11. Correlation and Reliability PDF

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This document includes lecture note about correlation and reliability for PSY 571 class by Dr. Tate in Spring 2018....

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Bivariate Correlation (Pearson’s r) & Cronbach’s Alpha Looking for relationships between two variables & Internal Consistency for Measurement Tools May 9. 2018 - May 14 . 2018 I.

Bivariate Correlation

1. Reasons to Distinguishes Bivariate Correlation from Other Tests -

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With the bivariate correlation, there is a continuously called predictor (viz. IV) and a continuously scaled criterion (viz. DV)  The difference is the scaling of the predictor Nonetheless, we cannot argue for causality and we can only describe relationships between two variables Bivariate correlation is often called “zero-order” or “simple” because it ignores other variables besides the two being examined * When scales are incommensurate, two separate scales are used for the two variables (X and Y) and the variables have different meaning. So, we cannot use paired samples t-test for those variables.

2. Discussing Bivariate Correlation -

The two variables are discussed abstractly as variable-X (x-axis) and variable-Y (y-axis) The logic of the bivariate correlation is how X varies in relation to Y How X varies with Y is referred to as covariation and co-variability of X and Y

3. Characteristics of (All) Correlations 1) Direction - Direction in correlation refers to whether X and Y increase or decrease together or whether as one increases, the other decreases  In other words, it asks whether X and Y vary in the same direction, vary in opposite directions, or no direction at all; indexed by the sign (+, -, none) 1 Positive correlation (+): X and Y tend to move in same direction (both increase or both decrease) 2 Negative correlation (-): X and Y tend to move in opposite directions (one increases, the other decreases) 3 Zero correlation: X and Y move in no direction together * There is no predictable relationship and two variables are independent and in an uncorrelated relationship 2) Magnitude

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Ask how strongly (or closely) X and Y vary together; indexed by the numerals (from 0 to 1.00 absolute value) Correlations vary in the range between +1.00 and -1.00, and including 0 A correlation of zero means no relationship Correlations closer to +/-1.00 indicate stronger relationships  X and Y vary one-to-one

II. Pearson Correlation Coefficient 1. Pearson Correlation Coefficient (r) -

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This coefficient measures the magnitude and direction of the linear relationship between two variables  “Linear” means constant slope at every point across the measured ends of both variable * On the graph, scores are located perfectly on the line or scattered near the line enough to represent * Yerkes-Dodson’s law (Curve-Linear relationship) : r=0 Pearson r therefore measures how closely X and Y vary together (magnitude) and whether they vary in the same or opposite directions (or in no direction together)

2. The Logic of r -

The conceptual formula for the Pearson correlation coefficient (r):

Degree Degree  r= ¿ which X ∧Y vary together ¿ which X ∧Y vary separately ¿ ¿ Covariability of X ∧Y  r= Variability of X ∧Y Separately -

From the conceptual formula, for a perfect correlation, the degree to which X and Y covary will be identical to the degree to which they vary separately. (r = +/- 1.00) If the covariability of X and Y = 0, there is zero correlation (r = 0)

3. Sum of Products of Deviations -

We need the sum of products (SP) of deviations of deviations from the means in order to calculate covariation of X and Y SP is much like SS in that we are examining deviations, but SP gives us a measure of covariability of X and Y  Definitional formula of SP: SP = ∑ (X−M X )(Y −M Y )  Computational formula of SP: SP =

∑ XY

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∑ X∑ Y n

4. Calculation of Pearson’s r

Degree SP ( Degree of covariability )

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r=

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SS is calculated just as before but separately for variables X and Y

√ SS X SSY (¿ which X∧Y vary seperately )

5. Effect Size for Correlation: r 2 -

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Effect size for correlation is measured with the familiar r 2  Also called the “coefficient of determination”  It is still proportion of variability accounted for, this time for one variable in the other and vice versa  There is no direction of causality Cohen’s generalized criteria is used for interpretation in the absence of literature-specific information

6. Hypothesis Testing with Pearson r 1) Hypothesis - One can examine whether observed correlations are statistically significant (or different from a standard)  One estimates the population parameter - The logic of a hypothesis test with r  Null hypothesis: H 0 : ρ =0  There is no correlation in the population  Alternative hypothesis: H A : ρ≠ 0  There is a correlation in the population - There are critical r values, and observed values more extreme than critical r will reject H 0 2) Degrees of freedom - For Pearson r, df = n-2 *n = number of pairs *This is because there are two variables and two different meanings which will be combined 3) Power and Pearson’s r - The formula for calculating power for a correlation coefficient: δ = ρ √ n−1

7. Assumptions of the Pearson r as a NHST

1) Linearity of Relationship - The Pearson r is based on the idea that a linear relationships exists between the variables  If a non-linear relationship exists, - If linearity is violated, one has to do another test  The class of these non-linear tests is curvilinear correlation - Linearity is diagnosed with a scatter plot 2) Absence of Outliers - The Pearson r is based on deviations from means, so it is influenced by outliers - If outliers exist, then the Spearman rho() or Kendall’s tau () are better  These techniques convert all the data to ranks...