Study design

The ODIN trial involved nine study centres in Finland (2), the Republic of Ireland (2), Norway (2), Spain (1) and the UK (2). The trial was designed to compare the outcomes of problem-solving treatment or a depression prevention course (psychoeducation) with outcome in a control group receiving no intervention. Within each centre patients were allocated randomly to receive either one of the two types of treatment (the treatment group) or no intervention (the control group). Problem-solving (but not psychoeducation) was available in Spain, Finland (both centres) and the UK (one centre). Psychoeducation (but not problem-solving) was available in Ireland (both centres) and Norway (both centres). The second UK centre was the only one in the trial in which patients could be allocated randomly to any of the three treatment arms. The main implication of this complex design is that the formal analysis should involve stratification by centre (to ensure that the treatment groups are being compared with the appropriate controls). Further details of the design, including detailed descriptions of the interventions offered, are provided in Dowrick et al (Reference Dowrick, Dunn and Ayuso-Mateos2000). Note that the results from 427 randomised patients are analysed in the present report; Dowrick et al (Reference Dowrick, Dunn and Ayuso-Mateos2000) used 426, as one patient had been inadvertently missed from the previous analysis owing to clerical error. Because of the small number of patients in the two centres from Ireland, in the present analysis the two Irish centres are treated as one.

For the purpose of the present paper there were three measured outcomes of treatment allocation (randomisation): how well the patient adhered to (complied with) the allocation treatment; whether or not the patient was lost to follow-up (six months after randomisation); and, if available, a measure of the severity of depression at follow-up. The latter was assessed using the total score of the Beck Depression Inventory (BDI; Reference Beck, Ward and MendelsonBeck et al, 1961). Adherence to the allocated treatment was measured on a four-point nominal scale: ‘Attended’, ‘Refused’, ‘Discontinued’ and ‘Did not attend’. In order to proceed with further analyses, this scale was dichotomised in one of two ways: for compliance A, ‘Attended’ was coded 1 and the rest 0; for compliance B, ‘Attended’ and ‘Discontinued’ were both coded 1 and the rest 0. A patient was deemed to have received treatment if he or she was in the allocated treatment group and the relevant compliance code was 1 (patients did not have access to treatment if they had been allocated to the control group).

Analysis strategy

In-depth analysis

Assumptions concerning non-compliance. We start by assuming that the patients taking part in the trial belong to one of two potentially latent classes: compliers and non-compliers. In the treatment group the non-compliers are those who fail to receive treatment when they are offered it. In the control group they are those patients who would have failed to receive treatment had they been offered it. Compliers are those who received treatment in the treatment group and those in the control group who would have received treatment had they been offered it. We can observe compliance status in the treatment group but it is latent or unobservable in the control group.

Randomisation ensures that, on average, the proportion of compliers in the control group is the same as that in the treatment group (Reference BloomBloom, 1984; Reference Sommer and ZegerSommer & Zeger, 1991). This means that we can estimate the proportion of unobserved compliers in the control group (or, equivalently, the proportion of compliers in the trial as a whole, π _c ) from the proportion observed in the treatment group (P _c ).

Definitions of treatment effects. We define the average causal effect (ACE) of treatment as the difference between the 6-month average BDI score for the treatment group and that for the control group (regardless of compliance status or whether the outcome is actually observed). An alternative term is the ‘average treatment effect’ (Reference Angrist, Imbens and RubinAngrist et al, 1996). This is the treatment effect that we are trying to estimate in a so-called ITT analysis. It is the difference in outcomes between the two treatment groups as randomised, as opposed to treatment actually received.

We define the complier average causal effect (CACE) as the difference between the 6-month average BDI score for the compliers in the treatment group and that for the compliers in the control group (regardless of whether the outcome is actually observed). An alternative term is the ‘local average treatment effect’ (Reference Angrist, Imbens and RubinAngrist et al, 1996). For reasons clearly explained by Sheiner & Rubin (Reference Sheiner and Rubin1995) and by Frangakis & Rubin (Reference Frangakis and Rubin1999), we do not consider effects estimated by methods involving analysis ‘per protocol’ or ‘as treated’ (the former compares the compliers in the treatment group with all of the controls, and the latter compares those who receive treatment with those who do not, regardless of random allocation) – neither being estimates of valid treatment effects described in this paper.

