Aronchpt3correlation

Aron,
Aron, Coups, & Aron

Chapter 3
Correlation and P di i
C l i d Prediction

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

Correlations
Can be thought of as a descriptive statistic for
the relationship b
h l i hi between two variables
i bl
Describes the relationship between two equal-
interval numeric variables
◦ e.g., the correlation between amount of time
studying and amount learned
y g
◦ e.g., the correlation between number of years
of education and salary


Graphing a Scatter Diagram
To make a scatter diagram:
Draw the axes and decide which variable goes on which axis
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
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
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-
vertical
axis variable and mark a clear dot.
Keep going until you have marked a dot for each person.


Linear Correlation
A linear correlation
◦ relationship between two variables that shows
up on a scatter diagram as dots roughly
approximating strai ht
a ro imatin a straight line

Curvilinear Correlation
Curvilinear correlation
◦ any association between two variables other
than a linear correlation
◦ relationship between two variables that shows
up on a scatter diagram as dots following a
systematic pattern that is not a straight line

No Correlation
No correlation
◦ no systematic relationship between two
variables


Positive and Negative Linear
Correlation
Positive Correlation
High scores go with high scores.
Low scores go with low scores.
Medium scores go with medium scores
scores.
When graphed, the line goes up and to the right.
e.g., level of education achieved and income
Negative Correlation
g
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


Importance of Identifying the
Pattern of Correlation
Use a scatter diagram to examine the pattern, direction,
and strength of a correlation
correlation.
◦ First, determine whether it is a linear or curvilinear relationship.
◦ If linear, look to see if it is a positive or negative
correlation.
l i
◦ 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.


The Correlation Coefficient
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
t diff t i bl


The Correlation Coefficient
(r)
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

Strength of Correlation Coefficients

Correlation Coefficient Value Strength of Relationship
+/- .70-1.00 Strong
g
+/- .30-.69 Moderate
+/- .00-.29 None (.00) to Weak

The value of a correlation defines the strength of the
correlation regardless of the sign
sign.
e.g., -.99 is a stronger correlation than .75

Formula for a Correlation
Coefficient
r = ∑ZxZy
N
Zx = Z score for each person on the X variable
Zy = Z score for each person on the Y variable
f h h bl
ZxZy = cross-product of Zx and Zy
∑ZxZy = sum of the cross-products of the Z scores over all
participants in the study


Steps for Figuring the Correlation
Coefficient
C ffi i

Change all scores to Z scores.
◦ Figure the mean and the standard deviation of each variable.
◦ Change each raw score to a Z score
score.
Calculate the cross-product of the Z scores
for each person.
p
◦ 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
cross products scores.
Divide by the number of people in the
study.
y

Calculating a Correlation Coefficient
g
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
0 61 2 ‐1.05
1 05 0.64
0 64

6 ‐0.61 3 ‐0.52 0.32

10 1.84 6 1.05 1.93

MEAN= 7 MEAN= 4 5.14 ΣZXZY

SD= 1.63
SD 1 63 SD= 1.91
SD 1 91 r=5.14/6
5 14/6 ΣZXZY
r=ΣZXZY

r=.85

Issues in Interpreting the
Correlation Coefficient
Direction of causality
y
◦ path of causal effect (e.g., X causes Y)
You cannot determine the direction
of causality just because two
variables are correlated.


Reasons Why We cannot Assume
Causality
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


Ruling Out Some Possible
Directions of Causality
Longitudinal Study
◦ a study where people are measured at two or
more points in time
e.g., evaluating number of hours of sleep at one time point and
then evaluating their levels of stress at a later time point
True Experiment
◦ a study in which participants are randomly
assigned to a particular level of a variable and
then measured on another variable
h d h i bl
e.g., exposing individuals to varying amounts of sleep in a
laboratory environment and then evaluating their stress levels


The Statistical Significance of a Correlation
Coefficient
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.
p
◦ If the probability (p) is less than some small degree
of probability (e.g., 5% or 1%), the correlation is
considered statistically significant.

Prediction
Predictor Variable (X)
variable being predicted from
e.g., level of education achieved
Criterion Variable (Y)
variable being predicted to
e.g.,
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 b i
h it i i bl ld be income.


Prediction Using Z Scores
Prediction Model
A person’s predicted Z score on the criterion
variable is found by multiplying the standardized
regression coefficient (β) by that person s Z score
person’s
on the predictor variable.
Formula for the prediction model using Z scores:
Predicted
P di t d Zy = (β)(Zx)
Predicted Zy = predicted value of the particular person’s Z
score on the criterion variable Y
Zx = particular person’s Z ’ score in the predictor
variable X


Steps for Prediction Using Z Scores
Determine the standardized regression
g
coefficient (β).
Multiply the standardized regression
u t p y t e sta a e eg ess o
coefficient (β) by the person’s Z score on
the predictor variable.
p


How Are You Doing?
So, let’s say that we want to try to predict a
person’s oral presentation score b d on a
’ l t ti based
known relationship between self-confidence
and presentation ability.
p y
Which is the predictor variable (Zx)? The
criterion variable (Zy)?
If r = .90 and Zx = 2 25 th Zy = ?
90 d 2.25 then

So what? What does this predicted value
tell us?

Prediction Using Raw Scores
Change the person’s raw score on the predictor
person s
variable to a Z score.
Multiply the standardized regression coefficient (β)
by the person’s Z score on the predictor variable.
Multiply β by Zx.
This gives the predicted Z score on the criterion variable.
Predicted Zy = (β)(Zx)
Change the person’s predicted Z score on the
g p p
criterion variable back to a raw score.
Predicted Y = (SDy)(Predicted Zy) + My


Example of Prediction Using Raw
Scores: Change Raw Scores to Z
S Ch R S t
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.
p
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

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

Scores: Find the Predicted Z Score
o t C t o a ab
on the Criterion Variable
Multiply the standardized regression coefficient
(β) by the person’s Z score on the predictor
person s
variable.
◦ Multiply β by Zx.
py y
This gives the predicted Z score on the criterion variable.
Predicted Zy = (β)(Zx) = (.85)(-1.84) = -1.56


p g
Scores: Change Raw Scores to Z
Scores
Change the person’s predicted Z score on the
criterion variable to a raw score
score.
◦ Predicted Y = (SDy)(Predicted Zy) + My
◦ Predicted Y = (1.92)(-1.56) + 4 = -3 00 + 4 =
(1 92)(-1 56) -3.00
1.00


The Correlation Coefficient and the
Proportion of Variance Accounted for
P fV A df
Proportion of variance accounted for (r2)
◦ To compare correlations with each other, you
have to square each correlation
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= .04
r
◦ Where, a r= .4, you have an r2= .16
◦ So, relationship with r = .4 is 4x stronger than
, p g
r=.2

Aronchpt3correlation

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