1. Hypothesis of Association: Correlation
Chapter 11

2. Difference vs. Association
Hypothesis of Association
relationship between two sets of variables are examined to determine whether they are associated or correlated
-IV is not manipulated but assigned
-can not determine cause and effect
-Correlation
Hypothesis of Difference
A deliberate manipulation of a variable to see if a difference in behavior occurs
-IV is not assigned but manipulated
-can determine cause-and-effect relationships
-Experiment

3. Remember: CORRELATION IS NOT CAUSATION
Correlation: a measure of the relationship between two variables -correlation coefficient (r): a number calculated from the formula for measuring a correlation *indicating strength & direction *ranges from +1 to -1 -strong relationship if the correlation coefficient is close to +1 or -1 *knowing the relationship between two variables allows us to make predictions -EX: if you study X amount of hours, a score of X is predicted -positive correlation *represented by a positive # (0 to +1) *as the value of one variable increases, the other variable also increases -EX: study time goes up, test scores go up OR as study time goes down, test scores go down -negative correlation *represented by a negative # (0 to -1) *as the value of one variable increases, the other variable decreases -EX: as study time goes up, party time goes down -zero correlation *represented by 0 *no relationship between variables -EX: study time & height
Remember: CORRELATION IS NOT CAUSATION
-does not tell us about cause & effect because of unaccounted variables

4. Correlation: Scatter Plots
Positive Correlation (Linear)
Negative Correlation (Linear)
r =0.8273
r =
Exam Scores
Exam Scores
Study Time
Party Time
*scatterplots usually take an oval shape
-there will be a bulge in the middle because this is where you will find the most scores
-just like the normal curve
Usually scatter plots take a linear pattern: draw a straight line through the dots to show pattern
-called the “regression line”
-regression line goes one direction (up or down)
**NOTE: SPSS figures out where to put the regression line, you don’t need to know how to do it exactly, just eyeball it for now
Curvilinear pattern: line curves
-this happens when a correlation exists in one direction but then levels off and starts going the other direction
-EX: drinking & fun….first couple shots you’re having a good time but 20 shots and you’re in a nightmare!!
Note: It doesn’t matter which variable you put on the x-axis and which you put on the y-axis
Curvilinear
Zero Correlation
Exam Scores
Height

5. Correlation: Pearson r
Formula:
Calculation
Step 1: Calculate the means
Step 2: Calculate the standard deviations
Step 3: Plug all values into the formula
Conceptually, when Karl Pearson based his equation on taking each raw score turning into a z-score, multiplying the z-score pairs together and summing them. Then the product sum is divided by the number of pairs to get the mean.
-This takes too long so we are going to use a shortcut formula!
-Pearson named this equation the product-moment correlation coefficient
*it was later renamed “Pearson R” in his honor
Pearson R Worksheet
Pearson R Homework due Next Class

6. Correlation: Pearson r
Testing a hypothesis using correlation
-r refers to the sample correlation and ρ (rho) refers to the population correlation
Ho: ρ = 0 there is no correlation in the population
Ha: ρ ≠ 0 there is a correlation in the population
Critical Values (Table E)
-df are N-2 (number of pairs of score minus 2)
-if the calculated r value is greater than or equal to the table r value then reject Ho
-NOTE: as sample size increases, really small correlations become significant
**EX: look at df=400
Guilford’s Interpretation for significant r values
Guilford’s Interpretations also apply to negative numbers

7. Correlation: Pearson r
Coefficient of Determination (r 2)
The percentage of variance in Y that you could expect to be associated with the variance in X
-ie. The difference between the Y scores that is explained by the difference between the X scores
To calculate: square the correlation coefficient
EXAMPLE: What are the reasons that people have different incomes?
Well, research shows education explains 20% of the variance in income
**r =.45 so r 2 =.20
**.20/100=20%

8. Correlation: Pearson r
Requirements for using Pearson r
The sample has been randomly selected from the population
Measurement for both variables must be in the form of interval and/or ratio data
The variables being measured must not depart significantly from normality
-variable data should take the shape of the normal curve if you measured the whole population
The assumption of homoscedasticity is reasonable
-points are fairly equally distributed above & below the regression line
The association is between X & Y is linear (not curvilinear)
-plot your data & make sure it takes an oval shape

9. Correlation: Spearman rs
Use Spearman rs when you can’t meet the requirements to use the Pearson
-when both sets of data are not interval and/or ratio
-when the data are skewed/non-normal distributions
-note: the Spearman rs (unike the Pearson r) is considered a non-parametric test
**ie. it does not make assumptions about normality of the population including the parameter mean or the parameter standard deviation
Calculating Spearman rs (when you have ordinal data)
Formula:
Step 1: determine the rank of each subject on both variables
**if you have interval data (on one set) convert it to ordinal by ranking it
Step 2: Obtain the absolute difference, d, between each subject’s pair of ranks
Step 3: Square each difference, d2
Step 4: Calculate Σd2 by adding the squared differences
Step 5: Plug the values into the formula

10. Correlation: Spearman rs
Case of ties
-if you have the same score for two or more subjects (see worksheet for example):
*add the ranks (that the scores are tied for) and divide by the number of tied scores
*give all the subjects that same rank
Testing a hypothesis using correlation
-r refers to the sample correlation and ρs (rho) refers to the population correlation
Ho: ρs= 0 there is no correlation in the population
Ha: ρs≠ 0 there is a correlation in the population
Critical Values (Table F)
-Use N (not df)
-if the calculated rs value is greater than or equal to the table r value then reject Ho
Spearman Rs Worksheet
Spearman Rs Homework due Next Class