Correlation and
Regression
Quantitative Methods in HPELS
440:210
Agenda
Introduction
 The Pearson Correlation
 Hypothesis Tests with the Pearson
Correlation
 Regression
 Instat
 Nonparametric versions

Introduction


Correlation: Statistical technique used to
measure and describe a relationship between
two variables
Direction of relationship:
 Positive
 Negative

Form of relationship:
 Linear
 Quadratic

...
Degree of relationship:
 -1.0
 0.0  +1.0
Uses of Correlations
Prediction
 Validity
 Reliability

Agenda
Introduction
 The Pearson Correlation
 Hypothesis Tests with the Pearson
Correlation
 Regression
 Instat
 Nonparametric versions

The Pearson Correlation

Statistical Notation  Recall for ANOVA:
r
= Pearson correlation
 SP = sum of products of deviations
 Mx = mean of x scores
 SSx = sum of squares of x scores
Pearson Correlation

Formula Considerations  Recall for ANOVA:
 SP

= S(X – Mx)(Y – My)
SP = SXY – SXSY / n
= S(X – Mx)2
 SSy = S(Y – My)2
 r = SP / √SSxSSy
 SSx
Pearson Correlation
Step 1: Calculate SP
 Step 2: Calculate SS for X and Y values
 Step 3: Calcuate r

Step 1  SP
SXY = (0*1)+(10*3)+(4*1)+(8*2)+(8*3)
SXY = 0 + 30 + 4 + 16 + 24
SXY = 74
SP = SXY – SXSY / n
SP = 74 – [30(100)]/5
SP = 74 - 60
SP = 14
SX=30 SY=10
SP = S(X – Mx)(Y – My)
SP = (-6*-1)+(4*1)+(-2*-1)+(2*0)+(2*1)
SP = 6 + 4 + 2 + 0 + 2
SP = 14
Step 2  SSx and SSy
Step 3  r
r = SP / √SSxSSy
 r = 14 / √(64)(4)
 r = 14 / √256
 r = 14/16
 r = 0.875

Interpretation of r


Correlation ≠ causality
Restricted range
data does not represent the full range of scores –
be wary
 If

Outliers can have a dramatic effect
 Figure

16.9
Correlation and variability
 Coefficient
of determination (r2)
Agenda
Introduction
 The Pearson Correlation
 Hypothesis Tests with the Pearson
Correlation
 Regression
 Instat
 Nonparametric versions

The Process

Step 1: State hypotheses

Non directional:



Directional:



a = 0.05
Step 3: Collect data and calculate statistic


H0: ρ ≤ 0 (no positive population correlation)
H1: ρ < 0 (positive population correlation exists)
Step 2: Set criteria


H0: ρ = 0 (no population correlation)
H1: ρ ≠ 0 (population correlation exists)
r
Step 4: Make decision

Accept or reject
Example
Researchers are interested in determining
if leg strength is related to jumping ability
 Researchers measure leg strength with
1RM squat (lbs) and vertical jump height
(inches) in 5 subjects (n = 5)

Step 1: State Hypotheses
Non-Directional
H0: ρ = 0
H1: ρ ≠ 0
Critical value = 0.878
Step 2: Set Criteria
Alpha (a) = 0.05
Critical Value:
Use Critical Values for Pearson
Correlation Table
Appendix B.6 (p 697)
Information Needed:
df = n - 2
Alpha (a) = 0.05
Directional or non-directional?
0.878
Step 3: Collect Data and Calculate Statistic
Data:
S
Calculate SP
SP = SXY – SXSY / n
SP = 27135 – [1065(126)]/5
SP = 27135 - 26838
SP = 297
X
Y
XY
200
25
5000
180
22
3960
225
27
6075
300
27
8100
160
25
4000
X
X-Mx
(X-Mx)2
1065
126
27135
200
-13
169
180
-33
1089
225
12
144
300
87
7569
160
-53
2809
Calculate SSx
M
213
S
11780
M
Step 3: Collect Data and Calculate Statistic
Calculate SSy
Y
Y-My
(Y-My)2
X
X-Mx
(X-Mx)2
25
-0.2
0.04
200
-13
169
22
-3.2
10.24
180
-33
1089
27
1.8
3.24
225
12
144
27
1.8
3.24
300
87
7569
25
-0.2
0.04
160
-53
2809
25.2
S
16.8
M
213
S
Calculate r
Step 4: Make Decision
r = SP / √SSxSSy
0.667 < 0.878
r = 297 / √11780(16.8)
Accept or reject?
r = 297 / √197904
r = 297 / 444.86
r = 0.667
11780
Agenda
Introduction
 The Pearson Correlation
 Hypothesis Tests with the Pearson
Correlation
 Regression
 Instat
 Nonparametric versions

