Noncentrality Parameter Research Articles

The Relation Among Fit Indexes,Power, and Sample Size in Structural Equation Modeling

The relation among fit indexes, power, and sample size in structural equation modeling is examined. The noncentrality parameter is required to compute power. The 2 existing methods of computing power have estimated the noncentrality parameter by specifying an alternative hypothesis or alternative fit. These methods cannot be implemented easily and reliably. In this study, 4 fit indexes (RMSEA, CFI, McDonald's Fit Index, and Steiger's gamma) were used to compute the noncentrality parameter and sample size to achieve certain level of power. The resulting power and sample size varied as a function of (a) choice of fit index, (b) number of variables/degrees of freedom, (c) relation among the variables, and (d) value of the fit index. However, if the level of misspecification were held constant, then the resulting power and sample size would be identical.

Power and sample size calculations for multivariate linear models with random explanatory variables

This article considers the problem of power and sample size calculations for normal outcomes within the framework of multivariate linear models. The emphasis is placed on the practical situation that not only the values of response variables for each subject are just available after the observations are made, but also the levels of explanatory variables cannot be predetermined before data collection. Using analytic justification, it is shown that the proposed methods extend the existing approaches to accommodate the extra variability and arbitrary configurations of the explanatory variables. The major modification involves the noncentrality parameters associated with the F approximations to the transformations of Wilks likelihood ratio, Pillai trace and Hotelling-Lawley trace statistics. A treatment of multivariate analysis of covariance models is employed to demonstrate the distinct features of the proposed extension. Monte Carlo simulation studies are conducted to assess the accuracy using a child’s intellectual development model. The results update and expand upon current work in the literature.

Combined linkage and association mapping of quantitative trait loci by multiple markers.

Using multiple diallelic markers, variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL) based on nuclear families. The objective is to build a model that may fully use marker information for fine association mapping of QTL in the presence of prior linkage. The measures of linkage disequilibrium and the genetic effects are incorporated in the mean coefficients and are decomposed into orthogonal additive and dominance effects. The linkage information is modeled in variance-covariance matrices. Hence, the proposed methods model both association and linkage in a unified model. On the basis of marker information, a multipoint interval mapping method is provided to estimate the proportion of allele sharing identical by descent (IBD) and the probability of sharing two alleles IBD at a putative QTL for a sib-pair. To test the association between the trait locus and the markers, both likelihood-ratio tests and F-tests can be constructed on the basis of the proposed models. In addition, analytical formulas of noncentrality parameter approximations of the F-test statistics are provided. Type I error rates of the proposed test statistics are calculated to show their robustness. After comparing with the association between-family and association within-family (AbAw) approach by Abecasis and Fulker et al., it is found that the method proposed in this article is more powerful and advantageous based on simulation study and power calculation. By power and sample size comparison, it is shown that models that use more markers may have higher power than models that use fewer markers. The multiple-marker analysis can be more advantageous and has higher power in fine mapping QTL. As an application, the Genetic Analysis Workshop 12 German asthma data are analyzed using the proposed methods.

Comparison of low angle radar clutter models

Three low angle clutter models, the compound-K, the noncentral gamma-gamma (NCGG), both of which model speckle, and the Weibull distributions are compared. The derivations of the first two densities are surprisingly similar, differing on only one critical point. The compound-K assumes that the Gaussian random variables that are the input to the square law detector have zero means, while the NCGG assumes that the Gaussian random variables have nonzero means. The compound-K must then assume that it is the output variance that is fluctuating to produce speckle, while the NCGG assumes that it is the noncentrality parameter, resulting from the addition of the nonzero means that is fluctuating. The difference in the resulting models is significant. The Weibull density has long been used to model clutter successfully, and it too will be considered for comparison, but its use is not based upon a physically motivated development. Rather, its use is based on comparisons of clutter output densities being well matched by the Weibull density with appropriate parameters.

