Searches identified 1,671 references (Figure 1, Additional files 1). Of these, 40 complete studies presenting economic evaluation results, 13 published protocols and one prematurely-terminated study met the inclusion criteria. Additional file 2 gives details of all included studies.
Of the completed studies, 23% (9/40) allowed for interactions between factors when analysing the primary clinical endpoint, 53% (21/40) assumed no interaction, while 25% (10/40) did not clearly state their methods (Table 1). Twenty studies (50%) used regression methods for the primary endpoint, of which five included interaction terms, seven did not and eight did not clearly describe their methods. Four studies used inside-the-table analysis and 14 used at-the-margins. Only three studies (8%) observed statistically significant interactions for the primary endpoint, although nine others (23%) observed large or qualitative interactions that did not reach statistical significance or for which significance was not reported. Interaction results were not clearly reported for 15 studies.
By contrast, 53% (21/40) of completed studies allowed for interactions in their base case economic evaluation: more than twice the number allowing for interactions in the primary endpoint. Studies were also more likely to report sufficient information to identify whether interactions were taken into account for cost-effectiveness than primary endpoints, although in most cases it was necessary to infer the methods used from the tables reported. Only five studies analysed economic results using regression analyses, while two used event-based cost-effectiveness analysis, 17 inside-the-table and 14 at-the-margins; this may reflect the difficulties associated with regression-based economic evaluation identified previously .
Fifteen completed studies (38%) presented the probability of treatment being cost-effective within the text or as cost-effectiveness acceptability curves. Of these: nine studies presented pair-wise comparisons giving the probability that one treatment is cost-effective compared with a single comparator; three studies presented figures showing how the probability of each treatment evaluated in the trial having highest NMB varies with ceiling ratio; and a further three studies presented acceptability curves for both pair-wise and multiple comparisons. Six further studies quantified uncertainty in other ways (e.g. scatter graphs or confidence intervals). One study also presented the value of information [10-12].
Sixteen studies (40%) reported results inside-the-table in sufficient detail that interactions for both costs and health benefits could be directly evaluated (See Additional file 3). Large interactions arose frequently: 33% (24/72) of interactions had an absolute magnitude larger than one or more simple effect (interaction:effect ratios >1 or <-1; Table 2). Interaction:effect ratios varied between -44 and 232. Overall, 33% of interactions were super-additive (23/72), 49% (35/72) were sub-additive or qualitative, while 17% (12/72) were mixed (Table 2). Large and qualitative interactions occurred at least as commonly for health benefits as for costs and NMB. Among the studies measuring health in units other than QALYs, 50% (7/14) of interactions were larger than simple effects. However, although 29% (7/24) of studies had qualitative interactions for NMB, the interaction changed the treatment adoption decision in only one case .
Six studies (reporting nine interactions) reported standard deviations around both costs and health benefits in each group [13-18]. Within these studies, 56% (5/9) of interactions for cost were statistically significant (p<0.05), although there were no statistically significant interactions for health benefits or NMB.
The six studies reporting standard deviations for each group [13-18] were used in simulation work to evaluate the different criteria for identifying which interactions should be included in economic analyses. Using simulated data means that: (a) whereas for a real trial, we only see one sample, for simulated data, we can generate multiple samples and see how performance varies; (b) we specify the true data-generating mechanism and can compare the conclusions of each individual sample against the true answer; (c) we can vary the characteristics of the data-generating mechanism (e.g. interaction size and sample size) and see the impact on the results. For simplicity, simulations focused on balanced 2x2 full factorial designs with no covariates or missing data. We therefore only included the first two levels for each factor evaluated by Hollis et al  and the Alexander Technique, Exercise And Massage (ATEAM) trial .
In addition to the original studies, five variants of each trial were simulated using interaction terms that were 0%, 50% or 200% of the size observed in the original study, and using double the sample size with either the original interaction or zero interaction (See Additional file 3). The analysis used Stata version 12 (College Station, Texas) to simulate and analyse 300 samples of each of the 36 scenarios from the six trials. The data-generation methods and Stata code are shown in Additional file 3.
