CoVID-19 appeared in the UK in early 2020, with its first case, imported from China, reported on the 30th January, and the first CoVID-linked death announced on the 6th March. Over the last few months the pandemic has spread in the community, hospitals and care homes. As at 5th September 2020, around 300,000 cases have been lab-confirmed and of these over 40,000 have died. The lock-down introduced on the 23rd of March has drastically reduced transmission, although the process of easing it is proving very challenging. Underlying the governmental response is surveillance information on a number of epidemiological indicators. In particular, trends in the number of deaths have been crucial to monitoring the burden of CoVID-19 as well as to informing models reconstructing and predicting the pandemic. However, information on these trends is affected by delayed reporting. While the number of deaths in people with lab-confirmed CoVID-19 infection peaked on 9th April, the number of reports of such deaths peaked 13 days later, on 22nd April. This is because deaths are rarely reported to the authorities on the day they occur, and some deaths take weeks to be reported. This results in righttruncated data: deaths occurring on day t are only observed if their reporting delay d is such that day t + d is earlier than the day up to which data are available [3] . This right-truncation makes it difficult to establish the true pattern of the numbers of deaths occurring over time and, in particular, to identify quickly any increase in the most recent days. The process of estimating the number of deaths that occurred on each day from the numbers of deaths that have so far been reported is referred to as 'nowcasting'. In this article, we nowcast the number of deaths in people with CoVID-19 both for the whole of England and separately for seven English regions, and we break down these numbers by five age strata: 0-44 years, 45-54 years, 55-64 years, 65-74 years and ≥ 75 years. To produce these nowcasts, we jointly model the distributions of reporting delays, the observed numbers of reported deaths and the actual numbers of deaths in each of the age strata within a region.
We use data on deaths in people with a laboratory-confirmed CoVID-19 diagnosis reported to Public Health England (PHE). These reports come from three sources: National Health Service England (NHSE), the Demographics Batch Service (DBS) and Health Protection Teams (HPT). NHSE reports include deaths occurring in hospitals notified by NHS trusts through the CoVID-19 Patient Notification System and reported daily to PHE. DBS deaths result from the daily tracing of CoVID-19 positive tests reported to PHE through the Second Generation Surveillance System in the NHS Spine to identify individuals who might have died in the previous 24 hours in any setting. HPT data include deaths notified by PHE HPT during outbreak management, primarily in non-hospital settings (e.g. care homes). These three data sources are not independent; duplications are identified daily and only the earliest report date is considered.
We consider data from 22nd March up to 29th June. We chose 22nd March because the first reports from DBS became available on 21st March and some of the deaths reported on 21st would have been reported earlier had DBS been used earlier. At 29th June there were a total of 37529 confirmed reports of deaths that occurred between 22nd March and 29th June. Figure 1a shows the number of deaths that occurred on each day and had been reported by 29th June. The effect of reporting delay is particularly evident in the last three days: while the numbers of deaths per day in the week up to 25th June were between 50 and 100, the numbers for 27th, 28th and 29th June were 12, 9 and 0. Figure 1b shows the number of deaths that were reported on each day. It can be seen that the number of reports tends to be lower on Sundays and Mondays, probably due to some staff not working during the weekend. This is particularly so in more recent weeks. In addition, the number of reports over a single Tuesday-to-Saturday period varies considerably from day to day. It is particularly low, for example, on 26th May. Figure 2 shows a simple estimate (see [3] for method) of the distribution of reporting delay by age strata, calculated using data from the whole of England and assuming that the delay distribution is the same for all times of death. This distribution is conditional on the delay being at most 42 days. Forty-two days was chosen because 2 . CC-BY 4.0 International license It is made available under a perpetuity.
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The copyright holder for this this version posted September 16, 2020. only 61 of the 37529 reported deaths had delays longer than six weeks. Compared to the three strata of individuals aged 45-74, the delays tend to be slightly longer on average in the ≥ 75 years stratum and even more so in the 0-44 years stratum.
