Appendix: Description of the sensitivity analyses

The basis of the sensitivity analyses was the imputation of missing data using linear regression. In brief, a regression model provides a prediction of every patient�s one-year headache score on the basis of baseline headache score and the randomisation strata (age, sex, chronicity, diagnosis, site). Additional variables (such as post-treatment score) were used in the prediction for the different analyses; however, treatment group (acupuncture or control) was deliberately excluded from the model as a conservative measure. The difference between a patient�s true and predicted headache score is called a residual and the distribution of residuals can be calculated for any particular regression model. For patients with missing data for headache score, we imputed the headache score predicted by the regression model, but added a randomly drawn residual. Imputed and non-missing data were then combined and analysed by ANCOVA as described in the methods section. The process was repeated 100 times and the difference between groups with associated standard error recorded for each iteration. The results for headache score were then combined following Rubin�s rules,¹ using NORM statistical software²; the results for difference in response (that is, proportion improving by at least 35%) were combined by simple averaging.

The imputations were conducted as shown in table A. Sensitivity analyses (table B) were conducted hierarchically, so that, for example, sensitivity analysis 2 used the data from the complete cases and the imputations for groups 1 and 2; sensitivity analysis 3 included all patients. Two additional sensitivity analyses were conducted: sensitivity analysis 4 was the same as sensitivity analysis 3 with the exception of one patient who provided no follow up data and who gave ineffectiveness of acupuncture as a reason for withdrawal. For this patient, change from baseline was fixed at the 5th centile (close to the worst possible result). In the final sensitivity analysis, we used an unadjusted t test to compare change between baseline and follow-up, as planned in the original protocol.

It has been argued that the coefficients for the linear prediction should be randomly sampled from a plausible distribution.¹ This was attempted by using NORM, but the imputation had poor properties: the data augmentation algorithm did not converge unambiguously, and some imputations led to implausible results, such as the mean change in headache scores being a 20 point increase in both groups. Nonetheless, both the estimate for the difference between groups and the p value obtained from this method were close to those reported in table B3.66, P=0.001).

1. Little RJA, Rubin DB. Statistical analysis with missing data. New York: John-Wiley, 1987.

2. Schafer J. NORM: multiple imputation of incomplete multivariate data under a normal model. (2.03). Pennsylvania, Pennsylvania State University, 2000.

Table A Data used in the imputations

Group	Patient group	n	Data used
Complete cases	Patients providing follow up score at one year	301	Actual follow up score
1	Patients providing post-treatment score but no score at one year	31	Imputed from randomisation strata and post-treatment score
2	Patients providing global score but no follow up diaries	45	Imputed from randomisation strata and change in global estimate of headache severity
3	Patients providing no follow up data	24	Imputed from randomisation strata and whether patient completed the post-treatment or one year follow up diaries

Table B Results of the sensitivity analyses

Sensitivity analysis	Total No	Difference between groups: headache score	Difference between groups: response (%)
Principal analysis	301	4.60 (P=0.0002)	21.9
Sensitivity analysis 1	332	4.42 (P=0.0004)	20.8
Sensitivity analysis 2	377	4.16 (P=0.0007)	19.1
Sensitivity analysis 3	401	3.91 (P=0.001)	18.2
Sensitivity analysis 4	401	3.85 (P=0.002)	18.0
Sensitivity analysis 5	301	3.96 (P=0.004)	21.9