2 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. Figure1: 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
3 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).
4 6.The final regression model isBDNF=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,for20minutesofexercise,percentageincreasein BDNF=0.638∗20−4.036=8.72will 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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5 References Hanley,J.A.(2016).Simpleandmultiplelinearregression:Samplesize considerations.JournalofClinicalEpidemiology,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