A simple way to adjust the observed figures for reporting delay is to use the estimates in Figure 2 (or corresponding estimates based on data from a single region) to weight up the numbers of deaths reported so far [3] . For example, of the 12 deaths that occurred on 27th June and had been reported by 29th June, nine were in the ≥ 75 strata. The estimated proportion of deaths reported within two days in this strata is 0.60. Therefore, the actual number of deaths in this strata can be estimated as 9/0.60 = 15. However, this estimate does not account for any dependence of the delay distribution on the day of death. Delays may length or shorten over calendar time and may depend (as Figure 1b suggests) on the weekday on which the death occurred. Also, while the numbers of deaths occurring on each day are likely to change smoothly over calendar time, this estimate makes no use of information about the numbers of deaths that occurred on the preceding and succeeding days, with the result that it can be very different from the estimates for those days. In addition, simple confidence interval calculations [3] underestimate uncertainty, unless the delay distribution really is the same for all times of death.
The problem of adjusting numbers of reported events to account for reporting delay arises in several contexts, including insurance claims [6] and epidemiology. Much work was done in the 1980's in the context of HIV/AIDS, e.g. [2, 3] is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
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The copyright holder for this this version posted September 16, 2020. . https://doi.org/10.1101/2020.09.15.20194209 doi: medRxiv preprint model that they had first applied to data on severe acute respiratory disease in Brazil [8] to nowcast the numbers of CoVID-19 death over the period from 2nd April to 5th May in each of seven English NHS regions. They use reports from NHS England only, rather than also reports from DBS and HPT. Their hierarchical model is itself an elaboration of a non-hierarchical model they had first applied to Dengue Fever [10] .
The general approach of S&E has several advantages. It allows flexible modelling of weekly and longer-term changes in the reporting delay distribution. It accounts for overdispersion in the delay distribution, which ensures that variation in this distribution which cannot be explained by such changes does not lead to the estimates of the numbers of deaths having excessive variance [10] . By modelling the expected number of deaths per day in each stratum as a smooth function of time, it allows the estimate of the number of deaths that occurred on one day to borrow information about the numbers of deaths that occurred on the preceding and succeeding days. Also, information about the delay distribution and pattern of the epidemic in one strata can be borrowed to inform those quantities in the other strata. Uncertainty can be quantified by posterior credible intervals.
In this article, we build on S&E's hierarchical model to nowcast in each region, with stratification by age group. Our model improves on theirs in a number of ways. First, we model weekday reporting effects in a more realistic fashion. As mentioned above, fewer deaths are reported on Sundays and Mondays, and the strength of this weekend effect changes over time. Second, we account for correlation between the numbers of reports made on the same day about individuals who died on different days. If, for example, the data suggests that an unusually low proportion of those deaths that occurred between two and seven days ago have been reported today, it is likely that few of the deaths that occurred yesterday and today will have yet been reported. Third, we describe a computationally efficient way to constrain the delay distribution to be the same in multiple strata. This is particularly useful if the delay distribution can be expected to be the same across a wide range of ages. We also illustrate how to use the model to calculate a posterior probability that the number of deaths has been rising in recent days.
Denote the first day for which data on reporting delays will be used as day 0 and the most recent day for which data are available as day T .
We shall model deaths that are reported within D days. For the CoVID-19 data, we set D = 42, because, as noted in Section 1, very few deaths were reported with delays greater than six weeks. Let D t denote the longest delay that we could have observed in individuals who died on day t, excluding any individuals with delays greater than D days. So, D t = D if t ≤ T − D and D t = T − t otherwise. Let Z std denote the number of deaths that occurred on day t (t = 0, . . . , T ) in age stratum s (s = 1, . . . , S) and are reported with a delay of d days (d = 0, . . . , D). We observe Z std when d ≤ D t ; otherwise Z std is unobserved. We used S = 5 strata:
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The copyright holder for this this version posted September 16, 2020. We model the data in a Bayesian framework, by specifying three submodels: one for the marginal distribution of the number of deaths, Y st , occurring on each day in each stratum; one for the conditional distribution of the observed numbers of reports, Z std , given the numbers of deaths and the delay distribution; and one for the delay distribution. Figure S1 shows a directed acyclic graph for the model (all figures prefixed by 'S' are in the Supplementary Materials). We fit this Bayesian model using a Markov chain Monte Carlo (MCMC) algorithm.
The number of deaths, Y st , that occurred in stratum s on day t is unobserved for t > T − D. Assume that, for stratum s = 1, . . . , S and time
where λ s,T −D+1 , . . . , λ sT and θ are unknown parameters.
. We shall assume that λ st changes smoothly over time t.
