The following data set reflects a within-subjects design with two outcome variables. The data are presented in a format suitable for entry into statistical software.
| Outcome1 | Outcome2 | |
|---|---|---|
| 1 | .00 | 4.00 |
| 2 | .00 | 7.00 |
| 3 | 3.00 | 4.00 |
| 4 | 5.00 | 9.00 |
The following tables represent typical output from statistical software. Options, labels, and layout vary from program to program.
The table of descriptive statistics can be used to assist in calculating the correlation.
| Mean | Std. Deviation | N | |
|---|---|---|---|
| Outcome1 | 2.000 | 2.449 | 4 |
| Outcome2 | 6.000 | 2.449 | 4 |
The table of inferential statistics shows the key elements to be calculated.
| Model | R | R^2 |
|---|---|---|
| 1 | 0.500 | 0.250 |
| Predictor | Estimate | SE | t | p | Std. Estimate |
|---|---|---|---|---|---|
| Intercept | 5.000 | 1.785 | 2.801 | 0.107 | |
| Outcome1 | 0.500 | 0.612 | 0.816 | 0.500 | 0.500 |
Sum of Cross Products: The Sum of Cross Products (SCP) is not easily determined solely from the summary statistics of the output, but rather from the data.
\[SCP = \sum ( X - M_X ) ( Y - M_Y ) = ( 0 - 2.000 ) ( 4 - 6.000 ) + ( 0 - 2.000 )( 7 - 6.000 ) + ( 3 - 2.000 )( 4 - 6.000 ) + (5 - 2.000)(9 - 6.000) = 9.000\]
Covariance: The Covariance is a function of the Sum of Cross Products and the sample size:
\[COV = \frac{SCP}{(N - 1)} = \frac{9.000}{(4 - 1)} = 3.000\]
Unstandardized Regression Coefficients: The Unstandardized Regression Coefficients involve Covariance, the Standard Deviations of the variables, and the Means of the variables:
\[B_1 = \frac{COV}{(SD)^2} = \frac{3.000}{(2.449)^2} = 0.500\]
\[B_0 = M_Y - (B_1)(M_X) = 6.000 - (0.500)(2.000) = 5.000\]
Standardized Regression Coefficients: The Standard Regression Coefficients involve the Regression Coefficient (for the predictor) and the Standard Deviations of the variables:
\[\beta_1 = (B_1)\frac{SD_X}{SD_Y} = (0.500)\frac{2.449}{2.449} = 0.500\]
Multiple Correlation: In bivariate regression, the Multiple Correlation is the same as the bivariate Correlation:
\[r = \frac{COV}{(SD_X) (SD_Y)} = \frac{3.000}{(2.449) (2.449)} = .500\]
Proportion of Variance Accounted For: The Proportion of Variance Accounted For is a function of the Correlation:
\[R^2 = .0500^2 = 0.250\]