Correlation

Strength of a Correlation: A zero correlation (r = 0) means that

there is no linear relationship between two factors

Indicates that the two factors change in different directions

1.As X increases, Y decreases 2.As X decreases, Y increases

Positive correlation Indicates that the two factors change in the same direction

1.As X increases, Y increases 2.As X decreases, Y decreases

Covariance: If one variable changes, and the other stays the same...

they do not covary (r = 0)

limitations in interpretation: there are at least 3 things to keep in mind when interpreting correlations

1.Causality 2.Outliers 3.Restriction of Range

A correlation can be used to:

1.Describe the pattern of change in the value of two factors 2.Determine whether the pattern observed in a sample is also present in the population from which the sample was derived

hypothesis testing- Step 5: Effect size, if necessary

In our example, the effect was non-significant, so we would not typically calculate effect size

Effect Size for r: Coefficient of determination (R^2)

Mathematically equivalent to eta-squared Measures the proportion of variance of one factor (Y) that can be explained by known values of a second factor (X)

Assumptions of Tests for Linear Correlations: Homoscedasticity

Similar to homogeneity of variance in ANOVA The assumption that there is an equal ("homo") variance or scatter ("scedasticity") of data points dispersed along the regression line

Assumptions of Tests for Linear Correlations: Linearity

The best way to describe a pattern of data is using a straight line

Restriction of Range

When the range of sample data is smaller than the range of data in the general population

violation of linearity

a population with a non linear relationship, which violates the assumption of linearity.

The Regression Line: The strength of a correlation reflects how

consistently scores for each factor change

violation of homoscedasticity

this population of scores shows a scatter of data points with unequal variances

correlation can often suggest

which direction we should go to next for future inquiry

Direction of a correlation: the correlation coefficient (r) ranges from

-1.0 to +1.0 Values closer to ±1.0 indicate stronger correlations r = -1.0 is just as strong as r = +1.0

The regression line is a

straight line that best fits a set of data points -Scores are more consistent (i.e., stronger correlation) the closer they fall to their regression line

violation of normality

the table and the scatter plot showing the relationship between the number of fingers on the right and left hands of six people.

Strength of a Correlation: The closer a correlation coefficient is to 0,

the weaker the correlation and the less likely that the two factors are related

Hypothesis testing steps- Step 1: State hypotheses

•Population correlation coefficient is symbolized by ρ (rho) •H0: ρ = 0 •Mood is not related to eating in the population •H1: ρ ≠ 0 •Mood is related to eating in the population

Hypothesis Testing with r

•Step 1: State hypotheses •Step 2: Set criteria for decision •Step 3: Compute test statistic •Step 4: Make a decision •Step 5: Compute effect size, if necessary •Step 6: APA-style report

Positive or Negative?: A researcher reports that as the speed of a car accident increases, the vehicle damage increases

Positive

Positive or Negative?: I hypothesize that as the number of hours spent studying increases, test performance increases

Positive

Covariance: The closer data points fall to the regression line...

the more that the values of the two factors vary together

A correlation describes

the strength and direction of the linear relationship between two factors (variables)

Strength of a Correlation: Conversely, the closer a correlation coefficient is to ±1,

the stronger the correlation and the more likely that the two factors are related

Calculating Pearson's r : Step 1

1.Compute means for X and Y 2.Compute deviation scores (these should add up to 0) 3.Compute the sum of products (SSxy) - I will give this to you** 4.Compute SSx and SSy - - I will give this to you**

Hypothesis testing steps- Step 2: Set criteria for decision

1.Two-tailed test (α = .05) 2.df = n - 2 (for correlations) •All scores of X except one are free to vary, and all scores of Y except one are free to vary •In our example, n = 5 •df = 5 - 2 •df = 3 3.Locate critical value in Table B.5 in Appendix B 4.Critical Value:

Issue of reverse causality

A is related to B But...does A lead to B? Or does B lead to A? -Example: GPA and attendance

Which of the following indicates the strongest correlation?: A)r = -0.57 B)r = +0.78 C)r = -0.90 D)r = +.88

C

hypothesis testing- Make a decision

Compare the critical value to the obtained value

Causality

Correlation ≠ Causation

Assumptions of Tests for Linear Correlations: Normality

Data are normally distributed

Pearson Correlation Coefficient

Measures the direction and strength of the linear relationship of two factors in which the data for both factors are measured on an interval or ratio scale of measurement

Positive or Negative?: Carl hypothesizes that the more UA 'celebrates achievement,' the less students care when UA celebrates achievement

Negative

Outliers

Outliers obscure the relationship between two factors by altering the direction and strength of a correlation

Pearson Correlation Coefficient also called

Pearson product-moment coefficient

Calculating Pearson's r : Step 2

Plug values into formula r = SSxy / √(SSxSSy)

Calculating Pearson's r: A health psychologist measures the relationship between mood and eating. She measures mood using a 9-point rating scale, where higher ratings indicate better mood. She measures eating as the average number of daily calories that five participants consumed in the previous week.

Step 1: Compute preliminary calculations Step 2: Compute Pearson's r

Covariance

The extent to which the values of two factors vary together

Third variable problem

The relationship between A & B could be caused by C -Example: Ice cream sales & Shark attacks

correlation summary: very limited in our

interpretation (causality, outliers, restriction of range)

The sign of r indicates only the

direction/slope of the correlation Positive values = positive relationship Negative values = negative relationship

Outliers are scores that

fall substantially above or below most other scores in a data set

In correlational analyses, we do not

manipulate an IV, nor do we control for confounding variables

Correlations help us

observe important relationships -Strength and Direction

A positive correlation ranges from:

r = 0 to r = +1.00

A negative correlation ranges from

r = 0 to r = -1.00

hypothesis testing- Step 3: Compute the test statistic

r = SSxy / √(SSxSSy)

Pearson Correlation Coefficient formula

r = SSxy / √(SSxSSy) or... r = covariance of X and Y / variance of X and Y separately