The model described in the last paragraph does not use data on the observed numbers of deaths Y st that occurred in the period before day T − D + 1. Such data may be useful, as they indicate whether the number of deaths per day was rising or falling over that period. It could reasonably be expected that the trajectory of the number of deaths per day in the period shortly after day T − D + 1 would be a continuation of the trajectory immediately before it. To exploit this information, we shall assume that expression (1) also holds for t = T 0 , . . . , T − D, where T 0 is some constant between 0 and T − D. We chose T 0 = T − D − 13, so that we use the most recent two weeks for which Y st is observed.
We follow S&E in using restrictive penalised cubic splines to model the assumption that λ st changes smoothly over time t. Furthermore, we assume that λ sT 0 , . . . , λ sT may follow similar trajectories in different strata s, while allowing these trajectories to differ. More specifically, log λ st (t = T 0 , . . . , T ) is taken to be the sum of an unknown stratum-specific intercept term ι s and two restricted cubic splines, one of which is common to all strata and one which is specific to stratum s. That is, log
Here,μ |τ ∼ Normal(0, Vτ −1 ) and µ s | τ s ∼ Normal(0, V τ −1 s ), x j (j = 1, . . . , T − T 0 + 1) is the jth row of the design matrix X for the splines, and V is the corresponding variance matrix [11, 12] . The unknown parametersτ and τ s determine the smoothness of the splines. If τ s = ∞ for all s, then µ s = 0, and 6 . CC-BY 4.0 International license It is made available under a perpetuity.
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The copyright holder for this this version posted September 16, 2020. . so λ st = λ t , i.e. the patterns of mortality in the five strata are the same. The R function jagam [11] can be used to calculate X and V .
For stratum s and time t, we model Z st0 , . . . , Z stD as
where P std is the unknown probability that an individual in stratum s who died on day t is reported with a delay of d days.
We shall allow P std to depend on t through both a smooth function of calendar-time and weekday reporting effects: average reporting delays may gradually shorten or lengthen over calendar time and Figure 1 shows deaths are less likely to be reported on Sundays and Mondays. However, even after accounting for these systematic effects, the observed numbers of daily death reports Z std may be overdispersed compared to a multinomial distribution. As S&E (2019) showed, failure to allow for such overdispersion can lead to overdispersion in the numbers of deaths Y st , and thus to unnecessarily imprecise nowcasts. So, to allow for overdispersion in Z std , we assume that, for each s and t, (P st0 , . . . , P stD ) is drawn from a generalised Dirichlet (GD) distribution [4] . That is, the density of (P st0 , . . . , P stD ) is
where B(., .) denotes the beta function (which is a ratio of gamma functions) and α st = (α st0 , . . . , α st,D−1 ) and β st = (β st0 , . . . , β st,D−1 )) are unknown parameters that will be modelled as functions of s, t and d (note that β st,j−1 = 0 when j = 0). The GD distribution is a flexible generalisation of the Dirichlet distribution; it allows, for example, positive correlation between P std and P std for d = d , unlike the Dirichlet distribution [4, 10] .
The marginal distribution of (Z st0 , . . . , Z stD ) (given Y st , α st and β st ) obtained by integrating out (P st0 , . . . , P stD ) is the generalised-Dirichlet-multinomial (GDM) distribution, which has a closed-form expression [4, 10] . This GDM distribution can be factorised as the product over
This distribution has probability mass function
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Since Z std is unobserved for d > D t , we only need include the likelihood contribution of (Z st0 , . . . , Z stDt ), i.e. the product of expression (4) over d = 0, . . . , min(D t , D− 1). By integrating out the unknown random variables Z st,Dt+1 , . . . , Z stD in this way, the Markov chain Monte Carlo (MCMC) algorithm used to fit the model is simplified compared to the algorithm used by S&E, the computational load is reduced, and the mixing of the Markov chain should be improved. Note that the Z std ! term in expression (4) can be ignored when calculating full-conditional distributions, because it does not depend on any unobserved variables.
We reparameterise the GD distribution in terms of
j=d P stj is the expected hazard of a reporting delay of d days for an individual in stratum s who died on day t [4] . The variance of this hazard is
Note, in particular, that as φ std → ∞, this variance tends to zero, which makes the GDM distribution of (Z st0 , . . . ,
Correspondingly, the beta-binomial distribution of equation (4) reduces to a binomial distribution. Thus, the φ std parameters describe the overdispersion of the numbers of reports relative to a multinomial distribution.
It is easy to show that the expected hazard v std can also be written as
where q std = E d j=0 P stj is the expectation of the probability that the delay is at most d days for an individual in stratum s who died on day t.
We model weekday reporting effects by assuming for d = 0, . . . , D − 1 that
Here, I(.) is the indicator function, and Day(t + d) = 1 if day t + d is Monday, Day(t + d) = 2 if day t + d is Tuesday, and so on. The quantities q * std and η s1 , η s2 , η s3 , η s4 , η s5 , η s7 are unknown parameters; we constrain η s6 = 0 for identifiability (Saturday is our baseline). The function g W st is a smooth function of t, which 8 . CC-BY 4.0 International license It is made available under a perpetuity.
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The copyright holder for this this version posted September 16, 2020. . describes how the weekend effect (i.e. the effect of its being Sunday or Monday) changes over calendar time. As we did earlier with g M st , we assume g W st is the sum of two restricted cubic splines, one common to all strata and one specific to stratum s. Equation (6) means that, given the delay is at least d days, the odds of a death on day t being reported with a delay of d is exp{η sj + g W st I(j = 1 or 7)} times greater when day t + d is weekday j than it would be if day t + d were Saturday.
To model calendar-time effects, we assume for d = 0, . . . , D − 1 that
where q † sd is an unknown parameter and g C st is a smooth function of t. As with g M st and g W st , we assume g C st is the sum of two restricted cubic splines, one common to all strata and one specific to stratum s. (6) lies between 0 and 1, we reparameterise q † sd as q † sd = 1 − d j=0 ψ sj , where ψ s0 , . . . , ψ s,D−1 take any values between 0 and 1. Note that if η 1 = . . . = η 7 = g W st = g C st = 0, then ψ sd = 1 − v std and q † sd = q std . In Section 4, we contrast our approach to modelling weekday and calendar-time reporting effects with the approach used by S&E.
The weekday effects η sj and g W st are reporting-day effects: they reflect the phenomenon that deaths may be more likely to be reported on certain days of the week. In addition to these systematic reporting-day effects, there may be non-systematic reporting-day effects. That is, there may be individual days on which the numbers of reports is unpredictably higher or lower than would be expected given the day of the week and the number of deaths that have occurred in recent days. Figure 1b suggests this is true, for example, of 26th May. These reporting-day effects would alter the expected hazard v std on day t+d for delays of all lengths d = 0, . . . , D −1, and hence induce correlations between P std , P s,t+1,d−1 , P s,t+2,d−2 , P s,t+3,d−3 , . . .. Such correlations are not reflected by the GD distribution of equation (3), which instead induces correlations between P st0 , P st1 , P st2 , P st3 , . . .. To allow for these nonsystematic reporting-day effects, we add a random effect δ t+d to equation (6), i.e. we change (6) to
where δ 0 , . . . , δ T are i.i.d. Normal(0, ω 2 ) and ω is an unknown parameter.
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The model described so far assumes the delay distribution for individuals who die on day t is different in each stratum s. For the CoVID-19 data, it is reasonable to assume the delay distributions are equal in strata s = 2, 3 and 4, i.e. individuals aged 45-74. We considered two alternative adaptations of the model to impose the constraint that the delay distribution is the same in some group S of strata (where S = {2, 3, 4} for the CoVID-19 data). The first adapted model sets the parameters and splines η sj , ψ sd , g W st and g C st that determine the delay distribution in stratum s to be equal for all s ∈ S. We denote the common values by a dot, so η sj = η •j , ψ sd = ψ •d , g st = g W •t and g st = g C •t . Now v std , the expected hazard of reporting on day t + d for an individual who died on day t, is the same for strata s ∈ S. If, in addition, φ std = φ •td for all s ∈ S, then {(P st0 , . . . , P stD ) : s ∈ S} have the same GD distribution. The second adapted model assumes P std = P •td for all s ∈ S, i.e. delay distributions {(P st0 , . . . , P stD ) : s ∈ S} are exactly equal. Let Z •td = s∈S Z std , Y •t = s∈S Y st and W •t = s∈S W st . We show in the Appendix that when P std = P •td ,
Assuming (P •t0 , . . . , P •tD ) is drawn from a GD distribution, we have (as in expression (4)),
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Compared to the first adapted model, this second model has the advantage of involving less computation. This is because each time η •j , ψ •d , δ j or a parameter of the splines underlying g W •t and g C •t is updated in the MCMC algorithm the single expression (8) is evaluated, rather than the |S| expressions (4). The only time expression (9) is evaluated is when Y st is updated. Note that the Z •td ! and W •t !/ s∈S W st ! terms in expressions (8) and (9), respectively, can be ignored when calculating full-conditional distributions, because they are functions only of observed variables.
