WEEK SIX ASSIGNMENT MHA 610

RUNNING HEAD: Brain Size

Brain Size and Intelligence

Alban Evans

MHA 610 Introductions to Biostatistics (NDF1506A)

March 11, 2015

Instructor: Glenn Cummings

Regression

To determine if a variable is significant, we test the coefficients returned in the model at 5% level of confidence using the Student t-test. The hypothesis will be as follows:

H0: bi = o

Vs.

H1: bi ≠ 0

If the p value is greater than 0.05, we fail to reject the Null hypothesis and conclude that the coefficient has no effect on the dependent variable. If p-value is less than 0.05, we reject the null hypothesis and conclude that the variable is a significant predictor of the independent variable (Vittinghoff, 2012).

Model 1: IQ ~ CCSA

Regression Statistics
Multiple R	0.16
R Square	0.02
Adjusted R Square	-0.03
Standard Error	13.41
Observations	20

	Coefficients	Standard Error	t Stat	P-value
Intercept	85.22	23.70	3.59	0.00
CCSA	2.26	3.36	0.67	0.51

Model: IQ = 85.22 + 2.26CCSA

The CCSA has a p-value larger than 0.05. Therefore, the Corpus Callosum Surface Area is not a significant predictor of Intelligent Quotient.

Model 2: IQ ~ HC

Regression Statistics
Multiple R	0.14
R Square	0.02
Adjusted R Square	-0.04
Standard Error	13.44
Observations	20.00

	Coefficients	Standard Error	t Stat	P-value
Intercept	45.05	94.81	0.48	0.64
HC	1.00	1.69	0.59	0.56

HC has a p-value of 0.56(>0.05) hence it is not a significant predictor of IQ.

Model 2: 45.05 + HC

Model 3: IQ ~ TOTSA

Regression Statistics
Multiple R	0.29
R Square	0.08
Adjusted R Square	0.03
Standard Error	12.98
Observations	20.00

	Coefficients	Standard Error	t Stat	P-value
Intercept	142.96	32.61	4.38	0.00
TOTSA	-0.02	0.02	-1.29	0.21

Model: IQ = 142.96 – 0.02TOTSA

The coefficient TOTSA has a p value larger than 0.05 (0.21>0.05). Therefore, we conclude that total brain surface area is not significant predictor of Intelligent Quotient.

MODEL 4: IQ ~ TOTVOL

Regression Statistics
Multiple R	0.06
R Square	0.00
Adjusted R Square	-0.05
Standard Error	13.55
Observations	20.00

	Coefficients	Standard Error	t Stat	P-value
Intercept	108.55	28.17	3.85	0.00
TOTVOL	-0.01	0.02	-0.27	0.79

Model: 108.55 – 0.01TOTVOL

The p value has a value greater than 0.79 (0.79>0.05). Therefore, the total brain volume is not a significant predictor of Intelligent Quotient.

MODEL 5: IQ~WEIGHT

Regression Statistics
Multiple R	0.00
R Square	0.00
Adjusted R Square	-0.06
Standard Error	13.57
Observations	20.00

	Coefficients	Standard Error	t Stat	P-value
Intercept	101.14	12.46	8.11	0.00
WEIGHT	0.00	0.16	-0.01	0.99

MODEL: IQ = 101.14 where weight~ 0

The p-value is larger than 0.05 (0.99>0.05). We conclude that Weight is a significant predictor of Intelligent Quotient.

To get the best model we shall use the Multiple R which is a square root of the R-adjusted. It represents the correlation between the dependent variable and the regression model predictions (Carlberg, 2014). Therefore, the model with the highest Multiple R will be the best model. In our case, the model for this scenario is:

Model: IQ = 142.96 – 0.02TOTSA

The model has the highest multiple R of 0.29 and thus qualifies to be the best model.

Combined Model Output

Regression Statistics
Multiple R	0.417684
R Square	0.17446
Adjusted R Square	-0.12038
Standard Error	13.98339
Observations	20

	Coefficients	Standard Error	t Stat	P-value
Intercept	75.10	110.21	0.68	0.51
CCSA	3.29	5.31	0.62	0.55
HC	1.19	2.33	0.51	0.62
TOTSA	-0.03	0.02	-1.25	0.23
TOTVOL	-0.01	0.04	-0.16	0.88
WEIGHT	-0.02	0.17	-0.09	0.93

MODEL:

IQ = 75.10 + 3.29CCSA + 1.19HC -0.03TOTSA – 0.01TOTVOL – 0.02WEIGHT

The model has a multiple R of 0.42. Compared to the uni-variate model, it is higher. Therefore, the combined model is much better than the uni-variate models. The prediction is much better as a combination than individual.

Testing the Square cube law

Regression output

ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.029327	0.029327	17.15072	0.000613163
Residual	18	0.030779	0.00171
Total	19	0.060105

	Coefficients	Standard Error	t Stat	P-value
Intercept	-1.76	0.63	-2.80	0.01
log(TOTVOL)	0.85	0.21	4.14	0.00

Model: log (CCSA) = -1.76 + 0.85 log (TOTVOL)

α = Exponent (-1.76) =0.17

This implies that for increasing values of Total Volume, the surface area decreases which agrees with the square-cube law.

From the regression output (ANOVA) the p value is less than 0.05 thus we conclude that the coefficient is significantly different from 2/3.

References

Carlberg, C. (2014). Statistical Analysis: Microsoft Excel 2013. New York: Que Publishing. Cheng-few lee, J. C. (2013). Statistics for Business and Financial Economics. New York: Springer Science & Business Media. Fox, J. (2002). An R and S -Plus companion to Applied Regression. New York: SAGE Publications. Hucker, K. (2001). Research Methods in Health, Care and Early Years. Chicago: Heinemann. Jones, D. S. (2002). Pharmaceutical Statitics. New York: Pharmaceutical Press. Vittinghoff, E. (2012). Regression in Biostatitics: Linear, Logistic, Survival, and Repeated Measures Models . New York : Springer Science & Business Media.