High Resolution T2 Association Tests of Complex Diseases Based on Family Data

This paper proposes family based Hotelling's T(2) tests for high resolution linkage disequilibrium (LD) mapping or association studies of complex diseases. Assume that genotype data of multiple markers or haplotype blocks are available for a sample of nuclear families, in which some offspring are affected. Paired Hotelling's T(2) test statistics are proposed for a high resolution association study using parents as controls for affected offspring, based on two coding methods: haplotype/allele coding and genotype coding. The paired Hotelling's T(2) tests take not only the correlation between the haplotype blocks or markers into account, but also take the correlation within each parent-offspring pair into account. The method extends two sample Hotelling's T(2) test statistics for population case control association studies, which are not valid for family data due to correlation of genetic data among family members. The validity of the proposed method is justified by rigorous mathematical and statistical proof under the large sample theory. The non-centrality parameter approximations of the test statistics are calculated for power and sample size calculations. From power comparison and type I error calculations, it is shown that the test statistic based on haplotype/allele coding is advantageous over the test statistic of genotype coding. Analysis using multiple markers may provide higher power than single marker analysis. If only one marker is utilized the power of the test statistic based on haplotype/allele coding is nearly identical to that of 1-TDT. Moreover, a permutation procedure is provided for data analysis. The method is applied to data from a German asthma family study. The results based on the paired Hotelling's T(2) statistic tests confirm the previous findings. However, the paired Hotelling's T(2) tests produce much smaller P-values than those of the previous study. The permutation tests produce similar results to those of the previous study; moreover, additional marker combinations are shown to be significant by permutation tests. The proposed paired Hotelling's T(2) statistic tests are potentially powerful in mapping complex diseases. A SAS Macro, Hotel_fam.sas, has been written to implement the method for data analysis.

Limited Time Series with a Unit Root

This paper develops an asymptotic theory for integrated and near-integrated time series whose range is constrained in some ways. Such a framework arises when integration and cointegration analyses are applied to time series that are bounded either by construction or because they are subject to control. The asymptotic properties of some commonly used integration tests are discussed; the bounded unit root distribution is introduced to describe the limiting distribution of the sample first-order autoregressive coefficient of a random walk under range constraints. The theoretical results show that the presence of such constraints can lead to drastically different asymptotics. Because deviations from the standard unit root theory are measured through two noncentrality parameters that can be consistently estimated, simple measures of the impact of range constraints on the asymptotic distributions are obtained. Generalizations of standard unit root tests that are robust to the presence of range constraints are also provided. Finally, it is shown that the proposed asymptotic framework provides an adequate approximation to the finitesample properties of the unit root statistics under range constraints.

Bias-Corrected Estimation of Noncentrality Parameters of Covariance Structure Models

A bias-corrected estimator of noncentrality parameters of covariance structure models is discussed. The approach represents an application of the bootstrap methodology for purposes of bias correction, and utilizes the relation between average of resample conventional noncentrality parameter estimates and their sample counterpart. The bias-corrected bootstrap estimator can be viewed as a possible alternative to the traditionally used one that is presently implemented in popular covariance structure modeling programs, and is illustrated by means of a numerical example.

Distribution of some statistics in the presence of blunders in geodetic observations

In presence of gross errors (blunders) in the observations, we derive the distributions of some statistics defined after a least-squares (LS) adjustment. Quadratic forms of the LS-residuals and LS-estimates of the blunders are considered along with some their combinations. We demonstrate, in two different ways, that the difference between the quadratic form of the LS-residuals and the quadratic form of the LS-estimates of the blunders has a central chi-square distribution, while each of these two quadratic forms has a non-central chi-square distribution with the same non-centrality parameter. Finally we prove the noncentral Fisher distribution of the ratio between the quadratic form of the LS-estimates of the blunders and the difference of the above-mentioned quadratic forms.

Power and Sample Size Calculations for Genetic Case/Control Studies Using Gene-Centric SNP Maps: Application to Human Chromosomes 6, 21, and 22 in Three Populations

