Evaluating the statistical conclusion validity of weighted mean results in meta-analysis by analysing funnel graph diagrams.

The validity of weighted mean results estimated in meta-analysis has been criticized. This paper presents a set of simple statistical and graphical techniques that can be used in meta-analysis to evaluate common points of criticism. The graphical techniques are based on funnel graph diagrams. Problems and techniques for dealing with them that are discussed include: (1) the so-called 'apples and oranges' problem, stating that mean results in meta-analysis tend to gloss over important differences that should be highlighted. A test of the homogeneity of results is described for testing the presence of this problem. If results are highly heterogeneous, a random effects model of meta-analysis is more appropriate than the fixed effects model of analysis. (2) The possible presence of skewness in a sample of results. This can be tested by comparing the mode, median and mean of the results in the sample. (3) The possible presence of more than one mode in a sample of results. This can be tested by forming a frequency distribution of the results and examining the shape of this distribution. (4) The sensitivity of the mean to the possible presence of atypical results (outliers) can be tested by comparing the overall mean to the mean of all results except the one suspected of being atypical. (5) The possible presence of publication bias can be tested by visual inspection of funnel graph diagrams in which data points have been sorted according to statistical significance and direction of effect. (6) The possibility of underestimating the standard error of the mean in meta-analyses by using multiple, correlated results from the same study as the unit of analysis can be addressed by using the jack-knife technique for estimating the uncertainty of the mean. Brief examples, taken from road safety research, are given of all these techniques.