Question

Is there a relationship between brain size and intelligence? The Full Scale IQ scores (FSIQ) and brain sizes (in pixels, as measured by MRI scans) of 39 subjects were measured. Researchers wished to study the relationship between FSIQ and brain size, using brain size to predict FSIQ. However, the researchers believed that brain size is also dependent on body size and that some adjustment for body size might be necessary in order to understand the relation between brain size and intelligence. Therefore, the researchers also measured the heights (in inches) of the 39 subjects and used height as a measure of body size. They then used a multiple regression model to predict FSIQ from brain size and height. They obtained the following results. S = 20.95 R-square = 0.271 The researchers assume that the statistical model for the relation between FSIQ, brain size, and height is the multiple linear regression model FSIQi = β0 + β1(brain size)i + β2(height)i + εi for i = 1, 2, ... , 39. The deviations εi are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. The researchers decided to test the hypotheses H0: β1 = 0 , Ha: β1 > 0. The P-value of the test is

between 0.01 and 0.05. greater than 0.05. less than 0.01.

Is there a relationship between brain size and intelligence? The Full Scale IQ scores (FSIQ) and brain sizes (in pixels, as measured by MRI scans) of 39 subjects were measured. Researchers wished to study the relationship between FSIQ and brain size, using brain size to predict FSIQ. However, the researchers believed that brain size is also dependent on body size and that some adjustment for body size might be necessary in order to understand the relation between brain size and intelligence. Therefore, the researchers also measured the heights (in inches) of the 39 subjects and used height as a measure of body size. For simplicity, the researchers divided subjects into two groups depending on whether their FSIQ was below or above average. Subjects with FSIQ scores that were below average were coded as 0, and those with FSIQ scores that were above average were coded as 1. The coded FSIQ values were then used as the response variable. To analyze these data the researchers should

use multiple regression to predict the coded FSIQ values using both brain size and height as predictor variables. use simple linear regression to predict the coded FSIQ values using only brain size as a predictor variable. use logistic regression.

Answer 1

Case 1.

The multiple linear regression model

FSIQ_i = β₀ + β₁(brain size)_i + β₂(height)_i + ε_i

for i = 1, 2, ... , 39. The deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. The researchers decided to test the hypotheses H₀: β₁ = 0 , H_a: β₁ > 0. The P-value of the test is between 0.01 and 0.05.
At alpha=0.05, the p-value is less than 0.05 significance level. Hence, there is a relationship between brain size and intelligence.

Case 2.

For simplicity, the researchers divided subjects into two groups depending on whether their FSIQ was below or above average. Subjects with FSIQ scores that were below average were coded as 0, and those with FSIQ scores that were above average were coded as 1. The coded FSIQ values were then used as the response variable. To analyze these data the researchers should Ans: use logistic regression.

Note: Now, the dependent variable FSIQ is become a binary. Hence, FSIQ does not have a linear relationship with the slopes of the. Hence, linear regression model can not apply.