MULTIVARIATE DATA ANALYSIS
Analysis of covariance (ANCOVA)
The ANCOVA model; ANOVA + covariates
𝑌𝑖𝑗 = µ+ α𝑗 + 𝑏𝑤(𝐶𝑖𝑗 − 𝐶) + 𝑒𝑖𝑗
µ - the overall mean
α𝑗- the group effect of group j
𝑏𝑤 - the within-groups regression weight
𝐶𝑖𝑗- the covariate score of individual i in group j
𝐶- the mean value of the covariate
Is there a significant effect of the teaching method on posttest? (report test statistic, df, p
value, and effect size). If yes, interpret that effect using the estimated marginal means and
the post hoc tests?
2 2
Reportthe η a.k.a. the 𝑅 seen in the bottom of the table, then report the F(corrected
model, error)
Is there a significant correlation between pretest and posttest in all groups?
Look at the Within Group Correlations table
Do you think that adding the covariate might lead to reduction of error, reduction of bias,
neither, or both?
Large and significant within-group correlations (correlation between the pretest and
the posttest) could lead to a possible reduction of error
Significant differences between group means could lead to a possible elimination of
bias
Check the assumptions of linearity and parallelism of the regression lines using the figures
and the ANOVA table with the Pretest * Method interaction. Is the ANCOVA model a
reasonable approximation of the data?
Linearity - look for a horizontal line in the standardized residuals
, Parallelism - regression lines between the covariate and the dependent variable
have the same regression weight bw in each group (bw = the within-groups
regression weight); it is assumed that there is no interaction between the factor and
the covariate → parallel regression lines
If the interaction effect is non significant, there is no interaction between the factor
and the covariate; regression lines parallel in population
Is the covariate significant? If yes, what is the value of the within-groups regression weight
for the covariate (bw)? Interpret this regression coefficient.
For bw; look at the Parameter Estimates table; under the pretest B value
To check whether the covariate is significant; look at the pretest value in the Tests of
Between-Subjects Effects table
If the pretest is significant; higher pretest scores are associated with higher posttest
scores
Calculating the adjusted means Yj .
*
𝑌𝑗 = 𝑌𝑗 − bw (𝐶𝑗− 𝐶)
Analysis of covariance (ANCOVA)
The ANCOVA model; ANOVA + covariates
𝑌𝑖𝑗 = µ+ α𝑗 + 𝑏𝑤(𝐶𝑖𝑗 − 𝐶) + 𝑒𝑖𝑗
µ - the overall mean
α𝑗- the group effect of group j
𝑏𝑤 - the within-groups regression weight
𝐶𝑖𝑗- the covariate score of individual i in group j
𝐶- the mean value of the covariate
Is there a significant effect of the teaching method on posttest? (report test statistic, df, p
value, and effect size). If yes, interpret that effect using the estimated marginal means and
the post hoc tests?
2 2
Reportthe η a.k.a. the 𝑅 seen in the bottom of the table, then report the F(corrected
model, error)
Is there a significant correlation between pretest and posttest in all groups?
Look at the Within Group Correlations table
Do you think that adding the covariate might lead to reduction of error, reduction of bias,
neither, or both?
Large and significant within-group correlations (correlation between the pretest and
the posttest) could lead to a possible reduction of error
Significant differences between group means could lead to a possible elimination of
bias
Check the assumptions of linearity and parallelism of the regression lines using the figures
and the ANOVA table with the Pretest * Method interaction. Is the ANCOVA model a
reasonable approximation of the data?
Linearity - look for a horizontal line in the standardized residuals
, Parallelism - regression lines between the covariate and the dependent variable
have the same regression weight bw in each group (bw = the within-groups
regression weight); it is assumed that there is no interaction between the factor and
the covariate → parallel regression lines
If the interaction effect is non significant, there is no interaction between the factor
and the covariate; regression lines parallel in population
Is the covariate significant? If yes, what is the value of the within-groups regression weight
for the covariate (bw)? Interpret this regression coefficient.
For bw; look at the Parameter Estimates table; under the pretest B value
To check whether the covariate is significant; look at the pretest value in the Tests of
Between-Subjects Effects table
If the pretest is significant; higher pretest scores are associated with higher posttest
scores
Calculating the adjusted means Yj .
*
𝑌𝑗 = 𝑌𝑗 − bw (𝐶𝑗− 𝐶)
