Other Mixed-Effect Models

Whilst some researchers have investigated non-parametric inference, others have considered parametric modifications to model 3.1. Hemming et al [49]

gives an overview of many of these modifications. Here, all models are writ-ten as LMM, but can be exwrit-tended to GLMM for non-normally distributed outcomes.

Fixed period effects that vary within strata

Hemming et al [49] suggest fitting different period effects within some strata of clusters. This relaxed the assumption of a common period effect, but still requires that the period effects are common to all clusters within the defined strata. This model would also still assume exchangeability of observations within clusters. Moreover, identification of appropriate strata would be diffi-cult it practice.

A Cluster-period interaction model

Several authors have suggested including a random effect for a cluster-period interaction will relax the assumptions of a common period-effect and exchange-ability of observations within clusters [12, 14, 35, 49, 50]. The suggested model is as follows, herein referred to as the cluster-period interaction model:

y_ijk = µ + β_j + θX_ij + u_i+ q_ij + e_ijk (3.2)

Chapter 3. Current Recommendations on Design and Analysis

where u_i ∼ N (0, σ_u²), q_ij ∼ N^0, σ_q^2 is a random-effect for each cluster-period, and e_ijk ∼ N (0, σ_e²). The random effects are independent. In this model, observations in different periods are less correlated than observations in the same period and period effects are allowed to vary between clusters. The total variance is fixed to be the same in all periods of the trial. The ICC is now the correlation of two observations in the same cluster in the same period:

ρ^∗ = σ_u²+ σ²_q σ_u²+ σ_q²+ σ_e²

and the cluster-level autocorrelation, describing the correlation between obser-vations in the same cluster but different periods, is:

π = σ²_u σ_u²+ σ²_q

The cluster-level autocorrelation is the same for all pairs of periods, regardless of whether they are close in time to one another, or further apart.

The model has also been extended to allow for cohort designs [14, 50]:

y_ijk = µ + β_j+ θX_ij + u_i+ q_ij + d_ik+ e_ijk (3.3) where d_ik ∼ N (0, σ²_d) is a random effect for individuals, and now e_ijk is a random effect for an individual in a specific period.

Despite several papers suggesting this model [14, 35, 49, 50], no literature has assessed whether it had better properties than model 3.1 for SWTs.

While there is a lack of advice available on whether a random effect for a cluster-period is necessary or sufficient for the analysis of SWTs, there is a more definitive answer for a CRXO, and a CRT with baseline observations.

For these trial designs, mixed-effect models should include a random effect for cluster-period interactions, as well as for clusters [51, 52]. Turner et al [51]

and Morgan et al [52] ran simulation studies comparing analysis methods for a CRXO. Morgan et al [52] directly compared model 3.1 and model 3.2, and found that model 3.2 had a type-one error rate closer to 5% than model 3.1.

Despite advice from the the methodological literature, uptake of of the cluster-period interaction model is poor for CRXO designs [53]. In the field of CRTs with baseline observations, similar results have been found: Ukoumunne and Thompson [27] found that a model similar to model 3.1 gave invalid results.

However, these designs only have 2 periods. SWTs can have many more periods

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than this, and it may be unrealistic to assume the same correlation between all periods.

Random-period model

An alternative parameterisation fits a random effect for each period, herein referred to as the random-period model:

y_ijk = µ + β_j + v_ij + θX_ij + u_i+ e_ijk (3.4) where v_ij ∼ M V N (0, Λ_v) where Λ_v is a (J − 1) × (J − 1) covariance matrix with Λ_v11 = 0, and v_ij and u_i may be correlated. This is modified from a version given by Heuvel et al [54], which treated the period effect as linear.

This is a very flexible model that allows the total variability to change in each period, and allows the cluster-level autocorrelation to be different for each pair of periods. However, this model will grow in complexity as the number of periods increases, unless a linear period effect is assumed.

