It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.

In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain number p of observed variables (the p-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables is p + 1) and for those by deleting an internal variable (so that the number is p − 1), provided that the number of factors is held constant.

A program SEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.

The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.

The locally best invariant test statistic for testing sphericity of normal distributions is shown to be a simple function of the Box/Geisser-Greenhouse degrees of freedom correction factor in a repeated measures design. Because of this relationship it provides a more intuitively appealing test of the necessary and sufficient conditions for valid F-tests in repeated measures analysis of variance than the likelihood ratio test. The properties of the two tests are compared and tables of the critical values of the Box/Geisser-Greenhouse correction factor are given.

Tests for a change point (e.g., Chen and Gupta, Parametric statistical change point analysis (2nd ed.). Birkhuser, Boston, 2012; Hawkins et al., J Qual Technol 35:355–366, 2003) have recently been brought into the spotlight for their potential uses in psychometrics. They have been successfully applied to detect an unusual change in the mean score of a sequence of administrations of an international language assessment (Lee and von Davier, Psychometrika 78:557–575, 2013) and to detect speededness of examinees (Shao et al., Psychometrika, 2015). The differences in the type of data used, the test statistics, and the manner in which the critical values were obtained in these papers lead to questions such as “what type of psychometric problems can be solved by tests for a change point?” and “what test statistics should be used with tests for a change point in psychometric problems?” This note attempts to answer some of these questions by providing a general overview of tests for a change point with a focus on application to psychometric problems. A discussion is provided on the choice of an appropriate test statistic and on the computation of a corresponding critical value for tests for a change point. Then, three real data examples are provided to demonstrate how tests for a change point can be used to make important inferences in psychometric problems. The examples include some clarifications and remarks on the critical values used in Lee and von Davier (Psychometrika, 78:557–575, 2013) and Shao et al. (Psychometrika, 2015). The overview and the examples provide insight on tests for a change point above and beyond Lee and von Davier (Psychometrika, 78:557–575, 2013) and Shao et al. (Psychometrika, 2015). Thus, this note extends the research of Lee and von Davier (Psychometrika, 78:557–575, 2013) and Shao et al. (Psychometrika, 2015) on tests for a change point.

The ACE and ADE models have been heavily exploited in twin studies to identify the genetic and environmental components in phenotypes. However, the validity of the likelihood ratio test (LRT) of the existence of a variance component, a key step in the use of such models, has been doubted because the true values of the parameters lie on the boundary of the parameter space of the alternative model for such tests, violating a regularity condition required for a LRT (e.g., Carey in Behav. Genet. 35:653–665, 2005; Visscher in Twin Res. Hum. Genet. 9:490–495, 2006). Dominicus, Skrondal, Gjessing, Pedersen, and Palmgren (Behav. Genet. 36:331–340, 2006) solve the problem of testing univariate components in ACDE models. Our current work as presented in this paper resolves the issue of LRTs in bivariate ACDE models by exploiting the theoretical frameworks of inequality constrained LRTs based on cone approximations. Our derivation shows that the asymptotic sampling distribution of the test statistic for testing a single bivariate component in an ACE or ADE model is a mixture of χ2 distributions of degrees of freedom (dfs) ranging from 0 to 3, and that for testing both the A and C (or D) components is one of dfs ranging from 0 to 6. These correct distributions are stochastically smaller than the χ2 distributions in traditional LRTs and therefore LRTs based on these distributions are more powerful than those used naively. Formulas for calculating the weights are derived and the sampling distributions are confirmed by simulation studies. Several invariance properties for normal data (at most) missing by person are also proved. Potential generalizations of this work are also discussed.

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.

