The tables of descriptive statistics can be used to determine the inferential statistics.
$Level1
Summary Statistics for the Data
N M SD Skew Kurt
Outcome1 4.000 2.000 2.449 0.544 -2.944
Outcome2 4.000 6.000 2.449 0.544 -2.944
$Level2
Summary Statistics for the Data
N M SD Skew Kurt
Outcome1 4.000 6.000 2.449 0.544 -2.944
Outcome2 4.000 7.000 2.449 -0.544 -2.944
$Level1
Correlations for the Data
Outcome1 Outcome2
Outcome1 1.000 0.500
Outcome2 0.500 1.000
$Level2
Correlations for the Data
Outcome1 Outcome2
Outcome1 1.000 0.389
Outcome2 0.389 1.000
The tables of inferential statistics show the key elements to be calculated.
$`Between Subjects`
Source Table for the Model
SS df MS
Blocks 25.000 1.000 25.000
Subjects 52.000 6.000 8.667
$`Within Subjects`
Source Table for the Model
SS df MS
Measures 25.000 1.000 25.000
Measures:Blocks 9.000 1.000 9.000
Residual 20.000 6.000 3.333
$`Between Subjects`
Hypothesis Tests for the Model
F df1 df2 p
Blocks 2.885 1.000 6.000 0.140
$`Within Subjects`
Hypothesis Tests for the Model
F df1 df2 p
Measures 7.500 1.000 6.000 0.034
Measures:Blocks 2.700 1.000 6.000 0.151
$`Between Subjects`
Proportion of Variance Accounted For by the Model
Est LL UL
Blocks 0.325 0.000 0.600
$`Within Subjects`
Proportion of Variance Accounted For by the Model
Est LL UL
Measures 0.556 0.031 0.737
Measures:Blocks 0.310 0.000 0.591
Descriptive Statistics: The descriptive statistics are calculated separately for each group or condition.
Grand (or Total) Mean: A grand mean can be determined by taking the weighted average of all of the group means.
\[M_{TOTAL} = \frac{\sum n_{GROUP} (M_{GROUP})}{N} = \frac{ 4 (2.000) + 4 (6.000) + 4 (6.000) + 4 (7.000) }{( 4 + 4 + 4 + 4 )} = 5.250\]
Subject Means: Each subject in the study would have an average score across the time points.
\[M_{S1} = \frac{0.000 + 4.000}{2} = 2.000\] \[M_{S2} = \frac{0.000 + 7.000}{2} = 3.500\] \[M_{S3} = \frac{3.000 + 4.000}{2} = 3.500\] \[M_{S4} = \frac{5.000 + 9.000}{2} = 7.000\] \[M_{S5} = \frac{4.000 + 9.000}{2} = 6.500\] \[M_{S6} = \frac{7.000 + 6.000}{2} = 6.500\] \[M_{S7} = \frac{4.000 + 4.000}{2} = 4.000\] \[M_{S8} = \frac{9.000 + 9.000}{2} = 9.000\]
Marginal Means: A level (marginal) mean can be determined by taking the weighted average of the appropriate group means.
For Factor:
\[M_{FACTOR1} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (2.000) + 4 (6.000) }{( 4 + 4 )} = 4.000\] \[M_{FACTOR2} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (6.000) + 4 (7.000) }{( 4 + 4 )} = 6.500\]
For Time:
\[M_{TIME1} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (2.000) + 4 (6.000) }{( 4 + 4 )} = 4.000\] \[M_{TIME2} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (6.000) + 4 (7.000) }{( 4 + 4 )} = 6.500\]
Between-Subjects Error Statistics: Between-subjects error refers to average differences across the participants within each Factor level.
\[SS_{ERROR(BETWEEN)} = \sum (\text{number of Time points}) (M_{SUBJECT} - M_{FACTOR\;LEVEL})^2 = \left[2(2.000-4.000)^2 + 2(3.500-4.000)^2 + 2(3.500-4.000)^2 + 2(7.000-4.000)^2\right] + \left[2(6.500-6.500)^2 + 2(6.500-6.500)^2 + 2(4.000-6.500)^2 + 2(9.000-6.500)^2\right] = 52.000\] \[df_{ERROR(BETWEEN)} = (\text{# levels of Factor})(\text{# subjects per level} - 1) = (2)(4-1) = 6\] \[MS_{ERROR(BETWEEN)} = \frac{SS_{ERROR(BETWEEN)}}{df_{ERROR(BETWEEN)}} = \frac{52.000}{6} = 8.667\]
Within-Subjects Variability: The within-subjects variability reflects person-by-time variability.
\[SS_{SUBJECTS} = \sum (Y - M_{SUBJECT})^2 = (0-2)^2 + (4-2)^2 + (0-3.5)^2 + (7-3.5)^2 + (3-3.5)^2 + (4-3.5)^2 + (5-7)^2 + (9-7)^2 + (4-6.5)^2 + (9-6.5)^2 + (7-6.5)^2 + (6-6.5)^2 + (4-4)^2 + (4-4)^2 + (9-9)^2 + (9-9)^2 = 54.000\] \[df_{SUBJECTS} = (\text{number of subjects})(\text{# time points} - 1) = (8)(2-1) = 8\]
Between-Subjects Effect Statistics: The between-subjects effect (Factor) is a function of the marginal means and sample sizes.
\[SS_{FACTOR} = \sum n_{LEVEL} (M_{LEVEL} - M_{TOTAL})^2 = 8(4.000 - 5.250)^2 + 8(6.500 - 5.250)^2 = 25.000\] \[df_{FACTOR} = \text{# levels of Factor} - 1 = 2 - 1 = 1\] \[MS_{FACTOR} = \frac{SS_{FACTOR}}{df_{FACTOR}} = \frac{25.000}{1} = 25.000\]
Within-Subjects Effect Statistics: The within-subjects effects include the main effect of Time and the Factor × Time interaction.
