Multicentre trials

A multicentre trial is carried out at several centres to increase patient numbers or to assess treatments in different settings. Sometimes increase variability in treatment effects due to centre differences (most likely in surgical trials)- can be included in model as centre and centre.treatment effects. Need to decide whether these effects should be fitted as fixed or random, based on what population you want to make inference about.

Important to check for any outlying centres – make indicate lapse of protocol. In fixed effects model, spurious outliers may be caused by small sample variation – random effects model does not have this problem.

Both fixed

Treatment effects: estimated with equal weight given to each centre. Cannot be estimated unless every treatment recieved at every centre. Treatment se: $Var(t_i - t_j) = \sigma^2_r ( 1 / n_i + 1 / n_j)$ . Always $\le$ variance with centre.treatment effects random Outlying centres: Determine from centre and centre.treatment means. Small centres may appear outlying because of random variation. Inference: Only to centres in trial.

Centre effects fixed, centre.treatment effects omitted

Often used when fixed centre.treatment effect non-significant, but can be difficult to detect small centre.treatment effects, and variability in treatment effect within centres is often ignored.

Treatment estimates: Take account of differing centre sizes Treatment se: As above Outlying centres: As above Inference: As above, but extrapolation to other centres seems more reasonable if no centre.treatment effect

Both random

Centre.treatment effects retained, even if non-significant, provided $\sigma^2_ct \ge 0$ .

Treatment estimates: Take account of differing centre sizes. Estimated from centre.treatment and centre error strata. Estimable even if some treatments not recieved in all centres. Treatment se: Based on centre.treatment variation. If equal number of patients recieve treatment at each centre, $Var(t_i - t_j) = 2(\sigma^2_r / . Not easily specified if centre sizes unequal. Outlying centres: Determined using shrunken centre and centre.treatment estimates Inference: Over population of possible centres.

Centre effects random, centre.treatment effects omitted

Useful when local estimates are required, because will get smaller treatment se.

Treatment estimates: Take account of differing centre sizes. Additional info from centre strata. Treatment se: Based on residual (within-centre) variation. If balanced, se has same formula as fixed effects case. Otherwise, expected to be less. Outlying centres: As above. Inference: Related to sampled centres only – only under strong assumption that no centre.treatment effect exists can inference be extended to entire population of centres.

Practical considerations

Plausibility of centre.treatment interaction

One approach to analysing multi-centre drug trials is to assume that there is no centre.treatment interaction, and if there is it is a fault of the study design. However, a drug effect can vary between centres because of different underlying populations, thus centre.treatment effects are always plausible.

Generalisation

Strictly speaking, shouldn’t generalise unless centres have been drawn at random from a population of interest. Rarely the case in practice (as with random selection of patients) – but usually do anyway, along with subjective judgements. Generally, multicentre trials are more generalisable than single-centre.

Number of centres

Accuracy of variance components dependent on number of centres used – if $\lt 5$ centres used, $\sigma^2_ct$ and treatment effect se may not be accurate. Main conclusions should be based on local results from fixed effect model – usually omitting centre.treatment term

Centre size

Sometimes some centres pullout after only collecting a little information. Often combined into one in fixed effects analyses, but doesn’t make much difference with random effects models.

Negative variance components

As for the general case, can remove or set 0 (preferred). If both centre and centre.treatment are negative, then can drop both and analyse as a simple between patient trial.

Balance

If centre.treatment term is included, balance will only occur if same number of patients recieved treatment at every centre (unlikely!) – in this case treatment estimates will equal raw means. In more usual (unbalanced) situation means will differ between fixed and random models.

If centre.treatment term is omitted, balance will occur when same proportion of patients recieve treatment at each centre.

Sample size estimation

($c \times r \times t$ patients in total)

In balanced trial $var(t_i - t_j) = 2/c (\sigma^2_r /r + \sigma^2_ct)$ and estimates for $c$ and $r$ can be obtained from the usual sample size estimation equation: $\Delta = (t_{\alpha /2} + t_{1-\beta}) SE (t_i - . $DF = (c-1)(t-1)$ not always known, but can use normal-distribution instead of t to get initial value, then iterate.

One difficulty is that estimates of both $\sigma^2_R$ and $\sigma^2_ct$ are required. Can use results from previous (or trial) study, or make educated (conservative) guess.

If only know relative cost $g$ , the total cost will be $crt + cg$ which is minimised when $r = \sqrt{ \frac{ g\sigma^2_r}{t \sigma^2_ct}}$ which can then substituted into the above formula.