Data on Covid-19 deaths are often unreliable. In Germany the number of Covid-19 deaths given by the Statistisches Bundesamt is taken from the Robert Koch Institute which requires a blood test confirming the presence of the corona virus at the time of death, symptoms are not sufficient.
In the UK a death certificate signed by a doctor giving Covid-19 as the cause of death would be accepted even if the the judgment of the doctor was based purely on the symptoms. People may also die due to reasons only indirectly attributable to the disease: patients may be refused treatment because of the strain put on medical services, others may not even seek hospital treatment through fear of contracting the disease.
For these and other reasons it has been suggested that excess deaths may give a more reliable measure of the effect of the Covid-19. Excess deaths are defined as the difference between the number of present deaths and a baseline, typically defined historically as the average number of deaths over previous years. The number of deaths can be usually be accurately determined, the main problem is that of delays. For example on the 27th May Spain increased the number of deaths by 12000 backdated to early March. To avoid such problems this paper only uses data from the UK, and in particular England and Wales, Germany and Belgium. The data is accurate and up-to-date with Germany being the laggard with delays of about three weeks. The sources are the Office of National Statistics for the UK, the Statistisches Bundesamt for Germany and the Belgian statistical office Statbel for Belgium. However even with such data the there still remain problems of comparing excess deaths. The reason is the effect of past influenza epidemics on the excess deaths calculated today. In 2018 there was a major influenza epidemic in Germany. If this one year is removed from the baseline the number of excess deaths for the weeks 11-21 increases from 7602 to 11639. It is as though the influenza epidemic of 2018 masks about 4000 excess deaths.
The problem can largely be avoided by taking the baseline to be the 10% quantile rather than the mean of deaths of previous years.
One widely used statistic for comparing excess deaths across countries is the Z-score. In particular it is the main means of comparison used by Euromomo https://www.euromomo.eu/ Under the "Objectives" one reads
The aim of the EuroMOMO project (European monitoring of excess mortality for public health action) is to operate coordinated timely mortality monitoring and analyses in as many European countries as possible, using a standardized approach to ensure that signals are comparable between countries. and under the heading "What is a z-score?" Z-scores are used to standardize series and enable comparison mortality pattern between different populations or between different time periods. The standard deviation is the unit of measurement of the Z-score. It allows comparison of observations from different normal distributions.
Unfortunately comparison of Z-scores without knowledge of the population sizes is impossible. Such comparisons can distort the effects of an epidemic in different countries by a large factor.
We show why this is so and why the use of Z-scores should be abandoned. It can be replaced with the much more intuitive concept of a P-score.
Epidemics exhibit themselves in the data as peaks. Quantifying epidemics is then the problems of localizing and evaluating the sizes of peaks. In the context of epidemics we take this to mean the quantification of an epidemic through its start and finish, its peak and the total number of deaths it caused. A similar problem occurs in the X-Ray diffractograms and we adopt the approach taken in [Davies et al., 2008] .
The number of excess deaths in any week is defined as the number of deaths in that week minus the baseline for that week. Little attention has been given to the baseline, in particular there seem to be no comparisons of the baselines of different countries. To show how large the baseline effect can be we consider data for England and Wales, Germany and Belgium.
The top panel of Figure week later than those of England and Wales. The increase in deaths due to heat waves is also visible in the German baseline and to a lesser extent in the Belgium baseline.
The main difference between the the baselines in the top panel is that the German one is clearly higher than than the other two. This is due to the difference in age distributions. England and ew0+  152  150  149  152  de0+  189  191  193  195  be0+  158  158  158  159  ew65+  679  670  670  686  de65+  755  765  774  792  be65+  684  688  687  704  ew064-29  28  29  29  de064-38  37  37  37  be064-34  34 32 32 3 Z-scores and P-scores
Given data X n = (X 1 , . . . , X n ) with meanX n we model the data as i.i.d. N (µ + δ, σ 2 ) with known µ and σ. We require a confidence interval for δ. The Z-score is
Thus with probability 0.95
One notes that the Z-score is divided by √ n.
We now consider the case where X satisfies a Poisson distribution with mean λ(1 + δ) with known λ. Again a confidence interval is required for δ. The Z-score is
for large λ. For example for λ = 300 the Kolmogorov distance between distribution functions of (X − λ)/ √ λ and N (0, 1) is less than 0.01. Thus we may write
with Z ≈ N (0, 1) and hence
If λ is the number of weekly deaths in a large European population of size n then λ ≈ 0.01n/52 = 0.00019n so the confidence interval becomes
If there is over dispersion, that is the variance of X is σ 2 λ for some σ the confidence interval becomes δ : |δ − 72Z sc / √ n| ≤ 141σ (1 + δ)/n Without knowledge of n and σ no information about δ can be obtained simply from the Z-score.
The Z-scores published by Euromomo are more complicated than this although exactly how they are calculated is not known. Whatever the case they are simply misleading and consequently worthless.
We define the P-score for the ith week as (x(i) − bsln(i)/bslin(i) where x(i) is the number of deaths and bsln(i) the value of the baseline for that week (see [Aron and Muellbauer, 2020] ). In contrast to the Z-scores the P-scores do not depend on the size of the population. The P-scores for the three countries and three age groups considered here are shown in Figure 6 .