Exclusion restriction. Given the treatment received, we assume that outcome is independent of random allocation. That is, the offer of treatment, in itself, does not influence outcome (Reference BloomBloom, 1984; Reference Sommer and ZegerSommer & Zeger, 1991). This assumption is often referred to as an ‘exclusion restriction’ (Reference Angrist, Imbens and RubinAngrist et al, 1996). From this assumption we can assume that the mean BDI score for the non-compliers in the control group is, on average, the same as that for the non-compliers in the treatment group. This enables us to estimate the unobserved mean for non-compliers in the control group by the observed average for the non-compliers in the treatment group.

It is straightforward to show from the exclusion restriction assumption that

(1)

\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ ACE={\pi}_{c}{\times}CACE\ \] \end{document}

where π _c is the proportion of compliers in the trial (Reference Angrist, Imbens and RubinAngrist et al, 1996). Typically, we proceed by first estimating the CACE and then, if we require an estimate of the ACE, using this to get an estimate of the ACE from equation (1) together with our estimate of π _c (i.e. P _c ). In the simple situation where outcome measures are available for all trial participants (i.e. there is no loss to follow-up), then the required estimate for the ACE is the familiar ITT estimate. In this circumstance we can then simply estimate the effect of receiving treatment (equivalent to the CACE) from

(2)

\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ CACE=ACE{/}{\pi}_{c}\ \] \end{document}

Details can be found in Angrist et al (Reference Angrist, Imbens and Rubin1996). When we have both non-compliance and non-ignorable missing follow-up data (see below), however, the naïve naïve but frequently used ITT estimates are likely to be biased and we have to approach the analysis via CACE estimation by taking into account the missing data mechanism (Reference Frangakis and RubinFrangakis & Rubin, 1999).

Missing data mechanisms and simple methods of CACE estimation. If, in addition to non-compliance, we also have missing outcome data then we have to make further assumptions concerning the missing data mechanism. The first option is to assume that the missing data mechanism is ignorable. Here the data are either missing completely at random or missing at random, in the sense defined by Little & Rubin (Reference Little and Rubin2002). Looking ahead, it is clear from a glance at Table 2 that the outcome data are not missing completely at random (loss to follow-up is clearly related to compliance status). But suppose, for example, in the simple situation where there are no measured covariates, that the probability of being missing is determined by observed compliance status (complier, non-complier or a member of the control group) and that, conditional on observed compliance status, outcome is statistically independent of whether outcome is actually observed. Here, the outcome data are missing at random (MAR). Under these assumptions it is straightforward to show that

Table 1

Patterns of adherence to allocated treatment

Centre	Treatment	Attended	Refused	Discontinued	Did not attend	Total
1	PE	6	0	3	6	15
2	PS	12	7	0	0	19
3	PS	17	0	5	1	23
4	PS	20	3	4	1	28
5	PE	22	15	5	0	42
6	PE	17	12	5	2	36
7	PE	3	5	1	6	15
7	PS	16	5	7	4	32
8	PS	15	5	6	0	26
Total		128 (54%)	52 (22%)	36 (15%)	20 (8%)	236
Controls						191

Table 2

Summaries of compliance status and availability of outcome data (using compliance A)

Centre	Offered treatment	Received treatment	Patients n (%)1	Patients with outcome data n (%)2
1 (Ireland)3	No	No	23 (100)	12 (52)
	Yes	No	9 (60)	2 (22)
	Yes	Yes	6 (40)	6 (100)
2 (Spain)	No	No	11 (100)	7 (64)
	Yes	No	7 (37)	3 (43)
	Yes	Yes	12 (63)	12 (100)
3 (Finland)	No	No	24 (100)	17 (71)
	Yes	No	6 (26)	2 (33)
	Yes	Yes	17 (74)	17 (100)
4 (Finland)	No	No	22 (100)	20 (91)
	Yes	No	8 (29)	6 (75)
	Yes	Yes	20 (71)	18 (90)
5 (Norway)	No	No	25 (100)	17 (68)
	Yes	No	20 (48)	11 (55)
	Yes	Yes	22 (52)	20 (91)
6 (Norway)	No	No	25 (100)	18 (72)
	Yes	No	19 (53)	15 (79)
	Yes	Yes	17 (47)	17 (100)
7 (UK)	No	No	37 (100)	31 (84)
	Yes	No	28 (60)	16 (57)
	Yes	Yes	19 (40)	15 (79)
8 (UK)	No	No	24 (100)	18 (75)
	Yes	No	11 (42)	4 (36)
	Yes	Yes	15 (58)	13 (87)
Total	No	No	191 (100)	140 (73)
	Yes	No	108 (46)	59 (55)
	Yes	Yes	128 (54)	118 (92)

