The non-central chi-square chart with two-stage samplings

In this paper, we consider the non-central chi-square chart with two stage samplings. During the first stage, one item of the sample is inspected and, depending on the result, the sampling is either interrupted, or it goes on to the second stage, where the remaining sample items are inspected and the non-central chi-square statistic is computed. The proposed chart is not only more sensitive than the joint (X) over bar and R charts, but operationally simpler too, particularly when appropriate devices, such as go-no-go gauges, can be used to decide if the sampling should go on to the second stage or not. (c) 2004 Elsevier B.V. All rights reserved.

Monitoring process mean and variability with one non-central chi-square chart

Traditionally, an (X) over bar -chart is used to control the process mean and an R-chart to control the process variance. However, these charts are not sensitive to small changes in process parameters. A good alternative to these charts is the exponentially weighted moving average (EWMA) control chart for controlling the process mean and variability, which is very effective in detecting small process disturbances. In this paper, we propose a single chart that is based on the non-central chi-square statistic, which is more effective than the joint (X) over bar and R charts in detecting assignable cause(s) that change the process mean and/or increase variability. It is also shown that the EWMA control chart based on a non-central chi-square statistic is more effective in detecting both increases and decreases in mean and/or variability.

The no-central chi-square chart with two-stage samplings

Throughout this article, it is assumed that the no-central chi-square chart with two stage samplings (TSS Chisquare chart) is employed to monitor a process where the observations from the quality characteristic of interest X are independent and identically normally distributed with mean μ and variance σ2. The process is considered to start with the mean and the variance on target (μ = μ0; σ2 = σ0 2), but at some random time in the future an assignable cause shifts the mean from μ0 to μ1 = μ0 ± δσ0, δ >0 and/or increases the variance from σ0 2 to σ1 2 = γ2σ0 2, γ > 1. Before the assignable cause occurrence, the process is considered to be in a state of statistical control (defined by the in-control state). Similar to the Shewhart charts, samples of size n 0+ 1 are taken from the process at regular time intervals. The samplings are performed in two stages. At the first stage, the first item of the i-th sample is inspected. If its X value, say Xil, is close to the target value (|Xil-μ0|< w0σ 0, w0>0), then the sampling is interrupted. Otherwise, at the second stage, the remaining n0 items are inspected and the following statistic is computed. Wt = Σj=2n 0+1(Xij - μ0 + ξiσ 0)2 i = 1,2 Let d be a positive constant then ξ, =d if Xil > 0 ; otherwise ξi =-d. A signal is given at sample i if |Xil-μ0| > w0σ 0 and W1 > knia:tl, where kChi is the factor used in determining the upper control limit for the non-central chi-square chart. If devices such as go and no-go gauges can be considered, then measurements are not required except when the sampling goes to the second stage. Let P be the probability of deciding that the process is in control and P 1, i=1,2, be the probability of deciding that the process is in control at stage / of the sampling procedure. Thus P = P1 + P 2 - P1P2, P1 = Pr[μ0 - w0σ0 ≤ X ≤ μ0+ w 0σ0] P2=Pr[W ≤ kChi σ0 2], (3) During the in-control period, W / σ0 2 is distributed as a non-central chi-square distribution with n0 degrees of freedom and a non-centrality parameter λ0 = n0d2, i.e. W / σ0 2 - xn0 22 (λ0) During the out-of-control period, W / σ1 2 is distributed as a non-central chi-square distribution with n0 degrees of freedom and a non-centrality parameter λ1 = n0(δ + ξ)2 / γ2 The effectiveness of a control chart in detecting a process change can be measured by the average run length (ARL), which is the speed with which a control chart detects process shifts. The ARL for the proposed chart is easily determined because in this case, the number of samples before a signal is a geometrically distributed random variable with parameter 1-P, that is, ARL = I /(1-P). It is shown that the performance of the proposed chart is better than the joint X̄ and R charts, Furthermore, if the TSS Chi-square chart is used for monitoring diameters, volumes, weights, etc., then appropriate devices, such as go-no-go gauges can be used to decide if the sampling should go to the second stage or not. When the process is stable, and the joint X̄ and R charts are in use, the monitoring becomes monotonous because rarely an X̄ or R value fall outside the control limits. The natural consequence is the user to pay less and less attention to the steps required to obtain the X̄ and R value. In some cases, this lack of attention can result in serious mistakes. The TSS Chi-square chart has the advantage that most of the samplings are interrupted, consequently, most of the time the user will be working with attributes. Our experience shows that the inspection of one item by attribute is much less monotonous than measuring four or five items at each sampling.

