How fair is a coin toss?

The answer, according to a preprint paper by a team of University of Amsterdam researchers, is not the 50-50 odds that statistical theory would predict for a coin flip.

They found the answer after having 48 humans flip real coins from 46 different currencies and denominations a total of 350,757 times. They found the side of the coin that was originally facing up before being flipped would show up again as the result of the coin flip approximately 50.8% percent of the time, or in statistical terms, they found Pr(same side) = 0.508. They also found a 95% confidence interval for their of [0.506, 0.509].

Interestingly, they didn't find a bias in favor of either "Heads" or "Tails" when flipping coins. Those odds did come out even, as the probability of coming up heads was very close to 50%, or Pr(heads) = 0.500 with a 95% confidence interval of [0.498, 0.502].

Although the same-side bias odds are close to even, it's also enough of a bias that coin flip odds are skewed as much as some casino games are in favoring the house. Here's how corresponding author František Bartoš described it:

If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you 19$ on average. This is more than the casino advantage for 6deck blackjack against an optimal player (5$) but less than that for single-zero roulette (27$).

— František Bartoš (@BartosFra) October 9, 2023

The natural question to ask next is why are coin flips biased this way. The researchers discuss what they believe is behind the phenomenon:

This variability is consistent with D-H-M model, in which the same-side bias originates from off-axis rotations (i.e., precession or wobbliness), which can reasonably be assumed to vary between people. Future work may attempt to verify whether ‘wobbly tossers’ show a more pronounced same-side bias than ‘stable tossers’. The effort required to test this more detailed hypothesis appears to be excessive, as it would involve detailed analyses of high-speed camera recordings for individual flips.

The D-H-M model refers to a 2007 study by Persi Diaconis, Susan Holmes, and Richard Montgomery that identified the role of the laws of mechanics in determining the outcome of a coin toss based on its initial condition. They concluded in their study "coin tossing is 'physics' not 'random'".

The University of Amsterdam researcher identify a trick for how to make a coin toss fair for when its outcome makes a real world difference. The trick is to conceal which side of the coin is facing up from human observers before the coin is flipped.

The findings are surprising enough that we've built a tool that can replicate the results of human coin flipping based on the study's results. The following tool will let you set the probability of an outcome of "Heads" assuming that is always the side facing up before the coin is flipped, where we've set the default value to 50.8%. You're welcome to run through as many of these 'physics-free' simulations as you like to compare the results with physical coin flips. [If you're reading this post on a site that republishes our RSS news feed, you may need to click through to our site to access a working version of the tool.]

Percentage of Heads: Flip Coin

Note: Just as in real life, you may get several Heads or Tails in a row as you click the "Flip Coin" button before you see the result change.

Until other research says otherwise, this may be the most true-to-life coin flip simulator you can ever expect to find!

References

František Bartoš, et al. Fair coins tend to land on the same side they started: Evidence from 350,757 flips. [PDF document]. DOI: 10.48550/arXiv.2310.04153. 10 October 2023.

Persi Diaconis, Susan Holmes, and Richard Montgomery. Dynamical bias in the coin toss. SIAM Review. 2007; 49(2): 211–235. DOI: 10.1137/S003614450444643. [Ungated PDF document].

Bing Chat. Basic code for this tool was generated using the prompt: "Please write the JavaScript and HTML code for a calculator that can indicate the result of a coin flip. The calculator should let the user indicate the percentage of coin flips that will have "Heads" as a result. The calculator should provide an image of either the Heads or Tails side of the coin as part of its output." We made minor modifications to the generated code.

Image credit: U.S. Mint Coin Classroom. Half Dollar. [Online article]. United States Government works. Accessed 5 November 2023.

How different would New York City's experience during the first wave of 2020's coronavirus pandemic have been if public officials had better and more timely information about how many people were really being infected by it?

That's a fascinating question raised by the Empire Center's Bill Hammond's retrospective analysis of the pandemic's impact in New York. In that analysis, Hammond features a chart comparing the information public officials had on the number of confirmed COVID-19 cases with the Institute for Health Metrics and Evaluation's improved estimates of how extensive coronavirus infections likely were in reality in early 2020. Here's a slightly modified* interactive version of the chart:

Compared with the official count produced by the New York State Department of Health (NYSDOH) in early 2020, COVID-19 infections were much more numerous and peaked much earlier in the IHME's improved estimates Hammond describes what the chart shows:

As seen in Figure 1, the state’s outbreak likely began by early February, a full month before its first laboratory-confirmed case [2]. The estimated number of infections soared to more than 60,000 per day on March 19, which was six times higher and three weeks earlier than shown by the state’s testing data.