(3)

\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ CACE_{MAR}=\frac{{\pi}_{c}{\mu}_{11}+(1-{\pi}_{c}){\mu}_{10}-{\mu}_{0}}{{\pi}_{c}}\ \] \end{document}

where μ₁₁ is the mean outcome for the compliers in the treatment group, μ₁₀ is the corresponding mean for the non-compliers and μ ₀ is the mean for the controls. The CACE _MAR value can be estimated easily by replacing μ _c and the three μ terms by their corresponding values in equation (3). If there are no missing outcome data, equation (3) simplifies to the estimator first described by Bloom (Reference Bloom1984). The standard error of this so-called moments estimate can be obtained using the delta technique or a simple bootstrap (Reference Efron and TibshiraniEfron & Tibshirani, 1993).

The alternative missing data option is that they are non-ignorable (Reference Little and RubinLittle & Rubin, 2002). That is, whether a patient has a missing outcome is dependent on the value of that outcome, even after conditioning on observed variables such as compliance status and baseline covariates. This is a much more difficult problem to deal with and we refer the interested reader to a recent paper on this topic by Frangakis & Rubin (Reference Frangakis and Rubin1999). A less demanding discussion of the work of Frangakis & Rubin is provided by Dunn (Reference Dunn2002b ). In order to keep the technical details to a minimum, we do not pursue this option in any detail in the present paper.

Refinement of CACE estimation: incorporating baseline covariates. Although technically more difficult, if we have access to baseline covariates (including treatment centre) we can develop more efficient (i.e. precise) CACE estimation methods. We can also get more stable estimates of the average treatment effects within each of the centres. Maximum likelihood methods, based on the joint distribution of the binary compliance status and a normally distributed outcome measure, have been developed by Angrist et al (Reference Angrist, Imbens and Rubin1996), Little & Yau (Reference Little and Yau1998) and Yau & Little (Reference Yau and Little2001) – the latter incorporating data missing at random. These methods enable the incorporation of covariates in the model to predict jointly both the latent compliance status and the outcome (the outcome is also predicted by compliance status as well as by the covariates).

In the present study, CACE models incorporating the potential use of baseline covariates (initial BDI score and centre membership) to predict both compliance status and outcome were fitted via maximum likelihood estimation using the expectation maximisation algorithm (Mplus Version 2.12; Muthén & Muthén, 1998–2002). The use of the latter software package in the application of this methodology on randomised controlled trial data with non-compliance is illustrated in detail by Jo & Muthén (Reference Jo, Muthén, Marcooulides and Schumacher2001), although they do not consider problems arising from missing outcome data.

Sensitivity analysis. Rather precise assumptions (e.g. concerning the definition of compliance, the missing data mechanism and exclusion restriction) are vital components of the analytical approaches described above for the estimation of average treatment effects. Having to make these assumptions is both a strength and a weakness of these approaches. If we get the assumptions wrong we risk invalid inferences, but a thorough examination of the implications of the assumptions helps to understand what might be going on in a psychological treatment trial. They force us to think more about the trial process and to clarify what we are really interested in estimating. Another vital component of the analytical approach therefore is to attempt to evaluate the sensitivity of our treatment effect estimates to changes in these assumptions.

All preliminary analyses and checks of the sensitivity of the treatment effects to assumptions concerning the definition of compliance were carried out using Stata Version 7.0 (StataCorp, 2001). An exploration of the sensitivity of the CACE estimates to the validity of the main exclusion restriction assumption (treatment allocation does not influence outcome except through its effect on treatment received), using either of the two definitions of compliance, was carried out as described by Jo (Reference Jo2002a , Reference Jo b ). Readers are also referred to Heckman et al (Reference Heckman, Smith and Taber1998) and Hirano et al (Reference Hirano, Imbens and Rubin2000).

