Invited Review: Integrating Quantitative Findings from Multiple Studies Using Mixed Model Methodology

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Abstract
In animal agriculture, the need to understand complex biological, environmental, and management relationships is increasing. In addition, as knowledge increases and profit margins shrink, our ability and desire to predict responses to various management decisions also increases. Therefore, the purpose of this review is to help show how improved mathematical and statistical tools and computer technology can help us gain more accurate information from published studies and improve future research. Researchers, in several recent reviews, have gathered data from multiple published studies and attempted to formulate a quantitative model that best explains the observations. In statistics, this process has been labeled meta-analysis. Generally, there are large differences between studies: e. g., different physiological status of the experimental units, different experimental design, different measurement methods, and laboratory technicians. From a statistical standpoint, studies are blocks and their effects must be considered random because the inference being sought is to future, unknown studies. Meta-analyses in the animal sciences have generally ignored the Study effect. Because data gathered across studies are unbalanced with respect to predictor variables, ignoring the Study effect has as a consequence that the estimation of parameters (slopes and intercept) of regression models can be severely biased. Additionally, variance estimates are biased upward, resulting in large type II errors when testing the effect of independent variables. Historically, the Study effect has been considered a fixed effect not because of a strong argument that such effect is indeed fixed but because of our prior inability to efficiently solve even modest-sized mixed models (those containing both fixed and random effects). Modern statistical software has, however, overcome this limitation. Consequently, meta-analyses should now incorporate the Study effect and its interaction effects as random components of a mixed model. This would result in better prediction equations of biological systems and a more accurate description of their prediction errors.