Part Two: Cheating in DCPS

DC Education Reform Ten Years After, 

Part 2: Test Cheats

Richard P Phelps

Ten years ago, I worked as the Director of Assessments for the District of Columbia Public Schools (DCPS). For temporal context, I arrived after the first of the infamous test cheating scandals and left just before the incident that spawned a second. Indeed, I filled a new position created to both manage test security and design an expanded testing program. I departed shortly after Vincent Gray, who opposed an expanded testing program, defeated Adrian Fenty in the September 2010 DC mayoral primary. My tenure coincided with Michelle Rhee’s last nine months as Chancellor. 

The recurring test cheating scandals of the Rhee-Henderson years may seem extraordinary but, in fairness, DCPS was more likely than the average US school district to be caught because it received a much higher degree of scrutiny. Given how tests are typically administered in this country, the incidence of cheating is likely far greater than news accounts suggest, for several reasons: 

·      in most cases, those who administer tests—schoolteachers and administrators—have an interest in their results;

·      test security protocols are numerous and complicated yet, nonetheless, the responsibility of non-expert ordinary school personnel, guaranteeing their inconsistent application across schools and over time; 

·      after-the-fact statistical analyses are not legal proof—the odds of a certain amount of wrong-to-right erasures in a single classroom on a paper-and-pencil test being coincidental may be a thousand to one, but one-in-a-thousand is still legally plausible; and

·      after-the-fact investigations based on interviews are time-consuming, scattershot, and uneven. 

Still, there were measures that the Rhee-Henderson administrations could have adopted to substantially reduce the incidence of cheating, but they chose none that might have been effective. Rather, they dug in their heels, insisted that only a few schools had issues, which they thoroughly resolved, and repeatedly denied any systematic problem.  

Cheating scandals

From 2007 to 2009 rumors percolated of an extraordinary level of wrong-to-right erasures on the test answer sheets at many DCPS schools. “Erasure analysis” is one among several “red flag” indicators that testing contractors calculate to monitor cheating. The testing companies take no responsibility for investigating suspected test cheating, however; that is the customer’s, the local or state education agency. 

In her autobiographical account of her time as DCPS Chancellor, Michelle Johnson (nee Rhee), wrote (p. 197)

“For the first time in the history of DCPS, we brought in an outside expert to examine and audit our system. Caveon Test Security – the leading expert in the field at the time – assessed our tests, results, and security measures. Their investigators interviewed teachers, principals, and administrators.

“Caveon found no evidence of systematic cheating. None.”

Caveon, however, had not looked for “systematic” cheating. All they did was interview a few people at several schools where the statistical anomalies were more extraordinary than at others. As none of those individuals would admit to knowingly cheating, Caveon branded all their excuses as “plausible” explanations. That’s it; that is all that Caveon did. But, Caveon’s statement that they found no evidence of “widespread” cheating—despite not having looked for it—would be frequently invoked by DCPS leaders over the next several years.[1]

Incidentally, prior to the revelation of its infamous decades-long, systematic test cheating, the Atlanta Public Schools had similarly retained Caveon Test Security and was, likewise, granted a clean bill of health. Only later did the Georgia state attorney general swoop in and reveal the truth. 

In its defense, Caveon would note that several cheating prevention measures it had recommended to DCPS were never adopted.[2] None of the cheating prevention measures that I recommended were adopted, either.

The single most effective means for reducing in-classroom cheating would have been to rotate teachers on test days so that no teacher administered a test to his or her own students. It would not have been that difficult to randomly assign teachers to different classrooms on test days.

The single most effective means for reducing school administratorcheating would have been to rotate test administrators on test days so that none managed the test materials for their own schools. The visiting test administrators would have been responsible for keeping test materials away from the school until test day, distributing sealed test booklets to the rotated teachers on test day, and for collecting re-sealed test booklets at the end of testing and immediately removing them from the school. 

Instead of implementing these, or a number of other feasible and effective test security measures, DCPS leaders increased the number of test proctors, assigning each of a few dozen or so central office staff a school to monitor. Those proctors could not reasonably manage the volume of oversight required. A single DC test administration could encompass a hundred schools and a thousand classrooms.

Investigations

So, what effort, if any, did DCPS make to counter test cheating? They hired me, but then rejected all my suggestions for increasing security. Also, they established a telephone tip line. Anyone who suspected cheating could report it, even anonymously, and, allegedly, their tip would be investigated. 

