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. Perform the linear regression. Explain your results in terms of the hypotheses. (no need to write equation)
   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

Scatter plot is given as

There is weak negative  relationship exists

Because value of r=-0.30 which is negative

Let us perform the regression

Regressing subconcussion exposure on white matter change with below mentioned steps in excel

Data->Data Analysis->Regression and choose these as X and Y variables respectively>Ok

SUMMARY 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	37.87171	3.149854	0.091157025
Residual	20	240.4664735	12.02332
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	1.890894	0.073215	-0.234386557	4.778417071	-0.234386557	4.778417071
X Variable 1	-6.912959936	3.895102237	-1.77478	0.091157	-15.0380008	1.212080931	-15.0380008	1.212080931

Here X=exposure & Y=change considered as variable

So least squares regression line will be

Change (Y) = -6.913 *Exposure(X) + 2.272

Regression std error= 3.467

The low p-value of 0.0912 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 exposure and white matter change can be rejected in favor of the alternative hypothesis at a confidence level of 10%.

Let us remove the outlier 0.89 then regression output will be

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.104318418
R Square	0.010882332
Adjusted R Square	-0.041176492
Standard Error	3.471079833
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2.518586244	2.518586	0.209039	0.652706126
Residual	19	228.919509	12.0484
Total	20	231.4380952
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.475869209	1.451936943	1.016483	0.322169	-1.563069732	4.514808149	-1.563069732	4.514808149
X	-2.66664874	5.832463125	-0.45721	0.652706	-14.87413433	9.54083685	-14.87413433	9.54083685

Standard error is same

But the p value is 0.65

Exposure is not good predictor of Change in white matter.

Also, the Exposure coefficient has come down from -6.91 to -2.67,

which indicates  that the Change in white matter will not actually change as fast

when we will eliminate the outlier data points.

the Outlier will tend to increase the correlation between the two variables.