Multiple Linear Regression (MLR)
More than likely we will be interested in the relationship between some response variable and more than one explanatory variable. This is because a single independent variable rarely fully explains some dependent variable. In these cases, rather than fitting a simple linear regression (SLR) model for each explanatory variable we can employ MLR, an extension of SLR, to assess these relationships in one statistical model.
In this example we will be using this data set originally provided by the Example 5-1 of the STAT 501 Regression Methods course course that includes data on participant’s performance IQ score (PIQ), brain size (Brain) measured from MRI scans and given as counts/10,000, and two variables on body size, Height in inches and Weight in pounds.
Plotting each of the variables against one another we can see that Height, Weight, and Brain each appear to have some positive relationship with one another, but what about with PIQ? There may be some relationships there, particularly with Brain, however it is difficult to tell for certain observationally. Using MLR we can determine what relationships, if any, these three physical features have with the PIQ scores.
To fit an MLR model in JMP we can select Analyze -> Fit Model and put PIQ in the Y box and Brain, Height, and Weight into the Construct Model Effects Box. We can leave all of the options to their defaults and go ahead and select Run to fit the model.
Within the print out of the model results, in the Analysis of Variance table we can see that the overall model has a statistically significant fit, from which in the Summary of Fit table we note that the r2 is 0.295 to indicate that 29.5% of the variance in PIQ scores is explained by the MLR model. Note that we are also given other model diagnostics such as the Residual by Predicted Plot, which indicates that there may be one possible outlier but otherwise the residual errors are apparently random.
Next, we can check the Parameter Estimates table to note that Brain and Height are statistically significant predictors of PIQ, and from their estimated coefficients under Estimate we see that those relationships are positive and negative, respectively. Conversely, Weight did not have a statistically significant relationship with PIQ scores. We can conclude that for every unit increase in brain size the expected PIQ score increases by 2.06 while every inch increase in height reduces the expected PIQ score by -2.73.