A second attempt to model the first wave of New York’s pandemic estimated that it began on Jan. 19 and reached a peak infection rate of almost 100,000 per day on March 24 [3]. These estimates indicate that the curve had already begun to bend – that is, the rate of increase had begun to slow – before Cuomo issued his stay-at-home order effective March 22 – likely because individuals and businesses were spontaneously limiting their activities in reaction to official warnings and news coverage.

Hammond explains how better knowledge of the true picture for the spread of COVID-19 infections could have shaped the response of both New York's governor and the state's public health officials:

The virus’s rapid spread in February and early March of 2020 shows the importance of detecting outbreaks early and responding quickly. If officials had become aware of this surge even a week or two sooner – and notified the public – they almost certainly could have avoided swamping hospitals and saved thousands of lives.

If they had merely known when the wave reached its peak, they might have avoided mistakes in late March.

For example, Cuomo and his administration would have had less cause to worry about a looming shortage of hospital capacity. They could have avoided spending time and money to build emergency hospital facilities that went largely unused. And they might never have issued the March 25 directive transferring Covid-positive patients into nursing homes – a decision that likely added to the high death rate in those facilities and contributed to Cuomo’s political downfall. [6]

Here's an example of the official data and modeled projections they did have in early 2020. The following chart is taken from the Institute for Health Metrics and Evaluation (IHME)'s 25 March 2020 projections showing its estimates of the minimum, likely, and maximum number of additional hospital beds that would be needed in the state of New York to care for the model's expected surge of coronavirus patients.

This chart presents just one of several coronavirus models whose projections were being combined and presented to Governor Cuomo by consultants from McKinsey & Co. to assist their ad hoc public health policy making. Had New York state government officials instead known the daily number of new COVID-19 infections had already passed its peak, they almost certainly would not have reached the point of panic they did. Panic that resulted in their creating one of the worse public health outcomes in U.S. history.

Unlike those now mostly-former New York state officials, the IHME is at least learning from its mistakes in modeling 2020's coronavirus pandemic.

Previously on Political Calculations

• Governor Cuomo and the Coronavirus Models
• Coronavirus Deaths and Nursing Home Admission Policies
• The Biggest Math Story of 2020
• Visualizing ICL's Pandemic Model's Failures

References

Hammond, Bill. Behind the Curve: The Extreme Severity of New York City's First Pandemic Wave. Empire Center. [PDF Document]. 30 August 2023.

Institute for Health Metrics and Evaluation. COVID-19 estimate downloads. March 25, 2020. [ZIP folder]. Accessed 15 October 2023.

Footnotes from Behind the Curve

[2] https://www.healthdata.org/covid/data-downloads.

[3] David García-García et al., “Identification of the first COVID-19 infections in the US using a retrospective analysis (REMEDID),” Spatial and Spatio-temporal Epidemiology, Vol. 42, August 2022. https://www.sciencedirect.com/science/article/pii/ S1877584522000405#fig0001.

[6] For more on the Cuomo administration’s handling of the pandemic in nursing homes, see the Empire Center’s August 2021 report, “ ‘Like Fire Through Dry Grass’: Documenting the Cuomo Administration’s Cover-up of a Nursing Home Nightmare.” https://www.empirecenter.org/publications/like-fire-through-dry-grass/

Other Notes

* We altered the dimensions of the chart and the line thickness for the IHME estimate of infections. We also added the options for downloading a copy of the chart and sharing it on social media.

It used to be that when August ended, Major League Baseball fans had a pretty good idea of which teams were going to contend for the world championship. Those were the days before the number of divisions in each league increased from two to three and MLB also introduced the wild card, which gave a chance to the best teams in baseball that didn't win their division.

Is that still true? And if it is, with sports betting having become so popular, is it worth wagering on the teams with the best odds of winning the World Series at the end of August?

As we write this, we have reached that point in the 2023 season, making these timely questions. Better still, we had no idea what the answer to the questions might be before we took on the challenge of answering them as they had been posed to us.

Here's how we went about answering them. We tracked down the historic odds Las Vegas bookmakers had given each major league baseball team in each season from 2012 through 2022, omitting the coronavirus pandemic-impacted 2020 season that is too different from a regular season to be used for comparison. These years represent what we'll call the modern era for major league baseball's postseason competition.

