Fig. 4.
From: Meta-evaluation of meta-analysis: ten appraisal questions for biologists
Visualizations of the three main types of meta-analytic models and their assumptions. a The fixed-effect model can be written as y i  = b 0 + e i , where y i is the observed effect for the ith study (i = 1…k; orange circles), b 0 is the overall effect (overall mean; thick grey line and black diamond) for all k studies and e i is the deviation from b 0 for the ith study (dashed orange lines), and e i is distributed with the sampling variance ν i (orange curves); note that this variance is sometimes called within-study variance in the literature, but we reserve this term for the multilevel model below. b The random-effects model can be written as y i  = b 0 + s i  + e i , where b 0 is the overall mean for different studies, each of which has a different study-specific mean (green squares and green solid lines), deviating by s i (green dashed lines) from b 0, s i is distributed with a variance of τ 2 (the between-study variance; green curves); note that this is the conventional notation for the between-study variance, but in a biological meta-analysis, it can be referred to as, say, σ 2 [study]. The other notation is as above. Displayed on the top-right is the formula for the heterogeneity statistic, I 2 for the random-effects model, where \( \overline{v} \) is a typical sampling variance (perhaps, most easily conceptualized as the average value of sampling variances, ν i ). c The simplest multilevel model can be written as y ij  = b 0 + s i  + u ij  + e ij , where u ij is the deviation from s i for jth effect size for the ith study (blue triangles and dashed blue lines) and is distributed with the variance of σ 2 (the within-study variance or it may be denoted as σ 2 [effect size]; blue curves), e ij is the deviation from u ij , and the other notations are the same as above. Each of k studies has m effect sizes (j = 1…m). Displayed on the top-right is the multilevel meta-analysis formula for the heterogeneity statistic, I 2, where both the numerator and denominator include the within-study variance, σ 2, in addition to what appears in the formula for the random-effects model