Power and sample size calculations are critical parts of any research design for genetic association. We present a method that utilizes haplotype frequency information and average marker-marker linkage disequilibrium on SNPs typed in and around all genes on a chromosome. The test statistic used is the classic likelihood ratio test applied to haplotypes in case/control populations. Haplotype frequencies are computed through specification of genetic model parameters. Power is determined by computation of the test’s non-centrality parameter. Power per gene is computed as a weighted average of the power assuming each haplotype is associated with the trait. We apply our method to genotype data from dense SNP maps across three entire chromosomes (6, 21, and 22) for three different human populations (African-American, Caucasian, Chinese), three different models of disease (additive, dominant, and multiplicative) and two trait allele frequencies (rare, common). We perform a regression analysis using these factors, average marker-marker disequilibrium, and the haplotype diversity across the gene region to determine which factors most significantly affect average power for a gene in our data. Also, as a ‘proof of principle’ calculation, we perform power and sample size calculations for all genes within 100 kb of the PSORS1 locus (chromosome 6) for a previously published association study of psoriasis. Results of our regression analysis indicate that four highly significant factors that determine average power to detect association are: disease model, average marker-marker disequilibrium, haplotype diversity, and the trait allele frequency. These findings may have important implications for the design of well-powered candidate gene association studies. Our power and sample size calculations for the PSORS1 gene appear consistent with published findings, namely that there is substantial power (>0.99) for most genes within 100 kb of the PSORS1 locus at the 0.01 significance level.

Power and sample size calculations in the presence of phenotype errors for case/control genetic association studies

BackgroundPhenotype error causes reduction in power to detect genetic association. We present a quantification of phenotype error, also known as diagnostic error, on power and sample size calculations for case-control genetic association studies between a marker locus and a disease phenotype. We consider the classic Pearson chi-square test for independence as our test of genetic association. To determine asymptotic power analytically, we compute the distribution's non-centrality parameter, which is a function of the case and control sample sizes, genotype frequencies, disease prevalence, and phenotype misclassification probabilities. We derive the non-centrality parameter in the presence of phenotype errors and equivalent formulas for misclassification cost (the percentage increase in minimum sample size needed to maintain constant asymptotic power at a fixed significance level for each percentage increase in a given misclassification parameter). We use a linear Taylor Series approximation for the cost of phenotype misclassification to determine lower bounds for the relative costs of misclassifying a true affected (respectively, unaffected) as a control (respectively, case). Power is verified by computer simulation.ResultsOur major findings are that: (i) the median absolute difference between analytic power with our method and simulation power was 0.001 and the absolute difference was no larger than 0.011; (ii) as the disease prevalence approaches 0, the cost of misclassifying a unaffected as a case becomes infinitely large while the cost of misclassifying an affected as a control approaches 0.ConclusionOur work enables researchers to specifically quantify power loss and minimum sample size requirements in the presence of phenotype errors, thereby allowing for more realistic study design. For most diseases of current interest, verifying that cases are correctly classified is of paramount importance.

Sibship T2 association tests of complex diseases for tightly linked markers

For population case-control association studies, the false-positive rates can be high due to inappropriate controls, which can occur if there is population admixture or stratification. Moreover, it is not always clear how to choose appropriate controls. Alternatively, the parents or normal sibs can be used as controls of affected sibs. For late-onset complex diseases, parental data are not usually available. One way to study late-onset disorders is to perform sib-pair or sibship analyses. This paper proposes sibship-based Hotelling's T2 test statistics for high-resolution linkage disequilibrium mapping of complex diseases. For a sample of sibships, suppose that each sibship consists of at least one affected sib and at least one normal sib. Assume that genotype data of multiple tightly linked markers/haplotypes are available for each individual in the sample. Paired Hotelling's T2 test statistics are proposed for high-resolution association studies using normal sibs as controls for affected sibs, based on two coding methods: 'haplotype/allele coding' and 'genotype coding'. The paired Hotelling's T2 tests take into account not only the correlation among the markers, but also take the correlation within each sib-pair. The validity of the proposed method is justified by rigorous mathematical and statistical proofs under the large sample theory. The non-centrality parameter approximations of the test statistics are calculated for power and sample size calculations. By carrying out power and simulation studies, it was found that the non-centrality parameter approximations of the test statistics were accurate. By power and type I error analysis, the test statistics based on the 'haplotype/allele coding' method were found to be advantageous in comparison to the test statistics based on the 'genotype coding' method. The test statistics based on multiple markers can have higher power than those based on a single marker. The test statistics can be applied not only for bi-allelic markers, but also for multi-allelic markers. In addition, the test statistics can be applied to analyse the genetic data of multiple markers which contain double heterozygotes -- that is, unknown linkage phase data. An SAS macro, Hotel_sibs.sas, is written to implement the method for data analysis.

