Experimental Psychology: ANCOVAs and Correlations

Introductory Experimental Psychology Overview

• Dr. Robert M. O'Hara teaches Introductory Experimental Psychology which focuses on ANCOVAs and Correlations.

Agenda Topics

• The agenda includes ANCOVA topics such as adjusting group means, interpreting outputs, and effect sizes.
• Measuring relationship topics include covariance and correlation coefficients.

Adjusting Group Means (ANCOVA)

• Adjusting group means is an ANCOVA procedure.
• Graph of Adjusted Group Means includes SPSS, Lecture, Group and Online learners: -Adjusted means for SPSS are 27.1 -Adjusted means for online are 22.5 -Adjusted means for group are 18.5 -Adjusted means for lecture are 14.1
• Raw means for SPSS learners are 29.4
• Raw means for online learners are 25.1
• Raw means for the group are 17.9
• Raw means for the lecture are 9.8.

Dog Therapy Example (ANCOVA)

• The dependent variable in the dog therapy example is student happiness.
• The independent variable is the dose of dog therapy, which includes control, 15-minute, and 30-minute intervals.
• Covariates account for the influence the student's love of dogs has on their happiness.
• Participant's happiness with control dose recorded 3.22 (1.79).
• Participant's happiness with 15 minute dose recorded 4.88 (1.46)
• Participant's happiness with 30 minute dose recorded 4.85 (2.12)
• Average love of dogs recorded: 2.73 (1.86)
• Love of dogs with control dose recorded 3.44 (2.07).
• Love of dogs with 15 minute dose recorded 3.12 (1.73)
• Love of dogs with 30 minute dose recorded 2.00 (1.63)

Interpreting Output Analysis (ANCOVA)

• "Dog_love" is the covariate, F(1, 26) = 4.96, p = .035, meaning students' happiness is influenced by their love of dogs.
• "Dose" is the independent variable with the effect of the covariate removed
• "Dose" is significant, F(2, 26) = 4.14, p = .027.
• Mean differences between the groups should be examined.

Estimated Marginal Mean/Adjusting Means (ANCOVA)

• Adjusted mean comparisons show all levels of all groups and are the same as ANOVA output.
• Covariates in the model are evaluated at 2.73 for "love of dogs".

Effect Size

• ANCOVAs use a partial eta-squared.
• Partial eta-squared is interpreted as the proportion of variance a variable explains that those other variables do not in the analysis.
• An independent variable (IV) effect size is .24
• A covariate effect size is .160
• Eta squared (𝜼 2 p ) is calculated as
  • 𝑆𝑆𝑆𝑆𝐸𝑓𝑓𝑒𝑒𝑐𝑡 (variable) / 𝑆𝑆𝑆𝑆𝐸𝑓𝑓𝑒𝑒𝑐𝑡 (total).

Independence and Treatment of Covariate

• For total variance in the dependent variable (DV), variance explained by the independent variable (IV) is separate from unexplained variance, and variance explained by the covariate.
• Violation of an assumption occurs if variance explained by the IV includes portions that overlap with variance explained by the covariate and unexplained variance.

Homogeneity of Regression Slopes

• Displays homogeneity and heterogeneity of regression slopes through graphs with regression lines
  • Homogeneity has regression lines, all with a similar slope, that shift up the Y axis depending on group
  • Heterogeneity has regression lines for different groups that have different slopes
• It assumes that the relationship between the covariate and the outcome is the same for all groups.

Wrapping up ANOVA & ANCOVA

• Conceptually, ANOVA & ANCOVA are the same except ANCOVA controls certain effects of variables.
• ANOVA helps decipher if groups differ and if they come from the same population.
• ANCOVA adjusts the DV to understand the relationship
• ANOVA is simple and applied broadly.
• ANCOVA requires a deep understanding of variables & their relationships.

Course Topic: Relating People With Continuous Variables

• Relationships are explored using a categorical variable (e.g., Group A vs. Group B).
• A relationship between a between-subjects design including groups of people is explored.
• A relationship between treatment & control groups, a basic experimental design, is explored.
• A relationship between within individuals/ a pre/post test design is explored.
• Psychology often looks to compare people with continuous variables such as the relationship between motivation and engagement.

Correlation Definition and Measures

• A correlation measures the extent to which two variables are related and their pattern of responses across variables.
• Variable A ↑, Variable B ↑
• Variable A ↑, Variable B ↓
• Variable A ↑, Variable B stays the same

Research with Psychological Correlations

• Gaining insights into human behavior and cognition relies on on correlations.
• Pearson's Correlation studies the relationship between two quantitative variables.

Correlation Information

• Variables relationships measured by form, strength, and direction
• Numerical correlations use quantification (r).

Correlations in Psychology

• Correlation examples include academic performance & sleep, social media & well-being, and stress & health outcomes.
• Further examples include parenting styles & child behavior, exercise & mental health, and technology use & attention span, as well as income & happiness.

Measuring Relationships in Variables

• Variance averages the amount that data vary from the mean of that variable.
  • V a r i a n c e = ∑ ( x i – x ) 2/𝑁−1= ∑ ( x i– x )( xi– x ) /𝑁−1
• Covariance is the sum of combined deviations
• Covariance is found using this formula: ∑(xi – x )(yi – y )/(N-1)
• The numerator is the cross-product deviations.

Covariances

• Positive covariances means variables moving in + way
• Negative covariances means variables moving in − way
• No linear relationship can be found among inconsistencies
• Uses measurement scale to compare in objective way
• Variables must convert to a standard set of units.

Standardizing Covariance

• Standardize the units to fix and standardize measurements
• It is possible to remove measurement scale by dividing by the SD of the variables
• The SD is the average deviation from the mean.
• Use the following formulas to determine the product of both Standard Deviations:
  • Covxy = 14.00
  • Sx = 3.162
  • Sy = 4.550
  • SxSy = 14.390
  • Covxy/SxSy = 14.00/14.390 = .973 ≈ .97
• Pearson's correlation coefficient is a standardized version of covariance.

Pearson Correlation Coefficient

• The Pearson correlation coefficient is denoted by r.
• 'r' values range from -1 to 1
• Interpreting 'r' should be done in context
• 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
• Test significance of correlation to determine if it is different than 0 or just by chance?

Correlation and Causality

• Correlation≠Causation
  • Correlation = statistical indicator of a relationship
  • Causation = change in one variable brings about changes in the other; cause & effect
• Reasons for this:
• Third-variable problem there may be other confounding variables
• Direction of causality impossible to conclude which variable is causing the change
• Coincidence sometimes relationships don’t exist

Correlation Anaysis

• Conducting a Correlation analysis includes:
  • Correlation between Anxiety and Exam Performance
  • Measures
• Time Spent Revising
• Score on exam
• Anxiety
• Gender
  • Analyzed by: Analyze>Correlate>Bivariate

Types of Correlations

• But what happens when we don't have two quantitative variable or when the relationship isn't linear nor follows a normal distribution which effects nonparametric tests
• Spearman Rho: When same as "Pearsons" Correlation but on ranked data
• Kendall's Tau: With More robust than rho then should use when have outliers
• Conduct it similarly to Pearson's in SPSS: Just uncheck Pearson and check either Spearman or Kendall

Partial Correlations

Measure the relationship between two variables adjusting for the effect that a third variable has on both of them