The intercept term ι s in the model for log λ st was assigned a non-informative normal prior with mean zero and variance 100. For the priors on the precision parameters τ and τ s of the splines for g M st we used the same inverse-gamma with shape and rate 0.5 that S&E (2020) used. We also used this prior for the corresponding precision parameters of the splines for g W st and g C st . The prior on the dispersion parameter θ of the negative binomial distribution for Y st was an exponential distribution with rate 2. This assigns considerable mass to values close to zero (no overdispersion), while still allowing considerable overdispersion.
For the delay distribution, we used a non-informative uniform prior on ψ sd and, following S&E (2019), used an exponential distribution with rate 0.01 for the prior on φ d . Weekday effects η sj were given non-informative normal priors with mean zero and variance 100. The prior on the precision, ω −2 , of the reporting-day random effects was a gamma distribution with shape 1 and rate 0.005. This implies a prior mean of 0.125 on the standard deviation ω, with 90% prior credible interval (0.04, 0.31). The value ω = 0.125 corresponds to 95% of the odds ratios exp(δ t+d ) lying between 0.78 and 1.28; ω = 0.31 corresponds to them lying between 0.54 and 1.84.
The model was fitted using a MCMC algorithm, implemented in R using the NIM-BLE package [5] . R code provided by S&E (2020) was used as a template for our code. Our R code is provided in the supplementary materials. The unknown random variables that appear (and are updated) in our MCMC algorithm are: the intercepts ι s ; the weekday effects η s1 , . . . , η s7 ; the parameters ψ s0 , . . . , ψ s,D−1 of the delay distribution; the numbers of deaths Y s,T −D+1 , . . . , Y sT ; the reporting-day random effects δ 0 , . . . , δ T and their standard deviation ω; the parametersμ, µ s , τ and τ s of the splines underlying g M st ; and the corresponding parameters of the splines underlying g W st and g C st . See Figure S1 for a directed acyclic graph of the model. We generated 300,000 iterations, of which the first 50,000 were discarded as burn-in.
In this section, we describe results from analysing data from the whole of England. Nowcasts produced by analysing the data from each of the seven NHS regions 11 . CC-BY 4.0 International license It is made available under a perpetuity.
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The copyright holder for this this version posted September 16, 2020. . separately are provided in the Supplementary Materials. We take the delay distribution to be the same in the three strata aged 45-74, using the second adapted model described in Section 2.3.4.
For these data, we made two modifications to the model described in Section 2.
First, as Figure 2 shows, only 3% of delays are longer than 14 days. We grouped these into weeks. That is, delays of 15-21 days were grouped together, as were 22-28, 29-35 and 36-42 days. We also grouped delays of zero days and one day, because only 445 (4.6%) of the 9719 delays of zero or one day were equal to zero. Second, when fitting the model by MCMC, we struggled to achieve good mixing of g W st and g C st , the weekend and calendar-time effects. This task may have been made more difficult by the small number of age groups (3) and large imbalance in the numbers of deaths in these groups (461, 100042 and 27020 in the 0-44, 45-74 and ≥ 75 groups, respectively; we excluded six deaths in people with unknown age). To improve the mixing of g W st and g C st , we removed, for both the weekend and the calendar-time effects, the spline specific to the ≥ 75 group, leaving one spline for the effect in the ≥ 75 stratum, one for the difference between the effects in the 0-44 and ≥ 75 strata, and one for the difference between the 45-74 and ≥ 75 groups.