From 2012 through 2021, that format awarded ten teams in both the American and National Leagues with the opportunity to compete to win the World Series in postseason play, which expanded to twelve teams in 2022. In this format, the three teams from each league that win their divisions get an automatic invitation to post-season play. Then the leagues award several wild card slots to additional teams in each league with the highest regular season winning percentages that didn't win a division title. From 2012 through 2021, that was two extra teams in each league, which expanded to three extra teams in each league starting in 2022.

For our analysis, we only care about the teams Las Vegas bookmakers thought had the best odds of going on to win the World Series as of 1 September during these seasons. In the following table, we've presented the lines for the top three ranked teams in each full season from 2012 through the 2023 season-to-date, only omitting 2020's pandemic-lockdown shortened season that isn't really comparable with the others. We've then indicated which team actually won the World Series and how much money a bettor would have if they wagered $100 on each of the top three teams, who then may or may not have gone on to win the World Series. Here's the table:

The odds in the table represent how much a bettor would win if they placed a simple $100 money line bet on 1 September for the indicated team going on to later win the season's World Series title. If one of these teams win, they would get the $100 they bet on it back plus the payout indicated by the odds. For example, a winning $100 bet on team with a payout line of (+450) would mean our gambler would have $550 (getting their $100 back plus $450 for winning) after the World Series. But since our hypothetical gambler can't predict the future, we've assumed they bet $100 on each of these three teams in each season, so even if one of these teams wins, they would lose the other $200 they bet on the other two teams.

How did our gambler do? First, we found that each of these teams reached the postseason in each full season from 2012 through 2022, so that already gave them a leg up over other teams. But making it into the postseason is only the first hurdle of the challenge. Of these teams, only 4 of Las Vegas' top-ranked teams qualified as wild card contenders, the other 26 were division winners.

Did any of Las Vegas' top-ranked teams go on to win the World Series? Over these 10 full seasons, we find 5 teams managed to hang on and win enough games to claim baseball's world championship. That happened in 2015, 2016, 2017, 2018, and 2022. The winning teams in these years were either ranked first or second by Las Vegas' bookmakers. In only one of these seasons, 2017, did any of the top ranked teams by Las Vegas' odds play each other for baseball's crown.

How did betting on Las Vegas' Top 3 ranked teams work out? In the five seasons where one of Las Vegas' Top 3 teams won the World Series, our gambler would have placed bets totaling $1,500, but would recover $500 and gain $1,965, leaving them with $2,465 in their pocket. That gain seems like it would be a good payout, but that requires ignoring the five other seasons where all of their bets were losing ones. Overall, of the $3,000 they wagered in total, they would have $2,465 left after accounting for all their winnings and losses. That's a net loss of $535 over these ten seasons.

Looking at World Series winners, we find 8 of the 10 teams won their respective divisions, even if the odds of their winning the World Series at the end of August put them outside the Top 3. Only two wild card teams, 2014's San Francisco Giants and 2019's Washington Nationals have won a world championship.

Of the ten World Series that took place over this period, only 2014's fall classic featured two teams that won wild card slots for the postseason. Two additional seasons, 2019 and 2022, featured a regular season division winner against a wild card team. The remaining seven other World Series were strictly between teams that won their divisions during the regular season.

Overall, the expansion of baseball's wild card playoffs have made the outcome of baseball championship race much less wild than we might have guessed. But if you think about it, the bookmakers are the only consistent winners when it comes to gambling on who will win the World Series.

Notes

* Lines from BetMGM on 26 August 2023. We assume these will be reasonably close to the odds that will be recorded for 1 September 2023.

Image credit: Photo by Joshua Peacock on Unsplash.

It was just a little over five months ago when the biggest banking crisis since the Great Financial Crisis of 2008-09 started. It began with the failures of Silicon Valley Bank and Signature Bank. These were the second and third-largest bank failures in U.S. history.

The week after they failed, famed hedge fund manager Michael Burry posted a remarkable chart on his Twitter account, saying it was "a good chart/guide", which he's since deleted. Here's a screenshot of that chart captured by Investing Whisperer:

The chart illustrates the relative risk of failure that various U.S. banks had at the time. It did that by showing how they compared to each other based on their percentage unrealized losses of capital that had been put at risk by the Federal Reserve's then year-long series of interest rate hikes (on the horizontal axis) and the percentage of accounts at the banks with large deposits (on the vertical scale). The chart is divided into four zones by the median percentages for each measure.

