Hypothesis test
The following hypotheses are used to test whether the group means are all equal:
H0 : µi = µj for
all i and j
HA: µi ≠ µj for
at least some i, j
We will describe some of the steps for this test, but cannot justify them here.
Mean sums of squares
The three sums of squares are first divided by values called their degrees of freedom:
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The mean total sum of squares is the sample variance of the response (ignoring groups). |
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The mean within-group sum of squares is the pooled estimate of the variance within groups. |
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The mean between-group sum of squares is harder to directly interpret. |
The numerators in these ratios add up:
SSTotal = SSBetween + SSWithin
and the same relationship holds for their denominators (degrees of freedom):
dfTotal = dfBetween + dfWithin
F ratio and p-value
The test statistic is an F-ratio,
This test statistic compares between- and within-group variation. The further
apart the group means, the larger SSBetween and the larger the F-ratio.
Large values of F suggest that H0 does not hold — that the group means are not the same.
The p-value for the test is the probability of such a high F ratio if H0 is true (all group means are the same). It is based on a standard distribution called an F distribution and is interpreted in the same way as other p-values.
The closer the p-value to zero, the stronger the evidence that H0 does not hold.
Analysis of variance table
An analysis of variance table (anova table) describes some of the calculations above: