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Relationships, cont. There IS a relationship, but its not Linear. R=0.0, but that DOESN'T mean that the two variables are.
Typology: Lecture notes
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Correlation: Do you have a relationship? Between two Quantitative Variables (measured on Same Person) (1) If you have a relationship (p<0.05)? (2) What is the Direction (+ vs. -)? (3) What is the Strength (r: from –1 to +1)? Regression: If you have a Significant Correlation: How well can you Predict a subject’s y-score if you know their X-score (and vice versa) Are predictions for members of the Population as good As predictions for Sample members?
No Relationship: r=0.0 Y-scores do not have a Tendency to go up or down as X-scores go up You cannot Predict a person’s Y-value if you know his X- Value any better than if you Didn’t know his X-score Positive Linear Relationship: Y-scores tend to go up as X-scores go up
Coefficient of Determination – r 2
Square of r-value r 2
If the Coefficient of Determination between height and weight Is r 2 =0.3 (r=0.9): •30% of variability in peoples weight can be Related to their height •70% of the difference between people in their of weight Is Independent of their height •Remember: This does not mean that weight is partially Caused by height Arm and leg length have a high coefficient of Determination but a growing leg does not cause Your arm to grow
Quasi-Independent Variable:
Dependent Variable: Physical Endurance The fatter the field, the weaker the correlation r=-0. r=-0. r=-0. r=-0. r=-0. r=-0.50 r=-
Restriction of Range cases an artificially low (underestimated) value of r. E.G. using just high GRE scores represented by the open circles. Common when using the scores to determine Who is used in the correlational analysis. E.G.: Only applicants with high GRE scores get into Grad School.
Raw Scores Deviation Scores
X d i^ **d ix
100 100 - - 100 - - 100 100 X-bar= Y-bar= Y-bar= SUM SUM SUM 400 400 400
X d i^ **d ix
Creates a line of “Best Fit” running through the data Uses Method of Least Squares The smallest Squared Distances between the Points and The Line Y-hat = a +bX and y= a +bX-hat a=intercept b=slope The Regression Line (line of best fit) give you a & b Plug in X to predict Y, or Y to predict X
Method of Lest Squares: •Minimizes deviations from regression line •Therefore, minimizes Errors of Prediction
Using several measures to predict a measure or future measure Y-hat = a + b 1
1
2
3
4 •Y-hat is the Dependent Variable •X 1
2
3
4 are the Predictor (Independent) Variables College GPA-hat = a + b 1 H.S.GPA + b 2 SAT + b 3 ACT + b 4 HoursWork R = Multiple Correlation (Range: -1 - 0 - +1) R = Coefficient of Determination (R*R * 100; 0 - 100%) Uses Partial Correlations for all but the first Predictor Variable
The relationship (shared variance) between two variables when the variance which they BOTH share with a third variable is removed Used in multiple regression to subtract Redundant variance when Assessing the Combined relationship between the Predictor Variables And the Dependent Variable. E.G., H.S. GPA and SAT scores.