What the Silicon Valley Bank and Signature Bank failures quickly demonstrated is that banks in the upper right zone of the chart were the most at risk of failing if they were to experience a run, having both more than 60% of their accounts with more than $250,000 in deposits and more than 30% in unrealized losses of capital from the Fed's rate hikes.

Soon, a third bank that falls within this apparent danger zone was cleared from it. First Republic Bank (FRC) was acquired by J.P. Morgan Chase on 1 May 2023. That leaves just Comerica (NYSE: CMA) as the only remaining bank meeting the double criteria to fall within the danger zone.

In the months since however, it appears the danger zone has expanded to cover all banks at risk of unrealized losses of their capital invested in U.S. Treasuries because of the Federal Reserve's rate hikes. on 25 July 2023, PacWest (PACW) was acquired by the smaller Banc of California (NYSE: BANC) in an action to rescue it.

Since then, both Fifth Third Bank (NASDAQ: FITB) and USBancorp (NYSE: USB) were downgraded by Moody's because of their elevated risks. Both banks are positioned on the right side of Burry's chart and are near the median threshold for large depositors that puts them at risk of the kind of run that Silicon Valley Bank experienced. Moody's put several other banks positioned at or near the median lines edging the danger zone, Huntington Banc (NYSE: HBAN), KeyCorp (NYSE: KEY), Truist (NYSE: TFC), and Regions Financial (NYSE: RF) on its watchlist for potential future downgrades.

So far, only the "too-big-too-fail" banks whose stock symbols are indicated in purple fonts seem to be immune from the risk of downgrade or unexpected fire sale for falling on the high unrealized loss side of the chart.

Or had been, until this week when Fitch Ratings hinted they may no longer be safe. It seems one way or another, the perceived danger zone will be cleared.

How much health risk do you have from carrying too much mass around your midsection?

That question arises because studies point to the Waist-to-Height Ratio (WHtR) as a better indicator of early health risk than the Body Mass Index (BMI). As a general rule of thumb, if the circumference of your waist is greater than half your height, you have an elevated risk for developing chronic conditions like hypertension, diabetes mellitus, hypercholesterolemia, joint and low back pains, hyperuricemia, and obstructive sleep apnea syndrome.

The Waist-to-Height Ratio is also reported to be better than BMI in predicting heart attacks, especially for women, with higher ratios corresponding to higher risk.

That sounds like good bit of information to have, so we've built a tool to calculate your Waist-to-Health Ratio. Since you probably already know your height, the hard part will be finding out your waist circumference. Here's a video showing how to measure it.

Once you've done that for yourself, you're ready to go. If you're accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version of the tool. Here it is:

In using the tool, be sure to use the same units of measurement for both waist circumference and height. You'll get accurate results so long as you don't start mixing and matching inches and centimeters together....

According to documents leaked in February 2022, starting in July 2022, U.S. Air Force personnel will have their Waist to Height Ratio assessed. Individuals with waists that measure at more than half their height will be reassessed six months later, with those whose waists exceed that threshold at the later measurement date separated from service. "Seperated from service" meaning "discharged from the Air Force". Here's the chart the Air Force will be using to make that determination:

The thresholds shown on this chart for low, moderate, and high risk are those we've built into the tool's feedback. We've also made a point of giving the answer to the same two-decimal place results as would be used by Air Force medical personnel in their assessments, so there are no surprises for what to expect.

Previously on Political Calculations

• What's Your Body Fat Percentage?
• How Do American Adults Measure Up?

References

Margaret Ashwell and Sigrid Gibson. Waist-to-height ratio as an indicator of ‘early health risk’: simpler and more predictive than using a ‘matrix’ based on BMI and waist circumference. BMJ Open 2016:6:3010159. [DOI: 10.1136/bmjopen-2015-010159 | NIH: PDF Document]. 14 March 2016.

Sanne A.E. Peters, Sophie H. Bots and Mark Woodward. Sex Differences in the Association Between Measures of General and Central Adiposity and the Risk of Myocardial Infarction: Results From the UK Biobank. Journal of the American Heart Association. Vol. 7, No. 5. [DOI: 10.1161/JAHA.117.008507]. 28 February 2018. American Heart Association. Waist size predicts heart attacks better than BMI, especially in women. [Online Article]. 28 February 2018.

Darsini Darsini, Hamidah Hamidah, Hari Basuki Notobroto, and Eko Agus Cahyono. Health risks associated with high waist circumference: A systematic review. Journal of Public Health Research. Vol. 9, No. 2: Papers from the 4th International Symposium of Public Health (4th ISOPH), Brisbane, Australia. 29-31 October 2019. [DOI: 10.4081/jphr.2020.1811 | NIH: PDF Document]. 2 July 2020.