Weekday reporting effects Figure 3 shows the posterior distribution of the weekday effects η sj in the ≥ 75 age stratum. As expected, the hazard of reporting is lower on Sundays and Mondays than on other days. It is also somewhat higher on Fridays than on the remaining days. Figure 4 shows the posterior mean and pointwise 95% posterior credible interval (CI) of g W st , the change in the weekend reporting effect over calendar time. The change is substantial: the posterior mean of g W st decreases from around 0.7 at the end of March to around −0.9 at the end of June. Since the posterior mean of η 57 is −1.2, this means that the odds ratio of reporting on a Sunday relative to a Saturday is estimated to be around exp(−1.2+0.7) = 0.6 in late March but around exp(−1.2 − 0.9) = 0.12 in late June. Figures S2-S5 show the corresponding results for the 0-44 and 45-74 age groups. These are similar to those in the ≥ 75 stratum, except that the credible intervals for the 0-44 stratum are wider, reflecting the smaller numbers of deaths, and hence less information, in this stratum. Figure 5 shows the posterior mean and pointwise 95% posterior CI of the calendartime effect on the delay distribution in the ≥ 75 age stratum. The reporting delays become shorter between 22nd March and the middle of April (the peak of the epidemic), and then lengthen again. In the most recent period, they appear to be shortening again. The pattern is similar in the 45-74 group (see Figure S7 ). In the 0-44 stratum, the delays also shorten between 22nd March and mid-April, but do not lengthen again after mid-April ( Figure S6 ). is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
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The copyright holder for this this version posted September 16, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
The copyright holder for this this version posted September 16, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
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The copyright holder for this this version posted September 16, 2020. . https://doi.org/10.1101/2020.09.15.20194209 doi: medRxiv preprint we see that the number of reports on Tuesday 26th May is much lower than the numbers on each of the preceding Tuesday to Saturday and following Wednesday to Friday. Although it is similar to the numbers on the preceding two days, these are Sunday and Monday, when the number of reports is expected to be low. The posterior mean of ω, the standard deviation of the random effects, is 0.23, with 95% posterior CI (0.19, 0.29).
The standard deviation of the reporting hazard P std / D j=d P std is {v std (1−v std )} 1/2 ×(φ d + 1) −1/2 (equation (5)). Figure S8 shows the posterior distribution of (φ d + 1) −1/2 for each delay d. The posterior means of (φ d + 1) −1/2 are between 0.10 and 0.15 for 1 ≤ d ≤ 7, and between 0.25 and 0.50 for 7 ≤ d ≤ 14. The posterior 95% CIs of (φ d + 1) −1/2 for d = 1, . . . , 14 do not include values much lower than 0.1. This suggests that a simpler model that replaced the GDM distribution for (Z st0 , . . . , Z stD ) given Y st with a multinomial distribution would be inadequate for these data, as such a model would effectively set (φ d + 1) −1/2 = 0.
The posterior mean of θ, the overdispersion parameter for Y st , is 0.30, with 95% posterior CI (0.08, 0.55). This suggests mild Poisson overdispersion: the variance of Y st is around 1.3 times as large as its expectation. Figure 7 shows the nowcasts for the five age strata. For ease of viewing, we show only the most recent 21 days (see Figures S9-13 for the whole period from March 31st). In the ≥ 75 stratum, nine and eight deaths occurred on 27th and 28th June, respectively, and were reported by 29th June. The corresponding estimated numbers of deaths on those two days are 63 and 66, respectively. Hence, it is estimated that only 9/63 = 14% of the deaths that occurred on Saturday 27th June had been reported within two days, and only 12% of deaths that occurred on Sunday 28th June had been reported within one day. These numbers may seem low, considering that Figure 2 shows that, on average, around 26% of deaths in this stratum are reported within one day and around 60% are reported within two days. There are likely to be two reasons for this. First, reporting is much lower on Sundays and Mondays than during the rest of the week, especially in the most recent period. Second, the estimated numbers of deaths for the days before 27th June were all above 60, and it is unlikely that the number of deaths would suddenly drop far below 60. The 95% posterior CIs for these two days are particularly wide, extending from around 45 to 90 deaths.
Finally, it is useful to be able to detect early signs of an increasing trend in the number of deaths in recent days. Figure 8 shows, for each of the strata, the posterior probability that the number of deaths in the most recent x days was greater than the number in the preceding x days (x = 1, . . . 7). There is no convincing evidence of an increase in any of the strata. However, many of the probabilities are in the region of 0.5, meaning that deaths are as likely to be increasing as they are to be decreasing or remaining stable.
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The copyright holder for this this version posted September 16, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
The copyright holder for this this version posted September 16, 2020. Timely monitoring of the number of CoVID-19-related deaths relies on methods to adjust the observed data for reporting delay. These methods need to be flexible enough to deal with the peculiar reporting patterns of these data and the way these patterns change over time. Here we built on the approach of S&E and derived agespecific delay-adjusted estimates of the number of deaths occurring in each of the seven English NHS regions.