The costs and benefits for each sample were analysed using four mixed models with different combinations of interaction terms: no interactions; interaction for costs only; interaction for health benefits only; and interactions for costs and benefits. The mixed models implemented seemingly-unrelated regression allowing for correlations between costs and benefits by predicting outcomes (which could be either costs or benefits) with random effects by patient. However, separate constants, treatment effects and (where appropriate) interactions were estimated for costs and benefits and unstructured residuals were used. This approach gives identical results to the sureg command . The log-likelihood, degrees of freedom and coefficients and their standard errors were recorded for each model.
The coefficients estimated in mixed models were used to calculate NMB. For simplicity, all costs were interpreted as though they were in pounds Sterling. Results focus on ceiling ratios of £20,000/QALY  for the five studies measuring benefits in QALYs, and £5,000 per unit of benefit for other studies.
We evaluated 15 criteria for determining which interactions should be taken into account (Table 3) and applied these to each simulated trial sample. We compared the results of each analysis against the “true” results for each dataset, which (for the purposes of this simulation study) were assumed to equal the mean values for treatment effects and interactions shown in Additional file 3, Table 3.3. The sensitivity and specificity for identifying interactions, the probability of adopting the best treatment and the opportunity cost of making the wrong decision  were evaluated for each of the 15 criteria (Table 4).
We used the opportunity cost as the primary measure of which criterion works best, since it focuses on the central question of economic evaluation: namely maximising health gains from a finite budget. Coverage, statistical power and bias were also calculated (Additional file 4).
The 15 criteria differed in the proportion and type of interactions that were correctly identified (Table 5). Other than the "always include interactions" criterion (criterion 1), including interactions where p<0.25 (criterion 5) and including interactions that are significant or greater than simple effects (criterion 10) resulted in the largest number of cost interactions being included. By contrast, criteria 5 and 9-12 included the largest number of benefit interactions. In general, specificity and sensitivity were inversely proportional; measures based on information criteria or statistical significance at alpha=0.05 tended to have high specificity and low sensitivity.
Averaging across all 36 scenarios from the six trials, including interactions ≥0.25 or ≥£250 minimised the opportunity cost from adopting treatments that do not in fact maximise true NMB, while the opportunity cost of “always include interactions” was £0.04 larger (Table 5). “Never include interactions” performed worst, while criteria 3-7 (based on statistical significance and information criteria) also performed poorly.
However, the criterion with lowest opportunity cost differed between individual scenarios (See Additional file 4). As expected, “never include interactions” was, on average, the best criterion for the scenarios that did not have qualitative interactions, although no criteria had high opportunity costs when interactions were zero. Across the 13 scenarios with qualitative interactions, “always include interactions” performed best, although criteria 11-13 also performed well (including qualitative interactions, including interactions >simple effects or including interactions ≥0.25 or ≥£250).
Across all scenarios, criterion 9 (including interactions >simple effects) had the highest probability of adopting the treatment that has highest true NMB (Table 5). “Never include interactions” performed worst overall on this measure, but performed best in scenarios without qualitative interactions for NMB. “Always include interactions” performed best when there were qualitative interactions. However, results differed substantially between scenarios (not shown).
Doubling sample size reduced the opportunity cost and the probability of adopting the wrong treatment for all criteria. However, criteria based on statistical significance or information criteria (which explicitly take account of sample size) did not appear to perform any better relative to other criteria in larger studies. Furthermore, criterion 11 (including qualitative interactions for cost, benefits or NMB) performed best in scenarios with double the original sample size, whereas “always include interactions” performed best with a smaller sample size.
Including all interactions was also the only criterion for which the 95% confidence intervals gave 95% coverage and also had no bias (See Additional file 4). Excluding all interactions had lowest coverage and highest bias. Including all interactions had lowest statistical power, while criteria 2, 8, 14 and 15 had highest statistical power (never include interactions, include qualitative interactions, include interactions ≥0.5 or ≥£500 and include interactions ≥1 or ≥£1000).
 One study meeting inclusion criteria was terminated early due to poor recruitment but was published as a monograph without analysis of economic results; this is considered in the review alongside protocols.
 Since four of these studies were larger than 2x2 or reported results by subgroup, this gave 24 interactions for each of three outcomes (cost, QALYs and NMB): 72 interactions in total.