Question

This is an exploratory problem intended to introduce the idea of curvilinear regression. Personally, I was a bit shocked to discover that multiple LINEAR regression is the main vehicle to calculate regressions for data with nonlinear relationships...sounds a bit counter-intuitive. However, if we think of the higher-power terms (quadratic, cubic, etc.) as distinct variables, the ideas work well together.

Here is a data set for students in a gifted program. The first score (X1=GPAX1=GPA) is the students’ math grade from last year, and the second score (Y=SATY=SAT) is their SAT-M score. As this is a non-representative group (when considering the population of all students taking math classes in high school), it is not unexpected to see range-restriction effects (generally all high performing, few lower performing representatives) or ceiling effects (maximum score on the SAT-M is 800). In data such as this, it is not uncommon to see non-linear trends.

GPA	SAT
3.2	760
3.8	775
3	760
2.8	745
4	770
3.5	760
3.1	760
3.2	770
3.3	765
3.5	765
3.5	755
3.3	760
3.6	765
2.9	750
2.1	725
3.2	765
3.4	770
3.8	765
2.2	720
2.8	760
2.8	755
3.6	755
3.6	770
3.5	765
3.4	770

Step 1: Copy the data into your prefered statistical software program. Change the variable names to GPA and SAT if need be. Before doing any analysis, look at a scatterplot of the data with GPA on the horizontal axis and SAT on the vertical axis. Be sure to note any trends.

The following includes information for Excel users. If you are not using Excel, please disregard.

Step 2: Run a regression (Data Analysis > Regression) with SAT as the X variable. Again, be sure to note what evidence supports the assumptions for a regression analysis. Report the regression equation and the requested statistics:

SAT=SAT=  +  ×GPA×GPA
(Report regression coefficients accurate to 3 decimal places.)

R2adj=Radj2=
(Report accurate to 3 decimal places.)

Step 3: Create a third variable called GPAsq (for squared GPA). In Excel, use a formula, something like =B1^2 and fill down the rest of the column.

Step 4: Run the quadratic regression by adding the independent variable GPAsq to the model. Report the regression equation and the requested statistics:

SAT=SAT=  +  ×GPA×GPA +  ×GPA2×GPA2
(Report regression coefficients accurate to 3 decimal places.)

R2adj=Radj2=
(Report accurate to 3 decimal places.)

Step 5: Notice how the adjusted coefficient of multiple determination changed from the bivariate regression to the quadratic (multiple) regression. The next step is to determine if this more complicated model is statistically significantly better than the more parsimonious linear model.

For the multiple regression model, what was the F-ratio and the resulting P-value?
Fmodel=Fmodel=
(Report accurate to 2 decimal places.)
P=P=
(Report accurate to 3 decimal places.)

Answer 1

Here, y = SAT, x = GPA. We input the given data in MS Excel and, as directed, use the "Regression" option under Data > Data Analysis to compute the regression equation and answer the given questions. The screenshot of the data and the output is given below.



From the scatterplot, we can say that there is an increasing trend.
The regression equation is coming out to be as: = 682.352 + 23.689 .
The adjusted R square value = 0.685.

Now, we introduce a third variable named "GPAsq" and then redo the calculation to find the new regression equation. The regression equation is: = 534.348 + 123.314 + (-16.331) .
The adjusted R square value = 0.801.
The F(model) value = 49.22 and the P value = 0.000.