Testing has grossly underestimated Covid-19 cumulative cases
Peter Roderick, Alison Macfarlane, and Allyson Pollock highlight multiple failings of the partly privatised testing regime for Covid-19. Their argument is strengthened by current evidence on the scale to which official data  for laboratory-confirmed cases underestimates the true numbers infected in England over the course of the pandemic.
As of 28 June 2020, cumulative confirmed cases are reported as 160,150 for England, a figure which excludes “tests carried out by commercial partners”, and 310,250 for the UK, including the commercial tests. Three results from the Medical Research Council Biostatistics Unit, University of Cambridge  suggest that the cumulative number of infections in England is over 4 million. Separate results from the most recent Office for National Statistics antibody sampling  indicate 3 million people in England have had the disease.
The MRC estimates the cumulative number of cases in England at 4,730,000 with 95% Credible Interval (4,270,000–5,240,000). They also estimate an Attack Rate of 8% (8%--9%), which implies 4.5 million infections in a population of 56.3m . Finally, they estimate an Infection Fatality Rate of 1.1% (0.9%--1.4%). Using the latest ONS data for death registrations  which shows 46,425 Covid-19 related deaths in England registered by 20 June, the estimated IFR implies a cumulative case total of 46,425 / 0.011 = 4.22 million.
The latest ONS antibody sampling estimated 5.4% (4.3%--6.5%) of people in England tested positive as of 13 June, which implies 3 million infections. However, antibodies only develop two weeks after infection; sampling was limited to the community, excluding hospitals, care homes, and other communal establishments; and the estimate was not reweighted to be representative.
Whilst the MRC and ONS estimates involve models and sampling, the confirmed cases are an empirical result. However, it is hard to believe that the discrepancy reflects a massive modelling error by the MRC and, independently, faulty sampling by the ONS. Rather, it shows the extent to which the testing regime has failed to find the cases, over the course of the pandemic.
The problem is compounded at local authority level, where the government data excludes the commercial testing results (which are not tied to postcode). The local authority data is then scaled by population to produce local rates, displayed on a map, uncorrected for local variation in extent or accuracy of testing. The inadequate data is reproduced on the BBC  with the mild caveat “However, these numbers only include people who have been tested”. Media reports may omit the caveat, as in “Official figures show Liverpool registered 544 coronavirus-related deaths up to 12 June and 1,677 cases up to 25 June.” 
With an IFR of 1.1%, 544 deaths implies around 50,000 cases in Liverpool, a figure which will come as an unwelcome surprise.
Competing interests: GD is a member of Keep Our NHS Public.