A Markov chain with transition probabilities pij(θ) that are functions of a parameter vector θ is defined. One of t input values is delivered to the chain on each trial. Under the hypothesis H0 the parameter vector is independent of the input; under the hypothesis H1 the vector is in general different for each different input. A likelihood ratio test for a single observation on a chain of great length is given for testing H0 against H1, given that the distribution of the inputs depends at most on the previous input and the present state of the chain. The test is therefore one of stationarity of the transition probabilities against a specific alternative form of nonstationarity. Application of the approach to a statistical test of lumpability of the states of a chain is indicated. Tests for other related hypotheses are suggested. Application of the test to “within-subject effects” for individual subjects is considered. Finally, some applications of the results in psychophysical contexts are suggested.

This paper refers to the exponential family of probability distributions and the conditional maximum likelihood (CML) theory. It is concerned with the determination of the sample size for three groups of tests of linear hypotheses, known as the fundamental trinity of Wald, score, and likelihood ratio tests. The main practical purpose refers to the special case of tests of the class of Rasch models. The theoretical background is discussed and the formal framework for sample size calculations is provided, given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds.

The likelihood ratio test is widely used in exploratory factor analysis to assess the model fit and determine the number of latent factors. Despite its popularity and clear statistical rationale, researchers have found that when the dimension of the response data is large compared to the sample size, the classical Chi-square approximation of the likelihood ratio test statistic often fails. Theoretically, it has been an open problem when such a phenomenon happens as the dimension of data increases; practically, the effect of high dimensionality is less examined in exploratory factor analysis, and there lacks a clear statistical guideline on the validity of the conventional Chi-square approximation. To address this problem, we investigate the failure of the Chi-square approximation of the likelihood ratio test in high-dimensional exploratory factor analysis and derive the necessary and sufficient condition to ensure the validity of the Chi-square approximation. The results yield simple quantitative guidelines to check in practice and would also provide useful statistical insights into the practice of exploratory factor analysis.

Van den Wollenberg stated a theorem specifying the conditions for a test, which is composed of two Rasch homogeneous subscales, as also behaving Rasch homogeneously. In our note, simple numerical counterexamples are given for which this theorem fails.

When some of observed variates do not conform to the model under consideration, they will have a serious effect on the results of statistical analysis. In factor analysis the model with inconsistent variates may result in improper solutions. In this article a useful method for identifying a variate as inconsistent is proposed in factor analysis. The procedure is based on the likelihood principle. Several statistical properties such as the effect of misspecified hypotheses, the problem of multiple comparisons, and robustness to violation of distributional assumptions are investigated. The procedure is illustrated by some examples.

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate ε correction factor is derived. The ε correction factor generalizes Box's ε and can be used to construct an adjusted MMM test.

The germination percentage (GP) is commonly employed to estimate the viability of a seed population. Statistical methods such as analysis of variance (ANOVA) and logistic regression are frequently used to analyse GP data. While ANOVA has a long history of usage, logistic regression is considered more suitable for GP data due to its binomial nature. However, both methods have inherent issues that require attention. In this study, we address previously unexplored challenges associated with these methods and propose the utilization of a likelihood ratio test as a solution. We demonstrate the advantages of employing the likelihood ratio test for GP data analysis through simulations and real data analysis.

Crop scientists occasionally compute sample correlations between traits based on observed data from designed experiments, and this is often accompanied by significance tests of the null hypothesis that traits are uncorrelated. This simple approach does not account for effects due to the randomization layout and treatment structure of the experiments and hence statistical inference based on standard procedures is not appropriate. The present paper describes how valid inferences accounting for all relevant effects can be obtained using bivariate mixed linear model procedures. A salient feature of the approach is that the bivariate model is commensurate with the model used for univariate analysis of individual traits and allows bivariate correlations to be computed at the level of effects. Heterogeneity of correlations between effects can be assessed by likelihood ratio tests or by graphical inspection of bivariate scatter plots of effect estimates. if heterogeneity is found to be substantial, it is crucial to focus on the correlation of effects, and usually, the main interest will be in the treatment effects. If heterogeneity is judged to be negligible, the marginal correlation can be estimated from the bivariate model for an overall assessment of association. The proposed methods are illustrated using four examples. Hints are given to alternative routes of analysis accounting for all treatment and design effects such as regression with groups and analysis of covariance.

