We compiled a list of the interventions implemented by each LTLA over time between 1 July 2020 and 5 November 2020. Data were obtained by monitoring the following government sources:

The interventions and tiers that LTLAs were subject to were also amended accordingly.

Since the tier classification came into place on 14 October 2020, a strict evaluation of the effects of tiers would limit our analysis to only 21 days until the second national lockdown on 5 November. However, the interventions that comprise a given tier were in place earlier. We therefore retrospectively assign a tier rating to LTLAs from 1 July 2020, when restrictions of the first national lockdown were lifted for the majority of the UK. The challenge is that the exact interventions that were implemented under tiers 1, 2 and 3 varied first between LTLAs, and second over time. We determine the interventions that comprise a given tier (and thus which tier a given LTLA would have been in before the creation of the tier system) using the number of LTLAs that implemented a given intervention on the day tiers 1–3 were applied. Our study period ends before the second national lockdown and the implementation of tier 4.

Figure 1 shows the number of LTLAs that implemented a given intervention when nominally under tiers 1, 2 and 3, at the earliest date they were introduced. Four interventions are implemented in all 310 LTLAs when under tier 1: (1) limiting indoor gatherings to at most six people; (2) limiting outdoor gatherings to at most six people; (3) curfew of 22:00 for hospitality venues; and (4) instruction to work from home where possible. A further three interventions (‘travel discouraged’, ‘no indoor mixing’, ‘overnight stays discouraged’) are considered to be part of tier 1 for 30 LTLAs. However, as this is minority of LTLAs, and as these interventions are always present under tier 2, we include them in the definition of tier 2. Tier 3 is the most consistently defined set of interventions and adds the following two interventions to tier 2: ‘residents cannot leave the local area’ and ‘pubs and bars closed table service only’. Closure of gyms is included in tier 3 for six LTLAs, but this is insufficiently frequent to warrant inclusion in the backdating of tier 3.

Full details of the R_t estimation can be found in Mishra et al.7 Briefly, however, the method estimates transmission by calculating backwards from observed deaths (day of death), cases and serological survey data while simultaneously allowing for the time lag between infection and death. Infection fatality ratios and infection ascertainment rates are also calibrated nationally. The method accounts for noise in the data, noise in the stochastic process, lags in reporting and day of the week variation.

With R_t estimates for weekly time points t=1, …, τ (τ=19 weeks between 1 July 2020 and 4 November 2020) and for each individual LTLA indexed by m=1,...M (M=310 individual local authorities), we use a regression model to estimate the effect size of the introduction of tiers. Specifically, for each LTLA m, we have a response variable ym∈Rτ of R_t values, and a binary design matrix xm∈1Rτ×3, where each column corresponds to the backdated tier 1, 2 or 3, and the indicator function is a binary toggle for whether an LTLA is in a given tier on a given date. Defining tiers cumulatively (eg, if an LTLA is in tier 2 then it is also in tier 1) makes little difference to our results. We propose a latent factor model that estimates the effect of tiers, but that also accounts for confounding from other factors. We name these other factors secular trends, and they represent changes in transmission due to individual or wider societal behavioural changes along with LTLA idiosyncrasies and demographics, which are not part of the formal tier system.

The linear model specification is:

ym∼Normal(μm,σ)μm=exp⁡(λαmT−βxmT)β∼Normal+(0,ϕ1)αm∼Normal+(0,ϕ2)λ∼Normal+(0,ϕ3)ϕ1,ϕ2,ϕ3∼Normal+(0,2)σ∼Normal+(0,2)

In the absence of LTLA-mandated interventions, a given LTLA m has a trend λαmT in its reproduction number over time R_t. The above model is multiplicative: tiers reduce the R_t trend λαmT by a factor of exp(−βxmT), for β∈R3 . The secular trends λαmT can be factorised into two basis functions with internal dimension b, that is, a national trend λ∈Rτ×b and a regional scaling αm∈Rb . This factorisation enforces shared trends across all LTLAs. The internal dimension b can be interpreted as the number of factors or categories that each LTLA has (to varying extents), in addition to the tiers, that may influence transmission intensity. b can also be interpreted as the number of ways in which regional R_t values can differ from national values. Model flexibility and complexity increase with b, and we set b=2. Note that the national trend λ has the same values for all LTLAs, while each αm basis differs by LTLA m. The first column in the basis function λ is an intercept, and the second are τ independent parameters. Normal shrinkage priors are used for λ to balance overfitting and underfitting. Conceptually, the intercept component of the basis matrix λ accounts for different LTLA starting values in R_t, which could be due to different demographics, behaviour or contact patterns.

