Point Estimation on Relative Risk under Inverse Sampling

When the underlying disease is rare, to control the coefficient of variation for the sample proportion of cases, we may wish to apply inverse sampling. In this paper, we derive the uniformly minimum variance unbiased estimator (UMVUE) of relative risk and its variance in closed form under inverse sampling. On the basis of a Monte Carlo simulation, we demonstrate that using the UMVUE of relative risk can substantially reduce the mean-squared-error of using the maximum likelihood estimator, especially when the number of index cases in both comparison samples is small. For a given fixed total cost, we include a program that can be used to find the optimal allocation for the number of index cases to minimize the variance of the UMVUE as well.     

Spatial models for spatial statistics: some unification

A general statistical framework is proposed for comparing linear models of spatial process and pattern. A spatial linear model for nested analysis of variance can be based on either fixed effects or random effects. Greig-Smith (1952) originally used a fixed effects model, but there are also examples of random effects models in the soil science literature. Assuming intrinsic stationarity for a linear model, the expectations of a spatial nested ANOVA and two term local variance (TTLV, Hill 1973) are functions of the variogram, and several examples are given. Paired quadrat variance (PQV, Ludwig & Goodall 1978) is a variogram estimator which can be used to approximate TTLV, and we provide an example from ecological data. Both nested ANOVA and TTLV can be seen as weighted lag-1 variogram estimators that are functions of support, rather than distance. We show that there are two unbiased estimators for the variogram under aggregation, and computer simulation shows that the estimator with smaller variance depends on the process autocorrelation.     

An alternative derivation of Harville's restricted log likelihood function for variance component estimation

Estimation of variance components in linear mixed models is important in clinical trial and longitudinal data analysis. It is also important in animal and plant breeding for accurately partitioning total phenotypic variance into genetic and environmental variances. Restricted maximum likelihood (REML) method is often preferred over the maximum likelihood (ML) method for variance component estimation because REML takes into account the lost degree of freedom resulting from estimating the fixed effects. The original restricted likelihood function involves a linear transformation of the original response variable (a collection of error contrasts). Harville's final form of the restricted likelihood function does not involve the transformation and thus is much easier to manipulate than the original restricted likelihood function. There are several different ways to show that the two forms of the restricted likelihood are equivalent. In this study, I present a much simpler way to derive Harville's restricted likelihood function. I first treat the fixed effects as random effects and call such a mixed model a pseudo random model (PDRM). I then construct a likelihood function for the PDRM. Finally, I let the variance of the pseudo random effects be infinity and show that the limit of the likelihood function of the PDRM is the restricted likelihood function.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

Variance matters: The shape of a datum

In the quantitative analysis of behaviour, choice data are most often plotted and analyzed as logarithmic transforms of ratios of responses and of ratios of reinforcers according to the generalized-matching relation, or its derivatives such as conditional-discrimination models. The relation between log choice ratios and log reinforcer ratios has normally been found using ordinary linear regression, which minimizes the sums of the squares of the y deviations from the fitted line. However, linear regression of this type requires that the log choice data be normally distributed, of equal variance for each log reinforcer ratio, and that the x (log reinforcer ratio) measures be fixed with no variance. We argue that, while log transformed choice data may be normally distributed, log reinforcer ratios do have variance, and because these measures derive from a binomial process, log reinforcer ratio distributions will be non-normal and skewed to more extreme values. These effects result in ordinary linear regression systematically underestimating generalized-matching sensitivity values, and in faulty parameter estimates from non-linear regression to assume hyperbolic and exponential decay processes. They also lead to model comparisons, which assume equal normally distributed error around every data point, being incorrect. We describe an alternative approach that can be used if the variance in choice is measured.     

A New Algorithm for Explicit Determination of Variance Component Estimations in Mixed Models and Mixed Classifications of Balanced ANOVA

This paper deals with the balanced case of the analysis of variance. The use of a classification function leads to an easy determination of all possible sources of variation of any mixed classification. For mixed models a new method is derived, which allows to represent explicit the ANOVA-estimations of the variance components respectively the estimation of the mean sum of squares of the fixed effects for all sources of variation. Thereby the corresponding F-quotients and the approximate confidence intervals of variance components are received in a simple way.     

