Meta-analysis of the published effects of HGP use on beef palatability in steers as measured by objective and sensory testing

The author is grateful to two anonymous referees for their thorough reviews and helpful criticisms.

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Consider a collection of k studies, each of which measures the effect of a particular intervention. It can be assumed that from each of these studies a point estimate of the effect of intervention and a standard error of that estimate are available. This Appendix considers two standard methods by which these studies may be combined in order to obtain an estimate and confidence interval for the overall effect of the intervention of interest.

For each of the k studies let (i = 1,2,…,k) denote the estimated effect of intervention and θ_i the true effect of intervention. A general model is then specified by

and the e_i are assumed to be independent. The estimated effect size can be any measure of effect provided the assumption of normality is (at least approximately) appropriate.

In practice the σ_i² are unknown and so estimated variances are used. It is widely assumed, however, that the individual studies provide reasonable estimates of the σ_i². This assumption is almost universally made when the studies are analysed individually and should be the case provided the studies are at least moderate in size. In order to emphasize that the within study variances are estimated, the notation is used, where is the standard error of . The general model is therefore rewritten as

For the meta-analyses considered here, the parameter of interest is the overall effect of intervention, denoted μ. In the remainder of this Appendix outlines both the fixed and random effects models and describes how each relates μ to the θ_i. In both cases the estimates are point estimates of μ. Mention is also made of a test of homogeneity, widely used to select either the fixed or random effects model.

The fixed effects model for meta-analysis assumes that the k studies are homogeneous—each having the same true effect of intervention. Therefore θ_i = μ for all i = 1,2,…,k, giving the model

This model only allows for within-study variation.

The overall effect μ is commonly estimated using a weighted average. Ignoring the sampling error in the , it is optimal to use weights proportional to giving

The notation is used to indicate that the weights use estimates , although the variation associated with these estimates is ignored in practice. Under the assumptions of independence and normality,

and an approximate 95% confidence interval for μ is given by

Under the random effects model a distribution is assumed for the θ_i giving the two-stage model

and where the e_i and ε_i are assumed to be independent. In this model, the true effect for study i is centred around the overall effect μ, allowing the individual studies to vary in both estimated and true effect. The between-study variance parameter, τ², is a measure of the heterogeneity between studies and clearly this model permits both within and between-study variation.

This model can be written

giving

under the assumptions of normality and independence. The variance of includes both within and between-study variance components. The fixed effects model is a special case of the random effects model with τ² = 0. Selection of either the fixed effects or random effects models can therefore be carried out by testing the hypothesis τ² = 0 against a one-sided alternative. If τ² = 0 the studies are considered to be homogeneous. If the σ_i² are known, a test of the homogeneity between studies can be carried out using a statistic defined by Cochran (1937):

which has a χ²_k−1 distribution under the hypothesis τ² = 0. In practice however the σ_i² are not known and the statistic

is used. If τ² = 0 then approximately, and the hypothesis of homogeneity is rejected if .

This test is frequently used to determine whether the fixed or random effects model should be adopted (for example Touloumi et al. 1997; Danesh et al. 1998). It has been suggested, however, that the power of this test can be low (Thompson and Pocock 1991; Hardy and Thompson 1998).

In order to estimate μ under the random effects model an estimate of τ², the between-study variance parameter, is required.

Under the random effects model it is assumed that . As for the fixed effects method a weighted average is generally used to estimate μ where the weights are derived from the variances of the , i = 1,2,…,k. Therefore

where

Here the sampling error in the is ignored and it is assumed that τ² is known. Under these assumptions

In practice however, τ² must be estimated. The most widely-used estimate of τ² is one proposed by DerSimonian and Laird (1986):

where denotes the observed value of the homogeneity statistic and

This estimate is obtained by equating the expected value of with the observed value, and truncating to ensure that .

The estimate of τ² is then directly incorporated into the random effects weights, giving

This yields

Confidence intervals for μ are calculated under the assumption of normality; thus a 95% confidence interval for μ is given by

In cases where the random effects method estimate and interval for μ are the same as those for the fixed effects method.