  Non-adherence to assigned treatment is a common issue in cluster randomised
trials (CRTs). In these settings, the efficacy estimand may be also of
interest. Many methodological contributions in recent years have advocated
using instrumental variables to identify and estimate the local average
treatment effect (LATE). However, the clustered nature of randomisation in CRTs
adds to the complexity of such analyses.
  In this paper, we show that under certain assumptions, the LATE can be
estimated via two-stage least squares (TSLS) using cluster-level summaries of
outcomes and treatment received. Implementation needs to account for this, as
well as the possible heteroscedasticity, to obtain valid inferences.
  We use simulations to assess the performance of TSLS of cluster-level
summaries under cluster-level or individual-level non-adherence, with and
without weighting and robust standard errors. We also explore the impact of
adjusting for cluster-level covariates and of appropriate degrees of freedom
correction for inference.
  We find that TSLS estimation using cluster-level summaries provides estimates
with small to negligible bias and coverage close to nominal level, provided
small sample degrees of freedom correction is used for inference, with
appropriate use of robust standard errors. We illustrate the methods by
re-analysing a CRT in UK primary health settings.
