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 table of descriptive 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 |
The table of inferential statistics shows the key elements to be calculated.
| Source | SS | df | MS | F | p | Eta^2 |
|---|---|---|---|---|---|---|
| Between | 56.000 | 2 | 28.000 | 4.667 | 0.041 | 0.509 |
| Within | 54.000 | 9 | 6.000 | |||
| Total | 110.000 | 11 |
Within Groups Statistics: Within-groups error statistics are a function of the within group variabilities.
\[SS_1 = ( SD_1^2 ) ( df_1 ) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_2 = ( SD_2^2 ) ( df_2 ) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_3 = ( SD_3^2 ) ( df_3) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_{WITHIN} = SS_1 + SS_2 + SS_3 = 18.000 + 18.000 + 18.000 = 54.000\] \[df_{WITHIN} = df_1 + df_2 + df_3 = 3 + 3 + 3 = 9\] \[MS_{WITHIN} = \frac{SS_{WITHIN}}{df_{WITHIN}} = \frac{54.000}{9} = 6.000\]
Grand (or Total) Mean: A grand mean can be determined by taking the weighted average of all of the group means.
\[M_{TOTAL} = \frac{\sum n_{GROUP} (M_{GROUP})}{N} = \frac{ 4 (2.000) + 4 (6.000) + 4 (7.000) }{( 4 + 4 + 4 )} = 5.000\]
Between Groups Statistics: The between-groups effect statistics are a function of the group (level) means and sample sizes.
\[SS_{BETWEEN} = \sum n_{GROUP} (M_{GROUP} - M_{TOTAL})^2 = 4 ( 2.000 - 5.000 )^2 + 4 ( 6.000 - 5.000 )^2 + 4 ( 7.000 - 5.000 )^2 = 56.000\] \[df_{BETWEEN} = \text{# groups} − 1 = 3 − 1 = 2\] \[MS_{BETWEEN} = \frac{SS_{BETWEEN}}{df_{BETWEEN}} = \frac{56.000}{2} = 28.000\]
Statistical Significance: The F statistic is the ratio of the between- and within-group variance estimates.
\[F = \frac{MS_{BETWEEN}}{MS_{WITHIN}} = \frac{28.000}{6.000} = 4.667\]With dfBETWEEN = 2 and dfWITHIN = 9, FCRITICAL = 4.256
Because F > FCRITICAL, p < .05
Confidence Intervals: For ANOVA, calculate the confidence intervals around (centered on) each mean separately.
Because each group has 3 df, tCRITICAL = ±3.182
\[CI_{M_1} = M_1 \pm (t_{CRITICAL}) (SE_{M_1}) = 2.000 \pm (3.182) (1.225) = [ −1.898, 5.898 ]\] \[CI_{M_2} = M_2 \pm (t_{CRITICAL}) (SE_{M_2}) = 6.000 \pm (3.182) (1.225) = [ 2.102, 9.898 ]\] \[CI_{M_3} = M_3 \pm (t_{CRITICAL}) (SE_{M_3}) = 7.000 \pm (3.182) (1.225) = [ 3.102, 10.898 ]\]
Effect Size: The Eta-Squared statistic is a ratio of the between group and the total group variability (Sum of Squares) estimates.
\[\eta^2 = \frac{SS_{BETWEEN}}{( SS_{BETWEEN} + SS_{WITHIN} )} = \frac{56.000}{( 56.000 + 54.000 )} = .509\]