ANCOVAs and Correlations

ANCOVAs and Correlations Overview

• This lesson will cover ANCOVAs, correlations, adjusting group means, data interpretation, effect sizes, and measuring relationships.

ANCOVA

• Adjusts group means to account for the influence of a covariate.
• Analyzes the main effects and covariates.
• Requires a deep understanding of the variables and relationships being tested.
• Conceptually the same as ANOVA, the application is different
• Controls for the effect of certain variables

Adjusting Group Means

• Raw means are adjusted using a covariate to provide a more accurate comparison between groups.

Dog Therapy Example

• This studies the impact of dog therapy on student happiness while considering the extent to which the student loves dogs.
• Dependent variable: student happiness.
• Independent variable: dose of dog therapy either control, 15 or 30 mins.
• Covariate: a student’s love of dogs rated from 0-7.
• The group means subsequently adjust to account for the influence of the dog's love on a student's happiness.
• The results reflect the therapy accurately, and the remaining influence of dog therapy beyond the influence of the love of dogs.

Interpreting ANCOVA Output

• "dog_love" covariate; F(1, 26) = 4.96, p = 0.035 meaning Students' happiness is influenced by their love of dogs.
• "Dose" is the IV w/effect of the covariate removed.
• "Dose" had significance (F(2, 26) = 4.14, p = 0.027).
• Mean differences between the groups need to be examined.

Estimated Marginal Mean

• This is also known as adjusting the means.
• Estimates are subsequently adjusted based on the covariate.
• The adjusted mean comparisons are conducted between all levels of all groups, which is the same as an ANOVA output.

Effect Size in ANCOVA

• Partial eta-squared (𝜼𝜼2p) determines the proportion of variance that a variable explains that other variables in the analysis do not explain.
• IV: 25.185 / (25.185+79.047) = .24
• Covariate: 15.076 / (15.076+79.047) = .160
• The IV has an effect size of .24, explaining the variance that is attributed to the covariate.

Independence of Covariate and Treatment

• Illustrates that if there is violation of assumption there is some covariance with the IV treatment impacting the variance explained.

Homogeneity of Regression Slopes

• Group 1, 2, 3 have different slope of regression as the covariate or independent variable impacts the mean difference DV (Y).

Key comparisons Between ANCOVA and ANOVA Methods

• ANOVA helps decipher difference between the groups averages, and the population.
• ANOVA’s are broadly applied, and are simple
• ANOVA can’t control for affect of certain variables
• ANCOVA adjusts the DV to remove the influence of additional variables, while providing a better understanding of the relationship between the DV and IV.
• ANCOVA requires a deep understanding of the variables and relationships.
• ANCOVA can control for the effect of certain variables

The Remainder of the Course

• Relationships are explored using a categorical variable: Group A vs. Group B.
• Between groups with the between-subjects design.
• Between treatment and control groups with the basic experimental design.
• Within individuals with pre/post test design.
• Psychology compares people considering continuous variables.

Correlation

• Measures the extent to which two variables are related.
• Responses pattern across variables.
• Key insight for human behavior and cognition.
• Pearson’s Correlation showcases the relationships between two quantitative variables.

What Correlations Tell Us

• Form is what kind of relationship variables have.
• Strength is how closely data points align around a pattern.
• Direction represents if the variables are in the same or opposing direction.
• Quantification is the numerical value to represent everything, denoted by 'r'.

Correlation Types and Use in Psych

• Expressed as a coefficient.
• Ranges from -1 to +1
• Positive Correlation: both Variable A and B increase.
• Variable A increases, variable B increases.
• Negative Correlation: Variable A increases and Variable B decreases.
• Variable A increases and variable B decreases.
• Zero Correlation: there is no correlation or linear relationship.
• Academic Performance & Sleep
• Social Media & Well-Being
• Stress & Health Outcomes
• Parenting Styles & Child Behavior
• Exercise & Mental Health
• Technology Use & Attention Span
• Income & Happiness

Measuring Relationships

• Variance refers to the average amount data vary from the mean of that variable, the formulas are as follows:
• Variance = Σ(xi – x)²／N-1 = Σ(xi – x)(xi – x)／N-1
• Covariance: Sum of combined deviations.
• Covariance(x, y) = Σ(xi – x)(yi – y)/ N-1
• Numerator: cross-product deviations.

Measuring Relationships

• Positive covariances: variables moving in + way.
• Negative covariances: variables moving in - way.
• No linear relationship if inconsistency in covariances.
• Depends on scale of measurement used.
• Can't compare in objective way.
• Convert to a standard set of units.

Standardizing the Covariance

Covxy = 14.00 Sx = 3.162 Sy = 4.550 SxSy = 14.390

• Pearson's correlation coefficient is essentially a standardized version of covariance. Covxy／SxSy = 14.00／14.390 =.973 ≈ .97

Pearson Correlation Coefficient

• Denotation 'r' always gets a value that can range from -1 to 1
• Interpretation should be done in the context of the study for practical significance.
• r=0: no linear correlation
• 0.1 to 0.3 (-0.1 to -0.3)= Small Weak Correlation
• 0.3 to 0.5 (-0.3 to -0.5)= Moderate Medium Correlation
• 0.5 to 0.7 (-0.5 to -0.7)= Large/Strong Correlation
• 0.7 to 0.9 (-0.7 to -0.9)= Very Large/Very Strong Correlation
• +1/(-1) = Perfect Positive/Negative Correlation
• A test of significance of correlation is if it different than 0 or just by chance: The null hypothesis says correlation does not have significant difference from 0

Correlation and Causality

• Correlation is not causation, even though they are related!
• Correlation: statistical indicator of relationship.
• Causation: a change in one variable brings about changes in the other; cause & effect.
• Reasons for this: -Third-variable problems with confounding
• Direction of causality makes it impossible to conclude which variable is causing the change.
• Coincidence: sometimes there are relationships that do not actually exits..

How to Conduct a Correlation Analysis

• Correlation between Anxiety and Exam and Performance 103 Participants.
• Analyze > Correlate > Bivariate.

Correlation Measures

• Time sent Revising.
• Exam performance.
• Exam anxiety.
• Gender.

Other Types of Correlations

• Nonparametric tests exist if there two variables with a non linear relationship.
• Spearman's rho same as Pearson's correlation but on ranked data.
• Kendall's Tau more robust than rho, so it is good when there are outliers.
• Conduct it similarly to Pearson's in SPSS - just uncheck Pearson and check either Spearman or Kendall.
• Measure the relationship between two, adjust for that effect what a third, as it has both.
• Partial Correlations.