# Exercise: Regression analyses

## Why do old men have big ears?

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.

The mean age of the patients was 53.75 years (range 30 - 93) and the mean ear length was 67.5 mm (range 52.0 - 84.0 mm). 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)

### Questions about 'Why do old men have big ears?'

1. What is a regression equation? What are the interpretations of the numbers 55.9 and 0.22 in the regression equation?

2. Can we conclude that the mean ear size at birth is 55.9 mm?

3. What assumptions about the data are required for regression analysis and do you think they satisfied here?

4. What are the conclusions and are they justified by the data?

## Physical fitness in children

In a study of physical fitness and cardiovascular risk factors in children, blood pressure and recovery index (post exercise recovery rate, an indicator of fitness) were measured (Hoffman and Walter 1989). Multiple regression was used to look at the relationship between systolic blood pressure and recovery index, adjusted for age, race, area of residence and ponderal index (wt/ht2). For the boys, the adjusted regression coefficient of systolic blood pressure on recovery index was given as follows: b = –0.086, SE(b) = 0.039, 95% CI = –0.162 to –0.010.

### Questions about physical fitness in children

1. What is meant by ‘multiple regression analysis’?

2. What is meant by the terms ‘b’, ‘SE(b)’ and ‘95% CI’?

3. What assumptions about the variables are required for these analyses to be valid?

4. Why was the regression adjusted and what does this mean?

5. What would be the effect of adjusting for race if systolic blood pressure were related to race and recovery index were not?

6. Why was the regression adjusted and what does this mean? What would be the effects of adjusting for ponderal index if blood pressure and recovery index were both related to ponderal index?