For Time:
\[SS_{TIME} = \sum n_{LEVEL} (M_{LEVEL} - M_{TOTAL})^2 = 8(4.000 - 5.250)^2 + 8(6.500 - 5.250)^2 = 25.000\] \[df_{TIME} = \text{# levels of Time} - 1 = 2 - 1 = 1\] \[MS_{TIME} = \frac{SS_{TIME}}{df_{TIME}} = \frac{25.000}{1} = 25.000\]
For the Interaction:
\[SS_{INTERACTION} = \sum n_{GROUP} (M_{GROUP} - M_{FACTOR} - M_{TIME} + M_{TOTAL})^2 = 4(2.000 - 4.000 - 4.000 + 5.250)^2 + 4(6.000 - 4.000 - 4.000 + 5.250)^2 + 4(6.000 - 6.500 - 4.000 + 5.250)^2 + 4(7.000 - 6.500 - 6.500 + 5.250)^2 = 9.000\] \[df_{INTERACTION} = (\text{# levels of Factor} - 1)(\text{# levels of Time} - 1) = (2 - 1)(2 - 1) = 1\] \[MS_{INTERACTION} = \frac{SS_{INTERACTION}}{df_{INTERACTION}} = \frac{9.000}{1} = 9.000\]
Within-Subjects Error Statistics: After removing the Time effect and the Interaction effect from the total within-subjects variability, the remaining variation is the within-subjects error term.
\[SS_{ERROR(WITHIN)} = SS_{SUBJECTS} - SS_{TIME} - SS_{INTERACTION} = 54.000 - 25.000 - 9.000 = 20.000\] \[df_{ERROR(WITHIN)} = df_{SUBJECTS} - df_{TIME} - df_{INTERACTION} = 8 - 1 - 1 = 6\] \[MS_{ERROR(WITHIN)} = \frac{SS_{ERROR(WITHIN)}}{df_{ERROR(WITHIN)}} = \frac{20.000}{6} = 3.333\]
Statistical Significance: Each F statistic is the ratio of an effect mean square to its corresponding error mean square in the correct stratum.
For the Factor Main Effect:
\[F_{FACTOR} = \frac{MS_{FACTOR}}{MS_{ERROR(BETWEEN)}} = \frac{25.000}{8.667} = 2.885\]With dfFACTOR = 1 and dfERROR(BETWEEN) = 6, p = .140
This would not be considered a statistically significant finding.
For the Time Main Effect:
\[F_{TIME} = \frac{MS_{TIME}}{MS_{ERROR(WITHIN)}} = \frac{25.000}{3.333} = 7.500\]With dfTIME = 1 and dfERROR(WITHIN) = 6, p = .034
This would be considered a statistically significant finding.
For the Interaction:
\[F_{INTERACTION} = \frac{MS_{INTERACTION}}{MS_{ERROR(WITHIN)}} = \frac{9.000}{3.333} = 2.700\]With dfINTERACTION = 1 and dfERROR(WITHIN) = 6, p = .152
This would not be considered a statistically significant finding.
Effect Size: The partial eta-squared statistic is a ratio of each effect Sum of Squares and the remaining variability after that effect’s corresponding error term has been partialled out.
For the Factor Main Effect:
\[\text{Partial} \; \eta^2 = \frac{SS_{FACTOR}}{( SS_{FACTOR} + SS_{ERROR(BETWEEN)} )} = \frac{25.000}{( 25.000 + 52.000 )} = 0.325\]Thus, 32.5% of the variability among the scores is accounted for by Factor.
For the Time Main Effect:
\[\text{Partial} \; \eta^2 = \frac{SS_{TIME}}{( SS_{TIME} + SS_{ERROR(WITHIN)} )} = \frac{25.000}{( 25.000 + 20.000 )} = 0.556\]Thus, 55.6% of the variability among the scores is accounted for by Time.
For the Interaction:
\[\text{Partial} \; \eta^2 = \frac{SS_{INTERACTION}}{( SS_{INTERACTION} + SS_{ERROR(WITHIN)} )} = \frac{9.000}{( 9.000 + 20.000 )} = 0.310\]Thus, 31.0% of the variability among the scores is accounted for by the Factor × Time interaction.
Confidence Intervals: For Mixed ANOVA, calculate the confidence intervals around (centered on) each mean separately (not shown here).
The mixed ANOVA provides statistics for the main effects and interaction in a mixed design. Each effect is summarized below in APA style, using the actual R output values:
A 2 (Factor) × 2 (Time) mixed ANOVA showed that the small main effect of Factor was not statistically significant, F(1,6) = 2.89, p = .140, partial η2 = .33, nor was the moderately sized interaction, F(1,6) = 2.70, p = .152, partial η2 = .31. However, the large main effect of Time was statistically significant, F(1,6) = 7.50, p = .034, partial η2 = .56.
Typically, the means, standard deviations, and confidence intervals would be presented in a table or figure associated with this text.