Correlation and Regression

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Correlation and Regression

• Quantitative Methods in HPELS
• 440210

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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 correlation
• SP sum of products of deviations
• Mx mean of x scores
• SSx sum of squares of x scores

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Pearson Correlation

• Formula Considerations ? Recall for ANOVA
• SP S(X Mx)(Y My)
• SP SXY SXSY / n
• SSx S(X Mx)2
• SSy S(Y My)2
• r SP / vSSxSSy

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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
SXY (01)(103)(41)(82)(83) SXY 0 30
4 16 24 SXY 74
SP SXY SXSY / n SP 74 30(100)/5 SP
74 - 60 SP 14
SP S(X Mx)(Y My) SP (-6-1)(41)(-2-1)
(20)(21) SP 6 4 2 0 2 SP 14
SX30
SY10
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Step 2 ? SSx and SSy
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Step 3 ? r

• r SP / vSSxSSy
• r 14 / v(64)(4)
• r 14 / v256
• 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 ? 0 (no population correlation)
• H1 ? ? 0 (population correlation exists)
• Directional
• H0 ? 0 (no positive population correlation)
• H1 ? lt 0 (positive population correlation
  exists)
• Step 2 Set criteria
• a 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 ?
0 H1 ? ? 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)
0.878
Information Needed df n - 2 Alpha (a)
0.05 Directional or non-directional?
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Step 3 Collect Data and Calculate Statistic
Data
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
Calculate SSx
X X-Mx (X-Mx)2
200 -13 169
180 -33 1089
225 12 144
300 87 7569
160 -53 2809
1065 126 27135
S
213
M
11780
S
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Step 3 Collect Data and Calculate Statistic
Calculate SSy
X X-Mx (X-Mx)2
200 -13 169
180 -33 1089
225 12 144
300 87 7569
160 -53 2809
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
213
M
11780
S
25.2
M
16.8
S
Step 4 Make Decision 0.667 lt 0.878 Accept or
reject?
Calculate r
r SP / vSSxSSy r 297 / v11780(16.8) r 297 /
v197904 r 297 / 444.86 r 0.667
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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
• Prediction
• Validity
• Reliability
• 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 value
• b slope of line
• X any X value
• a 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
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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 S(X Mx)(Y My)
• SSx S(X Mx)2
• b SP / SSx
• a My - bMx

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Example 16.14, p 557
Mx5
My6
SP S(X Mx)(Y My) SP 16
SSx S(X Mx)2 SP 10
b SP / SSx b 16 / 10 1.6
a My - bMx a 6 1.6(5) -2
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 gt 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 dichotomous
• Male vs. female
• Success 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)