Correlations. Distinguishing Characteristics of Correlation Correlational procedures involve one...
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Transcript of Correlations. Distinguishing Characteristics of Correlation Correlational procedures involve one...
CorrelationsCorrelations
Distinguishing Distinguishing Characteristics of Characteristics of CorrelationCorrelation Correlational procedures involve Correlational procedures involve
one sample containing all pairs of one sample containing all pairs of X and Y scoresX and Y scores
Neither variable is called the IV or Neither variable is called the IV or DVDV
Use the individual pair of scores Use the individual pair of scores to create a scatterplotto create a scatterplot
Correlation CoefficientCorrelation Coefficient
Describes three characteristics of Describes three characteristics of the relationship:the relationship:
1.1.DirectionDirection
2.2.FormForm
3.3.DegreeDegree
What Is A Large What Is A Large Correlation? Correlation?
Guidelines:Guidelines:– 0.00 to <±.30 – low0.00 to <±.30 – low– ±.30 to <±.50 – moderate±.30 to <±.50 – moderate– >±.50 – high>±.50 – high
While 0 means no correlation at all, While 0 means no correlation at all, and 1.00 represents a perfect and 1.00 represents a perfect correlation, we cannot say that .5 is correlation, we cannot say that .5 is half as strong as a correlation of half as strong as a correlation of 1.01.0
Pearson CorrelationPearson Correlation
Used to describe the linear Used to describe the linear relationship between two variables relationship between two variables that are both interval or ratio variablesthat are both interval or ratio variables
The symbol for Pearson’s correlation The symbol for Pearson’s correlation coefficient is coefficient is rr
The underlying principle of The underlying principle of rr is that it is that it compares how consistently each Y compares how consistently each Y value is paired with each X value in a value is paired with each X value in a linear mannerlinear manner
Calculating Pearson rCalculating Pearson r
There are 3 main steps to r:There are 3 main steps to r:1.1. Calculate the Sum of Products (SP)Calculate the Sum of Products (SP)
2.2. Calculate the Sum of Squares for X Calculate the Sum of Squares for X (SS(SSXX) and the Sum of Squares for Y ) and the Sum of Squares for Y (SS(SSYY))
3.3. Divide the Sum of Products by the Divide the Sum of Products by the combination of the Sum of Squarescombination of the Sum of Squares
1) Sum of Products1) Sum of Products To determine the degree to which To determine the degree to which
X & Y X & Y covarycovary (numerator) (numerator)– We want a score that shows all of the We want a score that shows all of the
deviation X & Y have deviation X & Y have in commonin common– Sum of ProductsSum of Products (also known as the (also known as the
Sum of the Sum of the Cross-productsCross-products))– This score reflects the This score reflects the shared shared
variabilityvariability between X & Y between X & Y– The degree to which X & Y deviate The degree to which X & Y deviate
from the meanfrom the mean together together
SP = ∑(X – MX)(Y – MY)
Sums of Product Sums of Product DeviationsDeviations Computational FormulaComputational Formula
n
YXXYPS
n in this formula refers to the number of pairs of scores
2) Sum of Squares X & 2) Sum of Squares X & YY
For the For the denominatordenominator, we need to take , we need to take into account the degree to which X & Y into account the degree to which X & Y vary separatelyvary separately
– We want to find all the variability that X & Y We want to find all the variability that X & Y do notdo not have in common have in common
– We calculate sum of squares We calculate sum of squares separatelyseparately (SS(SSXX and SS and SSYY))
– Multiply them and take the square rootMultiply them and take the square root
))(( YX SSSS
2) Sum of Squares X & 2) Sum of Squares X & YY The denominator:The denominator:
=
))(( YX SSSS
22 )()( YX MYMX
Hypothesis testing Hypothesis testing with rwith r Step 1) Set up your hypothesisStep 1) Set up your hypothesis
Step 2) Find your critical r-scoreStep 2) Find your critical r-score– Alpha and degrees of freedomAlpha and degrees of freedom
Hypothesis testing Hypothesis testing with rwith r Step 3) Calculate your r-obtainedStep 3) Calculate your r-obtained Step 4) Compare the r-obtained to r-Step 4) Compare the r-obtained to r-
critical, and make a conclusioncritical, and make a conclusion– If r-obtained < r-critical = fail to reject If r-obtained < r-critical = fail to reject
HoHo– If r-obtained > r-critical = reject HoIf r-obtained > r-critical = reject Ho
Coefficient Of Coefficient Of DeterminationDetermination
The squared correlation (rThe squared correlation (r22) measures the proportion of variability in the data that is explained by the relationship between X and Y
Coefficient of Non-Determination (1-Coefficient of Non-Determination (1-rr22): percentage of variance not ): percentage of variance not accounted for in Yaccounted for in Y
Correlation in Correlation in Research ArticlesResearch Articles
Coleman, Casali, & Wampold (2001). Adolescent strategies for coping with cultural diversity. Journal of Counseling and Development, 79, 356-362
Other Types of Other Types of CorrelationCorrelation
Spearman’s Rank CorrelationSpearman’s Rank Correlation– variable X is ordinal and variable Y is ordinalvariable X is ordinal and variable Y is ordinal
Point-biserial correlationPoint-biserial correlation– variable X is nominal and variable Y is variable X is nominal and variable Y is
intervalinterval Phi-coefficientPhi-coefficient
– variable X is nominal and variable Y is also variable X is nominal and variable Y is also nominal nominal
Rank biserialRank biserial– variable X is nominal and variable Y is ordinalvariable X is nominal and variable Y is ordinal
Example #2Example #2
Hours (X)Hours (X) Errors (Y)Errors (Y)
00 1919
11 66
22 22
44 11
44 44
55 00
33 33
55 55