Hypothesis of Association: Correlation Chapter 11

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

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

Correlation: Scatter Plots Positive Correlation (Linear) Negative Correlation (Linear) r =0.8273 r =-0.6321 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

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

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

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%

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

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

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