Preliminary examination of the data

Table 1 shows the patterns of adherence to the offered treatment (i.e. excluding controls) in each of the nine centres (separately for the two types of treatment offered by centre 7). To illustrate the variation in the patterns of adherence in more detail, we look at patterns of compliance using compliance A (Table 2). Compliance rates vary greatly from one centre to another (ranging from 40% in centre 1 to 74% in centre 3). Some of the variation may be explained by the type of treatment being offered, but we do not stress this aspect of the results because the design of the trial leads to this source of variation being almost completely confounded with the centre effects. These compliance rates may appear to be rather low, but the reader must bear in mind that the participants in the ODIN trial were recruited through a case-finding exercise. They were not patients who had actively sought help.

Loss to follow-up (i.e. missing outcome data) varies from one centre to another but is also markedly dependent on compliance status. Loss to follow-up in the compliers in the treatment group is very infrequent. In four of the nine centres the compliers provide 100% of the required outcome data, with follow-up of those in the other five centres ranging from 79% (centre 7) to 91% (centre 5). However, loss to follow-up is both more variable and more common in the non-compliers of the treatment group; here, follow-up rates range from 22% (centre 1) to 75% (centre 4). In no case is the within-centre follow-up rate for the non-compliers as high as that for the corresponding compliers. As might be expected, the follow-up rates for the controls lies somewhere between those for the compliers and non-compliers in the treatment group.

Moving on to consider the severity of depression at outcome (the mean BDI score at 6 months) we see that, on average, patients offered treatment do better than the controls (bottom three rows of Table 3). However, this difference is not always apparent within each of the centres. On average, the compliers in the treatment group have very similar outcomes to those who do not comply with the offered treatment (last two rows of Table 3) but again there is a considerable amount of variability in this difference from one centre to another. In centres 2–5 the compliers appear to fare better than the non-compliers. In centres 1, 6, 7 and 8, however, the non-compliers fare better. Again, we do not concentrate on the differences in effects for the two types of psychological intervention because these differences are confounded by differences between centres. Returning to the data for the whole trial (bottom three rows of Table 3), the equality of the mean of the BDI scores for the compliers and non-compliers in the treatment group, together with the exclusion restriction (the assumption that the mean BDI score for the non-compliers in the control group is the same as that for those in the treatment group), implies that the compliers in the control group have a worse outcome than the corresponding non-compliers. Attempts to understand why this might be so are detailed in the Discussion.

Table 3

Observed Beck Depression Inventory scores at baseline and 6 months (using compliance A)

Centre	Offered treatment	Received treatment	Baseline mean (s.d.)1	Six-month mean (s.d.)
1 (Ireland)	No	No	25.74 (9.72)	9.33 (8.98)
	Yes	No	26.44 (10.03)	1.00 (1.41)
	Yes	Yes	19.00 (7.46)	10.67 (3.61)
2 (Spain)	No	No	22.27 (6.59)	16.29 (7.87)
	Yes	No	21.86 (5.73)	12.67 (5.13)
	Yes	Yes	22.42 (8.59)	7.17 (5.13)
3 (Finland)	No	No	20.83 (6.31)	13.12 (8.53)
	Yes	No	22.17 (6.18)	12.50 (6.36)
	Yes	Yes	21.76 (5.66)	9.18 (6.61)
4 (Finland)	No	No	19.91 (6.53)	8.70 (7.41)
	Yes	No	23.13 (10.19)	8.83 (3.06)
	Yes	Yes	21.90 (7.30)	7.67 (4.30)
5 (Norway)	No	No	20.64 (6.26)	13.12 (9.75)
	Yes	No	20.85 (8.99)	14.91 (9.70)
	Yes	Yes	22.82 (8.81)	12.00 (7.95)
6 (Norway)	No	No	19.32 (7.12)	15.44 (9.41)
	Yes	No	18.53 (5.17)	14.20 (8.62)
	Yes	Yes	21.76 (9.42)	18.29 (11.97)
7 (UK)	No	No	24.70 (8.88)	21.90 (11.21)
	Yes	No	24.61 (9.19)	14.81 (11.48)
	Yes	Yes	25.47 (9.12)	21.00 (12.48)
8 (UK)	No	No	26.04 (8.71)	17.72 (10.63)
	Yes	No	22.54 (8.44)	12.00 (12.36)
	Yes	Yes	27.73 (8.03)	20.15 (11.81)
Total	No	No	22.58 (8.06)	15.16 (10.42)
	Yes	No	22.36 (8.40)	13.22 (9.35)
	Yes	Yes	23.15 (8.27)	13.32 (10.14)

