chapter 15 correlation and regression
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Chapter 15 Correlation and Regression. PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau. Chapter 15 Learning Outcomes. Concepts to review. Sum of squares (SS) (Chapter 4) Computational formula - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 15Correlation and Regression
PowerPoint Lecture Slides
Essentials of Statistics for the Behavioral Sciences Seventh Edition
by Frederick J. Gravetter and Larry B. Wallnau
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Chapter 15 Learning Outcomes
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Concepts to review
• Sum of squares (SS) (Chapter 4)– Computational formula– Definitional formula
• z-Scores (Chapter 5)
• Hypothesis testing (Chapter 8)
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15.1 Introduction to Correlation and Regression
• Measures and describes a relationship between two variables.
• Characteristics of relationships– Direction (negative or positive)– Form (linear is most common)– Strength
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Figure 15.1 Scatterplot for correlational data
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Figure 15.2 Examples of positive and negative relationships
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Figure 15.3 Examples of different values
for linear relationships
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15.2 The Pearson Correlation
• Measures the degree and the direction of the linear relationship between two variables
• Perfect linear relationship – Every change in X has a corresponding
change in Y– Correlation will be –1.00 or +1.00
y separatelY and X of variablity
Y and X ofity covariabilr =
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Sum of Products (SP)
• Similar to SS (sum of squared deviations)
• Measures the amount of covariability between two variables
∑ −−= ))(( YX MYMXSP
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SP – Computational formula
• Definitional formula emphasizes SP as the sum of two difference scores
• Computational formula results in easier calculations
n
YXXYSP ∑ ∑∑ −=
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Calculation of the Pearson correlation
• Ratio comparing the covariability of X and Y (numerator) with the variability of X and Y separately (denominator)
YX SSSS
SPr =
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Figure 15.4 Example 15.3 Scatterplot
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Pearson Correlation and z-scores
• Pearson correlation formula can be expressed as a relationship of z-scores.
1
1
−=
−=
∑
∑
n
zzr :Population
n
zzr :Sample
YX
YX
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Learning Check
• A scatterplot shows a set of data points that are clustered loosely around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data?
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Learning Check
• A scatterplot shows a set of data points that are clustered loosely around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data?
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Learning Check TF
• Decide if each of the following statements is True or False.
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Answer TF
20402010
)20)(20(20 −=−=−=SP
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15.3 Using and Interpreting the Pearson Correlation
• Correlations used for prediction
• Validity
• Reliability
• Theory verification
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Figure 15.5 Number of churches and number of serious crimes
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Interpreting correlations
• Correlation does not demonstrate causation
• Value of correlation is affected by the range of scores in the data
• Extreme points – outliers – have an impact
• Correlation cannot be interpreted as a proportion.– To show the shared variability, need to square
the correlation
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Figure 15.6 Restricted range and correlation
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Figure 15.7 Influence of outlier on correlation
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Coefficient of determination
• Coefficient of determination measures the proportion of variability in one variable that can be determined from the relationship with the other variable.
2rionDeterminat of oefficientC =
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Figure 15.8 Three degrees of linear relationship
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15.4 Hypothesis Testing with the Pearson Correlation
• Pearson correlation is usually computed for sample data, but used to test hypotheses about the relationship in the population.
• Population correlation shown by Greek letter rho (ρ)
• Nondirectional: H0: ρ = 0 and H1: ρ ≠ 0
• Directional: H0: ρ ≤ 0 and H1: ρ > 0
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Figure 15.9 Correlation of sample and population
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Hypothesis Test for Correlations
• Sample correlation used to test population ρ• Degrees of freedom (df) = n – 2• Hypothesis test can be computed using
either t or F.• Critical Values have been computed
– See Table B.6– A sample correlation beyond ± Critical Value
is very unlikely– A sample correlation beyond ± Critical Value
leads to rejecting the null hypothesis.
