Another concept related to correlation is linear regression. This procedure is used to derive the actual equation of the best fitting line through the points on a scatterplot. Regression also allows you to determine how well one variable can be used to predict another.
Below are SPSS outputs for each strength test (arm and strength) predicting each of the two performance measures (ratings and simulations). There are several numbers that are particularly noteworthy. First, the R-Square indicates the proportion of variance in the dependent variable explained by the independent variable. Thus, for predicting Ratings from Arm strength, you can see that the linear equation predicts .048 or approximately 5% of the variance in ratings. Next, the Standard Error indicates how far off you would be, on average, if you were to use the independent variable to predict scores on the dependent variable. Thus, if you used Arm strength scores you could predict ratings with an average error of 8.34 (on a 60-point scale).
The specific equation for the line of best fit can be derived from the numbers under the "B" column. The first number indicates the slope of the line (.089 for the first example) and the second number indicates the intercept (33.97 for the first example). Thus, one could get a predicted Ratings score by plugging in a person's Arm score into the equation:
The regression outputs for the other strength scores and performance measures are presented below.
Regression Equation of ARM predicting RATINGS
Multiple R .22128 R Square .04896 Adjusted R Square .04241 Standard Error 8.33922 Analysis of Variance DF Sum of Squares Mean Square Regression 1 519.16642 519.16642 Residual 145 10083.67246 69.54257 F = 7.46545 Signif F = .0071 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T ARM .089331 .032694 .221280 2.732 .0071 (Constant) 33.974907 2.665032 12.748 .0000
What is the predicted SIMS score given an ARM score of 110? 1.659752 3 2.02538 1.90677
Regression Equation of ARM predicting SIMS
Multiple R .68601 R Square .47061 Adjusted R Square .46696 Standard Error 1.22582 Analysis of Variance DF Sum of Squares Mean Square Regression 1 193.68606 193.68606 Residual 145 217.88128 1.50263 F = 128.89808 Signif F = .0000 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T ARM .054563 .004806 .686007 11.353 .0000 (Constant) -4.095160 .391745 -10.454 .0000
Regression Equation of GRIP predicting RATINGS
Multiple R .18326 R Square .03358 Adjusted R Square .02692 Standard Error 8.40639 Analysis of Variance DF Sum of Squares Mean Square Regression 1 356.07735 356.07735 Residual 145 10246.76153 70.66732 F = 5.03878 Signif F = .0263 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T GRIP .066090 .029442 .183257 2.245 .0263 (Constant) 33.724714 3.318697 10.162 .0000
What is the proportion of varance in the Ratings variable explained by the Grip variable? .03358 .1826 8.04639 .029442
Regression Equation of GRIP predicting SIMS
Multiple R .63985 R Square .40940 Adjusted R Square .40533 Standard Error 1.29474 Analysis of Variance DF Sum of Squares Mean Square Regression 1 168.49674 168.49674 Residual 145 243.07060 1.67635 F = 100.51412 Signif F = .0000 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T GRIP .045463 .004535 .639846 10.026 .0000 (Constant) -4.809675 .511141 -9.410 .0000
If we use the Grip variable to predict the SIMS variable, how far, on average, would the predicted value be from the actual value? .63985 .40940 1.29474 .045463