The CACE estimation

We now look at simple CACE estimates (i.e. moment estimates based on Equation (3)), ignoring centre membership. These estimates are derived using either of the two definitions of compliance. A negative estimate implies that receipt of treatment works. Using compliance A, CACE _MAR =-3.47 (s.e.=2.22). Using compliance B, CACE _MAR =-2.73 (s.e.=1.65). The CACE estimates are smaller (i.e. closer to zero) using compliance B than compliance A. For comparison, the ITT effect is just under two units (i.e. the ITT estimate is -1.88). None of these differences appears, at this stage, to be statistically significant (the ratio of the estimate to its standard error is <2).

We now present the result of a more formal series of analyses (Table 4). We use maximum likelihood estimation (assuming normality of the outcome BDI scores) and allow for the baseline BDI score as a covariate. All 427 subjects are included in the analysis. They all have data for baseline BDI and centre membership, but 110 of them have a missing 6-month BDI score. Here, we again assume that these missing data are ignorable. All analyses presented in Table 4 are based on the exclusion restriction (allocation to the treatment group has no effect on the non-compliers). Section (a) of Table 4 gives the results of fitting a CACE model in which baseline BDI and centre membership are allowed to predict both compliance and outcome (BDI at 6 months). The model also allows for a treatment × centre interaction (i.e. CACE estimates are free to vary from one centre to another). There is variation between centres but note, again, that compliance A leads to greater estimated treatment effects than compliance B.

Table 4

Maximum likelihood estimation of the complier average causal effect using the expectation maximisation algorithm¹

	Compliance A	Compliance B
(a) Treatment effects by centre2
logL	-1268.73	-1246.15
AIC	2593.46	2548.31
1 (estimate (s.e.))	1.02 (3.42)	0.24 (3.55)
2 (estimate (s.e.))	-9.96 (3.44)	-10.06 (3.62)
3 (estimate (s.e.))	-7.26 (2.68)	-5.80 (2.52)
4 (estimate (s.e.))	-5.22 (2.34)	-3.07 (2.01)
5 (estimate (s.e.))	-6.88 (3.62)	-6.41 (3.62)
6 (estimate (s.e.))	-3.10 (3.94)	-2.42 (3.76)
7 (estimate (s.e.))	-4.25 (3.28)	-4.47 (2.99)
8 (estimate (s.e.))	-2.21 (3.63)	0.56 (3.02)
(b) Treatment effects by type3
logL	-1272.20	-1250.58
AIC	2588.40	2545.17
Problem-solving (estimate (s.e.))	-5.48 (2.99)	-3.89 (1.64)
Psychoeducation (estimate (s.e.))	-4.45 (2.83)	-3.45 (2.60)
(c) Common treatment effect4
logL	-1272.33	-1250.61
AIC	2586.66	2543.21
With covariates (estimate (s.e.))	-5.27 (2.05)	-3.79 (1.70)
(d) Intention-to-treat
With covariates (estimate (s.e.))	-2.34 (0.94)

In section (b) of Table 4 we present the results of separate estimations for problem-solving and psychoeducation. These were obtained by fitting a single model to the complete data-set in which baseline BDI score and centre membership were allowed to predict both compliance and the 6-month BDI score. There were no treatment × centre interactions in the model. Fitting a common treatment effect (Table 4, section (c)) indicates that, although problem-solving appears to be slightly more effective than psychoeducation, the difference is nowhere near statistically significant: twice the difference in logL, that is 2 × (1272.33-1272.20), is distributed as χ² with one degree of freedom under the null hypothesis that the two treatments are equally effective. A similar comparison of the 2logL values for the models in sections (a) and (c) also indicates that the treatment × centre interactions are not statistically significant. However, the common treatment effects (using either compliance A or B) in section (c) are statistically significant: by refitting the model after constraining the treatment effects to be zero, the change in 2logL is 9.32 and 8.06, each with one degree of freedom, for compliances A and B, respectively. Section (d) of Table 4 provides an estimate of the ITT effect obtained by direct estimation in Mplus, assuming that missing 6-month BDI scores are ignorable.

