EBP Lec 13: Correlations and Comparisons

Opening Remarks

• Good Morning Greeting: The speaker welcomes everyone to the lecture, expressing enthusiasm and encouragement for the day's content.

• Structure of the Day: Mention of a four-hour lecture stint, indicating that future lectures will involve more interactive elements, including presentations and revisions.

• Previous Topics Recap: Review of the chocolate experiment and fundamentals of inferential statistics including:

  • Taking a random sample from a population

  • Making inferences about the population based on sample data

  • Introduction to the Central Limit Theorem for statistical analysis.

Key Concept: Correlation

• Definition of Correlation: A relationship between two variables.

• **Types of Correlation:

  • Positive Correlation:**

  • As one variable increases, the other also increases; similarly, if one decreases, the other decreases.

  • Example: Increased chocolate consumption corresponds with the number of Nobel Prize laureates.

• Example Comic Reference:

  • The comic emphasizes that "correlation does not imply causation."

Illustrative Examples of Correlation vs. Causation

1. Ice Cream and Drowning Example:

   • An increase in ice cream sales correlates with an increase in drownings, but sunny weather is the true underlying cause.

   • The speaker emphasizes the common logical mistake of assuming causation from correlation.

2. Married vs. Single Men:

   • Men who live longer are often married; however, healthier men are more likely to get married.

   • This instance demonstrates how statistical relationships can be misinterpreted.

3. Short-Sighted Children:

   • A study finds a correlation between sleeping with lights on and increased short-sightedness, later debunked as a genetic issue.

   • -

4. Self-Esteem and Academic Performance:

   • Initial conclusions that self-esteem causes good grades were reversed; in reality, good grades increase self-esteem.

Important Distinction: Statistical Significance

• Understanding Correlation Mistakes: The speaker emphasizes the dangers of erroneously inferring causation from correlation.

• Hard Limitations: Illustrates caution while interpreting correlations, noting such correlations may hint at deeper issues but require additional analysis.

Types of Correlation

• Positive Correlation:

  • Example provided: Scores in anatomy and physiology increasing together.

• Negative Correlation:

  • Conceptually opposite where an increase in one variable results in a decrease in the other; example given: hearing threshold levels correlating with speech discrimination scores.

• No Correlation:

  • Random data with no direct relationship, exemplified using unrelated variables such as house numbers and age.

• Curvilinear Relationships:

  • Explanation that some data may show relationships that are not linear (like health metrics against weight).

Measuring Correlation

• Pearson’s Correlation Coefficient (r):

  • Represents the strength of the correlation:

  • Interpretation of r:

    • $ r = 1$: Perfect positive correlation

    • $ r = -1$: Perfect negative correlation

    • $ r = 0$: No correlation

• Strength of Correlation

  • Value threshold interpretations:

  • 0.00-0.30: Weak

  • 0.30-0.70: Moderate

  • 0.70-1.00: Strong

Reporting Correlation Statistics

• Range of required statistics includes direction, strength, and significance (p-value).

• Example of Reporting: "There is a significant strong positive correlation between anatomy and physiology scores."

Apparent Counterpoints to Correlation

• Examples of situations where correlation isn’t indicative of causation for clarity.

Regression Analysis Overview

• Correlation establishes relationships; regression is used for predictions based on correlated data.

• Regression Equation: The example provided: physiology score = 4.5 + (0.961 * anatomy score).

• Important distinction between independent variable (anatomy score) and dependent variable (physiology score).

Assumptions for Linear Regression Analysis

• Need to work with interval or ratio data.

• Assumption of independence of observations.

• Assumption of linear relationships and normal distribution.

• Homoscedasticity: Equal variance among data points.

Conducting Regression Analysis using Software (Minitab)

• Steps outlined for executing regression in Minitab.

• Interpretation includes:

  • R value and its explanation of variance accounted for.

  • All reported outputs including significance of regression statistics.

Conclusion and Transition to T-tests

• Transition into t-tests context:

  • The speaker explains the necessity for comparing two groups with categorical independent variables that can't be handled by correlation or regression methods.

• Introduction to different types of t-tests: one-sample, two-sample, and paired t-tests, while discussing methods of executing them in practical scenarios.

Summary Guidance for a Research Assignment

• Group assignments announced: exploration of a provided research question, consultation of literature, data analysis, and integration of findings, culminating in a report.

• Reminder of tools and community resources available for student use.

Correlation – Concept and Purpose

Definition

• Correlation describes the relationship between two variables — how one changes with respect to the other.

• Can be:

  • Positive correlation: both increase or decrease together.

  • Negative correlation: one increases while the other decreases.

  • No correlation: no consistent relationship.

  • Curvilinear: relationship exists but not linear.

Key Reminder

• Example (video):

  • Ice cream sales vs drownings — both increase in summer, but caused by weather, not each other.

  • Marriage and lifespan — healthier men more likely to marry, not vice versa.

  • Children’s night lights and myopia — short-sighted parents leave lights on; genetics, not lighting, causes short-sightedness.

  • Self-esteem and grades — good grades → high self-esteem, not the reverse.

