Correcting Inhomogeneity of Variance with Power Transformation Weighting

Box, George E. P.; Hill, William J.

doi:10.1080/00401706.1974.10489207

Cited by 147 publications

(45 citation statements)

References 5 publications

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“…The method of least squares is undoubtedly one of the most extensively applied estimation techniques in analysing regression models with independently and identically distributed errors. However, the ordinary least squares estimator, though it remains unbiased, can be very inefficient under heterogeneity of error variances as demonstrated by the real example of Box and Hill (1974). Under such circumstances the method of weighted least squares (WLS) is often a preferred technique of estimation.…”

Section: Introductionmentioning

confidence: 99%

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Estimation of Parameters in Heteroscedastic Linear Models

Mak

1992

Journal of the Royal Statistical Society Series B: Statistical Methodology

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This paper deals with linear regression models with non-homogeneous error variances. The common situation where the error variance is a smooth function of the values of the regressor variables is considered. The error variance function is represented by a function of known form, but its expression involves the vector of unknown regression coefficients {J and an additional vector parameter 8. Estimation of {J by iterative weighted least squares (lWLS) therefore also requires updating the estimate of 8 at each iteration. When the error variance function is misspecified, different methods of updating 8 would yield IWLS estimators of varying degrees of efficiency. Under such situations, a method of updating 8 is proposed, resulting in an iterative procedure for estimating {J that is asymptotically optimal in the class of IWLS estimators. When the form ofthe error variance function is correctly specified, the estimation procedure proposed is asymptotically equivalent to weighted least squares with known optimal weights. A real example of its application is given and the extension to general non-linear models is briefly indicated.

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Section: Introductionmentioning

confidence: 99%

“…where (J is a p x 1 vector of regression coefficients, Y is the response variable and x is a p x 1 vector of explanatory variables. The error e has zero expectation and var(e) = (Box and Hill, 1974) and…”

Section: Introductionmentioning

confidence: 99%

Estimation of Parameters in Heteroscedastic Linear Models

Mak

1992

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…The relationship between variances before and after transformation has been applied for stabilising variance [35, 36] or representing inhomogeneity variances in linear models with Box-Cox transformation weighting [37]. …”

Section: Methodsmentioning

confidence: 99%

A random effects meta-analysis model with Box-Cox transformation

Yamaguchi

Maruo

Partlett

et al. 2017

BMC Med Res Methodol

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BackgroundIn a random effects meta-analysis model, true treatment effects for each study are routinely assumed to follow a normal distribution. However, normality is a restrictive assumption and the misspecification of the random effects distribution may result in a misleading estimate of overall mean for the treatment effect, an inappropriate quantification of heterogeneity across studies and a wrongly symmetric prediction interval.MethodsWe focus on problems caused by an inappropriate normality assumption of the random effects distribution, and propose a novel random effects meta-analysis model where a Box-Cox transformation is applied to the observed treatment effect estimates. The proposed model aims to normalise an overall distribution of observed treatment effect estimates, which is sum of the within-study sampling distributions and the random effects distribution. When sampling distributions are approximately normal, non-normality in the overall distribution will be mainly due to the random effects distribution, especially when the between-study variation is large relative to the within-study variation. The Box-Cox transformation addresses this flexibly according to the observed departure from normality. We use a Bayesian approach for estimating parameters in the proposed model, and suggest summarising the meta-analysis results by an overall median, an interquartile range and a prediction interval. The model can be applied for any kind of variables once the treatment effect estimate is defined from the variable.ResultsA simulation study suggested that when the overall distribution of treatment effect estimates are skewed, the overall mean and conventional I 2 from the normal random effects model could be inappropriate summaries, and the proposed model helped reduce this issue. We illustrated the proposed model using two examples, which revealed some important differences on summary results, heterogeneity measures and prediction intervals from the normal random effects model.ConclusionsThe random effects meta-analysis with the Box-Cox transformation may be an important tool for examining robustness of traditional meta-analysis results against skewness on the observed treatment effect estimates. Further critical evaluation of the method is needed.Electronic supplementary materialThe online version of this article (doi:10.1186/s12874-017-0376-7) contains supplementary material, which is available to authorized users.

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“…If 0 is known, we form the estimated weightsp8(x;, Sp) and then apply weighted least squares to estimate d. If 0 is unknown it too must be estimated. Box and Hill (1974), Pritchard et al (1977), Jobson and Fuller (1980) and Carroll and Ruppert (1982b) study this model. Jobson andFuller (1980) andRuppert (1982b) prove that if (S, 0) are consistent estimates of (~, 0) then all generalized least squares estimates of~will have the same asymptotic distribution as weighted least squares with known weights.…”

Section: Estimation Proceduresmentioning

confidence: 99%

“…This model is used in applications where it is reasonable to expect a positive mean response. See Box and Hill (1974), Pritchard et al (1977) and Bates et al (1985) for examples using power-of-the-mean variance models.…”

Section: Introductionmentioning

confidence: 99%