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.
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 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 Statistics7: 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.
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