The modelling of weekday and calendar-time effects described in Sections 2.3.1 and 2.3.2 is different from the way that S&E (2019, 2020) modelled them. If the weekday effects are removed from our model, it becomes equivalent to what S&E (2019) called the 'Survivor Model' with a calendar-time effect but without weekday effects. Figure S14 provides some support for this way of modelling the calendartime effect, suggesting that it is indeed additive on the probit scale, at least for delays of two or more days. Whereas we apply the weekday effects to the hazard in equation (6), S&E's (2020) model replaces equations (6) and (7) with
where h st is a cyclic cubic spline with a period of seven days (i.e. h st = h s,t+7 ). We believe that applying the weekday effects in equation (6) more accurately reflects these effects. Suppose, for example, that deaths are never reported on Sundays. This would be captured by our model through η 7 = −∞, but would not be captured by the spline h st , because h ts is a function of the weekday on which the individual died, rather than the weekday on which this death might be reported (see Appendix for further explanation). That weekday affects hazard of reporting is demonstrated by Figure 9 , which shows the estimated distribution of delay by weekday of death. The probability of reporting within one day is approximately the same for deaths occurring on any of Monday to Friday, but is lower for deaths occurring on Saturday or Sunday. The probability of reporting within two days is the same for deaths occurring on Monday to Thursday, but lower for those occurring on Friday and Saturday (and to some extent Sunday). For reporting within three days, the probability is the same for deaths on Monday to Wednesday, but lower for Thursday to Saturday.
An alternative way to model the calendar-time effect would be to apply it, together with the weekday effect, to the hazard in equation (6) We used data from March 22nd onwards to estimate the delay distribution, with calendar-time effect splines and weekend effect splines to allow for change over time in this distribution. As data accumulates over a longer period of time, it may become desirable to use data only from the most recent period, e.g. the last T = 90 days. This would prevent the model-fitting computation time increasing over time and limit the danger that the calendar-time and weekend effect splines are 20 . CC-BY 4.0 International license It is made available under a perpetuity.
is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
The copyright holder for this this version posted September 16, 2020. . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint
The copyright holder for this this version posted September 16, 2020. . https://doi.org/10.1101/2020.09.15.20194209 doi: medRxiv preprint insufficient to deal with changes over time in the delay distribution. To generate the results described in Section 3, which used T = 99, the computation time required to fit the model was 5.0 hours using one core of a MacBook Pro with a 2.6 GHz 6-core Intel i7 processor. This time was for the second adapted model of Section 2.3.4 and represents a reduction compared to that needed for the first adapted model (7.7 hours).
If computation time were a concern, two other measures that could be taken to reduce this, in addition to the measures already described, would be to decrease the maximum delay D and to increase the grouping of longer delays. We adjusted for delays of up to D = 42 days. S&E adjusted for delays of up to only D = 14 days. Overall, around 97% of deaths that are reported within 42 days are reported within 14 days, and so this restriction may be reasonable. We distinguished between delays of d and d + 1 days all the way up to d = 14 days, grouping longer delays into weeks. S&E, on the other hand, grouped together all delays of more than seven days. This reduces the computation time, but has the drawback that some information is discarded. Specifically, data on deaths that occurred between 8 and 13 days ago and that have been reported with delays of more than seven days are ignored. As well as potentially reducing statistical efficiency, this discarding of information allows the estimate of the number of deaths that occurred on any of these days to be lower than the number so far observed to have occurred on that day. It was for these reasons that we waited until 15 days before grouping delays and thereafter grouped them into weeks rather than into a single group.
The definition of a CoVID-19 related death is changing, with a number of alternatives being proposed based on the interval between the date of first positive test and the death event (https://assets.publishing.service.gov.uk/ government/uploads/system/uploads/attachment_data/file/908781/Technical_ Summary_PHE_Data_Series_COVID-19_Deaths_20200812.pdf). Our approach would still be appropriate to look at trends in these new data.
Finally, delayed reporting is a feature of administrative datasets, so the same problem would also affect the reporting of new CoVID-19 positive test from the current pillar schemes (https://assets.publishing.service.gov.uk/government/uploads/ system/uploads/attachment_data/file/878121/coronavirus-covid-19-testingstrategy.pdf). Adjustment for reporting delays of these data is even more crucial to a prompt monitoring of the pandemic. Although it could be addressed through the approaches presented here, further modelling considerations as well as computational efficiency would be required.