Some forms of cheating are best investigated through interviews. Probably the most frequent forms of cheating at DCPS—teachers helping students during test administrations and school administrators looking at test forms prior to administration—leave no statistical residue. Eyewitness testimony is the only type of legal evidence available in such cases, but it is not just inconsistent, it may be socially destructive. 

I remember two investigations best: one occurred in a relatively well-to-do neighborhood with well-educated parents active in school affairs; the other in one of the city’s poorest neighborhoods. Superficially, the cases were similar—an individual teacher was accused of helping his or her own students with answers during test administrations. Making a case against either elementary school teacher required sworn testimony from eyewitnesses, that is, students—eight-to-ten-year olds. 

My investigations, then, consisted of calling children into the principal’s office one-by-one to be questioned about their teacher’s behavior. We couldn’t hide the reason we were asking the questions. And, even though each student agreed not to tell others what had occurred in their visit to the principal’s office, we knew we had only one shot at an uncorrupted jury pool. 

Though the accusations against the two teachers were similar and the cases against them equally strong, the outcomes could not have been more different. In the high-poverty neighborhood, the students seemed suspicious and said little; none would implicate the teacher, whom they all seemed to like. 

In the more prosperous neighborhood, students were more outgoing, freely divulging what they had witnessed. The students had discussed the alleged coaching with their parents who, in turn, urged them to tell investigators what they knew. During his turn in the principal’s office, the accused teacher denied any wrongdoing. I wrote up each interview, then requested that each student read and sign. 

Thankfully, that accused teacher made a deal and left the school system a few weeks later. Had he not, we would have required the presence in court of the eight-to-ten-year olds to testify under oath against their former teacher, who taught multi-grade classes. Had that prosecution not succeeded, the eyewitness students could have been routinely assigned to his classroom the following school year.

My conclusion? Only in certain schools is the successful prosecution of a cheating teacher through eyewitness testimony even possible. But, even where possible, it consumes inordinate amounts of time and, otherwise, comes at a high price, turning young innocents against authority figures they naturally trusted. 

Cheating blueprints

Arguably the most widespread and persistent testing malfeasance in DCPS received little attention from the press. Moreover, it was directly propagated by District leaders, who published test blueprints on the web. Put simply, test “blueprints” are lists of the curricular standards (e.g., “student shall correctly add two-digit numbers”) and the number of test items included in an upcoming test related to each standard. DC had been advance publishing its blueprints for years.

I argued that the way DC did it was unethical. The head of the Division of Data & Accountability, Erin McGoldrick, however, defended the practice, claimed it was common, and cited its existence in the state of California as precedent. The next time she and I met for a conference call with one of DCPS’s test providers, Discover Education, I asked their sales agent how many of their hundreds of other customers advance-published blueprints. His answer: none.

In the state of California, the location of McGoldrick’s only prior professional experience, blueprints were, indeed, published in advance of test administrations. But their tests were longer than DC’s and all standards were tested. Publication of California’s blueprints served more to remind the populace what the standards were in advance of each test administration. Occasionally, a standard considered to be of unusual importance might be assigned a greater number of test items than the average, and the California blueprints signaled that emphasis. 

In Washington, DC, the tests used in judging teacher performance were shorter, covering only some of each year’s standards. So, DC’s blueprints showed everyone well in advance of the test dates exactly which standards would be tested and which would not. For each teacher, this posed an ethical dilemma: should they “narrow the curriculum” by teaching only that content they knew would be tested? Or, should they do the right thing and teach all the standards, as they were legally and ethically bound to, even though it meant spending less time on the to-be-tested content? It’s quite a conundrum when one risks punishment for behaving ethically.

Monthly meetings convened to discuss issues with the districtwide testing program, the DC Comprehensive Assessment System (DC-CAS)—administered to comply with the federal No Child Left Behind (NCLB) Act. All public schools, both DCPS and charters, administered those tests. At one of these regular meetings, two representatives from the Office of the State Superintendent of Education (OSSE) announced plans to repair the broken blueprint process.[3]

The State Office employees argued thoughtfully and reasonably that it was professionally unethical to advance publish DC test blueprints. Moreover, they had surveyed other US jurisdictions in an effort to find others that followed DC’s practice and found none. I was the highest-ranking DCPS employee at the meeting and I expressed my support, congratulating them for doing the right thing. I assumed that their decision was final.