Pedigree linkage disequilibrium mapping of quantitative trait loci.

In this paper, we propose to use pedigrees of any size and any types of relatives in joint high-resolution linkage disequilibrium (LD) and linkage mapping of quantitative trait loci (QTL) by variance component models. Two or multiple markers can be simultaneously used in modeling association with the trait locus, instead of using one marker a time in the analysis. The proposed method can provide a unified result by using two or multiple markers in the modeling. This may avoid the complications of different results obtained from the separate analysis of marker by marker. The models simultaneously incorporate both linkage and LD information. The measures of LD are modeled by mean coefficients, and linkage information is modeled by variance-covariance matrix. Using analytical formulas to calculate the regression coefficients, the genetic effects are shown to be decomposed into additive and dominance components. The noncentrality parameter approximations of test statistics of LD are provided to make power calculations. Power and type I error rates are explored to investigate the merit of the proposed method by both the analytical formulas and simulations. Comparing with the association between-family and association within-family ('AbAw') approach of Fulker and Abecasis et al, it is evident that the method proposed in this article is more powerful. The method is applied to investigate the relation between polymorphisms in the angiotensin 1-converting enzyme (ACE) genes and circulating ACE levels, with a better result than that of the 'AbAw' approach. Moreover, two markers I/D and 4656(CT)3/2 can fully interpret association with the trait locus at a 0.01 significance level, which provides a unique result for the ACE data.

Power of the Classical Twin Design Revisited

Statistical power of the classical twin design was revisited. The approximate sampling variances of a least-squares estimate of the heritability in a univariate analysis and estimate of the genetic correlation coefficient in a bivariate analysis were derived analytically for the ACE model. Statistical power to detect additive genetic variation under the ACE model was derived analytically for least-squares, goodness-of-fit and maximum likelihood-based test statistics. The noncentrality parameter for the likelihood ratio test statistic is shown to be a simple function of the MZ and DZ intraclass correlation coefficients and the proportion of MZ and DZ twin pairs in the sample. All theoretical results were validated using simulation. The derived expressions can be used to calculate power of the classical twin design in a simple and rapid manner.

Power of the Classical Twin Design Revisited

Statistical power of the classical twin design was revisited. The approximate sampling variances of a least-squares estimate of the heritability in a univariate analysis and estimate of the genetic correlation coefficient in a bivariate analysis were derived analytically for the ACE model. Statistical power to detect additive genetic variation under the ACE model was derived analytically for least-squares, goodness-of-fit and maximum likelihood-based test statistics. The noncentrality parameter for the likelihood ratio test statistic is shown to be a simple function of the MZ and DZ intraclass correlation coefficients and the proportion of MZ and DZ twin pairs in the sample. All theoretical results were validated using simulation. The derived expressions can be used to calculate power of the classical twin design in a simple and rapid manner.

Evaluating the Noncentral Chi-Square Distribution for the Cox-Ingersoll-Ross Process

The conditional distribution of the short rate in the Cox-Ingersoll-Ross process can be expressed in terms of the noncentral χ2-distribution. The three standard methods for evaluating this function are by its representation in terms of a series of gamma functions, by analytic approximation, and by its asymptotic expansion. We perform numerical tests of these methods over parameter ranges typical for the Cox-Ingersoll-Ross process. We find that the gamma series representation is accurate over a wide range of parameters but has a runtime that increases proportional to the square root of the noncentrality parameter. Analytic approximations and the asymptotic expansion run quickly but have an accuracy that varies significantly over parameter space. We develop a fourth method for evaluating the upper and lower tails of the noncentral χ2-distribution based on a Bessel function series representation. We find that the Bessel method is accurate over a wide range of parameters and has a runtime that is insensitive to increases in the noncentrality parameter. We show that by using all four methods it is possible to efficiently evaluate the noncentral χ2-distribution to a relative precision of six significant figures.