Heuvel et al [54] assessed this model in a simulation study, and found slightly inflated type-one error. But, the simulation study used a specific scenario with a small sample and assumed a linear time effect, so the results are not easily generalisable.

Intervention effect lag and wane

Heuvel et al [54] also suggested a model that allowed the intervention effect to increase with time in the intervention:

y_ijk= µ + βt_j + θX_ij(t_j − S_i) + u_i+ e_ijk (3.5) where t_j is the time from the first period to period j, S_i is the time at which cluster i switched to the intervention. The period effect is treated as con-tinuous, and the intervention effect is assumed to increase with time in the intervention. This is one way of dealing with the lag in the intervention dis-cussed in section 3.1.3. A more flexible approach to modelling change to the intervention over time, would be to have an interaction term between the in-tervention effect and time since switch [3]:

y_ijk = µ + βt_j + θX_ij + θ⁰X_ij(t_j − S_i) + u_i+ e_ijk

Chapter 3. Current Recommendations on Design and Analysis

This model now allows the intervention effect to increase or decrease with time in the intervention, allowing for a waning of effect over time, in which case θ⁰ would be negative, or an increase in the intervention with time in the intervention, in which case θ⁰ would be positive. Hemming et al [49] suggest a similar model with an interaction with period to allow the intervention effect to vary between the periods.

Another option is to treat the intervention effect as fractional, so that in the first period after the intervention is rolled out the intervention is assumed to be at half efficacy and for all later periods it is at full efficacy [33], or excluding observations from the period following a cluster switching to the intervention.

However, the intervention effect estimate might be sensitive to the proportion of efficacy assigned to each period and this is difficult to know in advance. In addition, excluding observations is inefficient.

Other fixed-effect parameterisations can be used to explore different aspects of the intervention effect [49, 55].

Fixed intervention effects that vary within strata

Hemming et al [49] suggest modelling different intervention effects within spe-cified strata of clusters. This relaxes the assumption of model 3.1 of a common intervention effect, but continues to assume equal correlation of observations with clusters. In addition, it may be difficult in it practice to identify strata where the intervention effect is likely to differ. This may be useful as a sens-itivity analysis, but may have limited use as a primary, prespecified analysis.

Random-intervention model

All analysis methods mentioned so far have assumed that the intervention effect is common to all clusters and that observations are exchangeable between the control and intervention conditions. An alternative model that allows the intervention effect to vary between clusters, herein referred to as the random-intervention model is:

y_ijk = µ + β_j+ (θ + z_i) X_ij + u_i+ e_ijk (3.6)

Chapter 3. Current Recommendations on Design and Analysis

where z_i ∼ N (0, σ_z²) and z_i and u_i may be correlated [56]. This is a repara-meterisation of the model suggested by Hemming et al [49]:

y_ijk = µ + β_j+ θX_ij + u^∗_0iX_ij+ u^∗_1i(1 − X_ij) + e_ijk where u_0i ∼ N^0, σ_u²

0

 and u_1i ∼ N^0, σ²_u

1

 are random effects for clusters while in the control and intervention conditions respectively, which may be correlated.

In CRTs, ignoring variability in the intervention effect between clusters has been shown to bias the intervention effect estimate and underestimate the effect’s standard error [57].

Robust variance and fixed cluster effects

Other modifications to model 3.1 include using a robust variance estimate, and treating cluster effects as fixed. Hussey and Hughes suggested using the jack-knife variance estimator with their analysis [33]. Moulton et al [58] suggested using a robust variance estimator with the Cox model instead of a random effect for cluster: since the Cox model conditions on time, this model should only incorporate vertical comparisons.

It should also be possible to account for clustering using fixed effects rather than random effects because the intervention effect can be estimated within clusters (as horizontal comparisons). This has performed well in literature to date in terms of estimating an intervention effect and producing valid confid-ence intervals [44]. However, this method implies that the clusters included in the study are the only clusters relevant to the research question, and so estimates the intervention effect with greater precision than is appropriate for generalising the intervention effect to a larger population of clusters [59].