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Partial correlation
• A partial correlation measures the relationship between two variables while controlling the influence of a third variable by holding it constant
)1)(1(
)(22yzxz
yzxyxyzxy
rr
rrrr
−−
−=⋅
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Figure 15.10 Controlling the impact of a third variable
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15.5 Alternative to the Pearson Correlation
• Pearson correlation has been developed– for linear relationships– for interval or ratio data
• Other correlations have been developed for– non-linear data– other types of data
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Spearman correlation
• Pearson correlation formula is used with data from an ordinal scale (ranks)– Used when both variables are measured on
an ordinal scale– Used when relationship is consistently
directional but may not be linear
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Figure 15.11 Consistent nonlinear positive relationship
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Figure 15.12 Scatterplot showing scores and ranks
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Ranking tied scores
• Tie scores need ranks for Spearman correlation
• Method for assigning rank– List scores in order from smallest to largest– Assign a rank to each position in the list– When two (or more) scores are tied, compute
the mean of their ranked position, and assign this mean value as the final rank for each score.
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Special formula for the Spearman correlation
• The ranks for the scores are simply integers
• Calculations can be simplified– Use D as the difference between the X rank
and the Y rank for each individual to compute the rs statistic
)1(
61
2
2
−−= ∑
nn
Drs
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Point-Biserial Correlation
• Measures relationship between two variables– One variable has only two values
(dichotomous variable)
• Same situation as the independent samples t-test in Chapter 10– Point-biserial r2 has same value as the r2
computed from t-statistic– t-statistic evaluates the significance – r statistic measures its strength
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Phi Coefficient
• Both variables (X and Y) are dichotomous– Both variables are re-coded to values 0 and 1– The regular Pearson formulas is used
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Learning Check
• Participants were classified as “morning people” or “evening people” then measured on a 50-point conscientiousness scale. Which correlation should be used to measure the relationship?
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Learning Check - Answer
• Participants were classified as “morning people” or “evening people” then measured on a 50-point conscientiousness scale. Which correlation should be used to measure the relationship?
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Learning Check
• Decide if each of the following statements is True or False.
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Answer
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15.6 Introduction to Linear Equations and Regression
• The Pearson correlation measures a linear relationship between two variables
• The line through the data– Makes the relationship easier to see– Shows the central tendency of the relationship– Can be used for prediction
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Figure 15.13 Regression line
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Linear equations
• General equation for a line– Equation: Y = bX + a– X and Y are variables– a and b are fixed constant
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Figure 15.14 Graph of a linear equation
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Regression
• Regression is the method for determining the best-fitting line through a set of data– The line is called the regression line
• Ŷ is the value of Y predicted by the regression equation for each value of X
• (Y- Ŷ) is the distance of each data point from the regression line: the error of prediction
• Regression minimizes total squared error
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Figure 15.15 Distance between data point & the predicted point
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Regression equations
• Regression line equation: Ŷ = bX + a
• The slope of the line, b, can be calculated
• The line goes through (MX,MY) so
X
Y
X s
sr or
SS
SPb =
XY bMMa −=
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Figure 15.16 X and Y points and regression line
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Figure 15.17 Perfectly fit regression line and regression line for Example 15.13
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Correlation and the standard error
• Predicted variability in Y scores:SSregression = r2 SSY
• Unpredicted variability in Y scores: SSresidual = (1 - r2) SSY
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Standard error of estimate
• Regression equation makes prediction
• Precision of the estimate is measured by the standard error of estimate
2
)ˆ( 2
−
−= ∑
n
YY
dfSSresidual
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Testing significance of regression
• Analysis of Regression– Similar to Analysis of Variance– Uses an F-ratio of two Mean Square values– Each MS is a SS divided by its df
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Mean squares and F-ratio
residual
residualresidual df
SSMS =
regression
regressionregression df
SSMS =
residual
regression
MS
MSF =
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Figure 15.18 Partitioning of the SS and df in Analysis of Regression
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Figure 15.19 Plot of data in Demonstration 15.1
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Learning Check
• A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7?
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Learning Check - Answer
• A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7?
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Learning Check
• Decide if each of the following statements is True or False.
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Answer
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