Major Points Scatterplots The correlation coefficient –Correlations on ranks Factors affecting correlations Testing for significance Intercorrelation matrices Other kinds of correlations

The Problem 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 graphical representation of the degree of relationship?

Relation between father and son’s height: Pearson, (1896) l Reliability 1

Another dataset: Heart Disease and Cigarettes Landwehr & Watkins report data on heart disease and cigarette smoking in 21 developed countries Data have been rounded for computational convenience. The results were not affected.

Scatterplot of Heart Disease CHD Mortality goes on y axis Cigarette consumption on x axis What does each dot represent? Best fitting line included for clarity

Cigarette Consumption per Adult per Day CHD Mortality per 10, {X = 6, Y = 11} 2

Cigarette Consumption per Adult per Day CHD Mortality per 10, {X = 6, Y = 11} 3

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”

Example Scatterplots x x x x x x x x x x x x x x x x x x x x x x y x x x x x x x x x x x x x x x x x x x x x x y High correlationLow correlation 4

Scatter plots: r =.00 5

r =.15

r =.40 6

r =.81

r =.99 7

r = -.79 Guessing correlations: from Rice Universityfrom Rice University 10

Another way to visualize a correlation Variance in A Variance in b Variance in A Variance in b Covariance 11

What is a Correlation Coefficient A measure of degree of relationship. Sign refers to direction. Based on covariance Measure of degree to which large scores go with large scores, and small scores with small scores Pearson’s correlation coefficient is most often used

Cigarette Consumption and Coronary Heart Disease Mortality for 21 countries Cigarette Consumption: per adult per day Coronary Heart Disease: Mortality per 10,000 population

Covariance The formula Index of degree to which both list of numbers covary When would cov XY be large and positive? When would cov XY be large and negative?

Calculation Cov XY = s X = 2.33 s Y = 6.69

Correlation Coefficient Symbolized by r Covariance ÷ (product of st. dev.)

Correlation in a random sample Generated 6 sets of random numbers (100 each) The correlation Matrix

Factors Affecting r Range restrictions Outliers Nonlinearity e.g. anxiety and performance Heterogeneous subsamples Everyday examples

The effect of outliers on correlations Dataset: 20 cases selected from darts and pros DARTS Pros r =.80

Dataset: one case altered to give more extreme values DARTS Pros r =

Summary of effect of outliers A few extreme values can have extreme effects Especially when sample size is sample You cannot randomly toss out data! You need to have a theoretical or statistical justification

Restriction of range: Countries With Low Consumptions Data With Restricted Range Truncated at 5 Cigarettes Per Day Cigarette Consumption per Adult per Day CHD Mortality per 10,

R between between grades in high school and grades in college. Scatter plot for 250 students who vary on High School GPA Scatter plot for students who have GPA equal to or greater than 3.5

no effect on Pearson's correlation coefficient. Example: r between height and weight is the same regardless of whether height is measured in inches, feet, centimeters or even miles. This is a very desirable property since choice of measurement scales that are linear transformations of each other is often arbitrary. Effect of linear transformations of data

An example: Scores on the Scholastic Aptitude Test (SAT) range from to 800 is an arbitrary range. You could subtract 100 points from each score and multiply each score by 3. Scores on the SAT would then range from Test would remain the same. r between SAT and some other variable (such as college grade point average) would not be affected by this linear transformation.

Non linear relationships Example: Anxiety and Performance r =.07 13

The interpretation of a correlation coefficient Ranges from –1 to 1 No correlation in the data means you will get a is 0 r or near it Suffers from sampling error (like everything else!). So you need to estimate true population correlation from the sample correlation.

Correlations in the sample differ from the correlations in the population by some amount (sampling error) Sometimes it is higher than population correlation, sometimes it is lower, rarely on the target. How do you know when to accept and when to reject correlation?

Possible ways to decide Accept it if it fits your hypothesis, reject it otherwise! Toss a coin Democratically: Ask your officemates to vote.

Fisherian Statistics: Null and Alternative Hypothesis Sampling error implies that sometimes the results we obtain will be due to chance (since not every sample will accurately resemble the population) The null hypothesis expresses the idea that an observed difference is due to chance. For example: There is no difference between the norms regarding the use of and voice mail

The alternative hypothesis (the experimental hypothesis) is often the one that you formulate: there is a correlation between people’s perception of a website’s reliability and the probability of their buying something on the site Why bother to have a null hypothesis? –Can you reject the null hypothesis The alternative hypothesis

An Example Relationship between browsing and buying on an electronic commerce site Data gathered from server logs Hypothesis: Those who browse longer also tend to purchase Hypothesis can be framed in another way: There is no relationship between time spent browsing and likelihood of purchase (Null Hypothesis)

Testing the significance of a r Population parameter =  Null hypothesis H 0 :  = 0 What would a true null mean here? What would a false null mean here? Alternative hypothesis (H 1 )

Tables of Significance Table in Appendix E.2 For N - 2 = 19 df, r crit =.433 Our correlation >.433 Reject H 0 Correlation is significant. More cigarette consumption associated with more CHD mortality.

SPSS Printout SPSS Printout gives test of significance. Double asterisks with footnote indicate p <.01.

SPSS Printout

SPSS printout for scatterplot

OPTIM RELINFL RELINV RELHOPE A matrix of scatterplots Correlation is significant at the 0.01 level (2-tailed). ** **.167**.266**.272** **.419**.167**.449** **.266**.419**.544**1.000 OPTIM RELINFL RELINV RELHOPE OPTIMRELINFLRELINVRELHOPE

A review of Scatterplots next three slides Infant mortality and number of physicians Life expectance and health care expenditures Cancer rate and solar radiation