tests of correlation

Knowledge and Understanding
- Students should demonstrate knowledge and understanding of inferential testing.
- Familiarisation with the use of inferential tests is expected.
Correlation vs. Difference
- The tests discussed (Spearman's rho and Pearson's r) are used to determine correlation between co-variables, rather than differences between sets of scores.

Usage
- Can be used with ordinal or interval data.
- Selected for investigations that are correlational rather than experimental.
Justification
- Chosen here due to the correlational nature of the investigation.

Definition
- Pearson's r is a statistical test used to measure the correlation between two sets of values.
- Applicable exclusively when the data is at the interval level.
Design Nature
- The design type does not matter (correlational rather than experimental).
Comparison with Spearman's
- Spearman's, unlike Pearson's, can be used when one or both variables are at the ordinal level, although it can also accommodate interval data.

Aim
- Investigate the matching hypothesis proposed by Walster et al. (1966), stating that couples in long-term relationships tend to possess similar levels of physical attractiveness.
Study Details
- Twelve couples were selected.
- Each partner's photograph taken and randomized to prevent awareness of relationships.
- Twenty participants rated each person’s attractiveness on a scale of 1 to 20.
- The median attractiveness ratings determined.
Hypotheses
- Alternative Hypothesis (H1): There is a positive correlation between the ratings of physical attractiveness given to two partners in a relationship (directional, one-tailed).
- Null Hypothesis (H0): There is no correlation between the ratings provided.
Data Collection and Analysis
- Step 1: Table of Ranks
  - Rank each score within each condition from lowest to highest.
  - For tied scores, use the mean of their ranks.
- Step 2: Calculate Differences
  - Determine the difference between ranks of each pair and square the differences.
  - Total the squared differences.

Aim
- Explore the correlation between the length of biofeedback usage in days and the reduction in resting heart rate measured in beats per minute (bpm).
Study Details
- Ten participants with chronic stress used biofeedback for varying lengths.
- Medical records used to compare baseline heart rates with present rates to determine reduction.
Hypotheses
- Alternative Hypothesis (H1): Positive correlation exists between the number of days using biofeedback and reduction in resting heart rate (directional, one-tailed).
- Null Hypothesis (H0): No correlation exists.
Data Calculations
- Step 1: Table of Data
  - Calculate the sum of scores for x and y (length of usage and heart rate reduction).
  - Compute squares of x and y ( $x^2$ and $y^2$ ).
  - Multiply x and y for each participant and sum these products ( $Σ(xy)$ ).

Table Format Examples
- Table 1: Calculations table with sample data.
- Table 2: Critical values of rho at significance levels:
  - One-tailed test:
    - α = 0.05, N = 4, critical value = 1.000.
    - α = 0.10, N = 5, critical value = 0.900.
  - Two-tailed test:
    - α = 0.05, N = 4, critical value = 0.950.
    - α = 0.10, N = 5, critical value = 0.800.

Scenario
- Researcher examines the positive correlation between heat and aggression by noting daily temperatures and violence incidents reported over 52 days.
- Pearson's test is utilized to analyze the collected data.
- Calculated value of r was reported as 0.281.

Significance Assessment
- Discuss whether the calculated result is significant (3 marks).
- Draw conclusions based on the study (2 marks).
Statistical Formulas
- For rho calculation, the formula is $<br /> ho = 1 - \frac{6d^2}{N(N^2 - 1)}$ where d is the difference of ranks and N is the number of pairs.
- The calculated value of rho must be equal to or more than the critical value for significance.
Application and Methodology
- Describe when a researcher would prefer the Spearman's rho test, citing two factors (2 marks).