The following data set reflects a between-subjects design with one factor (with three levels). The data are presented in a format suitable for entry into statistical software.
| Factor | Outcome | |
|---|---|---|
| 1 | 1.00 | .00 |
| 2 | 1.00 | .00 |
| 3 | 1.00 | 3.00 |
| 4 | 1.00 | 5.00 |
| 5 | 2.00 | 4.00 |
| 6 | 2.00 | 7.00 |
| 7 | 2.00 | 4.00 |
| 8 | 2.00 | 9.00 |
| 9 | 3.00 | 9.00 |
| 10 | 3.00 | 6.00 |
| 11 | 3.00 | 4.00 |
| 12 | 3.00 | 9.00 |
The following tables represent typical output from statistical software. Options, labels, and layout vary from program to program.
The tables of statistics can be used to determine the inferential statistics.
| N | Mean | Std. Deviation | Std. Error Mean | |
|---|---|---|---|---|
| Level 1 | 4 | 2.000 | 2.449 | 1.225 |
| Level 2 | 4 | 6.000 | 2.449 | 1.225 |
| Level 3 | 4 | 7.000 | 2.449 | 1.225 |
| Source | SS | df | MS | F | p | Eta2 |
|---|---|---|---|---|---|---|
| Factor | 56.000 | 2 | 28.000 | 4.667 | 0.041 | 0.509 |
| Within | 54.000 | 9 | 6.000 | |||
| Total | 110.000 | 11 |
The table of inferential statistics shows the key elements to be calculated.
| Factor | Factor | Mean Difference | SE Difference | HSD | p | Lower CI | Upper CI |
|---|---|---|---|---|---|---|---|
| Level1 | Level2 | -4.000 | 1.732 | -2.309 | 0.106 | -8.836 | 0.836 |
| Level1 | Level3 | -5.000 | 1.732 | -2.887 | 0.043 | -9.836 | -0.164 |
| Level2 | Level3 | -1.000 | 1.732 | -0.577 | 0.835 | -5.836 | 3.836 |
Mean Differences: Mean Differences (raw effects) are the differences between the means for all pairs of groups. Half of the possible pairwise comparisons are redundant and do not need to be calculated (though the mean differences will have the opposite signs because of subtraction order if they were calculated).
\[M_1 - M_2 = 2.000 − 6.000 = −4.000\] \[M_1 - M_3 = 2.000 − 7.000 = −5.000\] \[M_2 - M_3 = 6.000 − 7.000 = −1.000\]
Standard Error of the Difference: These standard errors are for the difference between the two group means in each comparison. The values are a function of the MSWITHIN (from the ANOVA) and the sample sizes. [In this case, because all groups are of the same size, the standard error for each comparison is the same.]
\[SE_{DIFF} = \sqrt{ \left( \frac{MS_{WITHIN}}{n_{GROUP}} \right) + \left( \frac{MS_{WITHIN}}{n_{GROUP}} \right) } = \sqrt{ \left( \frac{6.000}{4} \right) + \left( \frac{6.000}{4} \right) } = 1.732\]
Statistical Significance: The HSD statistic is a ratio of the mean difference to the standard error of the difference. There is one statistic for each of the comparisons.
Because the ANOVA has dfBETWEEN = 2 and dfWITHIN = 9, HSDCRITICAL = 2.792
\[HSD_{1vs2} = \frac{( M_1 - M_2 )}{SE_{DIFF}} = \frac{-4.000}{1.732} = 2.309\]Because HSD < HSDCRITICAL, p > .05
\[HSD_{1vs3} = \frac{( M_1 - M_3 )}{SE_{DIFF}} = \frac{-5.000}{1.732} = 2.887\]Because HSD > HSDCRITICAL, p < .05
\[HSD_{2vs3} = \frac{( M_2 - M_3 )}{SE_{DIFF}} = \frac{-1.000}{1.732} = 0.577\]Because HSD < HSDCRITICAL, p > .05
Confidence Intervals: For HSD, calculate the confidence intervals around (centered on) each mean difference separately.
\[CI_{1vs2} = ( M_1 - M_2 ) \pm (HSD_{CRITICAL}) ( SE_{DIFF}) = -4.000 \pm (2.792) (1.732) = [ −8.836, 0.836 ]\] \[CI_{1vs3} = ( M_1 - M_3 ) \pm (HSD_{CRITICAL}) ( SE_{DIFF}) = -5.000 \pm (2.792) (1.732) = [ −9.836, −0.164 ]\] \[CI_{2vs3} = ( M_2 - M_3 ) \pm (HSD_{CRITICAL}) ( SE_{DIFF}) = -1.000 \pm (2.792) (1.732) = [ −5.836, 3.836 ]\]