In his book [Huber, 2011] Peter Huber points out that the word 'model' has 'a bewilderingly wide semantic range' but with a common feature a model is a representation of the essential aspects of some real thing in an idealized, exemplary form, ignoring the inessential ones.
EuroMOMO aims to detect and measure excess deaths related to seasonal influenza, pandemics and other public health threats.
Given this it is strange that EuroMOMO should choose the Poisson model which, with a single parameter, is too simple to enable EuroMOMO to accomplish its aims. Because of this EuroMOMO has to perform several operations on the original data before calculating the Z-scores:
Z-score(s) are computed on the de-trended and de-seasonalized series, after a 2/3 powers transformation according to the method described in Farrington et al. 1996 . This enables the computation of Z-scores for series that are originally Poisson distributed.
The aims and the calculation of the Z-scores seem to contradict each other. If you want to measure excess deaths including seasonal data it seems contra-productive to de-trend and de-seasonalize.
The 2/3 powers transformations of [Farrington et al., 1996] are to take into account the asymmetry of the Poisson distribution when calculating Z-values. The smallest value of λ for the 24 countries considered by EuroMOMO is about 350 for Northern Ireland. The maximum difference between the cumulative distribution functions of the Po(350) and its normal approximation is about 0.014 so it hardly seems worthwhile making this correction. Moreover you can calculate the Z-score exactly without the normal approximation. Suppose the data are x and the model is Po(λ). The P-value for x is p = P λ (X ≥ x) which can be calculated using any statistical software. The
Even all this is not sufficient. There is over dispersion which must be estimated, also local trends, restriction to spring and autumn months: see https://www.euromomo.eu/how-it-works/methods/ Excess deaths due to influenza, pandemics, heat-waves and other public health threats manifest themselves in the data as peaks in the data: see Figure 8 which gives the number of daily deaths in Germany from 01.01.2016-26.04.2020. A sharp peak in July 2019 is clearly visible caused by temperatures of around 40C. Thus the aims of EuroMOMO stated above can be translated into the statistical problem of estimating peaks in a set of data. This is a problem in shape constrained non-parametric regression. We discuss it in the next section.
Epidemics are peaks in the mortality data. Of interest could be the number of peaks, their beginning and end, their heights and locations, and their power. The same problem occurs in X-Ray diffractograms and was treated in [Davies et al., 2008] . The approach given here is a simplified version of that paper. It is based on the so called 'taut string' algorithm of [Davies and Kovac, 2001 ].
The expression 'taut string' is taken from [Barlow et al., 1972] . Thirdly, the version used here minimizes the sum of squared residuals between a local minimum and the next local maximum subject to the function being non-decreasing with the corresponding property between a local maximum and the next local minimum. This is known as isotone regression, see [Barlow et al., 1972] , but here with the extra property that the minimum and maximum values are given by the taut string. In particular it minimizes the sum of squared residuals subject to the shape constraints satisfied by the taut string. The resulting residuals will give a very good estimate of the random variability of the data.
The top panel of Figure 7 shows the Belgian daily data from 2009-2019 together with the taut string regressing function. The centre panel shows the residuals and the bottom panel a Poisson process with the taut string as the parameter. The taut string is piecewise constant as is isotonic regression. The location of local minima and maxima is taken to be the midpoint of the interval defining this local extreme value. For the Belgian data these are 203, 393, 499, 549, 593, 725, 880, 909, 954, 1145, 1275, 1304,1316 1509, 1698, 1863, 2049, 2230, 2348, 2375, 2417, 2639, 2741, 2795, 2812 2954, 3085, 3095, 3148, 3350,3465, 3500, 3552, 3683, 3833, 3858, 3895 starting with a local minimum at day 203.
It seems more consistent to take the start of an epidemic not as the midpoint but as the right The over dispersion is calculated as 25%.
The residuals look random but may still contain some residual structure not removed by the taut string. To examine this we consider periodicities and lags. The data length is 4017. We now generate sine and cosine functions sin(πj(1 : n)/n), cos(πj(1 : n)/n), j = 1, . . . , n/2 with n = 4017, regard these as covariates and use the Gaussian selection procedure ( [Davies and Dümbgen, 2020]) to select covariates. A cut-off P-value of 0.01 resulted four functions being selected. Three had periodicities of almost exactly seven days with a smallest P-value 1.1e-61. The fourth had a period of 365 days and a P-value 4.4e-07. In the case of lags we allowed for lags of up to 500 days. With a cut-off P-value of 0.01 10 lags were selected. As was to be expected the lag of order one was dominant with a P-value 1.3e-21. The P-values of the remaining nine all exceeded 2.0e-07.
The main disadvantage of the taut string method is that the resulting function is piecewise constant. To obtain smoother functions, that is one which are differentiable, is considerably more difficult as determining a derivative is an ill-posed problem. Some form of regularization is required. We do this by minimizing the total variation of the derivative subject to the multiscale and the monotonicity constraints of the taut string. This is accomplished here by solving a linear programming problem which causes a large increase in computing time; [Kovac, 2007] has a much faster but less stable version. The upper panel of Figure 9 shows the result of minimizing the total variation of the first derivative, the lower panel that for the second derivative. The first function is piecewise linear, the second piecewise quadratic.The computing times were about ten minutes.
The standard deviations of the residuals were respectively 106 and 106.5, much larger than the standard deviation of the taut string residuals.
I thank Lutz Dümbgen and John Muellbauer for instructive exchanges of emails. 