Decomposition of $\rm R\times C$ contingency table chi -square: Application to binned DNA fragment size data and population structure analysis

The purpose of this research is to develop a new statistical method to determine the minimum set of rows (R) in a R x C contingency table of discrete data that explains the dependence of observations. The statistical power of the method will be empirically determined by computer simulation to judge its efficiency over the presently existing methods. The method will be applied to data on DNA fragment length variation at six VNTR loci in over 72 populations from five major racial groups of human (total sample size is over 15,000 individuals; each sample having at least 50 individuals). DNA fragment lengths grouped in bins will form the basis of studying inter-population DNA variation within the racial groups are significant, will provide a rigorous re-binning procedure for forensic computation of DNA profile frequencies that takes into account intra-racial DNA variation among populations. ^

Extended table of percentage points of the chi-square distribution /

"NAVWEPS report 7770. NOTS TP 2749."

Statnote 7: chi-square contingency tables

When the data are counts or the frequencies of particular events and can be expressed as a contingency table, then they can be analysed using the chi-square distribution. When applied to a 2 x 2 table, the test is approximate and care needs to be taken in analysing tables when the expected frequencies are small either by applying Yate’s correction or by using Fisher’s exact test. Larger contingency tables can also be analysed using this method. Note that it is a serious statistical error to use any of these tests on measurement data!

Analysing energy detector diversity receivers for spectrum sensing

The analysis of energy detector systems is a well studied topic in the literature: numerous models have been derived describing the behaviour of single and multiple antenna architectures operating in a variety of radio environments. However, in many cases of interest, these models are not in a closed form and so their evaluation requires the use of numerical methods. In general, these are computationally expensive, which can cause difficulties in certain scenarios, such as in the optimisation of device parameters on low cost hardware. The problem becomes acute in situations where the signal to noise ratio is small and reliable detection is to be ensured or where the number of samples of the received signal is large. Furthermore, due to the analytic complexity of the models, further insight into the behaviour of various system parameters of interest is not readily apparent. In this thesis, an approximation based approach is taken towards the analysis of such systems. By focusing on the situations where exact analyses become complicated, and making a small number of astute simplifications to the underlying mathematical models, it is possible to derive novel, accurate and compact descriptions of system behaviour. Approximations are derived for the analysis of energy detectors with single and multiple antennae operating on additive white Gaussian noise (AWGN) and independent and identically distributed Rayleigh, Nakagami-m and Rice channels; in the multiple antenna case, approximations are derived for systems with maximal ratio combiner (MRC), equal gain combiner (EGC) and square law combiner (SLC) diversity. In each case, error bounds are derived describing the maximum error resulting from the use of the approximations. In addition, it is demonstrated that the derived approximations require fewer computations of simple functions than any of the exact models available in the literature. Consequently, the regions of applicability of the approximations directly complement the regions of applicability of the available exact models. Further novel approximations for other system parameters of interest, such as sample complexity, minimum detectable signal to noise ratio and diversity gain, are also derived. In the course of the analysis, a novel theorem describing the convergence of the chi square, noncentral chi square and gamma distributions towards the normal distribution is derived. The theorem describes a tight upper bound on the error resulting from the application of the central limit theorem to random variables of the aforementioned distributions and gives a much better description of the resulting error than existing Berry-Esseen type bounds. A second novel theorem, providing an upper bound on the maximum error resulting from the use of the central limit theorem to approximate the noncentral chi square distribution where the noncentrality parameter is a multiple of the number of degrees of freedom, is also derived.

