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Correlation
Chapter 6
Assumptions for Pearson r
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X and Y should be interval or ratio.
X and Y should be normally distributed.
Each X should be independent of other X’s.
Each Y should be independent of other Y’s.
Scores on Y should be linearly related to X.
X, Y scores should be bivariate normal,
In a bivariate normal
distribution when you
look at a single X
value (0), most of the
Y points are clustered
around the blue
(regression line) and
as you get further
away from the blue
line there are fewer Y
data points at a given
X.
If there is a relationship between two variables,
then as one variable deviates from its mean,
the other variable in the same way or directly
opposite way.
Covariance is the averaged sum of combined
deviations.
Each variables
deviation from
its mean is
about the
same
magnitude,
then
covariance is
high.
Correlation
• The correlation (r) is computed by dividing the
covariance by the product of x and y’s standard
deviation.
• The value of r ranges from −1 to +1.
• The correlation coefficient is a standard measure of
effect size:
Correlation and Cause?
±.1 is a small effect
There is a high r between shoe
±.3 is a medium effect
size and math performance for
±.5 is a large effect
grade school children.
Two Types of Correlation
• Two types of corr: bivariate and partial.
• Bivariate correlation is the correlation
between two variables.
• Partial correlation is the correlation between
two variables when controlling the effect of
one or more additional variables.
Pearson’s Product Moment Correlation
• Correlation measures the association between
two variables.
• Correlation quantifies the extent to which the
mean, variation & direction of one variable are
related to another variable.
• r ranges from +1 to -1.
• Correlation can be used for prediction.
• Correlation does not indicate the cause of a
relationship.
Scatter Plot
• Scatter plot gives a visual description of the relationship
between two variables.
• The line of best fit is defined as the line that minimized
the squared deviations from a data point up to or down
to the line.
Line of Best Fit Minimizes Squared Deviations from a Data
Point to the Line
Always do a Scatter Plot to Check the Shape of the
Relationship
Will a Linear Fit Work?
2
1
0
0
-1
-2
-3
-4
1
2
3
4
5
6
Will a Linear Fit Work?
y = 0.5246x - 2.2473
R2 = 0.4259
2
1
0
0
-1
-2
-3
-4
1
2
3
4
5
6
6th Order Fit?
y = 0.0341x6 - 0.6358x5 + 4.3835x4 - 13.609x3 + 18.224x2 - 7.3526x - 2.0039
R2 = 0.9337
2
1
0
0
-1
-2
-3
-4
1
2
3
4
5
6
Will Linear Fit Work?
Y
2
1
0
0
-1
-2
-3
-4
50
100
150
200
250
Linear Fit
y = 0.0012x - 1.0767
R2 = 0.0035
2
1
0
0
-1
-2
-3
-4
50
100
150
200
250
Evaluating the Strength of a Correlation
• For predictions, absolute value of r < .7, may
produce unacceptably large errors, especially if the
SDs of either or both X & Y are large.
• As a general rule
–
–
–
–
Absolute value r greater than or equal .9 is good
Absolute value r equal to .7 - .8 is moderate
Absolute value r equal to .5 - .7 is low
Values for r below .5 give R2 = .25, or 25% are poor, and
thus not useful for predicting.
Significant Correlation??
If N is large (N=90) then a .205
correlation is significant.
ALWAYS THINK ABOUT R2
How much variance in Y is X
accounting for?
r = .205
R2 = .042, thus X is accounting
for 4.2% of the variance in Y.
This will lead to poor
predictions.
A 95% confidence interval will
also show how poor the
prediction is.
Venn diagram shows (R2) the amount of variance in Y
that is explained by X.
Unexplained Variance in Y. (1-R2) =
.36, 36%
R2=.64 (64%) Variance in Y that
is explained by X
A partial correlation
is used to remove
the effects of
Revision Time on
both Exam Anxiety
and Exam
Performance.
Then the unique
contribution of
Exam Anxiety on
Exam Performance
can be analyzed.
A coach ranked athletes based on jumping ability and sprinting ability. The
ranks for each athlete are presented in the table below. Compute the
correlation between the ratings.
