Summary and study notes

Welke onderwerpen worden behandeld in het hoorcollege?

ANOVA. ANOVA is about comparing groups means on a continuous variable. It is called analysis of variance because the analysis separated between groups variation (variance explained by group) and within group variation (unexplained/residual variance). There is a difference when the between group variation is large compared to the within group variation. If the difference is significant is dependent on the unexplained variation, which is the within group variance. ANOVA test if the between group is big compared to the within group variation. If the within group variance is small, then the difference could be significant. 

ANCOVA = comparing groups means while controlling for a covariate. In an ANCOVA there is always a categorical predictor (factor) and a covariate, which is a continuous variable.

Assumptions of ANCOVA are:

• Homogeneity of regression slopes = the regression lines must be parallel. 
• Independence of the covariate and the factor = ANCOVA can be used when groups are randomized. This implies that any differences between the groups on control variables (covariates) are chance differences.

Welke onderwerpen worden besproken die niet worden behandeld in de literatuur?

In dit college worden geen andere onderwerpen besproken dit niet worden behandeld in de literatuur.  

Welke recente ontwikkelingen in het vakgebied worden besproken? 

Er worden geen recente ontwikkelingen besproken. 

Welke opmerkingen worden er tijdens het college gedaan door de docent met betrekking tot het tentamen?

Er worden geen opmerkingen gedaan die betrekking hebben tot het tentamen. 

Welke vragen worden behandeld die gesteld kunnen worden op het tentamen? 

Er worden geen tentamenvragen behandeld. 

Hoorcollege aantekeningen

ANCOVA = comparing groups means while controlling for a covariate. ANOVA is about comparing groups means on a continuous variable. It is called analysis of variance because the analysis separated between groups variation (variance explained by group) and within group variation (unexplained/residual variance). There is a difference when the between group variation is large compared to the within group variation. If the difference is significant is dependent on the unexplained variation, which is the within group variance. ANOVA test if the between group is big compared to the within group variation. If the within group variance is small, then the difference could be significant. 

Two groups: teen moms and adult moms, question do the babies differ in their cognitive abilities. The variables are age of mother, which is categorized in teen and adult and the variable cognitive ability. But there are other things that influence cognitive ability of the babies, for example IQ. You don’t know exactly which caused the difference (if it’s IQ or the age of the moms). To tangle this problem, you can control by IQ through add it as a variable. This is analysis of covariance (ANCOVA) = there is another predictor that may can influence the outcome. 

In an ANCOVA there is always a categorical predictor (factor) and a covariate, which is a continuous variable. If you want to include IQ, what we have is a regression model. We assume a linear relationship between the two variables (IQ and cognitive ability). We have more groups, so a multiple regression model, so two linear relationships. You can summarize this in a regression equation: Y = b_o + b₁Di + b₂IQ_i (D = age group). The intercepts differ, but the slopes do not. This means that the regression lines are always parallel. 

The results of the ANOVA (not controlling for covariate) will be compared with the ANCOVA (control for covariate). When we are not controlling for covariate (so ANOVA), we talk about unadjusted means. In an ANOVA we compare the unadjusted means of the two groups on cognitive ability (we ignore IQ). In an ANCOVA the mean of the cognitive ability is corrected by the average IQ of the whole sample. Therefore, we use the regression line to see where the overall mean of IQ is for the whole sample and which score on cognitive ability belongs to that (for both groups). This is the covariate adjusted mean = the mean of the two groups adjusted by IQ (through pick the same IQ level for both groups, which is the overall mean IQ of the sample). The result is the ANCOVA. 

The key assumptions of the ANCOVA is that the regression lines must be parallel = homogeneity of regression slopes. There may not be interaction between the covariate and the variable. In the equation you can see that the slopes are always the same, but the intercept differ. You can check this assumption by add the interaction term and check for significant. If its significant, the assumption is violated. Because significance is not always reliable because of large samples (robust) you have to do one more check. Another check is to make a scatter plot with regression lines fitted per group and visually inspect the deviation from parallelism. 

Another ANCOVA assumption is the independence of the covariate and the factor. ANCOVA can be used when groups are randomized. This implies that any differences between the groups on control variables (covariates) are chance differences. ANCOVA should not be used on existing groups. Differences on the covariate can be associated within the grouping factor and disentangling these effects by an ANCOVA is not interpretable. Be careful with your interpretation with dealing with existing groups if there is a lot of overlap between the factor and the covariate. 

The F statistic = between group variance / within group variance. Adding the covariate will change the F-test if adjusted means change (this has effect on between group variance) or the covariate is related to Y (this has effect of within group variance). So, inclusion of a covariate can be useful when groups differ on the covariate, but also when they do not. To be a useful addition, the covariate must be related to Y. Before adding the covariate, you have to check the assumption of homogeneity of slopes. The interaction (between factor and variable) must not be significant. And after that you have to draw a plot and look for parallel regression lines. When the covariate is significant, then the covariate is related to the variable. 

We need information on the slope: direction and strength of the relation between the variable and the covariate to know what the relation is between the covariate and the variable. We need additional information to see which groups differ. We need effect size measures: partial eta-squared for age (covariate) and condition. 

• Parameter estimates are useful to form the regression equation (B).
• Estimates tells the total mean of the covariate (in the footnote). 
• Effect size (partial eta squared) tells the proportion variation that is explained by the model. It tells which variable has a stronger effect size/influence on the outcome. 

Follow up testing

We still don’t know which groups differ. You can use post hoc testing = figure out where the difference is after you know there is a difference (with correction to protect against inflated type I errors) or planned comparisons trough specific contrast-tests = before analysis you describe that you have different groups you want to compare. 

• Pairwise comparisons in SPSS compared every condition with the other conditions. Bonferroni correction protects against type I errors. It adjusts the p-value (p-value times the number of tests). 
• Custom hypothesis tests in SPSS compare specific groups you asked for. You don’t need to adjust for type I error.