I mentioned the decision to McGoldrick, who expressed surprise and speculation that it might have not been made at the highest level in the organizational hierarchy. Wasting no time, she met with other DCPS senior managers and the proposed change was forthwith shelved. In that, and other ways, the DCPS tail wagged the OSSE dog. 

* * *

It may be too easy to finger ethical deficits for the recalcitrant attitude toward test security of the Rhee-Henderson era ed reformers. The columnist Peter Greene insists that knowledge deficits among self-appointed education reformers also matter: 

“… the reformistan bubble … has been built from Day One without any actual educators inside it. Instead, the bubble is populated by rich people, people who want rich people’s money, people who think they have great ideas about education, and even people who sincerely want to make education better. The bubble does not include people who can turn to an Arne Duncan or a Betsy DeVos or a Bill Gates and say, ‘Based on my years of experience in a classroom, I’d have to say that idea is ridiculous bullshit.’”

“There are a tiny handful of people within the bubble who will occasionally act as bullshit detectors, but they are not enough. The ed reform movement has gathered power and money and set up a parallel education system even as it has managed to capture leadership roles within public education, but the ed reform movement still lacks what it has always lacked–actual teachers and experienced educators who know what the hell they’re talking about.”

In my twenties, I worked for several years in the research department of a state education agency. My primary political lesson from that experience, consistently reinforced subsequently, is that most education bureaucrats tell the public that the system they manage works just fine, no matter what the reality. They can get away with this because they control most of the evidence and can suppress it or spin it to their advantage.

In this proclivity, the DCPS central office leaders of the Rhee-Henderson era proved themselves to be no different than the traditional public-school educators they so casually demonized. 

US school systems are structured to be opaque and, it seems, both educators and testing contractors like it that way. For their part, and contrary to their rhetoric, Rhee, Henderson, and McGoldrick passed on many opportunities to make their system more transparent and accountable.

Education policy will not improve until control of the evidence is ceded to genuinely independent third parties, hired neither by the public education establishment nor by the education reform club.

The author gratefully acknowledges the fact-checking assistance of Erich Martel and Mary Levy.

Access this testimonial in .pdf format

Citation:  Phelps, R. P. (2020, September). Looking Back on DC Education Reform 10 Years After, Part 2: Test Cheats. Nonpartisan Education Review / Testimonials. https://nonpartisaneducation.org/Review/Testimonials/v16n3.htm


[1] A perusal of Caveon’s website clarifies that their mission is to help their clients–state and local education departments–not get caught. Sometimes this means not cheating in the first place; other times it might mean something else. One might argue that, ironically, Caveon could be helping its clients to cheat in more sophisticated ways and cover their tracks better.

[2] Among them: test booklets should be sealed until the students open them and resealed by the students immediately after; and students should be assigned seats on test day and a seating chart submitted to test coordinators (necessary for verifying cluster patterns in student responses that would suggest answer copying).

[3] Yes, for those new to the area, the District of Columbia has an Office of the “State” Superintendent of Education (OSSE). Its domain of relationships includes not just the regular public schools (i.e., DCPS), but also other public schools (i.e., charters) and private schools. Practically, it primarily serves as a conduit for funneling money from a menagerie of federal education-related grant and aid programs

COVID daily deaths around the world

Please bookmark this page: https://www.worldometers.info/coronavirus/#countries

It gives all sorts of data on infections, recoveries, testing, deaths, and so on from all over the world. If you look at it, you will see that none of MangoMussolini’s boasts are correct, and that some nations seem to have ‘beaten’ the virus — at least for now.

(Yes, I know, all data is somewhat suspect, and some countries are probably low-balling their numbers. But this is all the data we have.)

I will share some graphs I copied from that source, so you can see which nations appear to be doing a good job at shutting down the current pandemic. I will first show the world, then the USA, then about a dozen nations, arranged alphabetically. You will see that the US is very, very obviously not one of the countries whose leadership has been able to defeat this disease.

I will also share the number of deaths per million people.

0. THE WORLD:

This is for the entire world, and it’s deaths per day, as of today, August 4, 2020. Deaths are not going down. Worldwide, we have lost 90 people per million to this disease so far.
  1. THE UNITED STATES:
The vertical scale is obviously different from the one for the world. As you can see, deaths from COVID in the US are now about 1000 per day, and rising. The US has lost 481 people per million so far.

2. BRAZIL:

Brazil’s deaths never declined. They have lost 446 people per million.