Sensitivity of CACE estimates to assumptions

We now consider the results of our final series of sensitivity analyses. We start by replacing the exclusion restriction (effect of treatment allocation in the non-compliers is zero) by a series of alternative assumptions: the effect of treatment allocation in the non-compliers varies from -2.5 (beneficial to be allocated to treatment) to +2.5 (beneficial to be allocated to the control group). This procedure was carried out for data using either of the two compliance definitions. In each case the fitted model was equivalent to that in section (c) of Table 4. The rationale for the procedure is explained in detail by Heckman et al (Reference Heckman, Smith and Taber1998) and Jo (Reference Jo2002a ). Because the overall effect of allocation to treatment is a weighted average of the effect in the compliers (the CACE) and that in the non-compliers, we would expect that fixing the effect of allocation in the non-compliers to a negative value would bring the CACE estimate closer to zero. When the effect in the non-compliers is -2.5, for example, the modified CACE estimate is -3.18 (s.e.=3.66). On the other hand, when the effect in the non-compliers is fixed at +2.5 the CACE estimate is more marked, at -6.04 (s.e.=1.73). Because we set the fixed values of the effect in the non-compliers between -2.5 and +2.5, the CACE estimates (and their standard errors) move smoothly between these two extremes. Because our working model (section (c) of Table 4) has no treatment × centre or treatment × baseline BDI interactions, it is possible to relax the exclusion restriction and allow for the effect of treatment allocation to be estimated freely in the non-compliers (Reference JoJo, 2002b ). For compliance A, the estimated effect for the non-compliers was +1.43 (s.e.=5.83); using compliance B, it was +1.41 (s.e.=13.67). The corresponding CACE estimates were -5.81 (s.e.=3.75) and -4.13 (s.e.=5.07), respectively. Note that all four of these estimates are quite imprecise. In our final models, we constrained the effects of treatment allocation to be the same for compliers and non-compliers. This might seem strange but it is possible that offering treatment is beneficial but its receipt is not. The resulting joint estimates (-2.51 (s.e.=1.02) and -2.46 (s.e.=1.02) using compliances A and B, respectively) are very similar to the ITT estimate (with similar standard errors) in section (d) of Table 4. We conclude that the CACE estimates are reasonably robust to changes in assumptions and the effect of the receipt of treatment in those who get treated is likely to be somewhere between -5 and -4 points on the BDI scale.

Technical issues

We have presented methods for the estimattion of various average treatment effects in randomised controlled trials in which not everyone complies with the allocated treatment. The trial that we have used to illustrate these methods (ODIN) involved simply allocating patients to be offered psychological treatment or not. The control group were not given access to treatment and therefore the only form of non-compliance possible in this trial was for those offered treatment not to accept the offer or to discontinue treatment once it had been started. In other trials it might be possible for patients allocated to the control group, for example, to get access to treatment outside of the trial. Dunn (Reference Dunn2002b ) discusses an example like this. The simple methods of CACE estimation such as those involving the use of moments, based on equations (3) and (4) of the present paper, are quite straightforward to apply. The more sophisticated maximum likelihood procedures, however, need more technical expertise and experience. It is straightforward to apply similar statistical methods to binary (depressed/not depressed) outcomes and the simpler approaches are illustrated by Dunn (Reference Dunn2002a ).

