aron chpt 3 correlation

Chapter 3Correlation and Prediction

Copyright © 2011 by Pearson Education, Inc. All rights reserved

Aron, Coups, & Aron

Can be thought of as a descriptive statistic for the relationship between two variables

Describes the relationship between two equal-interval numeric variablese.g., the correlation between amount of time

studying and amount learned e.g., the correlation between number of years

of education and salary


Correlations

Scatter Diagram

0 5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

3.5

4

Number of Hours Studying

GP

A

To make a scatter diagram: Draw the axes and decide which variable goes on which axis.

The values of one variable go along the horizontal axis and the values of the other variable go along the vertical axis.

Determine the range of values to use for each variable and mark them on the axes.

Numbers should go from low to high on each axis starting from where the axes meet .

Usually your low value on each axis is 0. Each axis should continue to the highest value your measure can

possibly have. Make a dot for each pair of scores.

Find the place on the horizontal axis for the first pair of scores on the horizontal-axis variable.

Move up to the height for the score for the first pair of scores on the vertical-axis variable and mark a clear dot.

Keep going until you have marked a dot for each person.


Graphing a Scatter Diagram

A linear correlationrelationship between two variables that

shows up on a scatter diagram as dots roughly approximating a straight line

Linear Correlation

Curvilinear correlationany association between two variables other

than a linear correlationrelationship between two variables that

shows up on a scatter diagram as dots following a systematic pattern that is not a straight line

Curvilinear Correlation

No correlationno systematic relationship between two

variables


No Correlation

Positive Correlation High scores go with high scores. Low scores go with low scores. Medium scores go with medium scores. When graphed, the line goes up and to the right.

e.g., level of education achieved and income

Negative Correlation High scores go with low scores.

e.g., the relationship between fewer hours of sleep and higher levels of stress

Strength of the Correlation how close the dots on a scatter diagram fall to a

simple straight line


Positive and Negative Linear Correlation

Use a scatter diagram to examine the pattern, direction, and strength of a correlation. First, determine whether it is a linear or curvilinear

relationship. If linear, look to see if it is a positive or negative

correlation.Then look to see if the correlation is large, small, or

moderate.Approximating the direction and strength of a

correlation allows you to double check your calculations later.


Importance of Identifying the Pattern of Correlation

A number that gives the exact correlation between two variables

can tell you both direction and strength of relationship between two variables (X and Y)

uses Z scores to compare scores on different variables


The Correlation Coefficient

The sign of r (Pearson correlation coefficient) tells the general trend of a relationship between two variables. + sign means the correlation is positive. - sign means the correlation is negative.

The value of r ranges from -1 to 1.A correlation of 1 or -1 means that the variables

are perfectly correlated.0 = no correlation

The Correlation Coefficient ( r )

Correlation Coefficient Value Strength of Relationship

+/- .70-1.00 Strong

+/- .30-.69 Moderate

+/- .00-.29 None (.00) to Weak

Strength of Correlation Coefficients

The value of a correlation defines the strength of the correlation regardless of the sign.

e.g., -.99 is a stronger correlation than .75

r = ∑ZxZy

NZx = Z score for each person on the X variableZy = Z score for each person on the Y variableZxZy = cross-product of Zx and Zy ∑ZxZy = sum of the cross-products of the Z scores

over all participants in the study


Formula for a Correlation Coefficient

Change all scores to Z scores.Figure the mean and the standard deviation of

each variable.Change each raw score to a Z score.

Calculate the cross-product of the Z scores for each person.Multiply each person’s Z score on one variable

by his or her Z score on the other variable.Add up the cross-products of the Z scores.Divide by the number of people in the

study.


Steps for Figuring the Correlation Coefficient

Number of Hours Slept (X) Level of Mood (Y) Calculate r

X Zscore Sleep Y Zscore Mood Cross Product ZXZY

5 -1.23 2 -1.05 1.28

7 0.00 4 0.00 0.00

8 0.61 7 1.57 0.96

6 -0.61 2 -1.05 0.64

6 -0.61 3 -0.52 0.32

10 1.84 6 1.05 1.93

MEAN=7 MEAN=4 5.14 SZXZY

SD=1.63 SD=1.91 r=5.14/6 r=SZXZY

r=.85

Calculating a Correlation Coefficient

Direction of causalitypath of causal effect (e.g., X causes Y)

You cannot determine the direction of causality just because two variables are correlated.


