# One-way blocked analysis of variance (ANOVA)

One-way blocked analysis of variance (ANOVA) is used to test the null hypothesis that multiple population means are all equal, allowing for block effects.

(For two samples, one-way blocked ANOVA is equivalent to the two-sample paired t test.)

### Assumptions:

• The measurement errors are independent, and identically normally distributed with mean 0 and the same variance.
• The population (treatment) effect does not interact with the block effect. This means that blocks and treatments each have a simple additive (linear) effect on the measurement value, and that the mean of each observation is the sum of the treatment mean and the block mean (plus perhaps a "grand mean" that is constant for all measurements).
• The blocks may be considered either fixed or random, although they are usually considered random.
• If the blocks are fixed, then all the measurement values are independent, and normally distributed with the same variance.
• If the blocks are random, then all the measurement values are normally distributed with the same variance. The measurement errors are independent of the block effects. The block effects are identically normally distributed with mean 0. Values from the same treatment group have the same mean, after accounting for the block effect. Values from different blocks are independent.
• Measurements from the same random block will be positively correlated. It is assumed that the variance of the difference between the estimated means for any two treatments (groups) will be the same. This property is called sphericity. A slightly more restrictive assumption is that the covariances between observations within any block be the the same for any two different groups. This property is called compound symmetry. If compound symmetry exists, then sphericity also exists, but it is possible for sphericity to exist when compound symmetry does not. Tests or adjustments for lack of sphericity are usually actually based on possible lack of compound symmetry. (See Neter et al. for more details about sphericity and compound symmetry.)
• Once a particular block has been selected (i.e., the block effect has been accounted for), then observations in that block are independent.
• For a multiple comparisons test of the sample means to be meaningful, the treatment effect is viewed as fixed, so that the populations (treatment groups) in the experiment include all those of interest.

These assumptions imply that the variation within each block and the variation within each each sample (treatment) will be the same, since the variance is assumed to be the same for all the measurements.

A one-way blocked analysis of variance (ANOVA) tests whether any of the population means differ from each other. A multiple comparisons test may be used to answer the question of which population means differ from which other means, a question the ANOVA itself will not answer.

The purpose of the blocking factor is to account for a nuisance factor and/or to reduce the error term used in performing the test for the significance of the treatment effect. For this reason, the significance of the block effect itself is not tested, nor are multiple comparisons done between fixed blocks. Otherwise, a one-way blocked ANOVA is analyzed as a a two-way ANOVA with no interactions and no replications.

If there are only two treatments, the overall F test is equivalent to a paired t test, and compound symmetry and thus sphericity is guaranteed.

A one-way blocked ANOVA with random blocks is analyzed the same way as a repeated measures design with one repeated measures (one within) factor. The subjects are the blocks, and each subject either receives each treatment over time, or the same treatment evaluated at different times.

If the main goal of the analysis is simply to test the significance of the treatment effect, then the assumption of no interaction between blocks and treatments can be relaxed for a one-way blocked ANOVA with random blocks. The overall F test is the same as for the no-interaction case. Under compound symmetry, the correlation between two observations from the same block will still be constant, but will not be the same as in the no-interaction case.

### Guidance:

• Ways to detect before performing the one-way blocked ANOVA whether your data violate any assumptions.
• Ways to examine one-way blocked ANOVA results to detect assumption violations.
• Possible alternatives if your data or one-way blocked ANOVA results indicate assumption violations.

To properly analyze and interpret results of one-way blocked analysis of variance, you should be familiar with the following terms and concepts:

If you are not familiar with these terms and concepts, you are advised to consult with a statistician. Failure to understand and properly apply one-way blocked analysis of variance (ANOVA) may result in drawing erroneous conclusions from your data. Additionally, you may want to consult the following references:

• 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.
• Glen, W. A. and Kramer, C. Y. 1958. Analysis of variance of a randomized block design with missing observations Applied Statistics 7: 173-185.
• Miller, Rupert G. Jr. 1996. Beyond ANOVA, Basics of Applied Statistics. 2nd. ed. London: Chapman & Hall.
• 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.