ShapeFit. Waist to Height Ratio Calculator - Assess Your Lifestyle Risk. [Online Article and Tool]. 31 March 2015.

The reports of how COVID-19 changed U.S. life expectancy are grim.

The pandemic crushed life expectancy in the United States last year by 1.5 years, the largest drop since World War II, according to new Centers for Disease Control and Prevention data released Wednesday. For Black and Hispanic people, their life expectancy declined by three years.

U.S. life expectancy declined from 78.8 years in 2019 to 77.3 years in 2020. The pandemic was responsible for close to 74 percent of that overall decline, though increased fatal drug overdoses and homicides also contributed.

“I myself had never seen a change this big except in the history books,” Elizabeth Arias, a demographer at the CDC and lead author of the new report, told The Wall Street Journal.

The figures for just COVID-19's impact on U.S. life expectancy are roughly in line with the CDC's preliminary estimates from February 2021, which was based on the then-available data through the first half of 2020.

Unfortunately, the CDC's estimates are rather misleading. Dr. Peter Bach, the director of the Center for Health Policy and Outcomes at the Memorial Sloan Kettering Cancer Center, ran some back of the envelope calculations after the CDC released its preliminary estimates and came up with very different results.

The CDC reported that life expectancy in the U.S. declined by one year in 2020. People understood this to mean that Covid-19 had shaved off a year from how long each of us will live on average. That is, after all, how people tend to think of life expectancy. The New York Times characterized the report as “the first full picture of the pandemic’s effect on American expected life spans.”

But wait. Analysts estimate that, on average, a death from Covid-19 robs its victim of around 12 years of life. Approximately 400,000 Americans died Covid-19 in 2020, meaning about 4.8 million years of life collectively vanished. Spread that ghastly number across the U.S. population of 330 million and it comes out to 0.014 years of life lost per person. That’s 5.3 days. There were other excess deaths in 2020, so maybe the answer is seven days lost per person.

No matter how you look at it, the result is a far cry from what the CDC announced.

We built the following tool to do Bach's math, which checks out. You're welcome to update the figures with improved data or to replace them with other countries' data if you want to see the impact elsewhere in the world. If you're accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version of the tool.

So why is the CDC's estimate of the change in life expectancy estimate so different? As Bach explains, it is not because of either the data or the math, but rather, it is because of the CDC's assumptions in doing their math:

It’s not that the agency made a math mistake. I checked the calculations myself, and even went over them with one of the CDC analysts. The error was more problematic in my view: The CDC relied on an assumption it had to know was wrong.

The CDC’s life expectancy calculations are, in fact, life expectancy projections (the technical term for the measure is period life expectancy). The calculation is based on a crucial assumption: that for the year you are studying (2019 compared to 2020 in this case) the risk of death, in every age group, will stay as it was in that year for everyone born during it.

So to project the life expectancy of people born in 2020, the CDC assumed that newborns will face the risk of dying that newborns did in 2020. Then when they turn 1, they face the risk of dying that 1-year-olds did in 2020. Then on to them being 2 years old, and so on.

Locking people into 2020 for their entire life spans, from birth to death, may sound like the plot of a dystopian reboot of “Groundhog Day.” But that’s the calculation. The results: The CDC’s report boils down to a finding that bears no relation to any realistic scenario. Running the 2020 gauntlet for an entire life results in living one year less on average than running that same gauntlet in 2019.

Don’t blame the method. It’s a standard one that over time has been a highly useful way of understanding how our efforts in public health have succeeded or fallen short. Because it is a projection, it can (and should) serve as an early warning of how people in our society will do in the future if we do nothing different from today.

But in this case, the CDC should assume, as do we all, that Covid-19 will cause an increase in mortality for only a brief period relative to the span of a normal lifetime. If you assume the Covid-19 risk of 2020 carries forward unabated, you will overstate the life expectancy declines it causes.

In effect, the CDC's assumption projects the impact of COVID-19 in a world in which none of the Operation Warp Speed vaccines exist seeing as they only began rolling out in large numbers in the latter half of December 2020.

When the CDC repeats its life expectancy exercise next year, its estimates of the change in life expectancy should reflect the first year impact of the new COVID-19 vaccines, which will make for an interesting side by side comparison. Especially when comparisons of pre-vaccine case and death rates with post-vaccine data already look like the New Stateman's chart for the United Kingdom:

HT: Marginal Revolution

What percentage of the population needs to be vaccinated to usefully reduce the risk of dying from COVID-19?

