A simple analysis

If there are g treatments (or repeated measures) in the data, different pairs of treatments from the g can be analysed as paired data. For example, we can calculate the differences between the two selected treatments and find a 95% confidence interval for the difference between the treatment means, as described in the previous section.

There are g(g - 1)/2 possible pairs of treatments that can be compared and this quickly becomes unmanageable if the number of treatments, g, is large. However in some data sets, we can restrict attention to a smaller number of comparisons:

If one of the g treatments is a standard or baseline treatment, we can compare the other g - 1 treatments to this baseline.

Later in this section, we will explain how to simultaneously compare all treatments, but we first show two examples in which paired confidence intervals are used.

Codeine and acupuncture for dental pain relief

In a randomised block experiment, 32 subjects were grouped into blocks of four according to an initial assessment of their tolerance to pain. Four pain relief treatments were randomly allocated to the four patients in each group and pain relief scores were recorded from each subject two hours after dental treatment.

There is a control treatment (the placebo with no codeine and inactive acupuncture points) so we will compare the three other treatments to this one.

Incorrect analysis based on independent samples
The diagram initially shows 95% confidence intervals for the increase in mean pain relief score based on a naive analysis that treats the data as four independent samples — i.e. it shows two-sample confidence intervals to compare each pair of treatments.

  • With this incorrect analysis, all confidence intervals are wide. There seems to be considerable uncertainty about how much the treatments improve pain relief.
Correct analysis using paired differences
The samples are not independent — the 4 subjects in a block have similar pain relief scores. Click Show blocks of subjects then select Correct analysis using paired differences from the pop-up menu.

  • There is a similar pattern within each block of subjects.
  • The 95% confidence intervals are much narrower and give strong evidence that the three treatments do improve pain relief.

Effect of nicotine on tics

This repeated measures experiment involved 10 patients with Tourette's syndrome, a neurological disorder that results in tics. The number of tics was recorded during an initial 30-minute period (baseline), then during 30 minutes of chewing nicotine gum (containing 2 mg nicotine), from 0 to 30 minutes after chewing, and finally from 30 to 60 minutes after chewing.

There is again a baseline measurement for each patient against which the other measurements can be compared.

Incorrect analysis based on independent samples
The diagram initially shows 95% CIs for the difference between the mean number of tics at the three times after chewing nicotine gum and for the baseline, assuming that the data consist of four independent samples.

  • With this incorrect analysis, all confidence intervals are wide and include zero, so there seems to be little evidence that the gum has reduced the number of tics.
Correct analysis using paired differences
Click Show patients then select Correct analysis using paired differences from the pop-up menu.

  • There are considerable differences between patients in the severity of the condition.
  • The correct 95% CIs for the change in the mean number of tics are much narrower.