Correlation: Relationships between Variables

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2 Correlation Correlation: Relationships between Variables So far, nearly all of our discussion of inferential statistics has focused on testing for differences between group means However, researchers are often interested in graded relationships between variables, such as how well one variable can predict another Examples: How well do SAT scores predict a student s GPA? How is the amount of time a student takes to complete an exam related to her grade on that exam? How well do IQ scores correlate with income? How does a child s height correlate with his running speed? How does class size affect student performance?

3 Correlation: Relationships between Variables Correlation is a statistical technique used to describe the relationship between two variables. Usually the two variables are simply observed as they exist in the environment (with no experimental manipulation a correlational study) However, results from experimental studies (in which one of the variables is systematically manipulated) can also be analyzed using correlation

4 Mean Comparison Approach Height Weight Weights Short Tall

5 Correlation: Scatter Plots Height Weight

6 Scatter Plots Height Weight

7 Scatter Plots Height Weight

8 Scatter Plots Height Weight

9 Scatter Plots Height Weight

10 Characteristics of the Correlation A Correlation coefficient is a single number describing the relationship between two variables. This number describes: The direction of the relationship Variables sharing a positive correlation tend to change in the same direction (e.g., height and weight). As the value of one of the variables (height) increases, the value of other variable (weight) also increases Variables sharing a negative correlation tend to change in opposite directions (e.g., snowfall and beach visitors). As the value of one of the variables (amount of snowfall) increases, the value of the other variable (number of beach visitors) decreases. The strength of the relationship Variables that share a strong correlation (close to +1 or -1) strongly predict one another, while variables that share a weak correlation (near 0) do not.

11 Positive versus Negative Correlations Positive Correlation Negative Correlation

12 Strong versus Weak Correlations

13 Correlation is not Causation

14 Possible Sources of Correlation The relationship is causal. Manipulating the predictor variable causes an increase or decrease in the criterion variable. E.g., leg strength and sprinting speed The causal relationship is backwards (reverse causality). Manipulating the criterion variable causes changes in the predictor variable The two variables work together systematically to cause an effect The relationship may be due to one or more confounding variables Changes in both variables reflect the effect of a confounding variable E.g., intelligence as an explanation for correlated performance on different exams E.g., increasing density in cities increases the number of physicians and the number of crimes

15 Measuring Correlation: Pearson s r To compute a correlation you need a pair of scores, X and Y, for each individual in the sample. The most commonly used measure of correlation is Pearson s product-moment correlation coefficient, or more simply, Pearson s r. Conceptually, Pearson s r is a ratio between the degree to which two variables (X and Y) vary together and the degree to which they vary separately. r co-variability( XY, ) variability( X) variability( Y)

16 The Covariance The term in the numerator of Pearson s r is the covariance, an unnormalized statistic representing the degree to which two variables (X and Y) vary together. cov XY X M Y M X n 1 Y Mathematically, it is the average of the product of the deviations of two paired variables The covariance depends both on how consistently X and Y tend to vary together and on the individual variability of the variables (X and Y).

17 The Covariance Notice that the formula for covariance looks a lot like the formula for variance: s 2 X 2 X M X M X M n1 n1 X X X cov XY X M Y M X n 1 Y

18 The Covariance Moreover, they share a similar computational formula: s SS ; where SS n 1 X 2 X 2 X X 2 X X X XX n n cov XY SPXY ; where n 1 SP XY XY XY n

19 Computing Pearson s r Pearson s r is computed by dividing by the product of the standard deviations of each of the variables This removes the effect of the variability of the individual variables r cov XY s s SP XY SS SS X Y X Y

20 Computing Pearson s r: Example X Y

21 Computing Pearson s r: Example X Y XY Compute SS X, SS Y, & SP XY : Compute r: SS SS SP X Y XY X X N 5 Y Y N 5 X Y XY N r SPXY SS SS X Y

22 Computing Pearson s r: Example Hypothesis testing for r: The null hypothesis is that the population correlation coefficient ρ = 0 The alternative hypothesis is that ρ 0 tcrit ( df ) rcrit ( df ) ; df N 2 df t 2 crit

23 t-distribution Table α t One-tailed test α/2 α/2 -t t Two-tailed test Level of significance for one-tailed test Level of significance for two-tailed test df

24 Computing Pearson s r: Example tcrit ( df ) rcrit ( df ) ; df N df t ( df ) 2 crit tcrit (3) r crit tcrit df t crit

25 Critical values for Pearson s r Level of Significance for One-Tailed Test Level of Significance for Two-Tailed Test df = n

26 Computing Pearson s r: Example tcrit ( df ) rcrit ( df ) ; df N df t 2 crit tcrit (3) r crit tcrit df t crit ; accept H, the correlation is not significant. 0

27 Linear Correlation: Assumptions 1. Linearity Assumes that the relationship between the paired scores is best described by a straight line 2. Normality Assumes that the marginal score distributions, their joint distribution, and any conditional distributions are normally distributed 3. Homoscedasticity Assumes that the variability around the regression line is homogeneous across different score values

28 Other Correlation Coefficients Spearman s correlation coefficient (r s ) for ranked data As the name suggests, Spearman s correlation is used when the scores for both X and Y consist of (or have been converted to) ordinal ranks The point biserial correlation coefficient (r pb ) This correlation is used when one of the scores is continuous and the other is dichotomous, taking on one of only two possible values The phi correlation coefficient (r ϕ ) The phi correlation is used when both scores are dichotomous All of the above can be computed in the same manner as Pearson s correlation.

29 Converting Data for Spearman s Correlation Correlation Original Data Age Height r = 0.86

30 Converting Data for Spearman s Correlation Correlation Original Data Converted Scores Age Height Age Rank Height rank r = 0.86 r = 0.97

31 Converting Data for the Point Biserial Correlation

32 Converting Data for Phi Correlation