We're going to do a back-of-the-envelope calculation to estimate the answer to that question using Arizona's high quality COVID data in general, and the state's data for COVID-related hospital admissions and deaths in particular.

We're also going to build off our previous analysis that synchronized Arizona's figures for the number of positive COVID infection test results, hospitalizations, and deaths according to the approximate date of initial SARS-CoV-2 coronavirus exposure for the Arizonans who became infected and experienced these pandemic-related events. The following chart shows these three streams of data using a logarithmic scale, covering the period from 15 March 2020 through 30 April 2021.

We've annotated the chart to indicate two periods of "noise" in the data for deaths, which came into play when the daily number of COVID-related deaths of Arizonans dropped into the single digits. Because of the small numbers involved, having a relatively small change in the daily number can have an outsize effect on the appearance of the overall trend, which accounts for the "noisy" short-term trough that was recorded in mid-September 2020 and the short-term spike in late March 2021. We've added the dotted lines to these areas of the chart to indicate what the overall pattern would look like without the short term noise in the data.

Now to the bigger question. We're going to focus on the ratio of deaths to hospital admissions because these events represent the most serious classes of COVID infections. In Arizona, 75% of COVID-related deaths have occurred among the state's senior population, Age 65 or older. This same demographic has accounted for 46% of COVID-related hospital admissions in the state.

These figures confirm seniors are disproportionately vulnerable to both these outcomes if they become infected by the SARS-CoV-2 coronavirus. This fact is why this portion of the state's population was targeted for early COVID vaccinations once the vaccines became available.

Because the incidence of COVID-related deaths in concentrated in Arizona's senior population, we should see a sustained decline in the ratio of COVID deaths to hospital admissions corresponding to roughly when the population Age 65 or older achieved effective herd immunity. We can then identify what percentage of the state's elderly population had been received at least one vaccine dose at that point in time, which in turn, will give us a reasonable indication of what percentage of the population needs to be vaccinated for COVID to reduce its risk of death.

The next chart graphically shows the results when we combine these points of data together.

We find at least 55% of the population would need to have received at least one dose of the COVID-19 vaccines to provide the benefit of reduced risk of death from becoming infected by the coronavirus. That's the percentage of the Age 65 and older population of Arizonans who had been vaccinated as of 28 February 2021, which marks the point in time at which COVID-related deaths in the state began to plunge as a result of the Operation Warp Speed vaccination programs.

That's the low end for our estimate, because it does not consider the portion of the senior population who would have obtained natural immunity from having become infected with the SARS-CoV-2 coronavirus and who recovered from it. As of 28 February 2021, Arizona's senior population accounted for 109,897 known COVID infections, about 13.4% of the state's total at that time. Added to the 696,559 Age 65 or older Arizonans who had received at least one COVID vaccination dose at that date would put the high end of the estimate at 64%.

That upper level figure would explain why public health officials have set a target of 70% of the population for COVID vaccinations, but it seems strange they are not giving more weight to the potential contribution of natural immunity in achieving that goal. If they did, they could focus their limited resources for providing COVID vaccination more effectively.

Looking at Arizona's data, we would say the magic percentage for vaccinations to achieve useful COVID herd immunity is somewhere between 55% and 64% of the population. That's because there is almost certainly a good amount of overlap between those who have recovered from COVID and those who have been vaccinated. It would be more beneficial and less wasteful for public health officials to target the COVID vaccines to those who have not developed any antibodies to SARS-CoV-2 coronavirus infections.

References

Arizona Department of Health Services. COVID-19 Data Dashboard: Vaccine Administration. [Online Database]. Accessed 10 June 2021.

Three of the four high quality COVID-19 datasets we track for Arizona indicate a new adverse change in trends for the coronavirus pandemic in the state. Arizona's data for positive test results by date of sample collection, new hospital admissions, and ICU bed usage are confirming an increase in the incidence of COVID in the state. The fourth dataset for COVID-19 deaths by recorded date of death certificate does not as yet, but would not be expected to as yet because this data has the greatest lag from a change in the rate of incidence to the confirmation of a change in trend.

We've generated the following animated chart to cycle through each of the charts for these four datasets.