3. CHINA:

That big spike in the middle is when the Chinese regime discovered they had left out a lot of COVID deaths. Since that time, they have had very, very few. They have lost 3 (yes, THREE) people per million so far.

4. FRANCE:

Another country that successfully beat back the pandemic. They have lost 464 people per million so far.

5. GERMANY:

As did Germany. Their total dead work out to 110 per million people so far.

6. INDIA:

India, on the other hand, has not been successful. Deaths are increasing steadily; also, it would not be surprising if a lot of them are not even being counted. They report 28 people dead per million so far.

7. ITALY:

Italy was hit hard, and hit early. However, its daily death rates appear to be going in the right direction: down. Their toll is 582 dead per million.

8. MEXICO:

Mexico’s death rates do not appear to be going down. One might wonder if all of the COVID-19 deaths are even being counted. Their death total stands at 372 per million.

9. RUSSIA:

These figures are not going in the right direction. Plus, there are protests in Russia because folks in their Far East have evidence of serious undercounting. Their toll is 98 dead per million.

10. SOUTH AFRICA:

Not going in the right direction. Their toll is 144 per million.

11. SPAIN:

Like Italy, Spain was hit hard and early, but the daily death tolls now are approaching zero. Their death toll is 609 per million, one of the highest in the world.

12. SWEDEN:

Unlike the rest of Scandinavia, Sweden decided not to lock down at all. While the death rates are going down, they are doing so much more slowly than in most other European (and Scandinavian) nations.

Their death toll is 569 per million, one of the highest anywhere.

13: UNITED KINGDOM (BRITAIN)

The United Kingdom (aka Great Britain) was hit early, and its daily death toll is going down much more slowly than in other European nations.

Their death toll is 680 per million, which is, again, one of the highest anywhere.

===============================================================

I don’t think you will guess the nation with the highest total COVID death rate per million, so I’ll just tell you: it’s tiny San Marino at 1238. Next come Belgium (with 850) and the UK (with 680).

Here is a table listing the top 17 nations. Being in this group is not a good thing.

RankNationDeaths per million
1San Marino1,238
2Belgium850
3UK680
4Andorra673
5Spain609
6Peru600
7Italy582
8Sweden569
9Chile507
10USA481
11France464
12Brazil446
13Sint Maarten373
14Mexico372
15Netherlands359
16Ireland357
17Panama346

How to decide if anybody should listen to your ideas on how and whether to re-open schools, or maybe you should just hush.

Peter Greene has provided a nice flow chart to let you decide whether you should open your mouth with your ideas on how and whether to re-open the public schools, or whether you should just be quiet and listen.

So, should you just hush, or do you have something valuable to contribute to this subject?

My wife and I each taught for 30 years or so, and so we would be in the ‘speak right up’ category, but I don’t really know how the USA can get public education to work next year, especially since the danger is not going away, but apparently once more growing at an exponential clip.

Nobody should be listening to billionaires or their bought-and-paid-for policy wonks who once spent a whole two years in a classroom.

A few quotes from Greene’s column. (He is a much better writer than me, and much more original as well.)

==================================

To everyone who was never a classroom teacher but who has some ideas about how school should be reopened in the fall:

Hush.

Just hush.

There are some special categories of life experiences. Divorce. Parenthood. Deafness. Living as a Black person in the US. Classroom teacher. They are very different experiences, but they all have on thing in common.

You can read about these things. But if you haven’t lived it, you don’t know. You can study up, read up, talk to people. And in some rare cases that brings you close enough to knowing that your insights might actually be useful.

But mostly, you are a Dunning-Krueger case study just waiting to be written up.

The last thirty-seven-ish years of education have been marked by one major feature– a whole lot of people who just don’t know, throwing their weight around and trying to set the conditions under which the people who actually do the work will have to try to actually do the work. Policy wonks, privateers, Teach for America pass-throughs, guys who wanted to run for President, folks walking by on the street who happen to be filthy rich, amateurs who believe their ignorance is a qualification– everyone has stuck their oar in to try to reshape US education. And in ordinary times, as much as I argue against these folks, I would not wave my magic wand to silence them, because 1) educators are just as susceptible as anyone to becoming too insular and entrenched and convinced of their own eternal rightness and 2) it is a teacher’s job to serve all those amateurs, so it behooves the education world to listen, even if what they hear is 98% bosh.

But that’s in ordinary times, and these are not ordinary times.

