Correlation

Correlations

• Correlations measure the relationship between two variables.
• Correlation coefficients quantify the strength and direction of a relationship.
• Scatterplots graphically display the relationship between two variables.

Overview of Correlation Analysis

• Correlation coefficients: Numerical measures of the relationship between variables.
• Scatterplots: Visual representations of relationships by plotting data points.
• Calculating Pearson's r: Statistical method to determine the linear relationship between two variables.
• Interpreting correlation coefficients: Understanding the strength and direction of the relationship, ranging from +1 (perfect positive) to -1 (perfect negative) correlation.
• Coefficient of determination (r²): Representing the proportion of variance in one variable explained by the other.
• Statistical significance: Evaluating whether the observed correlation is likely due to chance or a true relationship.
• Spearman's correlation coefficient: Statistical method for measuring the monotonic relationship between two variables (useful for non-linear relationships).

Correlation

• Reflects the degree of relation between variables: Indicates the strength and direction of a relationship.
• Calculations of Correlation Coefficients:
  • Direction: Positive (+), negative (-): Positive indicates both variables tend to increase or decrease together; negative indicates one variable increases while the other decreases.
  • Strength: Closer to 0, weaker the relationship; further from 0, stronger the relationship.
• Form of Relationships: Scatterplots reveal the type of relationship (linear, non-linear).

Check Yourself

• Statements involving positive/negative relationships in various contexts (e.g., IQ and GPA, crime rates and population density, car weight and gas mileage, anxiety and task time).

Scatterplots

• Visualizations showing the relationship between two variables, with each point representing an individual.
• A visual representation of data helping determine correlational relationships.

Correlation & Scatterplots

• Data sets for examining the correlation between two sets of exam scores, for example.
• Use of correlation coefficient, for example r = 0.91
• Scatterplot examples displayed

Correlation & Scatterplots

• A correlation between the number of arrests and GPA, for example.
• Correlation coefficients displayed (-0.98)

Pearson's r

• Equation for calculating Pearson's r is displayed
• Different steps to calculate Pearson's r and associated formulas are shown.

Calculating SP (Sum of Products)

• Definitional and computational formulas for calculating Sum of Products: Different approaches to calculate the sum of products, including equations, to perform calculations for correlational data.
• Worked example for SP calculation: Example calculation using data.

Example #1 - Calculating SP - Definitional Formula

• Detailed steps of calculating SP (sum of products) for a particular data set.

Example #1 - Calculating SP - Computational Formula

• Alternate formula for the calculation of the sum of products.

Calculating Pearson's r

• Detailed steps to follow when implementing the Pearson's correlation method.
• Including formulas related to this measure.

Example #1 - Answers

• Complete example for solving for Pearson's correlation method.

Pearson's r

• Definition - A measure of the linear association between two variables.
• Covariability in X and Y in relation to the total variability in X and Y individually.

Using Pearson's r

• Pearson's r used in various statistical applications.
• Including prediction, validity, and reliability.

Verbal Descriptions of Correlations

• Example correlation values associated with real-world scenarios, including strengths and directions of relationships.

Interpreting Correlations

• Correlation does not equal causation: Even if two variables show a high correlation, it does not mean one causes the other.
• Directionality problem: Unable to determine which variable causes the other if there is a correlation.
• Third-variable problem: Unidentified variable could influence both variables and create a spurious correlation to the variables of interest.
• Restricted range: Analyzing data from a narrow range or a specific subject of study in a restricted scope. Correlational analyses may be obscured in these cases.
• Outliers and their impact: The effect of outliers on correlation coefficients, and how to analyze data with their presence.
• Coefficient of determination (r²): Evaluating the proportion of variability in one variable that can be predicted from the relationship with the other variable.

Hypothesis Testing for Correlations

• Making inferences based on sample data: Analyze sample data to determine if correlation data suggests a relationship that is statistically significant or random.
• Statistical significance vs. chance: Examining if the observed correlation is likely due to chance or a true relationship.

Degrees of Freedom for Correlations

• Sample size and degrees of freedom: Define the degrees of freedom for correlation analyses, in relation to sample sizes.
• Clarify when degrees of freedom are not applicable for correlational analyses, for instance when analyzing two sets of data points.

Using Table to Determine Significance

• Degrees of freedom (df) in correlation analysis: Using a statistical table, determine the significance of the observed correlation r, relative to a specific sample size and statistical level.
• Critical values: Significance level, such as α = .05, along with sample size utilized to find the critical value from statistical tables. Determine if the calculated "r" value is significant or not, relative to the critical value.
• APA format for reporting results: Correct formatting to document correlation results in a style appropriate for academic journal publications (APA).

Spearman Correlation

• Used for ordinal data: Explain the Spearman correlation method, including application scenarios where ordinal data applies.
• Ranking scores: Explain converting numerical to ranks using examples.
• Formula: Equation for calculating Spearman's rho. Explanation of the related variables in the equation.

Example #3 - Spearman Correlation

• Illustrative examples for practicing Spearman correlation analysis.

Example #3 Spearman Answers

• Illustrative solution to Spearman's correlation analysis problems.

Example #4 - Spearman Correlation

• Applying the Spearman correlation to movie rating data

Example #4 - Spearman Answers

• Illustrative solution to Spearman's correlation analysis problems.

Example #5 - Pearson's r Correlation

• Analyzing participant data to execute a Pearson's correlation analysis and interpret the results.
• Steps demonstrating how to conduct a Pearson's r analysis, including calculating a correlation coefficient, plotting a scatterplot, and determining whether a relationship is statistically significant at a .05 significance level.

This quiz covers the fundamental concepts of correlation analysis, including correlation coefficients, scatterplots, and Pearson's r. You will learn how to interpret correlation coefficients and assess statistical significance in the context of two variables. Test your understanding of the relationship between data points and the strength of these relationships.