Considering the respective lags that apply for each dataset, the likely timing of a significant change in the rate of incidence for SARS-CoV-2 coronavirus infections in the state between 26 March 2021 and 30 March 2021. This period coincides with contemporary news accounts of the Biden administration moving migrants from overloaded Border Patrol migrant facilities in Arizona counties bordering Mexico to facilities and small towns in Maricopa and Pinal Counties.

We think this activity is showing up in Arizona's COVID-19 statistics because these migrants have been exposed to Mexico's higher incidence of COVID-19 infections. While those cases peaked in late January 2021, several weeks after they peaked and began falling in Arizona, the relative difference in infection rates between Arizona's population and the entering migrants is enough to affect the trends for Arizona's COVID-19 data.

On 26 March 2021, Senator Mark Kelly (D-AZ) stated the Biden administration did not have an effective plan for the border. It took another month for President Biden to acknowledge the migration crisis at the U.S. border with Mexico as a crisis.

President Biden, talking to reporters after finishing golf today, concedes border issues are a “crisis,” saying refugee cap was linked to the “crisis that ended up on the border with young people.”

“We couldn’t do two things at once. And now we are going to increase the number.” pic.twitter.com/dwvrRM792k

— TV News HQ (@TVNewsHQ) April 18, 2021

On 21 April 2021, Arizona Governor Doug Ducey declared a state of emergency and sent Arizona National Guard troops to Arizona's border counties.

Coincidentally, that was the last date we updated our series on the COVID-19 pandemic in Arizona, where our analysis of the trends just before that point of time appeared very early in the morning....

Previously on Political Calculations

Here is our previous coverage of Arizona's experience with the coronavirus pandemic, presented in reverse chronological order.

• Improving COVID Trends Bottom and Flatline in Arizona
• Surge of Migrants, Lifting of Business Capacity Limits Change Arizona COVID Trends
• COVID-19 in Retreat in Arizona With Vaccines Gaining Traction
• The Ebb and Flow of COVID-19 in Arizona's ICUs
• Arizona's Plunging COVID-19 Caseloads and the Vaccines
• Arizona Enters Downward Trend for COVID-19 After Second Peak
• Arizona Passes Second COVID-19 Peak
• A Tale of Two States and the Coronavirus
• COVID-19 Questions, Answers, and Lessons Learned from Arizona
• The Deadly Intersection of Anti-Police Protests and COVID-19
• 2020 Campaign Events Drive Surge in Arizona COVID Cases
• Arizona Arrives at Critical Junction for Coronavirus Cases
• Arizona To Soon Reach A Critical Junction For COVID-19
• Getting More Than Care from Arizona's COVID ICU Beds
• Arizona's Decentralized Approach to Beating COVID
• Going Back to School with COVID-19
• Arizona Turns Second Corner Toward Crushing Coronavirus
• Arizona's Coronavirus Crest in Rear View Mirror
• The Coronavirus Turns a Corner in Arizona
• A Delayed First Wave Crests in the U.S. and a Second COVID-19 Wave Arrives
• The Coronavirus in Arizona
• A Closer Look at COVID-19 Deaths in Arizona
• The New Epicenter of COVID-19 in the U.S.
• How Long Does a Serious COVID Infection Typically Last?
• How Deadly is the COVID-19 Coronavirus?
• Governor Cuomo and the Coronavirus Models
• How Do False Test Outcomes Affect Estimates of the True Incidence of Coronavirus Infections?
• How Fast Could China's Coronavirus Spread?

References

We've continued following Arizona's experience during the coronavirus pandemic because the state's Department of Health Services makes detailed, high quality time series data available, which makes it easy to apply the back calculation method to identify the timing and events that caused changes in the state's COVID-19 trends. This section links that that resource and many of the others we've found useful throughout the coronavirus pandemic.

Arizona Department of Health Services. COVID-19 Data Dashboard: Vaccine Administration. [Online Database]. Accessed 25 April 2021.

Stephen A. Lauer, Kyra H. Grantz, Qifang Bi, Forrest K. Jones, Qulu Zheng, Hannah R. Meredith, Andrew S. Azman, Nicholas G. Reich, Justin Lessler. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Annals of Internal Medicine, 5 May 2020. https://doi.org/10.7326/M20-0504.

U.S. Centers for Disease Control and Prevention. COVID-19 Pandemic Planning Scenarios. [PDF Document]. Updated 10 September 2020.

More or Less: Behind the Stats. Ethnic minority deaths, climate change and lockdown. Interview with Kit Yates discussing back calculation. BBC Radio 4. [Podcast: 8:18 to 14:07]. 29 April 2020.