Notes in Case-Control Studies with Matched Pairs under Inverse Sampling

In case-control studies with matched pairs, the traditional point estimator of odds ratio (OR) is well-known to be biased with no exact finite variance under binomial sampling. In this paper, we consider use of inverse sampling in which we continue to sample subjects to form matched pairs until we obtain a pre-determined number (>0) of index pairs with the case unexposed but the control exposed. In contrast to use of binomial sampling, we show that the uniformly minimum variance unbiased estimator (UMVUE) of OR does exist under inverse sampling. We further derive an exact confidence interval of OR in closed form. Finally, we develop an exact test and an asymptotic test for testing the null hypothesis H₀: OR = 1, as well as discuss sample size determination on the minimum required number of index pairs for a desired power at α-level.     

Rapid screening for phenotype-genotype associations by linear transformations of genomic evaluations

Background

Currently, association studies are analysed using statistical mixed models, with marker effects estimated by a linear transformation of genomic breeding values. The variances of marker effects are needed when performing the tests of association. However, approaches used to estimate the parameters rely on a prior variance or on a constant estimate of the additive variance. Alternatively, we propose a standardized test of association using the variance of each marker effect, which generally differ among each other. Random breeding values from a mixed model including fixed effects and a genomic covariance matrix are linearly transformed to estimate the marker effects.

Results

The standardized test was neither conservative nor liberal with respect to type I error rate (false-positives), compared to a similar test using Predictor Error Variance, a method that was too conservative. Furthermore, genomic predictions are solved efficiently by the procedure, and the p-values are virtually identical to those calculated from tests for one marker effect at a time. Moreover, the standardized test reduces computing time and memory requirements.The following steps are used to locate genome segments displaying strong association. The marker with the highest − log(p-value) in each chromosome is selected, and the segment is expanded one Mb upstream and one Mb downstream of the marker. A genomic matrix is calculated using the information from those markers only, which is used as the variance-covariance of the segment effects in a model that also includes fixed effects and random genomic breeding values. The likelihood ratio is then calculated to test for the effect in every chromosome against a reduced model with fixed effects and genomic breeding values. In a case study with pigs, a significant segment from chromosome 6 explained 11% of total genetic variance.

Conclusions

The standardized test of marker effects using their own variance helps in detecting specific genomic regions involved in the additive variance, and in reducing false positives. Moreover, genome scanning of candidate segments can be used in meta-analyses of genome-wide association studies, as it enables the detection of specific genome regions that affect an economically relevant trait when using multiple populations.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-246) contains supplementary material, which is available to authorized users.     

Fixed‐effect variance and the estimation of repeatabilities and heritabilities: issues and solutions

Linear mixed‐effects models are frequently used for estimating quantitative genetic parameters, including the heritability, as well as the repeatability, of traits. Heritability acts as a filter that determines how efficiently phenotypic selection translates into evolutionary change, whereas repeatability informs us about the individual consistency of phenotypic traits. As quantities of biological interest, it is important that the denominator, the phenotypic variance in both cases, reflects the amount of phenotypic variance in the relevant ecological setting. The current practice of quantifying heritabilities and repeatabilities from mixed‐effects models frequently deprives their denominator of variance explained by fixed effects (often leading to upward bias of heritabilities and repeatabilities), and it has been suggested to omit fixed effects when estimating heritabilities in particular. We advocate an alternative option of fitting models incorporating all relevant effects, while including the variance explained by fixed effects into the estimation of the phenotypic variance. The approach is easily implemented and allows optimizing the estimation of phenotypic variance, for example by the exclusion of variance arising from experimental design effects while still including all biologically relevant sources of variation. We address the estimation and interpretation of heritabilities in situations in which potential covariates are themselves heritable traits of the organism. Furthermore, we discuss complications that arise in generalized and nonlinear mixed models with fixed effects. In these cases, the variance parameters on the data scale depend on the location of the intercept and hence on the scaling of the fixed effects. Integration over the biologically relevant range of fixed effects offers a preferred solution in those situations.     

