1Cal State NorthridgeΨ427AinsworthCorrelation and RegressionMajor Points - Correlation Questions answered by correlation Scatterplots An example The correlation coefficient Other kinds of correlations  Factors affecting correlations Testing for significanceThe Question Are two variables related?Does one increase as the other increases? e. g. skills and incomeDoes one decrease as the other increases? e. g. health problems and nutrition How can we get a numerical measure of the degree of relationship?2Scatterplots AKA scatter diagram or scattergram. Graphically depicts the relationship between two variables in two dimensional space.Scatterplot:Video Games and Alcohol Consumption024681012141618200 5 10 15 20 25Average Hours of Video Games Per WeekAverage Number of Alcoholic Drinks Per WeekDirect RelationshipScatterplot: Video Games and Test Score01020304050607080901000 5 10 15 20Average Hours of Video Games Per WeekExam ScoreInverse Relationship3An Example Does smoking cigarettes increase systolic blood pressure? Plotting number of cigarettes smoked per day against systolic blood pressureFairly moderate relationshipRelationship is positiveTrend?SMOKING3020100SYSTOLIC170160150140130120110100Smoking and BP Note relationship is moderate, but real. Why do we care about relationship?What would conclude if there were no relationship?What if the relationship were near perfect?What if the relationship were negative?4Heart Disease and Cigarettes Data on heart disease and cigarette smoking in 21 developed countries (Landwehr and Watkins, 1987)  Data have been rounded for computational convenience.The results were not affected.The DataCountry Cigarettes CHD111 2629 2139 2449 2158 1968 1378 1986 1196 23105 15115 13125 4135 18145 12155 3164 11174 15184 6193 13203 4213 14Surprisingly, the U.S. is the first country on the list--the country with the highest consumption and highest mortality.Scatterplot of Heart Disease CHD Mortality goes on ordinate (Y axis)Why? Cigarette consumption on abscissa (X axis)Why? What does each dot represent? Best fitting line included for clarity5Cigarette Consumption per Adult per Day12108642CHD Mortality per 10,0003020100{X = 6, Y = 11}What Does the Scatterplot Show? As smoking increases, so does coronary heart disease mortality. Relationship looks strong Not all data points on line.This gives us “residuals” or “errors of prediction” To be discussed laterCorrelation Co-relation The relationship between two variables Measured with a correlation coefficient Most popularly seen correlation coefficient: Pearson Product-Moment Correlation6Types of Correlation Positive correlationHigh values of X tend to be associated with high values of Y.As X increases, Y increases Negative correlationHigh values of X tend to be associated with low values of Y.As X increases, Y decreases No correlation No consistent tendency for values on Y to increase or decrease as X increasesCorrelation Coefficient A measure of degree of relationship. Between 1 and -1 Sign refers to direction. Based on covariance Measure of degree to which large scores on X go with large scores on Y, and small scores on X go with small scores on Y Think of it as variance, but with 2 variables instead of 1 (What does that mean??)187Covariance Remember that variance is: The formula for co-variance is: How this works, and why? When would covXYbe large and positive? Large and negative?2( ) ( )( )1 1XX X X X X XVarN NΣ − Σ − −= =− −1))((−−−Σ=NYYXXCovXYExampleCountry X (Cig.) Y (CHD) ( )X X− ( )Y Y− ( )X X−*( )Y Y− 1 11 26 5.05 11.48 57.97 2 9 21 3.05 6.48 19.76 3 9 24 3.05 9.48 28.91 4 9 21 3.05 6.48 19.76 5 8 19 2.05 4.48 9.18 6 8 13 2.05 -1.52 -3.12 7 8 19 2.05 4.48 9.18 8 6 11 0.05 -3.52 -0.18 9 6 23 0.05 8.48 0.42 10 5 15 -0.95 0.48 -0.46 11 5 13 -0.95 -1.52 1.44 12 5 4 -0.95 -10.52 9.99 13 5 18 -0.95 3.48 -3.31 14 5 12 -0.95 -2.52 2.39 15 5 3 -0.95 -11.52 10.94 16 4 11 -1.95 -3.52 6.86 17 4 15 -1.95 0.48 -0.94 18 4 6 -1.95 -8.52 16.61 19 3 13 -2.95 -1.52 4.48 20 3 4 -2.95 -10.52 31.03 21 3 14 -2.95 -0.52 1.53 Mean 5.95 14.52 SD 2.33 6.69 Sum 222.44 Example21.&( )( ) 222.4411.121 21 1cig CHDX X Y YCovNΣ − −= = =− − What the heck is a covariance?  I thought we were talking about correlation?8Correlation Coefficient Pearson’s Product Moment Correlation Symbolized by r Covariance ÷ (product of the 2 SDs) Correlation is a standardized covarianceYXXYssCovr =Calculation for Example CovXY= 11.12 sX= 2.33 sY= 6.69cov11.12 11.12.713(2.33)(6.69) 15.59XYX Yrs s= = = =Example Correlation = .713 Sign is positiveWhy? If sign were negativeWhat would it mean?Would not alter the degree of relationship.9Other calculations25 Z-score method Computational (Raw Score) Method1x yz zrN=−∑2 2 2 2( ) ( )N XY X YrN X X N Y Y−=   − −   ∑ ∑ ∑∑ ∑ ∑ ∑Other Kinds of Correlation Spearman Rank-Order Correlation Coefficient (rsp)used with 2 ranked/ordinal variablesuses the same Pearson formulaAttractiveness Symmetry3 24 61 12 35 46 5rsp =0.7726Other Kinds of Correlation Point biserial correlation coefficient (rpb)used with one continuous scale and one nominal or ordinal or dichotomous scale.uses the same Pearson formulaAttractiveness Date?3 04 01 12 15 16 0rpb =-0.492710Other Kinds of Correlation Phi coefficient (Φ)used with two dichotomous scales.uses the same Pearson formulaAttractiveness Date?0 01 01 11 10 01 1Φ =0.7128Factors Affecting r Range restrictionsLooking at only a small portion of the total scatter plot (looking at a smaller portion of the scores’ variability) decreases r.Reducing variability reduces r NonlinearityThe Pearson r (and its relatives) measure the degree of linear relationship between two variablesIf a strong non-linear relationship exists, r will provide a low, or at least inaccurate measure of the true relationship.Factors Affecting r Heterogeneous subsamplesEveryday examples (e.g. height and weight using both men and women) OutliersOverestimate CorrelationUnderestimate Correlation11Countries With Low ConsumptionsData