Athlete
1
2
3
4
5
6
7
8
9
10
Jump Rating
3
1
7
8
2
5
9
10
4
6
Sprint Rating
7
1
2
8
3
9
6
10
5
4
Level of
Measurement
is ordinal
Kendall’s tau is better if the data have several ties, if not
Spearman is fine.
Test-Retest Reliability (ICC)
and Day to Day Variation
Consistency of Measurements
• Reliability refers to the consistency of a test or
measurement.
• A test cannot be considered valid if it is not
reliable.
• You should know the day to day variation in
your dependent variable.
• How much does 1RM bench press change from
day to day?
• How much does VO2 max change from day to
day?
Subject
Trial 1
Trial 2
Difference
1
146
140
-6
2
148
152
+4
3
170
152
-18
4
90
99
+9
5
157
145
-12
6
156
153
-3
7
176
167
-9
8
205
218
+13
Mean
156
153
2.75
SD
32.8
32.9
10.7
In this experiment, we plan to average
trials, so we will use Averaged Measures
ICC.
Test-Retest
Reliability
• The 1RM squat strength
was measured on 8
subjects on two
separate days.
• In this experiment we
want to measure the
reliability of our
measurements.
• Here is the day to day
variation in 1RM bench
press strength.
Enter the
between day
variables.
Click Statistics
Click OK to run
SPSS Output
ANOVA
Between People
Within People
Total
Between Items
Res iduala
Total
Sum of
Squares
14689.750
30.250
399.750
430.000
15119.750
df
7
1
7
8
15
Mean Square
2098.536
30.250
57.107
53.750
1007.983
F
Sig
.530
Grand Mean = 154.6250
a. Tukey's test for nonadditivity is undefined for dichotomous data.
There is no difference between trials F(1,7) = .530, p = .49
If there is a difference between trials you may have a
learning effect or a fatigue effect and you must modify your
methods to control for learning and/or fatigue.
.490
Intraclass Correlation Coefficient
Single Meas ures
Average Meas ures
95% Confidence Interval
Intraclas s
a
Correlation
Lower Bound Upper Bound
b
.947
.761
.989
.973 c
.864
.995
F Tes t with True Value 0
Value
df1
df2
36.747
7.0
7
36.747
7.0
7
Sig
.000
.000
Two-way mixed effects model where people effects are random and measures effects are fixed.
a. Type C intraclass correlation coefficients using a cons is tency definition-the between-meas ure variance is
excluded from the denominator variance.
b. The estimator is the s ame, whether the interaction effect is present or not.
c. This estimate is computed as suming the interaction effect is abs ent, because it is not es timable otherwis e.
ICC = .973 [High Degree of Reliability]
Using Mixed Model, Type Consistency, Averaged Measures.
Subject
Trial 1
Trial 2
Difference
1
146
140
-6
2
148
152
+4
3
170
152
-18
4
90
99
+9
5
157
145
-12
6
156
153
-3
7
176
167
-9
8
205
218
+13
Mean
156
153
2.75
SD
32.8
32.9
10.7
Day to Day Variation
• If you train subjects for
8 weeks and they
improve their strength
by 2.8 Kg have you
actually done anything?
• Is the program
effective?
• Here is the day to day
variation in 1RM bench
press strength.
Manuscript Methods & Results Sections
METHODS
Measurement of Reliability
Prior to initiating the study, 8 subjects (not in the experiment), participated in a
test-retest assessment of measurement reliability. One RM bench press strength was
measured on two separate days for subjects in the reliability study. Reproducibility of
bench press strength was analyzed using SPSS (18.0 for Windows) to compute the
intraclass correlation coefficient (ICC) using a two factor mixed effects model and type
consistency (McGraw and Wong, 1996; Shrout and Fleiss, 1979).
RESULTS
A high degree of reliability was found between bench press strength measurements the
average measure ICC was .973 with a 95% confidence interval from .864 - .995. The
mean between day variation for 1RM strength was 2.75±10.7 Kg.
REFERENCES
McGraw KO and Wong SP. Forming inferences about some intraclass correlation
coefficients. Psychological Methods 1: 30-46, 1996.
Shrout PE and Fleiss JL. Intraclass correlations: Uses in assessing reliability. Psychol
Bull 86: 420-428, 1979.