PROPHET StatGuide: Two-factor factorial analysis of variance (ANOVA)

Two-factor factorial analysis of variance (ANOVA) is used to test the null hypotheses that each effect's level means are all equal, simultaneously for each of two factors/effects, and that the interactions between the two factors are all 0.

Assumptions:

• The effects for either factor fixed or random, although they are most commonly fixed.

• If both factors are fixed, then all the measurement values are independent, and normally distributed with the same variance. Measurement values that come from the same combination of levels of factor 1 and factor 2 have the same mean.

• If a single factor is random, then all the measurement values are normally distributed with the same variance. The measurement errors are independent of the random factor effects, and both are independent of the interaction effects. The random factor effects are identically normally distributed with mean 0. Values from the same level of the fixed factor have the same mean. Values from different levels of the random factor are independent. Measurements from the same level of the random factor will be positively correlated, with the same covariance for all levels of the random factor. However, once a particular level of the random factor has been selected (i.e., the random factor effect has been accounted for), then observations for that level of the random factor are independent. Two interaction effects involving different levels of the random factor are independent, while two interaction effects involving the same level of the random factor are negatively correlated.

• If both factors are random, then all the measurement values are normally distributed with the same mean and variance. The measurement errors are independent of the random factors' effects, and both are independent of the interaction effects. The effects for each factor are independent and identically normally distributed with mean 0. The interaction effects are independent and identically normally distributed with mean 0. Values from different levels of the random factor are independent. Measurements that share neither a common level of factor 1 nor a common level of factor 2 will be independent, while measurements from the same level combination of either factors will be correlated,

Prophet uses Type III sums of squares in performing multi-factor ANOVAs. This method requires that there be at least one observation for each possible combination of levels for factor 1 and factor 2 (i.e., no empty cells), although it is not required that each cell have the same number of observations.

If the effects for one or both factors are random, Prophet requires that each cell have the same number of observations.

When there is only one observation for each possible combination of the levels for factor 1 and factor 2 (1 value per cell), there is not enough information available to fit a two-factor factorial model that includes an interaction term, because there is no way to separate the variation of the measurements from the interaction between the two factors. In this case, Prophet performs a main-effects-only model, which assumes that there is no interaction between the two factors. This assumption may or may not be appropriate.

Guidance:

• Ways to detect before performing the two-factor factorial ANOVA whether your data violate any assumptions.

• Ways to examine two-factor factorial ANOVA results to detect assumption violations.

• Possible alternatives if your data or two-factor factorial ANOVA results indicate assumption violations.

To properly analyze and interpret results of two-factor factorial analysis of variance (ANOVA), you should be familiar with the following terms and concepts:

• multi-sample problem
• independent samples
• residuals
• Gaussian (normal) distribution
• equality of variance (homoscedasticity)
• transformations
• factors
• levels within factors
• two-way layout
• interaction

• Brownlee, K. A. 1965. Statistical Theory and Methodology in Science and Engineering. New York: John Wiley & Sons.
• Daniel, Wayne W. 1995. Biostatistics. 6th ed. New York: John Wiley & Sons.
• Lindman, Harold R. 1992. Analysis of Variance in Experimental Design. New York: Springer-Verlag.
• Miller, Rupert G. Jr. 1986. Beyond ANOVA, Basics of Applied Statistics. New York: John Wiley & Sons.
• Neter, J., Wasserman, W., and Kutner, M.H. 1990. Applied Linear Statistical Models. 3rd ed. Homewood, IL: Irwin.
• Rosner, Bernard. 1995. Fundamentals of Biostatistics. 4th ed. Belmont, California: Duxbury Press.
• Winer, B.J., Brown, D.R., and Michels, K.M. 1991. Statistical Principles in Experimental Design. 3rd ed. New York: McGraw Hill.
• Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed. New York: W. H. Freeman and Co.
• Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ: Prentice-Hall.

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Last modified: March 11, 1997