3. Types of Correlation

4. Pearson’s Correlation Coefficient (r)

Definition

• Quantifies strength and direction of linear relationship.

• Denoted by r (ranges from –1 to +1).

  • r = +1 → perfect positive linear relationship.

  • r = –1 → perfect negative linear relationship.

  • r = 0 → no relationship.

• Works with interval or ratio data only.

Comparison

• Spearman’s correlation (ρ) → used for ordinal data or non-parametric cases.

Strength Guidelines

5. Three Aspects to Report

1. Direction – Positive, Negative, or None.

2. Strength – Weak/Moderate/Strong.

3. Significance – p-value (< 0.05).

Important Note

• Strong correlation ≠ necessarily significant if sample size is too small.

  • e.g., r = 0.98 but n = 3 → p = 0.127 (not significant).

• Always report r and p together.

6. Pearson’s Correlation in Minitab

Procedure

1. Open dataset (e.g., Anatomy vs Physiology).

2. Go to Stat → Basic Statistics → Correlation.

3. Select both variables.

4. In “Graphs” options, tick “Show correlation and P value”.

Output Example

• r = 0.87, p = .001
  → Significant strong positive correlation between anatomy and physiology scores.

APA Reporting Style

• Omit leading zero (APA rule).

• Report as:

  r = .87, p < .001

7. Exercise: Height and Shoe Size

• Task: Run correlation for height and shoe size using class data.

• Result example:

  r = .87, p < .001
  → Strong, positive, significant correlation between height and shoe size.

• Report p < .001 (never p = 0).

• Use 2 decimal places (3 if needed for clarity).

8. Regression Analysis

Definition

• Explores prediction between two variables once correlation exists.

• Regression line (line of best fit):

  Y=A+BXY = A + BXY=A+BX

  • Y = dependent (predicted variable).

  • X = independent (predictor variable).

  • A = intercept.

  • B = slope (gradient).

Assumptions of Linear Regression

9. Output Interpretation

Key Metrics

Confidence vs Prediction Intervals

• Prediction interval always wider than confidence interval.

10. Regression in Minitab

Steps

1. Stat → Regression → Fitted Line Plot

2. Set:

   • Response (Y): Dependent variable (e.g., Physiology)

   • Predictor (X): Independent variable (e.g., Anatomy)

3. Under “Options,” tick:

   • Display confidence and prediction intervals.

Interpretation Example

• R² = .75, F(1,18) = 53.16, p = .001
  → Anatomy scores significantly predict Physiology scores.

APA Reporting Format

Example: Prediction

• Equation: Physiology = 4.46 + 0.9608 × Anatomy

• For Anatomy = 80 → Predicted Physiology = 81.32

11. Practice Example: Shoe Size Predicting Height

• Regression equation: Height = A + B × Shoe size

• R² = .76, F(1,55) = 176.09, p < .001
  → Shoe size strongly predicts height.

12. T-Tests Overview

Purpose

• Compare means between two groups or a group vs known value.

• Requires:

  • Categorical independent variable

  • Interval/ratio dependent variable

13. Types of T-Tests

14. One-Sample T-Test

Example

• Does the average Mars Bar weight differ from 16 g?

  • t-test result: t = 8.41, p < .001
    → Mean weight (17 g) significantly higher than standard.

Interpretation Steps:

1. Check significance (p < .05).

2. Compare means to determine direction.

3. Report in APA:

   t(19) = 8.41, p < .001
   “Average Mars Bar (M = 17.0 g, SD = 1.2) heavier than recommended 16 g.”

15. Two-Sample (Independent) T-Test

Example

• Do Cadbury chocolates differ in weight from Mars chocolates?

  • Cadbury (M = 14.4 g), Mars (M = 15.1 g)

  • t = –3.10, p = .002
    → Significant difference; Mars heavier.

APA Reporting

Notes:

• df = N – 2 (two samples).

• Sign of t doesn’t affect conclusion — only direction.

16. Paired (Dependent) T-Test

Concept

• Compares two related measures:

  • Same group tested twice (e.g., before/after intervention).

  • Two comparable measures from same participants.

Example

• Audiology students’ outdoor hours:

  • Summer (M = 5.3 h), Winter (M = 3.2 h)

  • p = .044 → significant difference.

APA Reporting

Notes

• Removes inter-subject variability.

• df = N – 1 (one sample measured twice).

17. One-Tailed vs Two-Tailed Tests

• Two-tailed = more conservative, standard approach.

• One-tailed = easier to reach significance but less rigorous.

18. Degrees of Freedom Summary

19. Minitab Procedures

20. Practical Activity Summary

• One-sample: Sangas R Us rating vs city average (75).

• Two-sample: Compare Sangas R Us vs Best Bagels ratings.

• Paired: Sangas R Us ratings in January vs June (improvement check).

Each output should include:

• Mean, SD

• t-value, df, p-value

• Interpretation of direction

• APA-style report.

21. Final Key Points

✅ Correlation = Relationship
✅ Regression = Prediction
✅ T-tests = Comparison

Always Report:

• Direction (positive/negative or group mean)

• Strength (r or R²)

• Significance (p-value)

• Interpretation (what it means in real context)