Exogeneity, weak identification and instrument selection in econometrics

La dernière décennie a connu un intérêt croissant pour les problèmes posés par les variables instrumentales faibles dans la littérature économétrique, c’est-à-dire les situations où les variables instrumentales sont faiblement corrélées avec la variable à instrumenter. En effet, il est bien connu que lorsque les instruments sont faibles, les distributions des statistiques de Student, de Wald, du ratio de vraisemblance et du multiplicateur de Lagrange ne sont plus standard et dépendent souvent de paramètres de nuisance. Plusieurs études empiriques portant notamment sur les modèles de rendements à l’éducation [Angrist et Krueger (1991, 1995), Angrist et al. (1999), Bound et al. (1995), Dufour et Taamouti (2007)] et d’évaluation des actifs financiers (C-CAPM) [Hansen et Singleton (1982,1983), Stock et Wright (2000)], où les variables instrumentales sont faiblement corrélées avec la variable à instrumenter, ont montré que l’utilisation de ces statistiques conduit souvent à des résultats peu fiables. Un remède à ce problème est l’utilisation de tests robustes à l’identification [Anderson et Rubin (1949), Moreira (2002), Kleibergen (2003), Dufour et Taamouti (2007)]. Cependant, il n’existe aucune littérature économétrique sur la qualité des procédures robustes à l’identification lorsque les instruments disponibles sont endogènes ou à la fois endogènes et faibles. Cela soulève la question de savoir ce qui arrive aux procédures d’inférence robustes à l’identification lorsque certaines variables instrumentales supposées exogènes ne le sont pas effectivement. Plus précisément, qu’arrive-t-il si une variable instrumentale invalide est ajoutée à un ensemble d’instruments valides? Ces procédures se comportent-elles différemment? Et si l’endogénéité des variables instrumentales pose des difficultés majeures à l’inférence statistique, peut-on proposer des procédures de tests qui sélectionnent les instruments lorsqu’ils sont à la fois forts et valides? Est-il possible de proposer les proédures de sélection d’instruments qui demeurent valides même en présence d’identification faible? Cette thèse se focalise sur les modèles structurels (modèles à équations simultanées) et apporte des réponses à ces questions à travers quatre essais. Le premier essai est publié dans Journal of Statistical Planning and Inference 138 (2008) 2649 – 2661. Dans cet essai, nous analysons les effets de l’endogénéité des instruments sur deux statistiques de test robustes à l’identification: la statistique d’Anderson et Rubin (AR, 1949) et la statistique de Kleibergen (K, 2003), avec ou sans instruments faibles. D’abord, lorsque le paramètre qui contrôle l’endogénéité des instruments est fixe (ne dépend pas de la taille de l’échantillon), nous montrons que toutes ces procédures sont en général convergentes contre la présence d’instruments invalides (c’est-à-dire détectent la présence d’instruments invalides) indépendamment de leur qualité (forts ou faibles). Nous décrivons aussi des cas où cette convergence peut ne pas tenir, mais la distribution asymptotique est modifiée d’une manière qui pourrait conduire à des distorsions de niveau même pour de grands échantillons. Ceci inclut, en particulier, les cas où l’estimateur des double moindres carrés demeure convergent, mais les tests sont asymptotiquement invalides. Ensuite, lorsque les instruments sont localement exogènes (c’est-à-dire le paramètre d’endogénéité converge vers zéro lorsque la taille de l’échantillon augmente), nous montrons que ces tests convergent vers des distributions chi-carré non centrées, que les instruments soient forts ou faibles. Nous caractérisons aussi les situations où le paramètre de non centralité est nul et la distribution asymptotique des statistiques demeure la même que dans le cas des instruments valides (malgré la présence des instruments invalides). Le deuxième essai étudie l’impact des instruments faibles sur les tests de spécification du type Durbin-Wu-Hausman (DWH) ainsi que le test de Revankar et Hartley (1973). Nous proposons une analyse en petit et grand échantillon de la distribution de ces tests sous l’hypothèse nulle (niveau) et l’alternative (puissance), incluant les cas où l’identification est déficiente ou faible (instruments faibles). Notre analyse en petit échantillon founit plusieurs perspectives ainsi que des extensions des précédentes procédures. En effet, la caractérisation de la distribution de ces statistiques en petit échantillon permet la construction des tests de Monte Carlo exacts pour l’exogénéité même avec les erreurs non Gaussiens. Nous montrons que ces tests sont typiquement robustes aux intruments faibles (le niveau est contrôlé). De plus, nous fournissons une caractérisation de la puissance des tests, qui exhibe clairement les facteurs qui déterminent la puissance. Nous montrons que les tests n’ont pas de puissance lorsque tous les instruments sont faibles [similaire à