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Bivariate Relationships

Bivariate Relationships. Are two variables related? Is IQ correlated with Education? Is IQ correlated with Income? Is IQ correlated with Shoe Size? The Scatterplot The Correlation Coefficient. Bivariate Relationships.

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Bivariate Relationships

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  1. Bivariate Relationships • Are two variables related? • Is IQ correlated with Education? • Is IQ correlated with Income? • Is IQ correlated with Shoe Size? • The Scatterplot • The Correlation Coefficient

  2. Bivariate Relationships Correlated variables have matching high and low scores. A subject who scores high in IQ should have a large shoe size.

  3. Bivariate Relationships Correlated variables have matching high and low scores. A subject who scores high in IQ should have a large shoe size.

  4. Bivariate Relationships Correlated variables have matching high and low scores. People with low Beginning Salaries should have low Current Salaries

  5. The Scatterplot: Does your starting salary predict your current salary?

  6. The Scatterplot for Perfect Correlation

  7. Ocular Correlation On a scale of 1 to 10, how large is this correlation?

  8. Ocular Correlation On a scale of 1 to 10, how large is this correlation?

  9. Ocular Correlation On a scale of 1 to 10, how large is this correlation?

  10. Ocular Correlation On a scale of 1 to 10, how large is this correlation?

  11. Pearson’s Correlation Coefficient (r) • A descriptive statistic that shows the magnitude of a linear relationship between two variables (r: regression). • Its range varies from: –1… to …+1 • r=1 and r=-1 show a perfect correlation. This means that the two variables exactly match. • The scatterplot of observations forms a perfect straight line. • We are not interested in such a relationship. Why?

  12. Direction of the relationship • Positive correlation: variables change in the same direction • Negative correlation: variables change in the opposite direction. r=.88 r=-.62

  13. How to calculate r: • Convert all scores to z-scores (for each variable). • Compute the cross-product (multiply pairs of z-scores). • Sum the cross-products. • Divide by the number of cases (“n”, i.e. observations). r= Sum ZXYY N

  14. Let’s practice. Occular Correlation • Open the correlate.sav file. • Make scatterplots for: Var001 with all the other variables • Compute Pearson’s correlation coefficients • What do the correlations suggest?

  15. Let’s practice • Open the renal.sav file. • Make scatterplots for: - length of stay in hospital vs. ICU length of stay - length of stay in hospital, ICU length of stay vs. Alive at Discharge • Compute Pearson’s correlation coefficients • What do the correlations suggest?

  16. Correlations reduce uncertainty and explain variance Imagine if you could predict the variance in coin tosses or the roll of dice

  17. Restricted Range makes for low correlations Download the data set called Job Salary Data Generate a histogram for the variable, Current Salary Generate a histogram for the variable Education Run a Pearson Correlation for the two variables Select Current Salary greater than (>) 30,000. Run the Pearson Correlation between Current Salary and Education again

  18. Significance of Correlations How large should the correlation be to conclude the variables are related? Is it possible this degree of relationship occurred by chance? Larger sample sizes allow you to reveal small significant correlations We will return to statistical significance after we study the t test.

  19. Correlation vs. Causation Which of these predicts performance on the final exam? Which causes a high grade on the exam? Hours of Study Hours of Sleep before the exam Mood on the day of the exam Parent’s Years of Education Student’s IQ Parent’s IQ A causes B; B causes A; C causes A and B

  20. Does Smoking Cause Cancer? Play cigcause.dv

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