MBA 8040 Assignment 3: Regression Analysis of BDNF and Exercise Data

This assignment presents a regression analysis exploring the relationship between exercise and BDNF (Brain-Derived Neurotrophic Factor) levels, considering the influence of gender and ethnicity. A scatterplot illustrates the positive correlation between exercise time and BDNF increase, with a linear trend line indicating the approximate increase in BDNF per minute of exercise. A multiple regression model was constructed, revealing the overall significance of the model (F=58.06, p<0.05). While gender was found to be insignificant, exercise and ethnicity (African and Asian) significantly predicted BDNF increase. The R-square value of 0.797 suggests that the independent variables explain nearly 80% of the variation in BDNF increase. The final regression model and interpretations of coefficients for ethnicity are provided, along with the adjusted R-square value, highlighting the importance of model fit and the potential impact of sample size and variable interactions.

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MBA 8040 Assignment 3: Regression

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1. Increase in BDNF is plotted on exercise in a scatterplot, and the plot is presented in Figure 1.
BDNF is found to have a positive increment with rise in exercise time. The correlation
between the two variables is found to be highly positive. Also, a linear relation is noted
between exercise and increase in BDNF. The linear trend line is plotted in the scatterplot.
The trend line indicated that for 1 minute increase in exercise BDNF will increase by
approximately by 0.72%. The coefficient of determination indicated that variation in
exercise time can explain almost 70% variation of BDNF increment, which is quite
remarkable.
Figure 1: Scatterplot presenting the linear relation between increase in BDNF and exercise time
2. The multiple regression model with prescribed dummy variables for gender and ethnicity has
been constructed in the excel file.
3. The regression model is overall significant ( F=58. 06 , p<0 . 05 ) as the p-value is less than 5%
level of significance, which signifies that the F-statistic is in the critical region. Hence, the
F-statistic is found to be statistically significant.
This means that the "F" and the "p" values show the total significance of the regression
model. In particular, they tested all null hypotheses with regression coefficients equal to
zero. It tests the complete model against a model without variables and uses the estimated

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value of the dependent variable as the average value of the dependent variable. The value of
F is the ratio of the sum of squares of the average regression divided by the sum of the
squares of mean error. The value of “p” is the probability that the null hypothesis of the
complete model is correct, where all regression coefficients are zero. This low value (p <
0.05) means that at least some of the regression parameters are not zero and the regression
equation has some cogency to match the data.
4. Gender is not a significant variable ( t=1. 52 , p=0 .133 ) in estimating the increase in BDNF at
5% level of significance. Males are assigned a dummy value = 1 and Females are assigned a
dummy value = 0 (baseline variable). This particular result implies that males do not have a
significantly higher increase in BDNF, compared to that of the females (controlling for
other predictors).
Exercise is a statistically significant predictor ( t=0 . 63 , p<0 . 05 ) at any level of significance.
It reflects the intensity of the positive impact which exercise exerts on increase in BDNF.
African people are noted to be significantly high ( t=6 . 81, p<0 . 05 ) in estimating increase in
BDNF compared to that of the Caucasian at 5% level of significance. Asian people are
noted to significantly ( t=12. 79 , p< 0 .05 ) in estimating increase in BDNF compared to that
of the Caucasian at any level of significance.
5. R-square or coefficient of determination value is 0.797. The implication of the value is that
deviation in the independent variables is able to explain approximately 79.7% variation in
the increase in BDNF. The value of R-square is acceptably high, which indicates that the
predictors are appropriate for estimating the increase of BDNF (Wiley, & Pace, 2015).

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6. The final regression model is BDNF=0. 638∗Exercise+6. 553∗African+12 . 860∗Asian−4 . 036 .
Caucasian ethnicity is taken as the baseline in the regression model (dummy value set to
zero). Hence, for 20 minutes of exercise, percentage increase in
BDNF=0. 638∗20−4 . 036=8. 72 will be noted for a Caucasian.
7. Coefficient of African in the model is 6.553 ( t =2. 5 , p <0 . 05 ) and that of the Asian is 12.86
( t=5 .11 , p< 0 .05 ) . Controlling for exercise and Asian ethnicity, percentage increase in
BDNF after exercise is noted to be approximately 6.55% greater in African people
compared that to the Caucasian people. Again, controlling for exercise and African
ethnicity, percentage increase in BDNF after exercise is noted to be approximately 12.86%
greater in Asian people compared that to the Caucasian individuals.
8. The adjusted R-square value is 0.779, and it is within 10% limit of the R-square value of
0.789. The adjusted R-square is used for the objective assessment of population dispersion,
which is explained by the regression equation. In general, increase in sample size decreases
the difference between the adjusted R-square and the expected R-square. In theory, this
happens as the expected R-square is less biased.
If sample size is increased for the final model, the standard error is reduced, but the model is
still affected by the omitted variables. Such a theoretical model is used to test assumptions,
and the best fit model is not always the appropriate model. No matter how small the
standard error is or how large the sample size is, incorrect regression model do not provide
the correct estimate.
In summary, increasing the sample size alone does not solve this problem. The model
should include the interaction of independent variables as predictors in the final regression
model (Hanley, 2016).

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References
Hanley, J. A. (2016). Simple and multiple linear regression: Sample size
considerations. Journal of Clinical Epidemiology, 79, 112-119.
doi:10.1016/j.jclinepi.2016.05.014
Wiley, J. F., & Pace, L. A. (2015). Chapter 14: Multiple Regression. In J. F. Wiley & L. A.
Pace (Eds.), Beginning R: An Introduction to Statistical Programming (pp. 139–161).
Berkeley, CA: Apress. https://doi.org/10.1007/978-1-4842-0373-6_14