# Chapter 9 Correlational Research Designs ```Chapter 9
Correlational Research Designs
• What are correlational research designs, and why are they used in
behavioral research.
• What patterns of association can occur between two quantitative
variables?
• What is the Pearson product-moment correlation coefficient? What
are its uses and limitations?
• How does the chi-square statistic assess association?
• What is multiple regression, and what are its uses in correlational
research design?
• How can correlational data be used to make inferences about the
causal relationships among measured variables? What are the
limitation of correlational designs in doing so?
• What are the best uses for correlational design?
Statistical Assessment of Relationships
Data
Are the data quantitative or nominal?
quantitative
nominal
Do you have more than two predictor variables?
No
Do you have more than two predictor variables?
Yes
Correlation Analysis: r
No
Yes
Chi-Square Analysis: 2
Regression Analysis: R
Log-Linear Analysis
Logistic Regression
The Correlation Coefficient
for Association among Quantitative Variables
Scatterplot
College
GPA
A graph in which the x axis indicates
the scores on the predictor variable
and the y axis represents the scores
on the outcome variable. A point is
plotted for each individual at the
intersection of their scores.
Regression Line
A line in which the squared distances
of the points from the line are
minimized. (least square methods)
4.0
3.0
2.0
1.0
1.0
2.0
3.0
4.0
High School
GPA
Y
Linear Relationships and Nonliniar Relationships
Y
Positive Linear
X
Y
Negative Linear
Y
Curvilinear
X
X
Y
Curvilinear
X
Independent
X
The Pearson Correlation Coefficient
Calculation
Esteem 1 Esteem 2
1
4 (4-3)/0.8 =1.67 4
2
4
3
3
3
2
4
2
2
5
2
1
Mean 3
2.4
Sesteem1 = 0.8 Sesteem2=1.04
r =  Z
ZY 
N 1
[(4-3)(4-2.4)]2 + ...
 ( X  X )(Y  Y )
[ ( X  X ) ][  (Y  Y ) ]
2
=
i
i
2
2
(4-2.4)2 + ...
(4-3)2 + ...
4+4+3+2+2
4 x 4 + 4 x 3 + ...
=
X
 XY 
 X  Y 
N
4+3+2+2+1
5
2
2








X
Y


2
2
 X 
  Y 

N
N

 

4 x 4 + 4 x 4 + 3 x 3 ...
4 x 4 + 3 x 3 + 2 x 2 ...
Interpretation of r
-1&lt; r &lt;1
If the relationship between X and Y are positive:
If the relationship between X and Y are negative:
0&lt;r&lt;1
-1 &lt; r &lt; 0
If p-value associated with the r is &lt; .05
The variable X and Y are significantly correlate to each other.
Positively: 0 &lt; r &lt; 1, Negatively -1 &lt; r &lt; 0
If p-value associated with the r is &gt;. 05
There is no significant correlation between X and Y
Reporting Correlations
r(Number of Participants) = Correlation Coefficient r, p &lt; p value.
“ As predicted by the research hypothesis, the variable of optimism
and reported health behavior were (significantly) positively correlated
in the sample (the data), r(20) = .52, p &lt; .01
Limitation
1. Cases in which the correlation between X and Y that have
curvilinier relationships
r=0
2. Cases in which the range of variables is restricted.
Example. SAT scores and college GPA
3. Cases in which the data have outliers
Restriction of Range
r &gt; |.99|
Limitation (visual)
Curviliniar
Small Range
Outlier
The Chi-square Statistic
for Association among nominal variables
Yes
No
30 (.15) 70 (.35) 100 (.50)
Northerner
45 (.225)
55 (.275)
60 (.30) 40 (.20) 100 (.50)
Southerner
45 (.225)
55 (.275)
90 (.45)110 (.55) 200 (1.00)
fe 
Row marginal X Column marginal
N
2 =
X=
( fo  fe )2
 f
e
(30  45) 2 (70  55) 2 (60  45) 2 (40  55) 2



22.5
27.5
22.5
27.5
Interpretation of 2
•Go to Table E in Appendix E.
•Degree of Freedom (df):
(Level of variable 1 - 1) X (Level of variable 2 -1)
•Number of Participants
•See the value at the intersection between Alpha p &lt; .05 and df
If 2 is greater than the value in Table E, the contingency table
is significantly differ from the expectation.
If 2 is greater than the value in Table E, the contingency table
is not significantly differ from the expectation.
Reporting Chi-Square Statistic
2 (degree of freedom (df), Number of Participants(N)) =
Chi value, p &lt; p value
As predicted by the research hypothesis, the southerners were more
likely to approve of a policeman striking an adult male citizen who
was being questioned as a suspect in a murder case, 2(1, N =30) =
34.23, p &lt; .01
```