Chapter 7 -Part 1 Correlation

Correlation Topics zCo-relationship between two variables. zLinear vs Curvilinear relationships zPositive vs Negative relationships zStrength of relationship

Mythical relationship between Baseball and Football performance Al Ben Chuck David Ed Frank George Baseball skill Very good Very poor Good Terrible Poor Average Excellent Football skill Very good Very poor Good Terrible Poor Average Excellent Is this a linear relationship? Baseball skill predicts football skill. Football skill predicts baseball skill. There is a strong relationship.

First we must arrange the scores in “order” Baseball skill Terrible Very Poor Poor Average Good Very Good Excellent Football skill Terrible Very Poor Poor Average Good Very Good Excellent David Ben Ed Frank Chuck Al George

Then we plot the scores * Ben * Ed * Frank * Chuck * Al * David * George Excellent Terrible Very Good Good Average Poor Very Poor ExcellentTerribleVery GoodGoodAveragePoorVery Poor Football Skill Baseball Skill This is definitely a linear relationship!

Let’s get more abstract? Excellent Terrible Very Good Good Average Poor Very Poor ExcellentTerribleVery GoodGoodAveragePoorVery Poor Football Skill Baseball Skill X Y

Linear or nonlinear? Let’s look at another set of values. Football skill Terrible Average Very Good Excellent Good Poor Baseball skill Terrible Very Poor Poor Average Good Very Good Excellent David Ben Ed Frank Chuck Al George Is this a linear relationship?

Is this linear? * Ben* Ed * Frank * Chuck * Al * David * George Excellent Terrible Very Good Good Average Poor Very Poor ExcellentTerribleVery GoodGoodAveragePoorVery Poor Football Skill Baseball Skill NO! It is best described by a curved line. It is a curvilinear relationship!

Positive vs Negative relationships zIn a positive relationship, as one value increases the other value tends to increase as well. Example: The longer a sailboat is, the more it tends to cost. As length goes up, price tends to go up. zIn a negative relationship, as one value increases, the other value decreases. Example: The older a sailboat is, the less it tends to cost. As years go up, price tends to go down.

Positive vs Negative scatterplot Negative relationship Positive relationship

Correlation Characteristics Linear vs Curvilinear

The strength of a relationship tells us approximately how the dots will fall around a best fitting line. zPerfect - scores fall exactly on a straight line. zStrong - most scores fall near the line. zModerate - some are near the line, some not. zWeak – lots of scores fall close to the line, but many fall quite far from it. zIndependent - the scores are not close to the line and form a circular or square pattern

Strength of a relationship Perfect

Strength of a relationship Strong

Strength of a relationship Moderate

Strength of a relationship Independent

What is this relationship?

What is this?

What is this?

What is this?

Comparing apples to oranges? Use t scores! zYou can use correlation to look for the relationship between ANY two values that you can measure of a single subject. zHowever, there may not be any relationship (independent). zA correlation tells us if scores are consistently similar on two measures, consistently different from each other, or have no real pattern

Comparing apples to oranges? Use t scores! zTo compare scores on two different variables, you transform them into t X and t Y scores. zt X and t Y scores can be directly compared to each other to see whether they are consistently similar, consistently quite different, or show no consistent pattern of similarity or difference

Similar t X and t Y scores = positive correlation. dissimilar = negative correlation. No pattern = independence. zWhen t scores are consistently more similar than different, we have a positive correlation. zWhen t scores are consistently more different than similar, we have a negative correlation. zWhen t scores show no consistent pattern of similarity or difference, we have independence.

Comparing variables zAnxiety symptoms, e.g., heartbeat, with number of hours driving to class. zHat size with drawing ability. zMath ability with verbal ability. zNumber of children with IQ. zTurn them all into t scores

Pearson’s Correlation Coefficient zcoefficient - noun, a number that serves as a measure of some property. zThe correlation coefficient indexes the consistency and direction of a correlation zPearson’s rho (  ) is the parameter that characterizes the strength and direction of a linear relationship (and only a linear relationship) between two population variables. zPearsons r is a least squares, unbiased estimate of rho.

Pearson’s Correlation Coefficient zr and rho vary from to zA negative value indicates a negative relationship; a positive value indicates a positive relationship. zValues of r close to or indicate a strong (consistent) relationship; values close to indicate a weak (inconsistent) or independent relationship.

r, strength and direction Perfect, positive+1.00 Strong, positive+.75 Moderate, positive+.50 Weak, positive+.25 Independent.00 Weak, negative -.25 Moderate, negative -.50 Strong, negative -.75 Perfect, negative -1.00

Calculating Pearson’s r zSelect a random sample from a population; obtain scores on two variables, which we will call X and Y. zConvert all the scores into t scores.

Calculating Pearson’s r zFirst, subtract the t Y score from the t X score in each pair. zThen square all of the differences and add them up, that is,  (t X - t Y ) 2.

Calculating Pearson’s r zEstimate the average squared distance between Z X and Z Y by dividing by the sum of squared differences by(n P - 1), that is,  ( t X - t Y ) 2 / (n P - 1) zTo turn this estimate into Pearson’s r, use the formula r =1 - (1/2  ( t X - t Y ) 2 / (n P - 1))

Note seeming exception zUsually we divide a sum of squared deviations around a mean by df to estimate the variance. zHere the sum of squares is not around a mean and we are not estimating a variance. zSo you divide  ( t X - t Y ) 2 by (n P - 1) zn P - 1 is not df for corr & regression (df REG = n P - 2)

Example: Calculate t scores for X DATA  X=30 N= 5 X=6.00 MS W = 40.00/(5-1) = 10 s X = 3.16 (X - X) X - X t x =(X-X)/ s SS W = 40.00

Calculate t scores for Y DATA  Y=55 N= 5 Y=11.00 MS W = 10.00/(5-1) = 2.50 s Y = 1.58 (Y - Y) Y - Y (t y =Y - Y) / s SS W = 10.00

Calculate r t Y t X t X - t Y (t X - t Y )  (t X - t Y ) 2 / (n P - 1)=0.200 r = (1/2 * (  (t X - t Y ) 2 / (n P - 1))) r = (1/2 *.200) = =.900  (t X - t Y ) 2 =0.80 This is a very strong, positive relationship.