Regression

 

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:

Ratings = .089*Arm + 33.97

 

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
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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
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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
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