|PART FOUR -- Essentials|--Analysis of Variance (ANOVA)
Purpose: To test for differences between/among two or more population means.
H0: μ1 = μ2 = μ3 . . .;
Population means are all equal.
Ha: not μ1 = μ2 = μ3 . . .; Population means are not all equal;
Note that Ha is not "all the population means are different."
Rejection of Ho means that there is a statistically significant difference between at
least two of the sample means.
Interval estimation of population means and differences between population means is
also possible.
Sums of squared deviations
TSS--total sum of squared deviations
SST--sum of squared deviations for treatments (between-group variation)
SSE--sum of squared deviations for error (within-group variation)
TSS = SST + SSE
Means of squared deviations--recall that a variance is a mean of squared deviations.
MST--mean of squared deviations for treatments (between-group variance)
MSE--mean of squared deviations for error (within-group variance)
Signal-to-noise analogy
Signal: between-group variance, MST
Noise: within-group variance, MSE
The more false Ho is (the larger the differences between/among population means),
the larger MST will be relative to MSE.
ANOVA table--standardized way of presenting computations and results
Calculated F ( test statistic, Fc ) is MST / MSE
Total degrees of freedom: the number of observations minus one
Degrees of freedom for treatments: number of treatments minus one
Degrees of freedom for error: the number of observations minus the number
of treatments
When there are only two groups and a t-test could be used, the Fc will be equal to the
square of the tc.
Reject Ho if Fc  Ft and if p  α.
Four assumptions (same as t-tests of chapter 9)
Samples
Random
Independent
Populations
Normally distributed
Equal variances
Moderate departures from the assumptions will not seriously affect validity (robust)
One-way ANOVA--completely randomized design
Two-way ANOVA--randomized block design
TSS = SST + SSB + SSE (B = "blocks")
Two calculated F's: treatments FT = MST / MSE and blocks FB = MSB / MSE
Total degrees of freedom: the number of observations minus one
Degrees of freedom for treatments: the number of treatments minus one
Degrees of freedom for blocks: the number of blocks minus one
Degrees of freedom for error: the number of observations minus the number of
treatments, minus the number of blocks, plus one
Estimation in One-Way ANOVA
tt in the following equations is based on the number of degrees of freedom for error.
Single population mean
 = X  t t ( ˆ x )
where
MSE

n
ˆ X =
MSE / n
Difference between two population means:
(  1 -  2 ) = ( x1 - x2 )  t t ˆ ( x1- x2 )
where
ˆ ( x - x ) = MSE x
1
2
1
+
n1
1
n2
Estimation in two-way ANOVA (randomized block design)
Two-way ANOVA estimation -- valid only for differences between population means.
Confidence intervals cannot be obtained for individual treatment means.
tt in the following equations is based on the number of degrees of freedom for error,
Difference between two population means:
(  1 -  2 ) = ( x1 - x2 )  t t ˆ ( x1- x2 )
where
ˆ ( x - x ) = MSE x
1
Three-way analysis of variance
"Latin square" design
2
1
n1
+
1
n2
Terminology--explain each of the following:
TSS--total sum of squared deviations, SST--sum of squared deviations for treatments
(between-group variation), SSE--sum of squared deviations for error (within-group
variation), variance, MST--mean of squared deviations for treatments (between-group
variance), MSE--mean of squared deviations for error (within-group variance), signal-tonoise ratio, ANOVA table, calculated F (MST / MSE), degrees of freedom (treatments,
blocks, error), four assumptions (same as t-tests of chapter 9), robust test--moderate
departures from the assumptions will not seriously affect validity, completely randomized
design, randomized block design, "Latin square" design
Skills and Procedures
 given appropriate data, conduct a one-way ANOVA and interpret the results; include
all possible 95% confidence intervals
 given appropriate data, conduct a two-way ANOVA and interpret the results; include
all possible 95% confidence intervals
Concepts
 explain why, when ANOVA deals with tests on means, it is called “analysis of
variance”
 explain the “signal-to-noise ratio” concept in the context of ANOVA
 describe the shortcoming that ANOVA shares with small-sample t-tests
 show where the variances are found in the ANOVA table
If the H0 is rejected:
“The difference between at least two of the sample means of the __________ is
statistically significant at the α level. The population means are probably not all equal.”
If the H0 is not rejected:
“The differences among the sample means of the __________ are not statistically
significant at the α level. All the population means could be equal.”