Transcript ANOVA - Duke University

ANOVA
One Way Analysis of Variance
ANOVA
Purpose: To assess whether there are
differences between means of multiple
groups. ANOVA provides evidence.
We compare variation among the means
of the groups with the variation within
groups.
Hence the name Analysis of Variance
ANOVA HYPOTHESES
Tests the null hypothesis that the
population means are all equal.
Ho: μ1 = μ2 = μ3 = μ4 = μ5
Ha: not all of the μi are equal.
ANOVA ASSUMPTIONS
The populations are normally distributed.
The standard deviations of the populations
are equal.
Moore and McCabe p. 755:
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Rule for examining standard deviation in
ANOVA
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If the largest standard deviation is less than twice the
smallest standard deviation, we can use methods based
on the assumption of equal standard deviations and our
results will be approximately correct.
Pooled Estimator of σ
When we assume that the population
standard deviations are equal, each
sample standard deviation is an estimate
of σ.
To combine into a single estimate use
pooled sample variance.
Sp = Root Mean Square Error (Root MSE)
ANOVA TABLE
Sum of Squares—represents variation in data, a
sum of squared deviations
SSTotal = SSGroups + SSError
SST - Measures variation around the overall
mean
SSG-Variation of the groups means around the
overall means
SSE-measures variation of each observation
around its group mean.
ANOVA Table Continued
Degrees of Freedom
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Degrees of Freedom Total = N – 1
Degrees of Freedom Group = I – 1
Degrees of Freedom Error = N – I
(I = number of groups)
DFT = DFG + DFE
Sp2 = MSE = SSE/DFE
MSG = SSG/DFG
F-Statistic
If use ANOVA to compare 2 populations,
the F-Statistic is exactly equal to the
square of the t-statistic.
F statistic = MSG/MSE
Use Notation F (I-1, N-I)