Correlation measures the strength of the relationship between 2 variables.

Transcription

1 Correlation measures the strength of the relationship between 2 variables.

2 Types of Correla-ons Positive: 2 or more variables move in the same direction Negative: 2 variables move in opposite directions No/zero: no relationship between 2 variables

3 Sca1erplots A special graph containing dots that represent all pairs of scores A positive correlation appears as a cluster of data points that goes from the lower left to the upper right A negative correlation appears as a cluster of data points that goes from the upper left to the lower right A no correlation has scattered dots in an irregular pattern

4 Prac-ce! Plot these data predicting GPA from the number of nights a person goes out each week on a scatterplot, What type of correlation is this? X (# nights) Y (GPA)

5 GPA and Nights out per week GPA (Criterion Variable) Nights out per week (Predictor Variable)

6 Perfect Correla-on When there is a perfect linear relationship, every change in the X variable is accompanied by a corresponding change in the Y variable.

7 Correla-on coefficient A number that describes the strength and direction of the relationship between your variables X and Y Pearson s r Describes the relationship in a scatter plot A number between - 1 and +1 Sign (- or +) indicates direction Number indicates strength Perfect Relationship No Relationship Perfect Relationship All great scientists have, in a certain sense, been great artists; the man with no imagination may collect facts, but he cannot make great discoveries. - Karl Pearson

8 Interpre-ng r: Effect size.1 = small/weak.3 = medium/moderate.5 = large/strong

9 Prac-ce! Small, medium, or large relationships? r = -.90 between miles you ride your bike and money you spend on gas r =.01 between anxiety and GPA r =.35 between creativity and intelligence r = -.25 between # of lies and # of friends r =.16 between money and problems r =.46 between stress and depression

10 Pu;ng things in perspec-ve FINDING r sample size Aspirin and reduced risk of death by heart attack.02 22,071 Chemotherapy and surviving breast cancer.03 9,069 Combat exposure in Vietnam and subsequent PTSD in 18 years.11 2,490 Post- high school grades and job performance.16 13,984 Clinical depression and suppressed immune functioning Viagra and improved male sexual functioning Increasing age & declining speed of information processing in adults.52 11,044 Nearness to the equator and daily temperature in the USA.60 19,724 Meyer G. J., et al., (2001). Psychological testing and psychological assessment: A review of evidence and issues. American Psychologist, 56,

11 Correla-on does not imply causa-on A variable may predict another well, but that does not mean that it CAUSES the other to occur Ice Cream & Polio? Vaccines & Autism?

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14 Calcula-ng Pearson s r Conceptually: r = degree to which X and Y vary together degree to which X and Y vary separately Mathematically: r = SP SS x xy SS y This is called the Sum of Products (SP) This is just the Sum of Squares (SS) for x and y Luckily you know this guy! (see ch 4, slide 3)

15 Toolbox Sum of Squares SS x = Σ(x- x ) 2 SS y = Σ(y- ˉy ˉ ) 2 Sum of Products SP xy = Σ(x- x )(y- ˉy ˉ) - or- SP xy = ΣXY (ΣX)(ΣY) n Remember: (1) Calculate the difference between each score & the mean (the difference score) (2) Add up each person s difference score (3) Square the total number Remember: (1) Calculate the difference score for each person s x and y variables (2) MULTIPLY each person s x and y difference scores together (3) Add up the whole sample This is just another formula. We are not going to focus on it, but it is explained in your book. Both are fine to use.

16 Lets do an example Does temperature (x) predict the number of arguments (y) a person has with their significant other? Person 1: 80, 2 arguments ; Person 2: 90, 4 arguments ; Person 3: 100, 9 arguments SS x SS y Temp x - x Diff Diff 2 Argue y - ˉy ˉ Diff Diff SSx = Σ(x- x ) 2 = (80-90) 2 + (90-90) 2 + (100-90) 2 = 200 SSy = Σ(y- ˉy ˉ) 2 = (2-5) 2 + (4-5) 2 + (9-5) 2 = 26 SPxy = Σ(x- x )(y- ˉy ˉ) = (80-90)(2-5) + (90-90)(4-5) + (100-90)(9-5) = 70 SP xy Diffx * Diffy Sum (-10)*(-3) 30 (0)*(-1) 0 (10)*(4) r = SP SS x xy SS y r = 70 (200)(26) 70 r = r =

17 Prac-ce! What is the correlation between the # of nights that someone goes out, and their GPA? Find SSx, Ssy, SPxy, Find r X (# nights) Y (GPA)

18 Prac-ce! What is the correlation between the # of nights that someone goes out, and their GPA? (nights, GPA) à (7, 1) (3, 2) (1, 4) SS x SS y Nights x - x Diff Diff 2 GPA y - ˉy ˉ Diff Diff Mean = Mean = SP xy Diffx * Diffy (3.33)*(-1.33) (-.67)*(-.33) (-2.67)*(1.67) SSx = Σ(x- x ) 2 = (7-3.67) 2 + (3-3.67) 2 + (1-3.67) 2 = SSy = Σ(y- ˉy ˉ) 2 = (1-2.33) 2 + (2-2.33) 2 + (4-2.33) 2 = 4.67 SPxy = Σ(x- x )(y- ˉy ˉ) = ( )(1-2.33) + (3-3.67)(2-2.33) + (1-3.67)(4-2.33) = Sum r = SP SS x xy SS y r = (18.67)(4.67) r = r =

19 Interpre-ng Pearson s r Values can be influenced by the range of scores.

20 Interpre-ng Pearson s r Values can be influenced by outliers.

21 Interpre-ng Pearson s r Correlation does not equal causation. Can tell you the strength and direction of a relationship between two variables but not the nature of the relationship. The directionality problem. A - > B or B - > A The third variable problem C - > A and B