SPSS 15.0|Correlation Analysis

Zhenting HE / 2024-01-03

Concept and Methods of Correlation Analysis #

Correlation analysis is a statistical method used to assess the degree of relationship between two or more variables. It helps to explore whether a dependency exists between the variables and measures the strength and direction of this relationship.

In correlation analysis, the relationship between variables is categorized into two types:

  • Functional Relationship: The value of one variable can be uniquely determined by another variable based on a known functional relationship (e.g., area and radius of a circle, sales quantity and revenue).
  • Statistical Relationship: The variables exhibit a relationship, but one variable cannot uniquely determine the value of the other (e.g., income and consumption, height and genetics).

Types of Correlation Relationships:

  1. Perfect Correlation: A perfect functional relationship exists between the variables (e.g., circumference and radius of a circle).
  2. Strong Correlation: A near-functional relationship exists between variables (e.g., family income and expenditure).
  3. Weak Correlation: A relationship exists, but it’s not obvious (e.g., farming area and crop yield).
  4. Zero Correlation: No relationship exists between the variables (e.g., students’ ages and their grades).

Types of Correlation Directions:

  • Positive Correlation: The variables increase or decrease together (i.e., both variables move in the same direction).
  • Negative Correlation: One variable increases while the other decreases (i.e., variables move in opposite directions).

Correlation Coefficient (r) and Its Interpretation: The correlation coefficient (r) is a measure that quantifies the degree of linear relationship between two variables and lies between -1 and +1.

  • r = 1: Perfect positive correlation (function relationship).
  • r = -1: Perfect negative correlation (function relationship).
  • r = 0: No linear relationship, but other forms of relationships may exist.
  • 0 < r ≤ 1: Positive correlation (variables increase or decrease together).
  • -1 ≤ r < 0: Negative correlation (one variable increases as the other decreases).

Classification of Correlation Strength:

  • r ≥ 0.8: Strong correlation.
  • 0.5 ≤ r < 0.8: Moderate correlation.
  • 0.3 ≤ r < 0.5: Weak correlation.
  • r < 0.3: Very weak correlation, often considered negligible.

Hypothesis Testing for Correlation: When conducting correlation analysis, hypothesis testing is used to assess whether a significant linear relationship exists between two variables:

  1. Null Hypothesis (H₀): There is no significant linear relationship between the variables (i.e., the correlation is zero).
  2. Alternative Hypothesis (H₁): There is a significant linear relationship between the variables.

Test Procedure:

  • Calculate the correlation coefficient.
  • Compute the associated probability value (p-value).
  • If the p-value ≤ significance level (typically 0.05), reject the null hypothesis (indicating a significant correlation).
  • If the p-value > significance level, fail to reject the null hypothesis (indicating no significant correlation).

Common Correlation Coefficients in Bivariate Analysis: #

The following are the most commonly used correlation coefficients for different types of variables:

  1. Pearson’s Correlation Coefficient: Measures the strength and direction of the linear relationship between two continuous variables (interval/ratio level).

    • Example: Correlation between height and weight.

  2. Spearman’s Rank Correlation Coefficient: Used for ordinal (ranked) variables or when the data does not meet the assumptions of Pearson’s correlation.

    • Example: Correlation between ranking of students and their exam scores.

  3. Kendall’s Tau-b: Another coefficient for ordinal variables, especially when dealing with ties in the data.

    • Example: Correlation between rankings of candidates by different judges.

Practical Operation: #

Let’s go through a practical example of performing correlation analysis using Pearson’s correlation coefficient in SPSS.

Case Study Example: Suppose we are interested in exploring the relationship between hours of study and exam scores. Both variables are continuous (interval or ratio level).

Variables:

  • X: Hours of study (continuous variable)
  • Y: Exam scores (continuous variable)

Steps:

  1. Data Input:

    • Enter the data for hours of study and exam scores in two separate columns in SPSS.

  2. Perform Pearson’s Correlation:

    • Navigate to Analyze > Correlate > Bivariate.
    • Select the two variables (hours of study and exam scores).
    • Choose Pearson under the correlation coefficient options.
    • Optionally, select two-tailed for testing the hypothesis of no correlation.

  3. Run the Analysis:

    • Click OK to run the correlation analysis.

  4. Interpret the Output: SPSS will provide the correlation coefficient (r) and the p-value.

Example Output:

Variable 1 Variable 2 Correlation (r) Sig. (p-value)
Hours of Study Exam Scores 0.85 0.001

Interpretation:

  • r = 0.85: This indicates a strong positive correlation between hours of study and exam scores. As the number of study hours increases, the exam score tends to increase as well.
  • p-value = 0.001: Since the p-value is less than 0.05, we reject the null hypothesis and conclude that there is a significant positive correlation between hours of study and exam scores.

Summary: #

Correlation analysis is a fundamental statistical tool for examining the strength and direction of relationships between variables. By calculating correlation coefficients, you can understand whether variables are related and, if so, how strongly. SPSS provides a straightforward way to compute these coefficients and test hypotheses about linear relationships. Always interpret the results in the context of the research question and the data, considering the direction and strength of the correlation to draw meaningful conclusions.

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Last modified on 2024-01-03