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Fixed-Effects Models
Fixed-effects models are a class of statistical models in which the levels (i.e., values) of independent variables are assumed to be fixed (i.e., constant), and only the dependent variable changes in response to the levels of independent variables. This class of models is fundamental to the general linear models that underpin fixed-effects regression analysis and fixed-effects analysis of variance, or ANOVA (fixed-effects ANOVA can be unified with fixed-effects regression analysis by using dummy variables to represent the levels of independent variables in a regression model; see the article by Andrew Gelman for more information); the generalized linear models, such as logistic regression for binary response variables and binomial counts; Poison regression for Poisson (count) response variables; as well as the analysis of categorical data using such techniques as the Mantel-Haenszel or Peto odds ratio. A common thesis in assuming a fixed-effects model among these analyses is that under conditions of similar investigation methods, similar measurements, and similar experimental or observational units, the mean response among the levels of independent variables should be comparable. If there is any discrepancy, the difference is caused by the within-study variation among the effects at the fixed levels of independent variables. This entry discusses the application of fixed effects in designed experiments and observational studies, along with alternate applications.
Designed Experiments
Fixed-effects models are very popular in designed experiments. The principal idea behind using these models is that the levels of independent variables (treatments) are specifically chosen by the researcher, whose sole interest is the response of the dependent variable to the specific levels of independent variables that are employed in a study. If the study is to be repeated, the same levels of independent variables would be used again. As such, the inference space of the study, or studies, is the specific set of levels of independent variables. Results are valid only at the levels that are explicitly studied, and no extrapolation is to be made to levels of independent variables that are not explicitly investigated in the study.
In practice, researchers often arbitrarily and systematically choose some specific levels of independent variables to investigate their effects according to a hypothesis and/or some prior knowledge about the relationship between the dependent and independent variables. These levels of independent variables are either of interest to the researcher or thought to be representative of the independent variables. During the experiment, these levels are maintained constant. Measurements taken at each fixed level of an independent variable or a combination of independent variables therefore constitute a known population of responses to that level (combination of levels). Analyses then draw information from the mean variation of the study to make inference about the effect of those specific independent variables at the specified levels on the mean response of the dependent variable. A key advantage of a fixed-effects model design is that important levels of an independent variable can be purposefully investigated. As such, both human and financial resource utilization efficiency may be maximized. Examples of such purposeful investigations may be some specific dosages of a new medicine in a laboratory test for efficacy, or some specific chemical compositions in metallurgical research on the strength of alloy steel, or some particular wheat varieties in an agriculture study on yields.
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