Issues in Interpreting the Correlation Coefficient

Variable X causes variable Y.e.g., less sleep causes more stress

Variable Y causes variable X.e.g., more stress causes people to sleep less

There is a third variable that causes both variable X and variable Y.e.g., working longer hours causes both stress

and fewer hours of sleep


Reasons Why We cannot Assume Causality

Longitudinal Studya study where people are measured at two or

more points in timee.g., evaluating number of hours of sleep at one

time point and then evaluating their levels of stress at a later time point

True Experimenta study in which participants are randomly

assigned to a particular level of a variable and then measured on another variablee.g., exposing individuals to varying amounts of

sleep in a laboratory environment and then evaluating their stress levels


Ruling Out Some Possible Directions of Causality

A correlation is statistically significant if it is unlikely that you could have gotten a correlation as big as you did if in fact there was no relationship between variables. If the probability (p) is less than some small

degree of probability (e.g., 5% or 1%), the correlation is considered statistically significant.

The Statistical Significance of a Correlation Coefficient

Predictor Variable (X)variable being predicted from

e.g., level of education achieved

Criterion Variable (Y)variable being predicted to

e.g., income

If we expect level of education to predict income, the predictor variable would be level of education and the criterion variable would be income.


Prediction

Prediction ModelA person’s predicted Z score on the criterion

variable is found by multiplying the standardized regression coefficient () by that person’s Z score on the predictor variable.

Formula for the prediction model using Z scores:Predicted Zy = ()(Zx) Predicted Zy = predicted value of the particular

person’s Z score on the criterion variable YZx = particular person’s Z score in the predictor

variable X


Prediction Using Z Scores

Determine the standardized regression coefficient ().

Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.


Steps for Prediction Using Z Scores

So, let’s say that we want to try to predict a person’s oral presentation score based on a known relationship between self-confidence and presentation ability.

Which is the predictor variable (Zx)? The criterion variable (Zy)?

If r = .90 and Zx = 2.25 then Zy = ?

So what? What does this predicted value tell us? Copyright © 2011 by Pearson Education, Inc. All

rights reserved

How Are You Doing?

Change the person’s raw score on the predictor variable to a Z score.

Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.Multiply by Zx.

This gives the predicted Z score on the criterion variable.Predicted Zy = ()(Zx)

Change the person’s predicted Z score on the criterion variable back to a raw score.Predicted Y = (SDy)(Predicted Zy) + My


Prediction Using Raw Scores


Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores

From the sleep and mood study example, we known the mean for sleep is 7 and the standard deviation is 1.63, and that the mean for happy mood is 4 and the standard deviation is 1.92.

The correlation between sleep and mood is .85.

Change the person’s raw score on the predictor variable to a Z score. Zx = (X - Mx) / SDx (4-7) / 1.63 = -3 / 1.63 = -1.84


Example of Prediction Using Raw Scores: Find the Predicted Z Score on the Criterion Variable

Multiply the standardized regression coefficient () by the person’s Z score on the predictor variable.Multiply by Zx.

This gives the predicted Z score on the criterion variable.Predicted Zy = ()(Zx) = (.85)(-1.84) = -1.56


Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores

Change the person’s predicted Z score on the criterion variable to a raw score.Predicted Y = (SDy)(Predicted Zy) + MyPredicted Y = (1.92)(-1.56) + 4 = -3.00 + 4

= 1.00

Proportion of variance accounted for (r2)To compare correlations with each other,

you have to square each correlation.This number represents the proportion of

the total variance in one variable that can be explained by the other variable.

If you have an r= .2, your r2= .04Where, a r= .4, you have an r2= .16 So, relationship with r = .4 is 4x stronger

than r=.2

The Correlation Coefficient and the Proportion of Variance Accounted for

aron chpt 3 correlation

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