Brain size and intelligence | Mathematics homework help

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Background: Is brain size a measure of intelligence? Brain size tends to vary with body size: for example, sperm whales and elephants have brains up to five times as massive as human brains. So across species, brain size is not a perfect measure of intelligence. And within species, the underlying organization (complexity of connections) and molecular activity of the brain are likely to be more directly associated with intelligence than mere size.

In this assignment, we will investigate relationships between physiological measures of the brain, and intelligence. Download and open the Excel workbook, MHA610_Week 5_Assignment_Brain_Data.xls. The workbook contains data on 20 youths, in rows two through 21. Eight variables (the columns) were recorded on each individual; the column headings are given in row one. The column headings are as follows:

IQ

the individual’s IQ

Order the birth order (1 = firstborn, 2 = not firstborn)
Pair marker for genotype
Sex gender, 1 = male, 2 = female
CCSA corpus callosum surface area (in cm2)
HC head circumference (in cm)
TOTSA total brain surface area (in cm2)
TOTVOL total brain volume (in cm3)
WEIGHT body weight (in kg)

        
The neuroanatomical measures CCSA, TOTSA, and TOTVOL were determined from magnetic resonance imaging (MRI) of the brains, followed by automated image analyses of the scans. The corpus callosum is a bundle of neural fibers beneath the cortex, connecting the left and right cerebral hemispheres of the brain; it is the communication highway between the two hemispheres. (The more lanes to the highway, the faster the traffic ought to flow.)

The following questions can be answered in Excel, StatDisk, or other statistics software you may have available.

  • Examine all of the pairwise correlations among the physiological measures CCSA, HC, TOTSA, TOTVOL, and WEIGHT. Which two variables have the strongest correlation? Report the correlation, and plot the scattergram for these two variables. Also, report the correlation and plot the scattergram for the two variables that have the weakest correlation.

 

  • Determine whether the physiological parameters CCSA, HC, TOTSA, TOTVOL, and WEIGHT are significant predictors of IQ. That is, run a sequence of univariate regressions, with IQ as the dependent variable, and the physiological parameters as the independent variables. Report the best univariate regression with statistics and a graph of the regression. Describe whether IQ can be accurately predicted from any of these brain measures individually or in combination.

BONUS. Power law distributions, that is, functional relationships between two variables in which one variable is roughly a power of the other, are often used to model physiological data. One of the oldest power laws, the square-cube law, was introduced by Galileo in the 1600’s: empirically, the square-cube law states that as a shape grows in size, its volume grows faster than its surface area. We shall investigate the square-cube law with two variables from our dataset, CCSA and TOTVOL. If CCSA varies with some power of TOTVOL, for example, CCSA = k * (TOTVOL) α (k is an unknown constant here), then a simple way of estimating the exponent α is via linear regression: take log(CCSA) as the dependent variable and log(TOTVOL) as the independent variable; the fitted regression coefficient (slope) is an estimate of the exponent. (Do you see why this is true?) Perform this linear regression, and report your results. Describe whether the regression coefficient is significantly different from 2/3. (The 2/3rd power law occurs often in nature.)

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