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Exercise: Head circumference, sex, and height

We have already seen that male students had a significantly bigger mean head circumference
than female students (Week 4 Exercise: Head circumference)
and also that, for females, head circumference and height were related
(Week 7 Exercise: Head circumference and height).
Could it be that the difference in head circumference between the sexes is explained
by the difference in height?

The following graph shows a plot of head circumference against height for males and females:

d

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Questions

1. What method could we use to answer the question:
is the difference in head circumference between the sexes explained by the difference in height?

Check suggested answer 1.

The regression equation is

Head (mm) = 396 + 0.0957 × height (mm) + 6.80 × sex

where sex = 1 for a female, 2 for a male.

2. What does 6.80 tell us?

Check suggested answer 2.

We can add confidence intervals and P values for the coefficients in this equation:

Head (mm) = 396
+ 0.0957 × height (mm) + 6.80 × sex

95% CI 334 to 459
0.0555 to 0.1360 -1.1 to 14.6

P<0.001
P<0.001
P=0.09

3. What can we conclude from this analysis?

Check suggested answer 3.

4. What assumptions about the sample and distributions are needed for this analysis
and what could we do to check them?

Check suggested answer 4.

These are the histogram and Normal plot for the residuals, as shown in Answer 4:

d

5. What can we conclude from these distribution graphs?

Check suggested answer 5.

This is the scatter plot of the residuals against the value predicted by the regression equation,
as shown in Answer 4:

d

6. What can we conclude from the scatter diagram?

Check suggested answer 6.

7. What other assumptions are we making about the relationship between head circumference,
height and sex?

Check suggested answer 7.

8. How could we check the assumption of linearity?

Check suggested answer 8.

The regression with the height squared term added is:

Head (mm) = 94.7 + 0.449×height (mm)
- 0.000104×height squared (mm^{2})
+ 7.23×sex

95% CI -621.9 to 811.4
-0.389 to 1.287
-0.000349 to 0.000142
-0.70 to 15.12

P=0.8
P=0.3
P=0.4

9. If we have only height in the regression model, height is a highly significant predictor.
If we have both height and height squared, neither term is significant.
Why is this?
What can we do about it?

Check suggested answer 9.

10. What can we conclude about the assumption of linearity?

Check suggested answer 10.

11. How could we check the assumption of no interaction between
the effects of sex and height?

Check suggested answer 11.

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Last updated: 20 March, 2007.

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