Conditional Rejection Probabilities of Student's t‐test and Design Adaptations

AbstractFor clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t‐test and show that it can be non monotone in the interim mean of the data and non monotone in the non‐centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t‐test and give a numerical example. Additionally, the conditional rejection probability given the interim t‐statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The probability to select the correct model using likelihood-ratio based criteria in choosing between two nested models of which the more extended one is true

The probability to select the correct model is calculated for likelihood-ratio-based criteria to compare two nested models. If the more extended of the two models is true, the difference between twice the maximised log-likelihoods is approximately noncentral chi-square distributed with d.f. the difference in the number of parameters. The noncentrality parameter of this noncentral chi-square distribution can be approximated by twice the minimum Kullback–Leibler divergence (MKLD) of the best-fitting simple model to the true version of the extended model. The MKLD, and therefore the probability to select the correct model increases approximately proportionally to the number of observations if all observations are performed under the same conditions. If a new set of observations can only be performed under different conditions, the model parameters may depend on the conditions, and therefore have to be estimated for each set of observations separately. An increase in observations will then go together with an increase in the number of model parameters. In this case, the power of the likelihood-ratio test will increase with an increasing number of observations. However, the probability to choose the correct model with the AIC will only increase if for each set of observations the MKLD is more than 0.5. If the MKLD is less than 0.5, that probability will decrease. The probability to choose the correct model with the BIC will always decrease, sometimes after an initial increase for a small number of observation sets. The results are illustrated by a simulation study with a set of five nested nonlinear models for binary data.

A Note on Extending Steiger's (1998) Multiple Sample RMSEA Adjustment to Other Noncentrality Parameter-Based Statistics

This article considers the implications for other noncentrality parameter-based statistics from Steiger's (1998) multiple sample adjustment to the root mean square error of approximation (RMSEA) measure. When a structural equation model is fitted simultaneously in more than 1 sample, it is shown that the calculation of the noncentrality parameter used in tests of approximate fit and in point and interval estimators of other noncentral fit statistics (except the expected cross-validation index) also requires a likeminded adjustment. Furthermore, it is shown that an adjustment is needed in multiple sample models for correctly calculating MacCallum, Browne, and Sugawara's (1996) approach to power analysis. The accuracy of these proposals is investigated and demonstrated in a small Monte Carlo study in which particular attention is paid to using appropriately constructed covariance matrices that give specified nonzero population discrepancy values under maximum likelihood estimation.

A Note on the Noncentrality Parameter and Effect Size Estimates for the F Test in ANOVA

The noncentrality parameter for the noncentral F is a precision-weighted sum of squares of treatment means, which is closely related to the test statistic and effect size. The two common effect size estimates are not based on the uniformly minimum variance unbiased (UMVU) estimate of the noncentrality parameter. The UMVU estimate of the noncentrality parameter implies a new and more conservative effect size estimate.

Sample size determination for comparing several survival curves with unequal allocations.

Ahnn and Anderson derived sample size formulae for unstratified and stratified designs assuming equal allocation of subjects to three or more treatment groups. We generalize the sample size formulae to allow for unequal allocation. In addition, we define the overall probability of death to be equal to one minus the censored proportion for the stratified design. This definition also leads to a slightly different definition of the non-centrality parameter than that of Ahnn and Anderson for the stratified case. Assuming proportional hazards, sample sizes are determined for a prespecified power, significance level, hazard ratios, allocation of subjects to several treatment groups, and known censored proportion. In the proportional hazards setting, three cases are considered: (1) exponential failures--exponential censoring, (2) exponential failures--uniform censoring, and (3) Weibull failures (assuming same shape parameter for all groups)--uniform censoring. In all three cases of the unstratified case, it is assumed that the censoring distribution is the same for all of the treatment groups. For the stratified log-rank test, it is assumed the same censoring distribution across the treatment groups and the strata. Further, formulae have been developed to provide approximate powers for the test, based upon the first two or first four-moments of the asymptotic distribution. We observe the following two major findings based on the simulations. First, the simulated power of the log-rank test does not depend on the censoring mechanism. Second, for a significance level of 0.05 and power of 0.80, the required sample size n is independent of the censoring pattern. Moreover, there is very close agreement between the exact (asymptotic) and simulated powers when a sequence of alternatives is close to the null hypothesis. Two-moment and four-moment power series approximations also yield powers in close agreement with the exact (asymptotic) power. With unequal allocations, our simulations show that the empirical powers are consistently above the target value of prespecified power of 0.80 when 50 per cent of the patients are allocated to the treatment group with the smallest hazard.