Unbiasedness of the Estimator of the Function of Expected Value in the Mixed Linear Model

The traditional method for estimating the linear function of fixed parameters in mixed linear model is a two-stage procedure. In the first stage of this procedure the variance components estimators are calculated and next in the second stage these estimators are taken as true values of variance components to estimating the linear function of fixed parameters according to generalized least squares method. In this paper the general mixed linear model is considered in which a matrix related to fixed parameters and or/a dispersion matrix of observation vector may be deficient in rank. It is shown that the estimators of a set of functions of fixed parameters obtained in second stage are unbiased if only the observation vector is symmetrically distributed about its expected value and the estimators of variance components from first stage are translation-invariant and are even functions of the observation vector.     

An adjusted coefficient of determination (R2) for generalized linear mixed models in one go

The coefficient of determination (R²) is a common measure of goodness of fit for linear models. Various proposals have been made for extension of this measure to generalized linear and mixed models. When the model has random effects or correlated residual effects, the observed responses are correlated. This paper proposes a new coefficient of determination for this setting that accounts for any such correlation. A key advantage of the proposed method is that it only requires the fit of the model under consideration, with no need to also fit a null model. Also, the approach entails a bias correction in the estimator assessing the variance explained by fixed effects. Three examples are used to illustrate new measure. A simulation shows that the proposed estimator of the new coefficient of determination has only minimal bias.     

Prediction error variance and expected response to selection,when selection is based on the best predictor – for Gaussian and threshold characters,traits following a Poisson mixed model and survival traits

In this paper, we consider selection based on the best predictor of animal additive genetic values in Gaussian linear mixed models, threshold models, Poisson mixed models, and log normal frailty models for survival data (including models with time-dependent covariates with associated fixed or random effects). In the different models, expressions are given (when these can be found – otherwise unbiased estimates are given) for prediction error variance, accuracy of selection and expected response to selection on the additive genetic scale and on the observed scale. The expressions given for non Gaussian traits are generalisations of the well-known formulas for Gaussian traits – and reflect, for Poisson mixed models and frailty models for survival data, the hierarchal structure of the models. In general the ratio of the additive genetic variance to the total variance in the Gaussian part of the model (heritability on the normally distributed level of the model) or a generalised version of heritability plays a central role in these formulas.     

Genetic heterogeneity of residual variance in broiler chickens

Aims were to estimate the extent of genetic heterogeneity in environmental variance. Data comprised 99 535 records of 35-day body weights from broiler chickens reared in a controlled environment. Residual variance within dam families was estimated using ASREML, after fitting fixed effects such as genetic groups and hatches, for each of 377 genetically contemporary sires with a large number of progeny (> 100 males or females each). Residual variance was computed separately for male and female offspring, and after correction for sampling, strong evidence for heterogeneity was found, the standard deviation between sires in within variance amounting to 15–18% of its mean. Reanalysis using log-transformed data gave similar results, and elimination of 2–3% of outlier data reduced the heterogeneity but it was still over 10%. The correlation between estimates for males and females was low, however. The correlation between sire effects on progeny mean and residual variance for body weight was small and negative (-0.1). Using a data set bigger than any yet presented and on a trait measurable in both sexes, this study has shown evidence for heterogeneity in the residual variance, which could not be explained by segregation of major genes unless very few determined the trait.     

Comparison of SANOVA Procedure of Simultaneous Testing Hypotheses to Others

In this paper a procedure SANOVA of simultaneous testing hypotheses is compared with others used in analysis of variance in a fixed linear model. The geometrical relation between SANOVA and Scheffé's confidence regions is discussed. It is shown that individual confidence intervals from SANOVA procedure are not longer than Scheffe's, Dunnett's and Tukey's ones. The cases, when they are the same are indicated. Theoretical considerations are illustrated by a practical example.     

