12/8/21, 3:40 PM Learn – ACN 6312.003 - Research Methods In Behavioral ... Omar Alshaikh 33 My eLearning ACN 6312.003 - Research Methods In Behavioral And Brain Sciences - Part I - F21 Week 11: (11/1 - 11/7): Correlations Course Homepage Learn Learn Lecture Outline 9: Correlation Contact Tracing Correlations Aims Measuring Relationships Scatterplots Covariance Pearson’s Correlation Coefficient Nonparametric measures Spearman’s Rho Kendall’s Tau Interpreting Correlations Causality Partial Correlations REMEMBER: Association Claims Association claims argue that one level of a variable is likely to be associated with a particular level of another variable. Association claims involve at TWO (or more) MEASURED VARIABLES. Variables that are associated are correlated. Ex: Correlational Research: Observing what naturally goes on in the world without directly interfering with it. Conducting correlation analysis: Basics What is a correlation? It is a way of measuring the extent to which two variables are related. It measures the pattern of responses across variables. Perfect Correlations No Correlation Negative Correlation Curvilinear https://elearning.utdallas.edu/webapps/blackboard/content/listContent.jsp?course_id=_210136_1&content_id=_4899087_1 1/5 12/8/21, 3:40 PM Learn – ACN 6312.003 - Research Methods In Behavioral ... Curvilinear Measuring relationships We need to see whether as one variable increases, the other increases, decreases or stays the same. This can be done by calculating the Covariance. We look at how much each score deviates from the mean. If both variables deviate from the mean by the same amount, they are likely to be related. Pearson Coefficient of Correlation Pearson Correlation Coefficient Revision of variance The variance tells us by how much scores deviate from the mean for a single variable. It is closely linked to the sum of squares. Covariance is similar – it tells is by how much scores on two variables differ from their respective means. A Sweet Example: Some data about toffees and advertisement A Sweet Example: Some data about toffees and advertisement Sum of Deviation Products When a correlation is positive Most high X values are paired with high Y values Most low X values are paired with low Y values Sum of Deviation Products When a correlation is negative Most high X values are paired with low Y values Most low X values are paired with high Y values Covariance Calculate the error between the mean and each subject’s score for the first variable (x). Calculate the error between the mean and their score for the second variable (y). Multiply these error values. Add these values and you get the cross product deviations. The covariance is the average cross-product deviations: Covariance: Plain Example Problems with covariance It depends upon the units of measurement. E.g. The covariance of two variables measured in Miles might be 4.25, but if the same scores are converted to km, the covariance is 11. One solution: standardise it! https://elearning.utdallas.edu/webapps/blackboard/content/listContent.jsp?course_id=_210136_1&content_id=_4899087_1 2/5 12/8/21, 3:40 PM Learn – ACN 6312.003 - Research Methods In Behavioral ... Divide by the standard deviations of both variables. The standardised version of covariance is known as the correlation coefficient. It is relatively affected by units of measurement. Pearson Coefficient of Correlation Computing the Pearson Coefficient of Correlation (interval or ratio data only!) The correlation coefficient The correlation coefficient: Sweet Example The correlation coefficient: Plain Example Pearson Coefficient of Correlation r Always between -1 and + 1 Pearson Coefficient of Correlation The sign of r shows the direction of the correlation - signifies a negative correlation + signifies a positive correlation The value of r shows the strength or magnitude of the correlation Absolute value Correlation Coefficients: Strength of Relationship Things to know about the correlation It varies between -1 and +1 0 = no relationship It is an effect size ±.1 = small effect ±.3 = medium effect ±.5 = large effect Coefficient of determination, r2 By squaring the value of r you get the proportion of variance in one variable shared by the other. Not cause and effect, but variation shared between two variables (we used this as effect size for t-tests) Correlation and causality The third-variable problem: In any correlation, causality between two variables cannot be assumed because there may be other measured or unmeasured variables affecting the results. Direction of causality: Correlation coefficients say nothing about which variable causes the other to change Correlation and Causation https://elearning.utdallas.edu/webapps/blackboard/content/listContent.jsp?course_id=_210136_1&content_id=_4899087_1 3/5 12/8/21, 3:40 PM Learn – ACN 6312.003 - Research Methods In Behavioral ... REMEMBER: How Do Experiments Support Causal Statements? Experiments can meet the three causal rules: 1. Covariance: Is there a relationship? 2. Temporal precedence: Time-Order Relationship - IV must precede DV 3. Internal validity: elimination of alternative explanations The effect should never occur without the presence of the cause. Correlations in SPSS Conducting correlation analysis: Assumptions An anxiety example Anxiety and Exam Performance Participants: 103 students Measures Time spent revising (hours) Exam performance (%) Exam Anxiety (the EAQ, score out of 100) Gender Violations of Assumptions: Normality Correlation using SPSS Statistics Reporting the results REMEMBER: Levels of measurement Categorical (entities are divided into distinct categories): Binary variable: There are only two categories Nominal variable: There are more than two categories Ordinal variable: The same as a nominal variable but the categories have a logical order Continuous (entities get a distinct score): Interval variable: Equal intervals on the variable represent equal differences in the property being measured Ratio variable: The same as an interval variable, but the ratios of scores on the scale must also make sense Nonparametric correlation Spearman’s rho Pearson’s correlation on the ranked data First Rank Data Lowest score a rank of 1, second lowest 2, etc (Section 7.4.1) Nonparametric correlation Kendall’s tau Better than Spearman’s for small samples Tied Ranks (multiple “3rd places” for example…) Nonparametric correlation World’s biggest liar competition https://elearning.utdallas.edu/webapps/blackboard/content/listContent.jsp?course_id=_210136_1&content_id=_4899087_1 4/5 12/8/21, 3:40 PM Learn – ACN 6312.003 - Research Methods In Behavioral ... 68 contestants Measures Where they were placed in the competition (first, second, third, etc.) Creativity questionnaire (maximum score 60) Correlation output Spearman’s rho Correlation output Kendall’s tau Partial and Semi-Partial Correlations Partial correlation: Measures the relationship between two variables, adjusting for the effect that a third variable has on them both Semi-partial correlation: A measure of the relationship between two variables while adjusting for the effect that one or more additional variables have on one of those variables. If we call our variables x and y, it gives us a measure of the variance in y that x alone shares Partial and Semi-Partial Correlations Partial correlation: Measures the relationship between two variables, adjusting for the effect that a third variable has on them both Semi-partial correlation: A measure of the relationship between two variables while adjusting for the effect that one or more additional variables have on one of those variables. If we call our variables x and y, it gives us a measure of the variance in y that x alone shares Doing partial correlation Partial correlation output Correlation Example Correlation Example Correlation Example We asked intro psychology students how many servings of fish they have a year and what their midterm exam grade was Correlation Example We asked intro psychology students how many servings of fish they have a year and what their midterm exam grade was Correlation Examples from Class https://elearning.utdallas.edu/webapps/blackboard/content/listContent.jsp?course_id=_210136_1&content_id=_4899087_1 5/5
