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=GPA) is the students’ math grade from last year, and the second score (Y=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
3.6	745
3.4	740
3.5	735
2.8	720
3.2	735
3.6	740
4	745
3.1	735
3.6	740
3	735
3.8	740
3.4	745
3.8	750
2.1	700
2.9	725
3.5	740
3.2	745
2.8	730
3.6	730
3.2	740
3.4	745
3.3	740
2.8	735
2.2	695
3.3	740

Step 1: Copy the data into your preferred 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=___+___×GPA
(Report regression coefficients accurate to 3 decimal places.)

R2adj=
(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=__+__ ×GPA + ___ ×GPA2
(Report regression coefficients accurate to 3 decimal places.)

R2adj=
(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=
(Report accurate to 2 decimal places.)
P=
(Report accurate to 3 decimal places.)

Answer 1

1)

2)

		SUMMARY OUTPUT
		Regression Statistics
	Multiple R	0.84951
	R Square	0.721668
	Adjusted R Square	0.709566
	Standard Error	7.06443
	Observations	25
	ANOVA
		df	SS	MS	F	Significance F
	Regression	1	2976.158	2976.158	59.63508	7.82E-08
	Residual	23	1147.842	49.90617
	Total	24	4124
		Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
	Intercept	656.447	10.24413	64.08029	1.97E-27	635.2554	677.6386	635.2554	677.6386
	GPA	24.15321	3.127691	7.722375	7.82E-08	17.68308	30.62333	17.68308	30.62333

SAT = 656.447 + 24.153*GPA

Adjusted R Square = 0.710
..............

3)

GPA^2	GPA	SAT
12.96	3.6	745
11.56	3.4	740
12.25	3.5	735
7.84	2.8	720
10.24	3.2	735
12.96	3.6	740
16	4	745
9.61	3.1	735
12.96	3.6	740
9	3	735
14.44	3.8	740
11.56	3.4	745
14.44	3.8	750
4.41	2.1	700
8.41	2.9	725
12.25	3.5	740
10.24	3.2	745
7.84	2.8	730
12.96	3.6	730
10.24	3.2	740
11.56	3.4	745
10.89	3.3	740
7.84	2.8	735
4.84	2.2	695
10.89	3.3	740

4)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.923376
R Square	0.852624
Adjusted R Square	0.839226
Standard Error	5.256076
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	2	3516.221	1758.11	63.63893	7.12E-10
Residual	22	607.7793	27.62633
Total	24	4124
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	500.8541	36.00674	13.91001	2.22E-12	426.1807	575.5276	426.1807	575.5276
GPA^2	-17.1422	3.877083	-4.42141	0.000216	-25.1827	-9.10158	-25.1827	-9.10158
GPA	128.804	23.78323	5.415747	1.94E-05	79.48056	178.1274	79.48056	178.1274

SAT = 500.854 + (-17.142)*GPA + 128.804*GPA^2

Adjusted R Square   0.839

.........

5)

ANOVA
	df	SS	MS	F	Significance F
Regression	2	3516.221	1758.11	63.63893	0.00000
Residual	22	607.7793	27.62633
Total	24	4124

F ratio = 63.64

p value = 0.000

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