Not much has changed over the last two weeks with respect to COVID trends in Arizona. Which is very good news.

That's because of what didn't happen after Arizona's economy fully reopened, which we mark from the lifting of capacity limits on businesses such as bars, restaurants, gyms, and others that would be considered prime territory for spreading coronavirus infections. What didn't happen is significant, because instead of developing a rising trend for COVID-19 cases, hospitalizations, ICU bed usage, and deaths as it did after its first reopening attempt following its first wave of COVID-19, Arizona has instead experienced steady, relatively flat levels for all these measures. What had looked like the potential stalling of a previous trend of improvement is really demonstrating the benefits of the widespread deployment of Operation Warp Speed's vaccines in Arizona. The protections offered by the vaccines are enabling the gains of improvement to be sustained. Even as the kind of activities that would have previously resulted in the reversal of a positive trend for coronavirus infections have expanded.

That's evident in the latest updates to the four charts tracking the trends for each of these measures based on Arizona's high quality data for the numbers of new cases by test sample collection date, hospitalizations by date of admission, deaths by date as recorded on death certificates, and ICU bed usage.

Since our last update, the trend for cases has continued moving sideways, with the trends for both hospitalizations and ICU bed usage now following suit. It will take a little more time to see if the dataset for deaths attributed to COVID-19 will follow the same pattern, but the early indicators are that it will.

The next time we look at Arizona's high quality COVID-19 data, we'll be looking to confirm the answer to a very different question.

Previously on Political Calculations

Here is our previous coverage of Arizona's experience with the coronavirus pandemic, presented in reverse chronological order.

• Improving COVID Trends Bottom and Flatline in Arizona
• Surge of Migrants, Lifting of Business Capacity Limits Change Arizona COVID Trends
• COVID-19 in Retreat in Arizona With Vaccines Gaining Traction
• The Ebb and Flow of COVID-19 in Arizona's ICUs
• Arizona's Plunging COVID-19 Caseloads and the Vaccines
• Arizona Enters Downward Trend for COVID-19 After Second Peak
• Arizona Passes Second COVID-19 Peak
• A Tale of Two States and the Coronavirus
• COVID-19 Questions, Answers, and Lessons Learned from Arizona
• The Deadly Intersection of Anti-Police Protests and COVID-19
• 2020 Campaign Events Drive Surge in Arizona COVID Cases
• Arizona Arrives at Critical Junction for Coronavirus Cases
• Arizona To Soon Reach A Critical Junction For COVID-19
• Getting More Than Care from Arizona's COVID ICU Beds
• Arizona's Decentralized Approach to Beating COVID
• Going Back to School with COVID-19
• Arizona Turns Second Corner Toward Crushing Coronavirus
• Arizona's Coronavirus Crest in Rear View Mirror
• The Coronavirus Turns a Corner in Arizona
• A Delayed First Wave Crests in the U.S. and a Second COVID-19 Wave Arrives
• The Coronavirus in Arizona
• A Closer Look at COVID-19 Deaths in Arizona
• The New Epicenter of COVID-19 in the U.S.
• How Long Does a Serious COVID Infection Typically Last?
• How Deadly is the COVID-19 Coronavirus?
• Governor Cuomo and the Coronavirus Models
• How Do False Test Outcomes Affect Estimates of the True Incidence of Coronavirus Infections?
• How Fast Could China's Coronavirus Spread?

References

We've continued following Arizona's experience during the coronavirus pandemic because the state's Department of Health Services makes detailed, high quality time series data available, which makes it easy to apply the back calculation method to identify the timing and events that caused changes in the state's COVID-19 trends. This section links that that resource and many of the others we've found useful throughout the coronavirus pandemic.

Arizona Department of Health Services. COVID-19 Data Dashboard: Vaccine Administration. [Online Database]. Accessed 20 April 2021.

Stephen A. Lauer, Kyra H. Grantz, Qifang Bi, Forrest K. Jones, Qulu Zheng, Hannah R. Meredith, Andrew S. Azman, Nicholas G. Reich, Justin Lessler. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Annals of Internal Medicine, 5 May 2020. https://doi.org/10.7326/M20-0504.

U.S. Centers for Disease Control and Prevention. COVID-19 Pandemic Planning Scenarios. [PDF Document]. Updated 10 September 2020.

More or Less: Behind the Stats. Ethnic minority deaths, climate change and lockdown. Interview with Kit Yates discussing back calculation. BBC Radio 4. [Podcast: 8:18 to 14:07]. 29 April 2020.