### Solution to question 1

#### (a)

The sample size required in a study will be smaller if:

• The difference we wish to detect between groups is large
• The standard deviation of the measurements is small
• We decide to run the study with a low power
• We decide to use a high level of significance (e.g.α = 0·1)

#### (b)

• Standard deviation = 0·8
• Difference in means = 0·4

Using, α = 0.05, Power =90%, we get 'magic number' = 10·5

Using formula supplied:

m = 2x0·82÷0·42x10·5 = 84

This is 84 per group so we need a total of 168 patients

This is a minimum number, assuming no dropouts. If we expect patients to drop out during the study we would expect to have to recruit more patients initially.

#### (c)

Bias is the distortion of estimated effects caused by a systematic difference between the groups being compared. For example, if we are investigating the effect of a toothpaste additive on caries experience and one group is comprised of children who eat sweets regularly and the other of children who are deprived of sweets then we can expect our results to be biased if eating sweets causes caries.

The best way to deal with the problem of bias in a study is to design the study properly so that all potential sources of bias are eliminated. So, when selecting subjects we would do so randomly. If we allow subjects to be selected by the experimenter or to be self-selecting we greatly increase the chance of a biased study. There is always a chance that particular types of subjects might prefer particular types of treatment or behave differently, perhaps sweet-eaters will be less likely to brush their teeth. If the experimenter is aware of which of her subjects are in a test group they may unconsciously give them better treatment.

Collection of information in questionnaire-type studies may be biased towards sufferers. Patients who have suffered badly from caries may recall eating too many sweets as children whereas those with healthy teeth may be not only likely to have eaten less sweets but also be less likely to remember eating too many if they did.

Failing to allow for confounders in either the design or analysis stage of a study can lead to bias. The example given at the start of the answer is one instance of a confounder: a variable that is associated with both the exposure and the outcome. We can control for confounders by such techniques as matching or stratification; or by using multivariate modelling techniques at the analysis stage of a study.

Finally if we cannot eliminate bias but we have some idea of the direction of the bias we may still get useful information. For example, if in the example above the new additive is given to the sweet-eating group and their caries experience is still reduced compared to the non-sweet-eating controls, then we can be fairly sure the additive worked as the effect was seen despite the bias.

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