correlation. definition the linear correlation coefficient r measures the strength of the linear...
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Correlation
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Definition
The linear correlation coefficient r measures the strength of the linear relationship between paired x- and y- quantitative values in a sample.
We can often see a relationship between two variables by constructing a scatterplot.
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Scatterplots of Paired Data
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Scatterplots of Paired Data
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Requirements
1. The sample of paired (x, y) data is a random sample.
2. Visual examination of the scatter plot must confirm that the points approximate a certain pattern.
3. The outliers must be removed if they are known to be errors.
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Notation for the Linear Correlation Coefficient
n represents the number of pairs of data present.
denotes the addition of the items indicated.
x denotes the sum of all x-values.
x2 indicates that each x-value should be squared and then those squares added.
(x)2 indicates that the x-values should be added and the total then squared.
xy indicates that each x-value should be first multiplied by its corresponding y-value. After obtaining all such products, find their sum.
r represents linear correlation coefficient for a sample.
represents linear correlation coefficient for a population.
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nxy – (x)(y)
n(x2) – (x)2 n(y2) – (y)2r =
The linear correlation coefficient r measures the strength of a linear relationship between the paired values in a sample.
Formula
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Datax
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Example: Calculating r
Using the simple random sample of data below, find the value of r.
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Example: Calculating r - cont
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nxy – (x)(y)
n(x2) – (x)2 n(y2) – (y)2r =
61 – (12)(23)
4(44) – (12)2 4(141) – (23)2r =
-32
33.466 r = = -0.956
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Example: Calculating r - cont
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Properties of the Linear Correlation Coefficient r
1. –1 r 1
2. The value of r does not change if all values of either variable are converted to a different scale.
3. The value of r is not affected by the choice of x and y. Interchange all x- and y-values and the value of r will not change.
4. r measures strength of a linear relationship.
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Interpreting r : Explained Variation
The value of r2 is the proportion of the variation in y that is explained by the linear relationship between x and y.
For Example if r = 0.926, we get r2 = 0.857.
We conclude that 0.857 (or about 86%) of the variation in Y can be explained by the linear relationship between X and Y. This implies that 14% of the variation in Y cannot be explained by X
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Formal Hypothesis Test
We wish to determine whether there is a significant linear correlation between two variables.
H0: = (no significant linear correlation)
H1: (significant linear correlation)
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Test statistic:
Critical values:
Use Tables with degrees of freedom = n – 2
1 – r 2
n – 2
rt =
Test Statistic is t
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P-value: Use Tables with degrees of freedom = n – 2
Conclusion: If the absolute value of t is > critical value reject H0 and conclude that there is a linear correlation. If the absolute value of t ≤ critical value, fail to reject H0; there is not sufficient evidence to conclude that there is a linear correlation.
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Test Statistic is t(follows format of earlier chapters)
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SlideSlide 17
CovarianceMeasure of linear relationship between variables
If the relationship between the random variables is nonlinear, the covariance might not be sensitive to the relationship
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SlideSlide 18
Pearson’s Correlation Coeff.Pearson's correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviations:
The above formula defines the population correlation coefficient, commonly represented by the Greek letter ρ (rho). Substituting estimates of the covariances and variances based on a sample gives the sample correlation coefficient, commonly denoted r :
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SlideSlide 19
Pearson correlation coefficient
Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.
The Spearman correlation coefficient is often thought of as being the Pearson correlation coefficient between the ranked variables. In practice, however, a simpler procedure is normally used to calculate ρ. The n raw scores Xi, Yi are converted to ranks xi, yi, and the differences di = xi − yi between the ranks of each observation on the two variables are calculated
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SlideSlide 20Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.
A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear. In contrast, this does not give a perfect Pearson correlation
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SlideSlide 21Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.
When the data are roughly elliptically distributed and there are no prominent outliers, the Spearman correlation and Pearson correlation give similar values
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SlideSlide 22Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.
The Spearman correlation is less sensitive than the Pearson correlation to strong outliers that are in the tails of both samples