Regression

Recall  Several uses of correlation:
 Prediction
 Validity
 Reliability
Regression attempts to predict one
variable based on information about the
other variable
 Line of best fit

Regression

Line of best fit can be described with the
following linear equation  Y = bX + a
where:
Y
= predicted Y value
 b = slope of line
 X = any X value
 a = intercept
25
5
Y = bX + a, where:
Y = cost (?)
b = cost per hour ($5)
X = number of hours (?)
a = membership cost ($25)
Y = 5X + 25
Y = 5X + 25
Y = 5(10) + 25
Y = 5(30) + 25
Y = 50 + 25 = 75
Y = 150 + 25 = 175
Line of best fit
minimizes
distances of points
from line
Calculation of the Regression Line
Regression line = line of best fit = linear
equation
 SP = S(X – Mx)(Y – My)
 SSx = S(X – Mx)2
 b = SP / SSx
 a = My - bMx

Example 16.14, p 557
Mx=5
My=6
SP = S(X – Mx)(Y – My)
SSx = S(X – Mx)2
b = SP / SSx
SP = 16
SP = 10
b = 16 / 10 = 1.6
a = My - bMx
Y = bX + a
a = 6 – 1.6(5) = -2
Y = 1.6(X) - 2
Agenda
Introduction
 The Pearson Correlation
 Hypothesis Tests with the Pearson
Correlation
 Regression
 Instat
 Nonparametric versions

Instat - Correlation

Type data from sample into a column.

Label column appropriately.




Choose “Manage”
Choose “Column Properties”
Choose “Name”
Choose “Statistics”

Choose “Regression”

Choose “Correlation”
Instat – Correlation



Choose the appropriate variables to be
correlated
Click OK
Interpret the p-value
Instat – Regression

Type data from sample into a column.

Label column appropriately.




Choose “Manage”
Choose “Column Properties”
Choose “Name”
Choose “Statistics”

Choose “Regression”

Choose “Simple”
Instat – Regression

Choose appropriate variables for:







Response (Y)
Explanatory (X)
Check “significance test”
Check “ANOVA table”
Check “Plots”
Click OK
Interpret p-value
Reporting Correlation Results

Information to include:




Examples:


Value of the r statistic
Sample size
p-value
A correlation of the data revealed that strength and
jumping ability were not significantly related (r = 0.667,
n = 5, p > 0.05)
Correlation matrices are used when
interrelationships of several variables are tested
(Table 1, p 541)
Agenda
Introduction
 The Pearson Correlation
 Hypothesis Tests with the Pearson
Correlation
 Regression
 Instat
 Nonparametric versions

Nonparametric Versions
Spearman rho  when at least one of the
data sets is ordinal
 Point biserial correlation  when one set
of data is ratio/interval and the other is
dichotomous

 Male
vs. female
 Success vs. failure

Phi coefficient  when both data sets are
dichotomous
Violation of Assumptions
Nonparametric Version  Friedman Test
(Not covered)
 When to use the Friedman Test:

 Related-samples
design with three or more
groups
 Scale of measurement assumption violation:

Ordinal data
 Normality

assumption violation:
Regardless of scale of measurement
Textbook Assignment

Problems: 5, 7, 10, 23 (with post hoc)