1. SADC Course in Statistics Comparing Means from Paired Samples (Session 13)

2. To put your footer here go to View > Header and Footer 2 Learning Objectives By the end of this session, you will be able to recognise whether two samples are paired or independent explain why there is a gain in precision with paired samples carry out a paired t-test and interpret results from such a test present and report on conclusions from paired t-tests

3. To put your footer here go to View > Header and Footer 3 Paired samples - aims In comparing two samples we would aim to improve the precision of the comparison wherever possible, i.e. reduce the standard error used in the test statistic. e.g. If the aim is to compare the average weight of males with that of females amongst malnourished children, it would be better to assess pairs of children of the same age, having one male and one female in each pair.

4. To put your footer here go to View > Header and Footer 4 Benefits of pairing The paired approach means that gender difference would be clearer within a pair of the same age, thus removing age to age variability from the comparison. The situation could be further improved by matching the children, not only by age, but other characteristics too, e.g. their location (rural or urban), levels of poverty, etc. Matching leads to a study of the differences between each pair, say d i for pair i.

5. To put your footer here go to View > Header and Footer 5 An example Suppose we have data for the mean annual income (in 1000s), of doctors and dentists in the UK, from 10 different regions. The data (fictitious) are given in the table. Region DoctorsDentists 136.1 36.8 237.637.0 339.238.6 438.136.0 537.334.7 637.935.5 735.434.0 836.235.3 936.834.1 1037.938.3

6. To put your footer here go to View > Header and Footer 6 Null and alternative hypotheses The hypotheses to be tested are: H 0 : 1 - 2 = 0 versus H 1 : 1 - 2 0 This is equivalent to H 0 : d = 0 versus H 1 : d 0 where d refers the mean of the population of differences between the two groups. Thus, the two-sample case reduces to a single sample situation. Hence ideas covered in Session 09 applies to the single sample formed from differences between each set of pairs.

7. To put your footer here go to View > Header and Footer 7 Test procedure The ten differences are found to be: -0.7 0.6 0.6 2.1 2.6 2.4 1.4 0.9 2.7 -0.4 Further, = 1.22, while = 1.497 Hence the t-statistic for testing H 0 is: t = = = 3.15 Comparing with t-tables with 9 d.f. shows this result is significant at the 2% significance level. The exact p-value = 0.012

8. To put your footer here go to View > Header and Footer 8 Conclusions and other issues Conclusion: There is some evidence of a difference between the mean salaries at a 2% significance level. Note: If we had ignored the pairing by region and conducted an independent samples (or two-sample t-test), the test statistic is t= 1.95 on 18 d.f. This is clearly non- significant, thus leading to a different conclusion from above. So take care to recognise pairing where it occurs.

9. To put your footer here go to View > Header and Footer 9 References Armitage, P., Matthews J.N.S. and Berry G. (2002). Statistical Methods in Medical Research. 4th edn. Blackwell. Clarke, G.M. and Cooke, D. (2004). A Basic Course in Statistics. 5th edn. Edward Arnold. Johnson, R.A. and Bhattacharyya, G.K. (2001). Statistics Principles and Methods. 4th edn. Wiley.

10. To put your footer here go to View > Header and Footer 10 Some practical work follows…