Question

Some female psychology students were investigating whether intelligence depends on brain size. They each took a standard test that measured verbal IQ and also underwent an MRI scan to measure their brain size. The resulting data are in the file IQBrain.csv. The variables are

IQ Result of the verbal IQ test. IQ is measured on an artificial index scale, usually meant to have an average value of 100.

BrainV Brain volume measured in thousands of ‘pixels’. The size of a pixel derives from the two-dimensional resolution of the MRI and the spacing of image ‘slices’. It can vary between scanners (and even between scans on the same scanner). The pixel size is the same for all scans in this study, but the standard volumetric equivalent (e.g., millilitres) is not known.

IQ	BrainV
132	816.932
132	951.545
90	928.799
136	991.305
90	854.258
129	833.868
120	856.472
100	878.897
71	865.363
132	852.244
112	808.02
129	790.619
86	831.772
90	798.612
83	793.549
126	866.662
126	857.782
90	834.344
129	948.066
86	893.983

(a) Obtain the following R output, treating IQ as the response variable:

1. (i) scatter plot [3]

2. (ii) standardised residual plot [2]

(iii) distributionofresiduals [3] (iv) regressionanalysis(summaryoflinearmodel). [2,total=10]

(b) What statistical model is assumed in your regression analysis?

[2]

(c) As far as possible, assess the extent to which the model is appropriate to the data. [4]

4. (d) Leaving aside any problems identified in part (c), evaluate the evidence that verbal IQ is related to brain volume. Use a 5% significance level, and describe any relationship you find.

   [3]

5. (e) Provide a 95% prediction interval for the verbal IQ of a female whose brain volume was 870,000 pixels. [4]

(f) One of the students showed the scatterplot to a friend who said, “Well, there is clearly a trend there. It looks like people with larger brains do have higher IQ.” Write a brief response to this comment.

Answer 1

a)

i) Read data into R as follows and run plot command:

> data = read.csv("input.csv") //assuming data is in input.csv file

> plot(x=data$BrainV, y=data$IQ)

800 850 900 950 data$BrainV dil

Carrying out regression analysis in R:

> model = lm(data$IQ ~ data$BrainV)

> summary(model)

Call:
lm(formula = data$IQ ~ data$BrainV)

Residuals:
Min 1Q Median 3Q Max
-38.718 -18.870 9.529 16.373 27.069

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 24.18350 76.38158 0.317 0.755
data$BrainV 0.09884 0.08837 1.119 0.278

Residual standard error: 21.53 on 18 degrees of freedom
Multiple R-squared: 0.06499,   Adjusted R-squared: 0.01305
F-statistic: 1.251 on 1 and 18 DF, p-value: 0.278

ii) Plot of std residuals:

> plot(model$residuals)

model$residuals 40 -30 -20 -10 0 10 20 0 8

iii) The residuals look evenly distributed around a central mean line.

iv) Summary of the regression analysis is shown before answer ii) above

b) Least squares regression analysis is the model used above

c) The high p-value of 0.278 of the coefficient of the predictor variable BrainV shows that the model is not quite appropriate for the data

d) As p-value of coefficient of the predictor variable BrainV > 0.05, it is not a significant predictor of IQ at the 5% significance level.

e) 95% prediction interval is given by:

n- 1)*s prediction interval, Pl-y+-tn-2, (a-a/2) Sy

where $\hat{y}$ = 0.09884 * 870 + 24.18350 = 110.1743

and t_{n-2, (1-alpha/2)} = t-score (two-tailed) for n-2 = 18 dof at 5% significance level = 2.101

Hence, PI = (63.5, 156.85)

f) The IQ seems higher at some lower as well as higher end of the brain sizes. Hence, there doesn't seem to be a clear increasing trend in IQ levels as Brain size increases. Hence, the comment is invalid.