1. BPK 304W Correlation

2. Correlation Coefficient (r)
Correlation Coefficient (r) is a measure of association between two variables
Varies from -1 to +1
r is a ratio of variability in X to that of Y.
0 = no relationship;
1 = perfect relationship
Correlation

3. Correlation

4. Linear Fit High Correlation Coefficient does not mean a linear fit

5. Correlation does not mean causation
Spurious Correlations – coincidental correlation between two unrelated variables
A study of boys aged 6 to 18 years produced correlations of standing broad jump with other measures.
Which had the highest correlation?
Correlation

6. Range of the Data affects the Correlation Coefficient

7. Correlation coefficient depends upon the orientation of the two groups

8. Significance of the Correlation Coefficient
The critical value of the correlation coefficient is determined by the sample size
Bigger sample size = lower critical value of r
Statistical significance of r does not infer “practical significance”
Correlation

9. Degrees
Probability
of Freedom
0.05
0.01 1
.997
1.000 24
.388
.496 2
.950
.990 25
.381
.487 3
.878
.959 26
.374
.478 4
.811
.917 27
.367
.470 5
.754
.874 28
.361
.463 6
.707
.834 29
.355
.456 7
.666
.798 30
.349
.449 8
.632
.765 35
.325
.418 9
.602
.735 40
.304
.393 10
.576
.708 45
.288
.372 11
.553
.684 50
.273
.354 12
.532
.661 60
.250 13
.514
.641 70
.232
.302 14
.497
.623 80
.217
.283 15
.482
.606 90
.205
.267 16
.468
.590
100
.195
.254 17
.575
125
.174
.228 18
.444
.561
150
.159
.208 19
.433
.549
200
.138
.181 20
.423
.537
300
.113
.148 21
.413
.526
400
.098
.128 22
.404
.515
500
.088
.115 23
.396
.505
1,000
.062
.081
Table 2-4.2: Critical Values of the Correlation Coefficient

10. Coefficient of Determination R squared (r2)
The circle represents the total variance in the measure
Weight
75% unexplained
Correlation of
Weight with Arm Girth
r = 0.5, r2 = 0.25
Therefore 25% of the variance in weight is explained by arm girth
25%
Arm Girth

11. Correlations between all variables
Correlation Matrix
Correlations between all variables
Weight vs Arm Girth
r = 0.5, r2 = 0.25
Weight vs Calf Girth
r = 0.6 , r2 = 0 .36
Arm Girth vs Calf Girth
r = 0.4 , r2 = 0 .16
Weight
Arm Girth
Calf Girth

12. BPK 304W Regression

13. Prediction Can we predict one variable from another?
Linear Regression Analysis
Y = mX + c
m = slope; c = intercept
Regression

14. Linear Regression Correlation Coefficient (r) how well the line fits
Standard Error of Estimate (S.E.E.)
how well the line predicts
Regression

15. Least Sum of Squares Curve Fitting
(Residual)
Regression

16. Assumptions about the relationship between Y and X
For each value of X there is a normal distribution of Y from which the sample value of Y is drawn
The population of values of Y corresponding to a selected X has a mean that lies on the straight line
In each population the standard deviation of Y about its mean has the same value

17. Standard Error of Estimate
measure of how well the equation predicts Y
has units of Y
true score 68.26% of time is within plus or minus 1 SEE of predicted score
Standard deviation of the normal distribution of residuals
Regression

18. Right Hand L. = 0.99Left Hand L. + 0.254 r = 0.94 S.E.E. = 0.38cm
Regression

19. How good is my equation? Regression equations are sample specific
Cross-validation Studies
Test your equation on a different sample
Split sample studies
Take a 50% random sample and develop your equation then test it on the other 50% of the sample
Regression

20. Multiple Regression More than one independent variable
Y = m1X1 + m2X2 + m3X3 …… + c
Same meaning for r, and S.E.E., just more measures used to predict Y
Stepwise regression
variables are entered into the equation based upon their relative importance
Regression

21. Building a multiple regression equation
X1 has the highest correlation with Y, therefore it would be the first variable included in the equation.
X3 has a higher correlation with Y than X2.
However, X2 would be a better choice than X3. to include in an equation with X1, to predict Y.
X2 has a low correlation with X1 and explains some of the variance that X1 does not. X3 Y X1 X2

22. Standardized Regression
The numerical value is of mn is dependent upon the size of the independent variable
Y = m1X1 + m2X2 + m3X3 …… + c
Variables are transformed into standard scores before regression analysis, therefore mean and standard deviation of all independent variables are 0 and 1 respectively.
The numerical value of zmn now represents the relative importance of that independent variable to the prediction
Y = zm1X1 + zm2X2 + zm3X3 …… + c
Regression