# Applied Biostatistics, Exercises - Mathematics - 5, Exercises for Mathematical methods. The University of York

## Mathematical methods

Description: Correlation and regression
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University of York
Department of Health Sciences
Applied Biostatistics
Exercise: Correlation and Regression
Question 1
In a study of blood pressure during pregnancy and foetal growth, 209 healthy women having their
first pregnancy had 24 hour blood pressure readings taken in mid-pregnancy. The size of the baby
was recorded at birth. The abstract included the following: ‘It was found that a 5 mm Hg increase
in mean 24 hour diastolic blood pressure at 28 weeks’ gestation was associated with a 68 g (95% CI
3 - 132) decrease in birth weight . . . Maternal mean 24 hour diastolic blood pressure at 28 weeks’
gestation was also inversely associated with the infant’s ponderal index (weight/height
3
) at birth . . .
(P=0.06).’ (Churchill and Beevers 1996). (N.B.: weight/height
3
is the usual ponderal index for
infants, rather than weight/height
2
as is used for adults and older children.)
a) What method would be used to calculate the 68g per 5 mm Hg?
b) What assumptions would the method require?
c) What is meant by ‘increase’ and ‘decrease’ here? Do they mean that when a woman’s blood
pressure went down her baby’s weight went up?
Question 2
A general practice based study sought to find out if people’s ears increase in size as they get older.
206 patients were studied with ear size being assessed by the length of the left external ear from the
top to the lowest part. Measurements were made simply, using a transparent plastic ruler. The
relation between the patient’s age and ear length (see graph below) was examined by calculating a
regression equation.
Length of ear (mm)
Age of patient (years)
20 40 60 80 100
50
60
70
80
90
The mean age of the patients was 53.75 years (range 30 - 93) and the mean ear length was 675mm
(range 520 - 840mm). The linear regression equation was
ear length = 55.9 + 0.22 × age
with the 95% confidence interval for the b coefficient being 0.17 to 0.27. The author concluded
that ‘It seems therefore that as we get older our ears get bigger (on average by 0.22 mm a year)’
(Heathcote 1995).
a) What are the interpretations of the numbers 55.9 and 0.22 in the regression equation?
b) Are the assumptions about the data are required for the regression analysis satisfied here?
c) Are the conclusions justified by the data?
Question 3
The birth weights of 1,333 fifty-year-old Swedish men were traced through birth records. Adult
height and birth weight were significantly correlated (r = 0.22, P<0.001) (Leon et al., 1996).
a) What is meant by ‘correlated’ and ‘r = 0.22’?
b) What assumptions are required for the calculation of the P value?
c) What can we conclude about the relationship between adult height and birth weight?
References
Churchill, D. and Beevers, D.G. (1996) Differences between office and 24-hour ambulatory blood
pressure measurement during pregnancy. Obstetrics and Gynecology 88, 455-61.
Heathcote, J.A. (1995) Why do old men have big ears? British Medical Journal 311, 1668.
Leon, D.A., Koupilova, I., Lithell, H.O., Berglund, L., Mohsen, R., Vagero, D., Lithell, U.B., and
McKeigue, P.M. (1996) (Failure to realise growth potential in utero and adult obesity in relation to
blood pressure in 50 year old Swedish men. British Medical Journal 312, 401-6.
Questions from Martin Bland and Janet Peacock: Statistical Questions in Evidence-based Medicine,
Oxford University Press, Oxford, 2000.