Intensive field surveys were conducted in eastern Nebraska to determine the frequency distribution model and associated parameters of broadleaf and grass weed seedling populations. The negative binomial distribution consistently fit the data over time (1992 to 1993) and space (fields) for both the inter and intrarow broadleaf and grass weed seedling populations. The other distributions tested (Poisson with zeros, Neyman type A, logarithmic with zeros, and Poisson-binomial) did not fit the data as consistently as the negative binomial distribution. Associated with the negative binomial distribution is a k parameter. k is a nonspatial aggregation parameter related to the variance at a given mean value. The k parameter of the negative binomial distribution was consistent across weed density for individual weed species in a given field except for foxtail spp. populations. Stability of the k parameter across field sites was assessed using the likelihood ratio test There was no stable or common k value across field sites and years for all weed species populations. The lack of stability in k across field sites is of concern, because this parameter is used extensively in the development of parametric sequential sampling procedures. Because k is not stable across field sites, k must be estimated at the time of sampling. Understanding the variability in it is critical to the development of parametric sequential sampling strategies and understanding the dynamics of weed species in the field.

A few studies have examined the association between vitamin D and telomere length, and fewer still have examined the relationship in black or male populations. We investigated the cross-sectional association between the vitamin D metabolite 25-hydroxyvitamin D (25(OH)D) concentration in plasma and relative leucocyte telomere length (LTL) in 1154 US radiologic technologists who were 48–93 years old (373 white females, 278 white males, 338 black females, 165 black males). Plasma 25(OH)D concentration was measured by the chemiluminescence immunoassay, and relative LTL was measured by quantitative PCR. Logistic regression was used to obtain OR and 95 % CI for long v. short (based on median) LTL in relation to continuous 25(OH)D, quartiles of 25(OH)D and 25(OH)D deficiency. We found no significant association between continuous 25(OH)D and long LTL in all participants (Ptrend=0·440), nor in white females (Ptrend=0·845), white males (Ptrend=0·636), black females (Ptrend=0·967) or black males (Ptrend=0·484). Vitamin D deficiency (defined as 25(OH)D<30 nmol/l), however, was significantly associated with short LTL in whites (P=0·024), but not in other groups. In this population, we found little evidence to support associations between 25(OH)D and long LTL over the entire range of 25(OH)D in the overall study population or by sex and race.

The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed in this paper. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for mixtures [X. Liu and Y. Shao, Ann. Stat. 31 (2003) 807–832] and hidden Markov chains [E. Gassiat, Ann. Inst. Henri Poincaré 38 (2002) 897–906]. First, we study the case of mixtures of autoregressive models (i.e. independent regime switches). In this framework, we give sufficient conditions to keep the LRTS tight and compute its the asymptotic distribution. Second, we consider the extension of the ideas in Gassiat [Ann. Inst. Henri Poincaré 38 (2002) 897–906] to autoregressive models with regimes switches according to a Markov chain. In this case, it is shown that the marginal likelihood is no longer a contrast function and cannot be used to select the number of regimes. Some numerical examples illustrate the results and their convergence properties.

This paper deals with the likelihood ratio test (LRT) for testing hypotheseson the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRTstatisticsunder some conditions that are proved to be almost necessary.A detailed solution is given for two testing problems: thetest of a single distribution against any mixture, with application to Gaussian, Poisson andbinomial distributions; the test of the number of populations in afinite mixture with or without structural parameter.

We study the LRT statistic for testing a single population i.i.d. model against a mixture of two populations with Markov regime.We prove that the LRT statistic converges to infinity in probability as the number of observations tends to infinity. This is a consequence of a convergence result of the LRT statistic for a subproblem where the parameters are restricted to a subset of the whole parameter set.

This paper considers the generalized likelihood ratio (GLR) test or its modification dealing with nested models. The algorithm for evaluating the critical values and the error-rates for the canonical tests are provided; a table of critical values of a class of GLR tests is also given. The test proposed in the paper has applications in time-series model selection.