One point that we should stress here is that all analyses, however simple, are vitally dependent on assumptions that might be difficult to justify for a given trial and often can be almost impossible to verify. Some of the assumptions will, however, be much more credible than others. This means that there is no one approach to the analysis that is obviously the best one. An important component of these estimation methods should be checking wherever possible the sensitivity of the results to various assumptions made. Unfortunately, sensitivity analyses are very rare in practice. In their systematic review of how 89 randomised controlled trials with missing follow-up data dealt with this problem in their estimation of ITT effects, Hollis & Campbell (Reference Hollis and Campbell1999) found that only one report included any attempt at a sensitivity analysis. However, our analysis strategy is presented as an informal suggestion and not a prescription. Our aim is to encourage trial statisticians and others to probe their data in more detail. We emphasise, however, that we are not suggesting that ITT methods be abandoned but that more care should be taken in their use and they should be supplemented by CACE-based methods as described above. The best method of analysis must be dependent on the characteristic of the trial under consideration.

The challenge of patient preference

One of the major challenges for psychological treatment trials is that the patients cannot be blinded. Therapists need the cooperation and often the active participation of their subjects for the success of the therapy. The preferences and other beliefs of the patients may have an important impact on compliance with an offered treatment and also on the efficacy of the treatment actually received. To date, there are only a few intervention studies that have evaluated whether patient preference for a specific treatment has an effect on treatment outcome (Reference Bedi, Lee and HarrisonBedi et al, 2000; Reference Ward, King and LloydWard et al, 2000). The interpretation of the results of a randomised controlled trial of a psychological intervention is particularly challenging in the presence of these preference effects (Reference Brewin and BradleyBrewin & Bradley, 1989; Reference McPherson and BrittonMcPherson & Britton, 2001). A statistical analysis strategy that highlights the effects of preferences, in the present case through concentration on the problems of non-compliance and subsequent loss to follow-up, may rest on challengeable assumptions but the process of making these assumptions and offering them to challenge will lead to a clearer understanding of what we need to concentrate on in interpreting the resulting estimates. It might be particularly helpful to consider the definition of compliance and what we think the separate effects of an offer of psychological intervention (or failure to offer in the case of the control group) on the compliers and non-compliers might be.

Does the mere offer of treatment have a therapeutic effect?

One of the key assumptions in the analyses presented in this paper is the exclusion restriction – the assumption that the offer of treatment in itself does not have any effect on outcome. This assumption is necessary to ensure the identifiability of the CACE estimates (i.e. can we get unique estimates from the data?) when we do not have access to baseline covariates. When we have access to covariates, which can be used to predict jointly the compliance and outcome, then when given an appropriate model (Reference JoJo, 2002b ) we can relax the restriction assumption and actually estimate the effect of offering treatment in the non-compliers. Unfortunately, in the present example the effect was only weakly identified (it was estimated with very large standard errors). Interestingly, however, the estimate of treatment allocation in the non-compliers was positive (i.e. it was slightly harmful to be offered treatment if you were then going to decline the offer). Similar findings were obtained by Jo (Reference Jo2002b ) in his reanalysis of the JOBS II trial (Reference Vinokur, Price and SchulVinokur et al, 1995; Reference Vinokur and SchulVinokur & Schul, 1997). The JOBS II was a randomised trial to prevent poor mental health and to promote high-quality re-employment among the unemployed. The overall level of compliance with the offered treatment (5 half-day training sessions) was similar to that in the ODIN trial. Jo (Reference Jo2002b ) argued that the offer of intervention to the non-compliers is likely to have led to demoralisation arising from their failure to take up the offered treatment. The non-compliers in the control group do not suffer this demoralisation, however, because they have not been offered anything.

In our compliance A, patients who initially accepted the offer of treatment but who subsequently failed to turn up for appointments or discontinued their treatment after having started it were classified along with the refusals as ‘non-compliers’. It could be argued, however, that those who discontinued their treatment were partial compliers who might have received some benefit from the offered intervention. Here it might be better to think of our complier/non-complier dichotomy as a comparison of patients with high compliance with those of low compliance (Reference JoJo, 2002b ). If this were indeed the correct interpretation, then we might expect the offer of treatment to have a small beneficial effect in the low compliers and a larger beneficial effect in the high compliers. In our compliance B, however, we put the discontinued patients in with those who attended a full course of treatment. The non-compliers in this case might be labelled accurately as non-compliers, whereas the compliers are a mix of high and low compliers. However, the effect of treatment allocation in the non-compliers was not seen to be beneficial using either compliance A or B, but the CACE estimate was more marked (further from zero) when using compliance A than compliance B. One possible explanation is that the treatment had no more benefit in those who discontinued than in those who refused or failed to turn up for any treatment. In this situation the CACE estimated using compliance A gives us the more realistic treatment effect because that obtained using compliance B will be attenuated towards zero by including the discontinued patients with those who fully complied with the offered therapy.

