Cohen’s Linearly Weighted Kappa is a Weighted Average of 2×2 Kappas

An agreement table with n∈ℕ_≥3 ordered categories can be collapsed into n−1 distinct 2×2 tables by combining adjacent categories. Vanbelle and Albert (Stat. Methodol. 6:157–163, 2009c) showed that the components of Cohen’s weighted kappa with linear weights can be obtained from these n−1 collapsed 2×2 tables. In this paper we consider several consequences of this result. One is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2×2 tables, where the weights are the denominators of the 2×2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.     

All Finitely Axiomatizable Tense Logics of Linear Time Flows Are CoNP-complete

We prove that all finitely axiomatizable tense logics with temporal operators for ‘always in the future’ and ‘always in the past’ and determined by linear fows time are coNP-complete. It follows, for example, that all tense logics containing a density axiom of the form ■ⁿ⁺¹_F p → ⁿ_F p, for some n ≥ 0, are coNP-complete. Additionally, we prove coNP-completeness of all ∩-irreducible tense logics. As these classes of tense logics contain many Kripke incomplete bimodal logics, we obtain many natural examples of Kripke incomplete normal bimodal logics which are nevertheless coNP-complete.     

Children Exposed to Intra-Familial Violence

Abstract

This investigation examined factors associated with attrition and retention within a sample of children with a history of intra-familial violence referred for outpatient services. The sample included 118 children, aged 4-17 years; of these 64% completed treatment, 20% did not engage in treatment and 16% terminated prematurely. Caregivers reporting high child-related parental distress (F(2, 32) = 3.9, p = 0.03) and high psychological distress (F(2, 70) = 3.3, p = 0.04) were less likely to engage in treatment. Making a child abuse report during assessment or treatment was not found to be associated with treatment engagement or attrition, χ² ₍₂₎= 0.4, p = 0.82. The implications of these findings for clinical practice and treatment outcome research are discussed.     

Correcting an analysis of variance for clustering

A great deal of educational and social data arises from cluster sampling designs where clusters involve schools, classrooms, or communities. A mistake that is sometimes encountered in the analysis of such data is to ignore the effect of clustering and analyse the data as if it were based on a simple random sample. This typically leads to an overstatement of the precision of results and too liberal conclusions about precision and statistical significance of mean differences. This paper gives simple corrections to the test statistics that would be computed in an analysis of variance if clustering were (incorrectly) ignored. The corrections are multiplicative factors depending on the total sample size, the cluster size, and the intraclass correlation structure. For example, the corrected F statistic has Fisher's F distribution with reduced degrees of freedom. The corrected statistic reduces to the F statistic computed by ignoring clustering when the intraclass correlations are zero. It reduces to the F statistic computed using cluster means when the intraclass correlations are unity, and it is in between otherwise. A similar adjustment to the usual statistic for testing a linear contrast among group means is described.     

Percentage points of Wilks'L mvc andL vc criteria

Likelihood ratio tests have been proposed by Wilks for testing the hypothesis of equal means, variances, and covariances (H _mvc) and the hypothesis of equal variances and covariances (H _vc) in ap-variate normal distribution. Using exact distributions of the appropriate likelihood ratio statistics, tables of the .05 and .01 points of these distributions are constructed forp = 4, 5, 6, 7 and sample sizen = 25 (5) 60 (10) 100. A correction factor is recommended for largern. Two numerical examples illustrate use of the tables. A nonparametric test is proposed forH _mvc when the multivariate parent population is known to be non-normal.This research was supported partly by the Office of Naval Research under Contract No. Nonr-855(06) and partly by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. 18(600)-83. Reproduction in whole or in part for any purpose of the United States Government is permitted.     