The second basis αm scales and moves the national secular trend λ for a given LTLA m. The secular trend broadly corresponds to changes over time due to behaviour and other national policies that are independent from the tiers (such as ‘the rule of six’). However, the absolute impact of the national trend can vary by LTLA. That is, the overall shape of the trend is shared nationally, but its magnitude and starting value can differ across LTLAs. As well as being relatively parsimonious, this formulation captures the interdependence of R_t values between LTLAs, while not assuming any a priori relationship between LTLAs.

It is possible to allow for a larger internal dimension value b, but given that our R_t estimates and backdated tiers consist of only 22 weeks of data for each LTLA, increasing b risks overfitting. With greater internal dimension, the model is too flexible and so the data will struggle to separate the effects of the tiers βxmT from the underlying trend in each LTLA’s transmission λαmT . A similar problem arises if the priors for the β parameters are not restricted to non-negative values. Such additional model complexity spuriously nullifies the effects of tiers. Alternative bases for λ such as b-splines could be used, but we opted for a non-structured basis to limit a priori assumptions. Partial pooling of β, whereby the effects of tiers would differ by LTLA, was investigated but did not produce significant differences between LTLAs.

We conducted a sensitivity analysis on the priors of the hyperparameters φ₁, φ₂, φ₃ and σ of the hierarchical model (which, respectively, inform the values for tier coefficients, the LTLA-specific scaling of the secular trend, the secular trend itself and the random noise around the resulting R_t values). In our main model these are given as positive normal distributions with SD of 2. However, allowing the SD to take values of 0.5, 1 or 2.5 made negligible difference to our results. Further, we investigated heavy tailed Cauchy distributions. While this resulted in a posterior distribution with greater curvature, it made negligible difference to estimated tier coefficients. Our results are therefore robust to changes in the prior, and we therefore opted for normal distributions with SD of 2 for our main model, as this prior is wide and uninformative.

We fit to R_t estimates by LTLA over time, and not to cases and deaths. NPIs and tiers ultimately affect transmission, which in turn affects cases and deaths only after a delay. Therefore, regressing on the time-varying reproduction number is preferable as such delays do not need to be incorporated into the model.

Fitting was performed using the Bayesian software platform Stan,8 via the R package ‘rstan’9 and R V.3.6.3,10 using 2000 iterations with a warm-up of 500 iterations across 10 chains. R-hat and other diagnostics indicated a fully converged model. All model code and data sets are available from either the authors, or alternatively at https://github.com/ImperialCollegeLondon/covid19model.

We estimated the effects of tiers 1, 2 and 3 between the first and second national lockdowns in the UK. We estimated of the local time-varying reproduction number R_t using our established semimechanistic model. We combined these estimates with data detailing the timing of NPIs and tiers at LTLA level to use as inputs in a Bayesian hierarchical model with a latent factor analysis.

Our approach aimed to account for broader secular trends at the national level in addition to the tiers at LTLA level. The model reproduced R_t estimates reasonably well and broadly captured their variation over time and variation between LTLAs. The national secular trend increased between early July and late September, notwithstanding regional differences and control measures applied. The national trend began to plateau from mid-September (before the introduction of the tier system) and declined shortly after, implying that transmission reduction was not solely attributable to tiers. We find that tier 1 has almost no effect on transmission beyond the secular national trend, and that tier 2 yielded only minor reductions in transmission. However, tier 3 was more effective, reducing transmission by an average of 23%. We estimated that, had tier 3 been in effect throughout all LTLAs, the real-time reproduction number R_t would have been reduced to below 1 in 93% of LTLAs (288 LTLAs) at the start of the second national lockdown, as opposed to the 29% of LTLAs (90 LTLAs) that actually did manage to reduce R_t to below 1 at this time.

Our approach has a number of limitations. While the latent factors account for confounding to some degree, the coefficients we estimate are non-causal and therefore provide only associative effects. We do not consider the interaction of interventions, but merely their joint effect as mandated through the tier system. In generating counterfactuals our model makes a ‘difference in differences’ counterfactual assumption, which has previously been shown to have limitations11 arising from the assumption of parallel trajectories. It is important to note that the effect sizes we model quantify the instantaneous and constant impact of the tiers on R_t, whereas the effects of tiers may vary over time, perhaps with a lag before they take effect or with a waning of efficacy.

Our backdating of tiers is imperfect: government announcement of a given tier may have an additional effect beyond that of the particular NPIs within that tier. However, it remains a reasonable approximation that enables the analysis of NPIs in finer detail than lockdown. It would be useful to measure the effects of the specific interventions that and comprise the tiers, as this would enable more targeted measures for COVID-19 control. However, the data are unfortunately insufficiently powered to make such inferences.

Further, our priors assume that tiers will either reduce transmission or be ineffective, but do not allow tiers to increase R_t. However, given that reduced human contact must reduce transmission, this is a reasonable assumption. We also do not propagate uncertainty in the reproduction number in the hierarchical model, as the high number of LTLAs renders this computationally intractable. The development of frameworks that can propagate uncertainty reliably is an important priority for future work.