Is there evidence of resentful demoralisation

In the ODIN trial the compliers in the control group (i.e. those who would have accepted the treatment if they had been offered it) do worse than the non-compliers. Why? One possible interpretation is that those people who would like help (and would have accepted treatment if offered it) but who are denied access to it because of allocation to the control group suffer from resentful demoralisation (Reference Brewin and BradleyBrewin & Bradley, 1989). They do worse than they would have done if they had never been recruited to the trial. This resentful demoralisation, if present, would lead to the CACE estimate being too optimistic. An alternative interpretation is that the non-compliers are patients who think (on the whole, correctly) that they will get better anyway and therefore do not need the offered treatment (the compliers, on the other hand, are sicker and feel more in need of help). These two interpretations cannot be distinguished from the present data. The design of trials to enable separate estimation of treatment and preference effects would need a lot of careful thought. A starting point might be the two-stage design proposed by Rücker (Reference Rücker1989) – first randomise patients to have a choice or not, and then randomise those without a choice to the competing treatments while, at the same time, allowing those allocated to the choice arm to select their own treatment. Rucker's design, however, is probably impractical because it takes little account of reality (i.e. the proposed analysis assumes complete compliance with the two random allocations and also that there will be complete follow-up data). The so-called patient preference design of Brewin & Bradley (Reference Brewin and Bradley1989), despite its popularity among some clinical researchers, would appear to be a blind alley – it has very little validity from a statistical viewpoint. A useful device might be to seek patient preferences prior to randomisation (Reference Torgerson, Klaber-Moffett and RussellTorgerson et al, 1996). This would not only provide important information on preference effects but also would lead to better prediction of compliance and more efficient (precise) CACE estimates. Interestingly, investigators in one of the Norwegian centres of the ODIN trial informed us after the above analysis that they had asked patients prior to randomisation about their interest in receiving the treatment (as suggested in Reference Torgerson, Klaber-Moffett and RussellTorgerson et al, 1996). Those patients who were allocated to the control condition but had expressed an interest in the treatment prior to randomisation appeared to do worse than those who had not (Reference Dalgard and BørveDalgard & Børve, 2000).

Concluding remarks

In the interpretation and evaluation of the results of a simple randomised controlled trial such as ODIN one can ask two related and complementary questions: ‘What is the effect of offering treatment?’ and ‘What is the effect of the receipt of treatment?’ The former is answered using an ITT estimate of the treatment effect (i.e. the impact of randomisation) and the latter through CACE estimation (i.e. adjusting for non-compliance). The answers to both questions are likely to be interesting and important and it is reasonably straightforward to obtain answers to both. We stress that in promoting the use of CACE estimation we are not advocating that trialists should abandon ITT. This should always be the primary analysis. What we are advocating is that trialists move beyond ITT in order to learn more from their data and search for explanations for their primary results.

CLINICAL IMPLICATIONS

1. ▪ Estimation of the complier average causal effect (CACE) enables one to evaluate the effect of receipt of treatment in a randomised controlled trial in which a proportion of patients do not comply with their allocated treatment.

2. ▪ Estimation of CACE should not be seen as an alternative to the pragmatic intention-to-treat (ITT) analysis but as a means of going beyond ITT estimates to seek an explanation for the pragmatic effects.

3. ▪ The CACE estimates present trial results in a way that is closer to the real world of the practicing clinician than ITT estimates of treatment effects, and therefore may be more clinically relevant.

LIMITATIONS

1. ▪ The CACE analysis presented here assumes that the specified intervention is not available to patients outside of the trial condition. However, the methods can be extended easily to cope with more complex situations.

2. ▪ Estimation of CACE assumes that compliance is a dichotomous (yes/no) condition, whereas patients may have differing degrees of compliance. Again, the methodology can be extended to cope with quantitative compliance–response relationships.

3. ▪ Estimation of CACE is dependent upon potentially challengeable assumptions that frequently cannot be tested using the data at hand. These challenges should, however, stimulate investigators to come up with more informative designs.