Equation 1

Note. Mean Squares follow a chi-square distribution. The ratio of two chi-square distributions follows an f-distribution. Therefore, the test statistics can be precisely called F-Statistics, f-tests or f-value.

Simply, the test statistics can be interpreted as “how strong the treatment is as compared to the random variations in the experimental units.” The stronger/larger the treatment effect, MS_Tretment is as compared to the natural variations in the experimental units, MSE; the stronger the possibility of observing a significant result. In the ANOVA model I, there is only one random error and that is the natural variations in the experimental units. However, in both the ANOVA Model II and III, there are more than just the random variations in the experimental units as random factors are included in the study. Therefore, the appropriate F-Statistics, especially for ANOVA model II & III, is not as straightforward to develop as compared to the ANOVA Model I. In some cases, an appropriate exact F-Statistics does not even exist. Therefore, many software packages, such as MINITAB, SAS, SPSS, Design Expert, and JMP will not automatically produce the F-Statistics for a complex design of experiment. Even if statistical software packages produce results, understanding and interpretation of the results in the context of the problems require the understanding of the correct/appropriate random error associated with the treatment/effect. In this module, we will show a very simple step by step technique to develop the correct F-Statistics flawlessly (Hicks, 1982; Kutner et al., 2005; Underwood, 1996). First, we will start with the fixed effect model, ANOVA Model I as we are familiar with it. Second, we will discuss the pure random effect model, ANVOA Model II. Finally, we will discuss the mixed model (ANOVA model III), including fixed, random and nested factors.

Understanding the process of developing the correct F-Statistics will help design any complex experiments, including, split-plot, split-split-plot, sampling, sub-sampling, repeated measure, nested, nested-nested, and hierarchical design of experiments. The design and analyses of experiments are guided by the factor types (e.g., fixed, random, and nested), rather than the name of the experiments. The experimental design names are more associated with the field of study, such as “repeated measure” in the human/animal subject study, while a similar design is known as “split-plot” in the agriculture field. Factor types determine the appropriate ANOVA Model (e.g., Model III) and the correct F-Statistics.

Equation 2