Critical Assessment of the C-Optimality Design Criteria for Estimating the Median Effective Dose in Quantal Dose-Response Curves

In this paper the properties of C-optimal designs constructed for estimating the median effective dose within the framework of two-parametric linear logistic models are critically assessed. It is well known that this design criterion which is based on the first-order variance approximation of the exact variance of the maximum likelihood estimate of the ED50 leads to a one-point design where the maximum likelihood theory breaks down. The single dose used in this design is identical with the true but unknown value of the ED50. It will be shown, that at this one-point design the asymptotic variance does not exist. A two-point design in the neighbourhood of the one-point design which is symmetrical about the ED50 and associated with a small dose-distance would be nearly optimal, but extremely nonrobust if the best guess of the ED50 differs from the true value. In this situation the asymptotic variance of the two-point design converging towards the one-point design tends to infinity. Moreover, taking in consideration, that for searching an optimal design the exact variance is of primary interest and the asymptotic variance serves only as an approximation of the exact variance, we calculate the exact variance of the estimator from balanced, symmetric 2-point designs in the neighbourhood of the limiting 1-point design for various dose distances and initial best guesses of the ED50. We compare the true variance of the estimate of the ED50 with the asymptotic variance and show that the approximations generally do not represent suitable substitutes for the exact variance even in case of unrealistically large sample sizes. Kalish (1990) proposed a criterion based on the second-order asymptotic variance of the maximum likelihood estimate of the ED50 to overcome the degenerated 1-point design as the solution of the optimization procedure. In fact, we are able to show that this variance approximation does not perform substantially better than the first–order variance. From these considerations it follows, that the C-optimality criterion is not useful in this estimation problem. Other criteria like the F-optimality should be used.     

On selection criteria and estimation of parameters when the variance is heterogeneous

Summary Procedures for ranking candidates for selection and for estimating genetic and environmental parameters when variances are heterogeneous are discussed. The best linear unbiased predictor (BLUP) accounts automatically for heterogeneous variance provided that the covariance structure is known and that the assumptions of the model hold. Under multivariate normality BLUP allowing for heterogeneous variance maximizes expected genetic progress. Examples of application of BLUP to selection when residual or genetic variances are heterogeneous are given. Restricted maximum likelihood estimation of heterogeneous variances and covariances via the expectation-maximization algorithm is presented.     

具有亲缘关系矩阵的混合模型的方差组分迭代估计——应用于计算BLUP的方差比值和遗传参数

论述的是来自非均街资料的混合模型中具有亲缘关系矩阵时利用迭代法估计方差组分问题。这篇文章表明计算程序是可行的,只要能够按照混合模型中固定效应的结构矩阵和Henderson方法3的固定效应的假设条件正确地计算二次型约化平方和,就可获得较为精确的方差组分估计值;而且表明方差初始比值k偏高或偏低,不影响迭代求解的最后结果,这是因为在迭代过程中可以通过结构矩阵x'x和x'x的控制而自行调整。这些方差组分不仅可应用于选种种畜用的BLUP计算,还可用来估计遗传参数。     

Increased power with modified forms of the Levene (Med) test for heterogeneity of variance

While the conventional Levene (Med) test is a widely used and robust test for detecting heterogeneity of variance, it does not take notice of either the linear dependencies among the residuals involved or the possibility of a mean (or median)-variance relationship. This paper explores the substantial improvements in power possible by investigating the benefits both of removing such linear dependencies (structural zeros) and of modeling (even roughly) such relationships.     

Multibreed analysis by splitting the breeding values

An equivalent model for multibreed variance covariance estimation is presented. It considers the additive case including or not the segregation variances. The model is based on splitting the additive genetic values in several independent parts depending on their genetic origin. For each part, it expresses the covariance between relatives as a partial numerator relationship matrix times the corresponding variance component. Estimation of fixed effects, random effects or variance components provided by the model are as simple as any model including several random factors. We present a small example describing the mixed model equations for genetic evaluations and two simulated examples to illustrate the Bayesian variance component estimation.