Statistical power in complex experimental designs

We show that power and sample size tables developed by Cohen (1988, pp. 289–354, 381–389) produce incorrect estimates for factorial designs: power is underestimated, and sample size is overestimated. The source of this bias is shrinkage in the implied value of the noncentrality parameter, λ, caused by using Cohen’s adjustment ton for factorial designs (pp. 365 and 396). The adjustment was intended to compensate for differences in the actual versus presumed (by the tables) error degrees of freedom; however, more accurate estimates are obtained if the tables are used without adjustment. The problems with Cohen’s procedure were discovered while testing subroutines in DATASIM 1.2 for computing power and sample size in completely randomized, randomized-blocks, and split-plot factorial designs. The subroutines give the user the ability to generate power and sample size tables that are as easy to use as Cohen’s, but that eliminate the conservative bias of his tables. We also implemented several improvements relative to “manual” use of Cohen’s tables: (1) Since the user can control the specific values of 1- β,n, andf used on the rows and columns of the table, interpolation is never required; (2) exact as opposed to approximate solutions for the noncentralF distribution are employed; (3) solutions for factorial designs, including those with repeated measures factors, take into account the actual error degrees of freedom for the effect being tested; and (4) provision is made for the computation of power for applications involving the doubly noncentralF distribution.     

Sex differences in means and variability on the progressive matrices in university students: A meta‐analysis

A meta‐analysis is presented of 22 studies of sex differences in university students of means and variances on the Progressive Matrices. The results disconfirm the frequent assertion that there is no sex difference in the mean but that males have greater variability. To the contrary, the results showed that males obtained a higher mean than females by between .22d and .33d, the equivalent of 3.3 and 5.0 IQ conventional points, respectively. In the 8 studies of the SPM for which standard deviations were available, females showed significantly greater variability (F_(882,656)=1.20, p<.02), whilst in the 10 studies of the APM there was no significant difference in variability (F_(3344,5660)=1.00, p>.05).     

Stuart’s tau measure of effect size for ordinal variables: Some methodological considerations

The reporting of measures of effect size has become increasingly important in psychology. A Monte Carlo resampling permutation procedure is introduced to find near-optimum maximum values for Stuart’s τ _c measure for two-way ordinal contingency tables, also termed Kendall’s τ _c since Kendall introduced τ _a and τ _b . Comparisons between resampling and exact procedures demonstrate the accuracy and utility of resampling measures of effect size for two-way ordinal contingency tables. The resampling procedure is shown to be more precise than the traditional method of standardizing τ _c .     

Some Results on Mean Square Error for Factor Score Prediction

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γ _p = Φ ^1/2 Λ _p ′Ψ _p ⁻¹ Λ _p Φ ^1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γ _p or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors. The author is obliged to Maarten Speekenbrink and Peter van Rijn for their assistance with plotting the figures. In addition, I am obliged to the referees for their stimulating remarks.     

Assessing Approximate Fit in Categorical Data Analysis

A family of Root Mean Square Error of Approximation (RMSEA) statistics is proposed for assessing the goodness of approximation in discrete multivariate analysis with applications to item response theory (IRT) models. The family includes RMSEAs to assess the approximation up to any level of association of the discrete variables. Two members of this family are RMSEA₂, which uses up to bivariate moments, and the full information RMSEA_n. The RMSEA₂ is estimated using the M₂ statistic of Maydeu-Olivares and Joe (2005, 2006), whereas for maximum likelihood estimation, RMSEA_n is estimated using Pearson's X² statistic. Using IRT models, we provide cutoff criteria of adequate, good, and excellent fit using the RMSEA₂. When the data are ordinal, we find a strong linear relationship between the RMSEA₂ and the Standardized Root Mean Squared Residual goodness-of-fit index. We are unable to offer cutoff criteria for the RMSEA_n as its population values decrease as the number of variables and categories increase.     

Effects of mindfulness-based stress reduction on prenatal stress,anxiety and depression

Low well-being is common among Chinese pregnant women but few effective interventions currently exist to improve prenatal stress and negative emotions. Mindfulness-based stress reduction (MBSR) has been proved to be effective in reducing stress and rarely studies were focused on Chinese pregnant women. The aim of the current paper is to investigate the effects of 8-week MBSR on prenatal stress, anxiety and depression among Chinese pregnant women. A sample of 66 pregnant women randomly allocated into either the MBSR group (n=34) or the control group (n=32). Participants in the MBSR group received a group 8-week, 90-min each time intervention. The results found a significant interaction between time and condition for prenatal stress (F=45.51, p<0.001, η ²=0.427), anxiety (F=19.30, p<0.001, η ²=0.240), while depression showed no time-by-group interaction (F=0.29, p=0.589, η ²=0.005). As for the sub-scale of state anxiety, while there was only no time effect (F=3.68, p=0.060, η ²=0.057). The findings of this study preliminary indicated effects of the MBSR intervention on self-reported prenatal stress and anxiety in comparison to a treatment-as-usual control. Effect on depression was not observed may due to the low level of depression of participants. This study provides preliminary evidence that MBSR is suitable for Chinese pregnant women and be effective in decreasing prenatal stress, anxiety.     

