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Correlation and Regression
Quantitative Methods in HPELS
440:210
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Agenda
Introduction The Pearson Correlation Hypothesis Tests with the Pearson
Correlation Regression Instat Nonparametric versions
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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
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Uses of Correlations
Prediction Validity Reliability
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Agenda
Introduction The Pearson Correlation Hypothesis Tests with the Pearson
Correlation Regression Instat Nonparametric versions
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The Pearson Correlation Statistical Notation Recall for ANOVA:
r = Pearson correlationSP = sum of products of deviationsMx = mean of x scores
SSx = sum of squares of x scores
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Pearson Correlation
Formula Considerations Recall for ANOVA:SP = (X – Mx)(Y – My)
SP = XY – XY / n
SSx = (X – Mx)2
SSy = (Y – My)2
r = SP / √SSxSSy
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Pearson Correlation
Step 1: Calculate SP Step 2: Calculate SS for X and Y values Step 3: Calcuate r
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Step 1 SP
SP = (X – Mx)(Y – My)SP = (-6*-1)+(4*1)+(-2*-1)+(2*0)+(2*1)SP = 6 + 4 + 2 + 0 + 2SP = 14
SP = XY – XY / nSP = 74 – [30(100)]/5SP = 74 - 60SP = 14
X=30 Y=10
XY = (0*1)+(10*3)+(4*1)+(8*2)+(8*3)XY = 0 + 30 + 4 + 16 + 24XY = 74
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Step 2 SSx and SSy
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Step 3 r
r = SP / √SSxSSy
r = 14 / √(64)(4) r = 14 / √256 r = 14/16 r = 0.875
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Interpretation of r
Correlation ≠ causality Restricted range
If data does not represent the full range of scores – be wary
Outliers can have a dramatic effect Figure 16.9
Correlation and variability Coefficient of determination (r2)
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Agenda
Introduction The Pearson Correlation Hypothesis Tests with the Pearson
Correlation Regression Instat Nonparametric versions
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The Process
Step 1: State hypotheses Non directional:
H0: ρ = 0 (no population correlation) H1: ρ ≠ 0 (population correlation exists)
Directional: H0: ρ ≤ 0 (no positive population correlation) H1: ρ < 0 (positive population correlation exists)
Step 2: Set criteria = 0.05
Step 3: Collect data and calculate statistic r
Step 4: Make decision Accept or reject
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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)
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Step 1: State Hypotheses
Non-Directional
H0: ρ = 0
H1: ρ ≠ 0
Step 2: Set Criteria
Alpha () = 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?
Critical value = 0.878
0.878
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Step 3: Collect Data and Calculate Statistic
Data:
X Y XY
200 25 5000
180 22 3960
225 27 6075
300 27 8100
160 25 4000
1065 126 27135
Calculate SPSP = XY – XY / nSP = 27135 – [1065(126)]/5SP = 27135 - 26838SP = 297
Calculate SSx
X X-Mx (X-Mx)2
200 -13 169
180 -33 1089
225 12 144
300 87 7569
160 -53 2809
213M 11780
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Calculate SSy
Y Y-My (Y-My)2
25 -0.2 0.04
22 -3.2 10.24
27 1.8 3.24
27 1.8 3.24
25 -0.2 0.04
25.2M 16.8
X X-Mx (X-Mx)2
200 -13 169
180 -33 1089
225 12 144
300 87 7569
160 -53 2809
213M 11780
r = SP / √SSxSSy
r = 297 / √11780(16.8)
r = 297 / √197904
r = 297 / 444.86
r = 0.667
Step 3: Collect Data and Calculate Statistic
Calculate r Step 4: Make Decision
0.667 < 0.878
Accept or reject?
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Agenda
Introduction The Pearson Correlation Hypothesis Tests with the Pearson
Correlation Regression Instat Nonparametric versions
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Regression Recall Several uses of correlation:
PredictionValidityReliability
Regression attempts to predict one variable based on information about the other variable
Line of best fit
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Regression
Line of best fit can be described with the following linear equation Y = bX + a where:Y = predicted Y valueb = slope of lineX = any X valuea = intercept
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Y = bX + a, where:
Y = cost (?)
b = cost per hour ($5)
X = number of hours (?)
a = membership cost ($25)Y = 5X + 25
Y = 5(10) + 25
Y = 50 + 25 = 75
Y = 5X + 25
Y = 5(30) + 25
Y = 150 + 25 = 175
5
25
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Line of best fit minimizes
distances of points from line
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Calculation of the Regression Line
Regression line = line of best fit = linear equation
SP = (X – Mx)(Y – My)
SSx = (X – Mx)2
b = SP / SSx
a = My - bMx
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Example 16.14, p 557
SP = (X – Mx)(Y – My)
SP = 16
SSx = (X – Mx)2
SP = 10
b = SP / SSx
b = 16 / 10 = 1.6
a = My - bMx
a = 6 – 1.6(5) = -2
Mx=5 My=6
Y = bX + a
Y = 1.6(X) - 2
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Agenda
Introduction The Pearson Correlation Hypothesis Tests with the Pearson
Correlation Regression Instat Nonparametric versions
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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”
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Instat – Correlation Choose the appropriate variables to be
correlated Click OK Interpret the p-value
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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”
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Instat – Regression
Choose appropriate variables for: Response (Y) Explanatory (X)
Check “significance test” Check “ANOVA table” Check “Plots” Click OK Interpret p-value
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Reporting Correlation Results Information to include:
Value of the r statistic Sample size p-value
Examples: 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)
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Agenda
Introduction The Pearson Correlation Hypothesis Tests with the Pearson
Correlation Regression Instat Nonparametric versions
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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 dichotomousMale vs. femaleSuccess vs. failure
Phi coefficient when both data sets are dichotomous
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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
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Textbook Assignment
Problems: 5, 7, 10, 23 (with post hoc)