Question

1. 23.38 Football and brain changes. American football is a contact sport that can expose players to repeated head impacts. While most impacts do not lead to a medically diagnosed concussion, an increasing body of evidence suggests that repetitive subconcussive head trauma can also result in neurocognitive deficits. Researchers recruited young football players (mean age, 12 years old). The players wore sensor-equipped helmets during all practice and game events for an entire football season to record subconcussion exposure (computed as an index value on a scale of 0 to 1). Each player also received a magnetic resonance imaging (MRI) scan at the beginning and at the end of the football season to measure changes in the brain white matter tract (computed as a percent of the original white matter volume). Here are the findings for 22 players:¹⁴

   Exposure	Change	Exposure	Change
   0.89	− 6.1	0.33	1.8
   0.05	0.0	0.11	1.9
   0.35	0.2	0.36	2.2
   0.37	− 1.2	0.08	2.3
   0.11	6.3	0.10	− 3.0
   0.19	− 4.6	0.06	8.4
   0.10	1.1	0.34	− 2.7
   0.45	1.3	0.12	3.0
   0.23	6.9	0.14	− 2.2
   0.46	1.7	0.21	− 1.4
   0.10	− 3.6	0.20	0.7

   1. Create a scatterplot of white matter change and subconcussion exposure, using subconcussion exposure as the explanatory variable. Describe the pattern.
   2. The researchers performed a linear regression analysis on the data. What is the equation for the least-squares regression line? What is the regression standard error? Test the null hypothesis of no relationship between subconcussion exposure and white matter change.
   3. One player had an unusually large subconcussion exposure. Obtain the equation for the least-squares regression line without this data point and test the null hypothesis of no relationship between subconcussion exposure and white matter change. What is the regression standard error?
   4. Use your work to describe the impact of the unusual observation on the regression analysis.

Answer 1

Regressing subconcussion exposure on white matter change (in Excel go to Data->Data Analysis->Regression and choose these as X and Y variables respectively), we obtain the following regression output:

Regression Statistics
Multiple R	0.368868063
R Square	0.136063648
Adjusted R Square	0.09286683
Standard Error	3.467466463
Observations	22
ANOVA
	df	SS	MS	F	Significance F
Regression	1	37.87170833	37.87170833	3.149853515	0.091157028
Residual	20	240.4664735	12.02332367
Total	21	278.3381818
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	2.272015257	1.201555964	1.890894244	0.073214557	-0.234386564	4.778417078	-0.234386564	4.778417078
Exposure	-6.912959936	3.895102237	-1.774782667	0.091157028	-15.03800083	1.212080954	-15.03800083	1.212080954

Hence, least squares regression line: Change (y) = -6.913 * Exposure(x) + 2.272

Regression std error (from above) = 3.467

The low p-value of 0.0912 ( < 0.1 ) of the Exposure coefficient, indicates that we can say with a confidence level of 90% (which has p-value of 0.1) that the Exposure is a significant predictor of white matter Change. Hence, the null hypothesis of no relationship between subconcussion exposure and white matter change can be rejected in favor of the alternative hypothesis at a confidence level of 10%.

Now, removing the player with Exposure of 0.89, we obtain the regression output as:

Regression Statistics
Multiple R	0.104318
R Square	0.010882
Adjusted R Square	-0.04118
Standard Error	3.47108
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2.518586244	2.518586244	0.209039146	0.652706126
Residual	19	228.919509	12.04839521
Total	20	231.4380952
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.475869	1.451936943	1.016482993	0.322169391	-1.563069738	4.514808	-1.56307	4.514808
Exposure	-2.66665	5.832463125	-0.45720799	0.652706126	-14.87413436	9.540837	-14.8741	9.540837

The std error stays nearly the same as 3.47 as seen above.

Looking at the high p-value of 0.65 for the slope coefficient, we can say that Exposure is no more a good predictor of Change in white matter. Hence, we lost the precision when we removed the big exposure data point.

Also, the Exposure coefficient has come down from -6.91 to -2.67, which suggests that the Change in white matter doesn't actually change as fast when we remove the outlier data points. Hence the outlier tended to enhance the degree of linear relationship between the two variables much beyond what it should normally be.