Does a self management intervention lower distress in woman diagnosed with breast cancer?1

Psychological distress is particularly common with any cancer diagnosis. This paper examined the psychological distress in a cohort of women newly diagnosed with breast cancer and postulates on the underlying theory. A longitudinal study of a cohort of women (n = 147) diagnosed with breast cancer (within the past 1 year), was conducted at University Malaya Medical Centre, Kuala Lumpur. Data were collected at baseline and at post 4‐week intervention. Analysis of variance was conducted to examine for any significant differences in the change‐scores between the experimental group (n = 69) and the control group (n = 78). Using the change scores, analyses of variance showed significant differences between groups for stress, F(1, 140) = 13.68, p < .0001, anxiety F(1, 140) = 8.44, p < .004, and depression, F(1, 140) = 11.57, p < .0001. Levels of stress, anxiety and depression generally decreased significantly in the experimental groups, p < .05, but either maintained or increased in the control group. This study indicates that the level of psychological distress of women with breast cancer can be ameliorated with a 4‐week self‐management intervention. Lower stress levels were also found in women who reported engagement in higher physical activity than women with low physical activity.     

Theories with the Independence Property

A first-order theory T{{\mathcal T}} has the Independence Property provided T   \vdash (Q)(FT F₁ ú. . .úF_n){{{\mathcal T} \, \, \vdash (Q)(\Phi \Rightarrow {\Phi_1} \vee.\,.\,.\vee {\Phi_n})}} implies T   \vdash (Q)(FT F_i){{{\mathcal T} \, \, \vdash (Q)(\Phi \Rightarrow {\Phi_i})}} for some i whenever F,F₁, . . . ,F_n{{\Phi,\Phi_1,\,.\,.\,.\,,\Phi_n}} are formulae of a suitable type and (Q) is any quantifier sequence. Variants of this property have been noticed for some time in logic programming and in linear programming.     

When can we trust theF-approximation of the box-test?

Consider a multivariate context withp variates andk independent samples, each of sizen. To test equality of thek population covariance matrices, the likelihood ratio test is commonly employed. Box'sF-approximation to the null distribution of the test statistic can be used to computep-values, if sample sizes are not too small. It is suggested to regard theF-approximation as accurate if the sample sizesn are greater than or equal to 1+0.0613p ²+2.7265p-1.4182p ^0.5+0.235p ^1.4* In (k), for 5p30,k20.This research was supported by the Deutsche Forschungsgemeinschaft through Ste 405/2-1.     

Empirically Corrected Rescaled Statistics for SEM with Small N and Large p

Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic T_rml proposed by Satorra and Bentler was recommended in the literature of SEM. However, T_rml has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to T_rml so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where T_rml may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.     

Note on construction of an item analysis table for the high-low-27-per-cent group method

This paper describes the construction of a new item analysis table for the high-low-27-per-cent group method. The table provides a ready means of translating the observed proportions of success in the two extreme groups (p _H ,p _L ) into measures of item difficulty and item discrimination (p, Δ, andr). The tabled values of both the difficulty index,p, and the discrimination index,r, have been derived from Karl Pearson's tables of the normal bivariate surface.     

Reasons and Entailment

What is the relation between entailment and reasons for belief? In this paper, I discuss several answers to this question, and I argue that these answers all face problems. I then propose the following answer: for all propositions p ₁,…,p _n and q, if the conjunction of p ₁,…, and p _n entails q, then there is a reason against a person’s both believing that p ₁,…, and that p _n and believing the negation of q. I argue that this answer avoids the problems that the other answers to this question face, and that it does not face any other problems either. I end by showing what the relation between deductive logic, reasons for belief and reasoning is if this answer is correct.