Guggenberger(2008)]. Cependant, la puissance existe tant qu’au moins un seul instruments est fort. La conclusion de Guggenberger (2008) concerne le cas où tous les instruments sont faibles (un cas d’intérêt mineur en pratique). Notre théorie asymptotique sous les hypothèses affaiblies confirme la théorie en échantillon fini. Par ailleurs, nous présentons une analyse de Monte Carlo indiquant que: (1) l’estimateur des moindres carrés ordinaires est plus efficace que celui des doubles moindres carrés lorsque les instruments sont faibles et l’endogenéité modérée [conclusion similaire à celle de Kiviet and Niemczyk (2007)]; (2) les estimateurs pré-test basés sur les tests d’exogenété ont une excellente performance par rapport aux doubles moindres carrés. Ceci suggère que la méthode des variables instrumentales ne devrait être appliquée que si l’on a la certitude d’avoir des instruments forts. Donc, les conclusions de Guggenberger (2008) sont mitigées et pourraient être trompeuses. Nous illustrons nos résultats théoriques à travers des expériences de simulation et deux applications empiriques: la relation entre le taux d’ouverture et la croissance économique et le problème bien connu du rendement à l’éducation. Le troisième essai étend le test d’exogénéité du type Wald proposé par Dufour (1987) aux cas où les erreurs de la régression ont une distribution non-normale. Nous proposons une nouvelle version du précédent test qui est valide même en présence d’erreurs non-Gaussiens. Contrairement aux procédures de test d’exogénéité usuelles (tests de Durbin-Wu-Hausman et de Rvankar- Hartley), le test de Wald permet de résoudre un problème courant dans les travaux empiriques qui consiste à tester l’exogénéité partielle d’un sous ensemble de variables. Nous proposons deux nouveaux estimateurs pré-test basés sur le test de Wald qui performent mieux (en terme d’erreur quadratique moyenne) que l’estimateur IV usuel lorsque les variables instrumentales sont faibles et l’endogénéité modérée. Nous montrons également que ce test peut servir de procédure de sélection de variables instrumentales. Nous illustrons les résultats théoriques par deux applications empiriques: le modèle bien connu d’équation du salaire [Angist et Krueger (1991, 1999)] et les rendements d’échelle [Nerlove (1963)]. Nos résultats suggèrent que l’éducation de la mère expliquerait le décrochage de son fils, que l’output est une variable endogène dans l’estimation du coût de la firme et que le prix du fuel en est un instrument valide pour l’output. Le quatrième essai résout deux problèmes très importants dans la littérature économétrique. D’abord, bien que le test de Wald initial ou étendu permette de construire les régions de confiance et de tester les restrictions linéaires sur les covariances, il suppose que les paramètres du modèle sont identifiés. Lorsque l’identification est faible (instruments faiblement corrélés avec la variable à instrumenter), ce test n’est en général plus valide. Cet essai développe une procédure d’inférence robuste à l’identification (instruments faibles) qui permet de construire des régions de confiance pour la matrices de covariances entre les erreurs de la régression et les variables explicatives (possiblement endogènes). Nous fournissons les expressions analytiques des régions de confiance et caractérisons les conditions nécessaires et suffisantes sous lesquelles ils sont bornés. La procédure proposée demeure valide même pour de petits échantillons et elle est aussi asymptotiquement robuste à l’hétéroscédasticité et l’autocorrélation des erreurs. Ensuite, les résultats sont utilisés pour développer les tests d’exogénéité partielle robustes à l’identification. Les simulations Monte Carlo indiquent que ces tests contrôlent le niveau et ont de la puissance même si les instruments sont faibles. Ceci nous permet de proposer une procédure valide de sélection de variables instrumentales même s’il y a un problème d’identification. La procédure de sélection des instruments est basée sur deux nouveaux estimateurs pré-test qui combinent l’estimateur IV usuel et les estimateurs IV partiels. Nos simulations montrent que: (1) tout comme l’estimateur des moindres carrés ordinaires, les estimateurs IV partiels sont plus efficaces que l’estimateur IV usuel lorsque les instruments sont faibles et l’endogénéité modérée; (2) les estimateurs pré-test ont globalement une excellente performance comparés à l’estimateur IV usuel. Nous illustrons nos résultats théoriques par deux applications empiriques: la relation entre le taux d’ouverture et la croissance économique et le modèle de rendements à l’éducation. Dans la première application, les études antérieures ont conclu que les instruments n’étaient pas trop faibles [Dufour et Taamouti (2007)] alors qu’ils le sont fortement dans la seconde [Bound (1995), Doko et Dufour (2009)]. Conformément à nos résultats théoriques, nous trouvons les régions de confiance non bornées pour la covariance dans le cas où les instruments sont assez faibles.

