Correlation and Regression

A BRIEF overview

Correlation Coefficients

Continuous IV & DV or dichotomous variables (code as 0-1)

mean interpreted as proportion Pearson product moment correlation

coefficient range -1.0 to +1.0

Interpreting Correlations

1.0, + or - indicates perfect relationship 0 correlations = no association between

the variables in between - varying degrees of

relatedness r2 as proportion of variance shared by two

variables which is X and Y doesn’t matter

Positive Correlation

regression line is the line of best fit With a 1.0 correlation, all points fall

exactly on the line 1.0 correlation does not mean values

identical the difference between them is

identical

Negative Correlation

If r=-1.0 all points fall directly on the regression line

slopes downward from left to right sign of the correlation tells us the

direction of relationship number tells us the size or magnitude

Zero correlation

no relationship between the variables a positive or negative correlation gives

us predictive power

Direction and degree

Direction and degree (cont.)

Direction and degree (cont.)

Correlation Coefficient r = Pearson Product-Moment Correlation Coefficient zx = z score for variable x

zy = z score for variable y N = number of paired X-Y values Definitional formula (below)

N

zzr yx )(

Raw score formula

])(][)([ 2222 YYNXXN

YXXYNr

Interpreting correlation coefficients comprehensive description of relationship direction and strength need adequate number of pairs more than 30 or so same for sample or population population parameter is Rho (ρ) scatterplots and r more tightly clustered around line=higher correlation

Examples of correlations

-1.0 negative limit -.80 relationship between juvenile street

crime and socioeconomic level .43 manual dexterity and assembly line

performance .60 height and weight 1.0 positive limit

Describing r’s Effect size index-Cohen’s guidelines:

Small – r = .10, Medium – r = .30, Large – r = .50

Very high = .80 or more Strong = .60 - .80 Moderate = .40 - .60 Low = .20 - .40 Very low = .20 or less

small correlations can be very important

Correlation as causation??

Nonlinearity and range restriction

if relationship doesn't follow a linear pattern Pearson r useless

r is based on a straight line function if variability of one or both variables is

restricted the maximum value of r decreases

Linear vs. curvilinear relationships

Linear vs. curvilinear (cont.)

Range restriction

Range restriction (cont.)

Understanding r

Simple linear regression

enables us to make a “best” prediction of the value of a variable given our knowledge of the relationship with another variable

generate a line that minimizes the squared distances of the points in the plot

no other line will produce smaller residuals or errors of estimation

least squares property

Regression line

The line will have the form Y'=A+BX Where: Y' = predicted value of Y A = Y intercept of the line B = slope of the line X = score of X we are using to predict Y

Ordering of variables

which variable is designated as X and which is Y makes a difference

different coefficients result if we flip them generally if you can designate one as

the dependent on some logical grounds that one is Y

Moving to prediction

statistically significant relationship between college entrance exam scores and GPA

how can we use entrance scores to predict GPA?

Best-fitting line (cont.)

Best-fitting line (cont.)

Calculating the slope (b)

N=number of pairs of scores, rest of the terms are the sums of the X, Y, X2, Y2, and XY columns we’re already familiar with

22 )()(

))(()(

XXN

YXXYNb

Calculating Y-intercept (a)

b = slope of the regression line the mean of the Y values the mean of the X values

Y

X

XbYa )(

Let’s make up a small example

SAT – GPA correlation How high is it generally? Start with a scatter plot Enter points that reflect the relationship

we think exists Translate into values Calculate r & regression coefficients