultra-fast outflows in radio-quiet AGNs

<i>Context. <i/>Blue-shifted Fe K absorption lines have been detected in recent years between 7 and 10 keV in the X-ray spectra of several radio-quiet AGNs. The derived blue-shifted velocities of the lines can often reach mildly relativistic values, up to 0.2–0.4c. These findings are important because they suggest the presence of a previously unknown massive and highly ionized absorbing material outflowing from their nuclei, possibly connected with accretion disk winds/outflows.<i>Aims. <i/>The scope of the present work is to statistically quantify the parameters and incidence of the blue-shifted Fe K absorption lines through a uniform analysis on a large sample of radio-quiet AGNs. This allows us to assess their global detection significance and to overcome any possible publication bias.<i>Methods. <i/>We performed a blind search for narrow absorption features at energies greater than 6.4 keV in a sample of 42 radio-quiet AGNs observed with <i> XMM-Newton<i/>. A simple uniform model composed by an absorbed power-law plus Gaussian emission and absorption lines provided a good fit for all the data sets. We derived the absorption lines parameters and calculated their detailed detection significance making use of the classical F-test and extensive Monte Carlo simulations.<i>Results. <i/>We detect 36 narrow absorption lines on a total of 101 <i> XMM-Newton<i/> EPIC pn observations. The number of absorption lines at rest-frame energies higher than 7 keV is 22. Their global probability to be generated by random fluctuations is very low, less than 3 × 10<sup>-8<sup/>, and their detection have been independently confirmed by a spectral analysis of the MOS data, with associated random probability <i><<i/>10<sup>-7<sup/>. We identify the lines as Fe XXV and Fe XXVI K-shell resonant absorption. They are systematically blue-shifted, with a velocity distribution ranging from zero up to ~0.3c, with a peak and mean value at ~0.1c. We detect variability of the lines on both <i>EW<i/>s and blue-shifted velocities among different <i> XMM-Newton<i/> observations even on time-scales as short as a few days, possibly suggesting somewhat compact absorbers. Moreover, we find no significant correlation between the cosmological red-shifts of the sources and the lines blue-shifted velocities, ruling out any systematic contamination by local absorption. If we define ultra-fast outflows (UFOs) those highly ionized absorbers with outflow velocities higher than 10<sup>4<sup/> km s<sup>-1<sup/>, then the majority of the lines are consistent with being associated to UFOs and the fraction of objects with detected UFOs in the whole sample is at least ~35%. This fraction is similar for type 1 and type 2 sources. The global covering fraction of the absorbers is consequently estimated to be in the range <i>C<i/> <i>∼<i/> 0.4-0.6, thereby implying large opening angles.<i>Conclusions. <i/>From our systematic X-ray spectral analysis on a large sample of radio-quiet AGNs we have been able to clearly assess the global veracity of the blue-shifted Fe K absorption lines at <i>E<i/> <i>><i/> 7 keV and to overcome their publication bias. These lines indicate that UFOs are a rather common phenomenon observable in the central regions of these sources and they are probably the direct signature of AGN accretion disk winds/ejecta. The detailed photo-ionization modeling of these absorbers is presented in a companion paper.

A SIMULATION STUDY ON TESTS FOR ONE-WAY ANOVA UNDER THE UNEQUAL VARIANCE ASSUMPTION

The classical F-test to compare several population means depends on the assumption of homogeneity of variance of the population and the normality. When these assumptions especially the equality of variance is dropped, the classical F-test fails to reject the null hypothesis even if the data actually provide strong evidence for it. This can be considered a serious problem in some applications, especially when the sample size is not large. To deal with this problem, a number of tests are available in the literature. In this study, the Brown-Forsythe, Weerahandiís Generalized F, Parametric Bootstrap, Scott-Smith, One-Stage, One-Stage Range, Welch and Xu-Wangís Generalized F-tests are introduced and a simulation study is performed to compare these tests according to type-1 errors and powers in different combinations of parameters and various sample sizes.

Statistical validation of megavariate effects in ASCA

BackgroundInnovative extensions of (M) ANOVA gain common ground for the analysis of designed metabolomics experiments. ASCA is such a multivariate analysis method; it has successfully estimated effects in megavariate metabolomics data from biological experiments. However, rigorous statistical validation of megavariate effects is still problematic because megavariate extensions of the classical F-test do not exist.MethodsA permutation approach is used to validate megavariate effects observed with ASCA. By permuting the class labels of the underlying experimental design, a distribution of no-effect is calculated. If the observed effect is clearly different from this distribution the effect is deemed significantResultsThe permutation approach is studied using simulated data which gave successful results. It was then used on real-life metabolomics data set dealing with bromobenzene-dosed rats. In this metabolomics experiment the dosage and time-interaction effect were validated, both effects are significant. Histological screening of the treated rats' liver agrees with this finding.ConclusionThe suggested procedure gives approximate p-values for testing effects underlying metabolomics data sets. Therefore, performing model validation is possible using the proposed procedure.

Testing for Covariate Effects in the Fully Nonparametric Analysis of Covariance Model

Traditional inference questions in the analysis of covariance mainly focus on comparing different factor levels by adjusting for the continuous covariates, which are believed to also exert a significant effect on the outcome variable. Testing hypotheses about the covariate effects, although of substantial interest in many applications, has received relatively limited study in the semiparametric/nonparametric setting. In the context of the fully nonparametric analysis of covariance model of Akritas et al., we propose methods to test for covariate main effects and covariate–factor interaction effects. The idea underlying the proposed procedures is that covariates can be thought of as factors with many levels. The test statistics are closely related to some recent developments in the asymptotic theory for analysis of variance when the number of factor levels is large. The limiting normal distributions are established under the null hypotheses and local alternatives by asymptotically approximating a new class of quadratic forms. The test statistics bear similar forms to the classical F-test statistics and thus are convenient for computation. We demonstrate the methods and their properties on simulated and real data.

Analysis of repeated measures under unequal variances

Problem of making inferences on a widely used repeated measures model is considered without the assumption of equal error variances. By taking the generalized approach to making statistical inference, we derive necessary formulae to compute exact generalized p -values for testing the equality of treatment effects, occasion effects, and their interactions. We also provide formulae for making inferences about the variance components of the model. Advantage of the generalized p -values over the classical F-test is demonstrated by means of an example.