O uso da estatística de qui-quadrado no controle de processos

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

STATISTICAL TEST FOR GENOTYPE AND ENVIRONMENT CONTRIBUTION IN THE GENOTYPES x ENVIRONMENTS INTERACTION MATRIX

The objective of the present work was to propose a method for testing the contribution of each level of the factors in a genotypes x environments (GxE) interaction using multi-environment trials analyses by means of an F test. The study evaluated a data set, with twenty genotypes and thirty-four environments, in a block design with four replications. The sum of squares within rows (genotypes) and columns (environments) of the GxE matrix was simulated, generating 10000 experiments to verify the empirical distribution. Results indicate a noncentral chi-square distribution for rows and columns of the GxE interaction matrix, which was also verified by the Kolmogorov-Smirnov test and Q-Q plot. Application of the F test identified the genotypes and environments that contributed the most to the GxE interaction. In this way, geneticists can select good genotypes in their studies.

Response probability distribution of built-up vibro-acoustic systems

The vibro-acoustic response of built-up structures, consisting of stiff components with low modal density and flexible components with high modal density, is sensitive to small imperfections in the flexible components. In this paper, the uncertainty of the response is considered by modeling the low modal density master system as deterministic and the high modal density subsystems in a nonparametric stochastic way, i.e., carrying a diffuse wave field, and by subsequently computing the response probability density function. The master system's mean squared response amplitude follows a singular noncentral complex Wishart distribution conditional on the subsystem energies. For a single degree of freedom, this is equivalent to a chi-square or an exponential distribution, depending on the loading conditions. The subsystem energies follow approximately a chi-square distribution when their relative variance is smaller than unity. The results are validated by application to plate structures, and good agreement with Monte Carlo simulations is found. © 2012 Acoustical Society of America.

The impact of perceived stage of cancer on carers’ anxiety and depression during the patients’ final year of life

Objective: This paper explores the effects of perceived stage of cancer (PSOC) on carers' anxiety and depression during the patients' final year. Methods: A consecutive sample of patients and carers (N=98) were surveyed at regular intervals regarding PSOC, and anxiety and depression using the Hospital Anxiety and Depression Scale. Means were compared by gender using the Mann-Whitney U-test. The chi-square was used to analyse categorical data. Agreement between carers' and patients' PSOC was estimated using kappa statistics. Correlations between carers' PSOC and their anxiety and depression were calculated using the Spearman's rank correlation. Results: Over time, an increasing proportion of carers reported that the cancer was advanced, culminating at 43% near death. Agreement regarding PSOC was fair (kappa=0.29-0.34) until near death (kappa=0.21). Carers' anxiety increased over the year; depression increased in the final 6 months. Females were more anxious (p=0.049, 6 months; p=0.009, 3 months) than males, and more depressed until 1 month to death. The proportion of carers reporting moderate-severe anxiety almost doubled over the year to 27%, with more females in this category at 6 months (p=0.05). Carers with moderate-severe depression increased from 6 to 15% over the year. Increased PSOC was weakly correlated with increased anxiety and depression. Conclusions: Carers' anxiety exceeded depression in severity during advanced cancer. Females generally experienced greater anxiety and depression. Carers were more realistic than patients regarding the ultimate outcome, which was reflected in their declining mental health, particularly near the end.

Characteristics and predictors of high-range speeding offenders

This paper examines an aspect of the data taken from a larger study evaluating the effect of speeding penalty changes on speeding recidivism in Queensland. Traffic offence data from May 1996 to August 2007 were provided to the research team for two cohorts of offenders: individuals who committed a speeding offence in May 2001; and individuals who committed a speeding offence in May 2003. Data included details of the offenders’ index offence, previous and subsequent traffic offences (speeding and other) and their demographic characteristics. Using this data the aim of this component of the research was to use demographic data and the previous traffic offences of these individuals to explore the characteristics and predictors of high-range speeding offenders. High-range offenders were identified as those individuals who committed two or more speeding offences with a recorded speed of 30 km/hr or more above the speed limit. For the purposes of comparison, low-range offenders (committed one speeding offence in the time-frame and that offence was less than 15 km/hr over the speed limit) and mid-range offenders (all other offenders) were identified. Using Chi-square and logistic regression analyses, characteristics and predictors of high-range speeding offenders were identified. The implications and limitations of this study are also discussed.

The relationship of loneliness and social anxiety with children's and adolescents' online communication

Children and adolescents are now using online communication to form and/or maintain relationships with strangers and/or friends. Relationships in real life are important for children and adolescents in identity formation and general development. However, social relationships can be difficult for those who experience feelings of loneliness and social anxiety. The current study aimed to replicate and extend research conducted by Valkenburg and Peter (2007b), by investigating differences in online communication patterns between children and adolescents with and without selfreported loneliness and social anxiety. Six hundred and twenty-six students aged 10-16 years completed a questionnaire survey about the amount of time they engaged in online communication, the topics they discussed, who they communicated with, and their purposes of online communication. Following Valkenburg and Peter (2007b), loneliness was measured with a shortened version of the UCLA Loneliness Scale (Version 3) developed by Russell (1996), whereas social anxiety was assessed with a sub-scale of the Social Anxiety Scale for Adolescents (La Greca & Lopez, 1998). The sample was divided into four groups of children and adolescents: 220 were “non-socially anxious and non-lonely”, 139 were “socially anxious but not lonely”, 107 were “lonely but not socially anxious”, and 159 were “lonely and socially anxious”. A one-way ANOVA and chi-square tests were conducted to evaluate the aforementioned differences between these groups. The results indicated that children and adolescents who reported being lonely used online communication differently from those who did not report being lonely. Essentially, the former communicated online more frequently about personal things and intimate topics, but also to compensate for their weak social skills and to meet new people. Further analyses on gender differences within lonely children and adolescents revealed that boys and girls communicated online more frequently with different partners. It was concluded that for these vulnerable individuals online communication may fulfil needs of self-disclosure, identity exploration, and social interactions. However, future longitudinal studies combining a quantitative with a qualitative approach would better address the relationship between Internet use and psychosocial well-being. The findings also suggested the need for further exploration of how such troubled children and adolescents can use the Internet beneficially.