There’s a whole lot of discussion about the issues involved in starting up school this fall. The discussion is made difficult by the fact that all options stink. It is further complicated by the loud voices of people who literally do not know what they are talking about.

Will these ‘lost’ months of school really matter?

David Berliner explains that the academic topics untaught during these months of coronavirus shutdowns of schools aren’t really all that much to worry about — as long as kids have been engaged in useful or imaginative projects of their own choosing. This first appeared on Diane Ravitch’s blog. I found it at Larry Cuban’s blog.

Worried About Those “Big” Losses on School Tests Because Of Extended Stays At Home? They May Not Even Happen,
And If They Do, They May Not Matter Much At All!

David C. Berliner
Regents Professor Emeritus
Mary Lou Fulton Teachers College
Arizona State University
Tempe, AZ.

Although my mother passed away many years ago, I need now to make a public confession about a crime she committed year in and year out. When I was young, she prevented me from obtaining one year of public schooling. Surely that must be a crime!

Let me explain. Every year my mother took me out of school for three full weeks following the Memorial Day weekend. Thus, every single year, from K through 9th grade, I was absent from school for 3 weeks. Over time I lost about 30 weeks of schooling. With tonsil removal, recurring Mastoiditis, broken bones, and more than the average ordinary childhood illnesses, I missed a good deal of elementary schooling.
How did missing that much schooling hurt me? Not at all!

First, I must explain why my mother would break the law. In part it was to get me out of New York City as the polio epidemic hit U.S. cities from June through the summer months. For each of those summers, my family rented one room for the whole family in a rooming house filled with working class families at a beach called Rockaway. It was outside the urban area, but actually still within NYC limits.

I spent the time swimming every day, playing ball and pinochle with friends, and reading. And then, I read some more. Believe it or not, for kids like me, leaving school probably enhanced my growth! I was loved, I had great adventures, I conversed with adults in the rooming house, I saw many movies, I read classic comics, and even some “real” literature. I read series after series written for young people: Don Sturdy, Tom Swift, the Hardy Boys, as well as books by Robert Louis Stevenson and Alexander Dumas.

So now, with so many children out of school, and based on all the time I supposedly lost, I will make a prediction: every child who likes to read, every child with an interest in building computers or in building model bridges, planes, skyscrapers, autos, or anything else complex, or who plays a lot of “Fortnite,” or “Minecraft,” or plays non-computer but highly complex games such as “Magic,” or “Ticket to Ride,” or “Codenames” will not lose anything measurable by staying home. If children are cared for emotionally, have interesting stuff to play with, and read stories that engage them, I predict no deficiencies in school learning will be detectable six to nine months down the road.
It is the kids, rich or poor, without the magic ingredients of love and safety in their family, books to engage them, and interesting mind-engaging games to play, who may lose a few points on the tests we use to measure school learning. There are many of those kinds of children in the nation, and it is sad to contemplate that.

But then, what if they do lose a few points on the achievement tests currently in use in our nation and in each of our states? None of those tests predict with enough confidence much about the future life those kids will live. That is because it is not just the grades that kids get in school, nor their scores on tests of school knowledge, that predict success in college and in life. Soft skills, which develop as well during their hiatus from school as they do when they are in school, are excellent predictors of a child’s future success in life.

Really? Deke and Haimson (2006), working for Mathmatica, the highly respected social science research organization, studied the relationship between academic competence and some “soft” skills on some of the important outcomes in life after high school. They used high school math test scores as a proxy for academic competency, since math scores typically correlate well with most other academic indices. The soft skills they examined were a composite score from high school data that described each students’ work habits, measurement of sports related competence, a pro-social measure, a measure of leadership, and a measure of locus of control.

The researchers’ question, just as is every teacher’s and school counselor’s question, was this: If I worked on improving one of these academic or soft skills, which would give that student the biggest bang for the buck as they move on with their lives?

Let me quote their results (emphasis by me [-not me! GFB])

Increasing math test scores had the largest effect on earnings for a plurality of the students, but most students benefited more from improving one of the nonacademic competencies. For example, with respect to earnings eight years after high school, increasing math test scores would have been most effective for just 33 percent of students, but 67 percent would have benefited more from improving a nonacademic competency. Many students would have secured the largest earnings benefit from improvements in locus of control (taking personal responsibility) (30 percent) and sports-related competencies (20 percent). Similarly, for most students, improving one of the nonacademic competencies would have had a larger effect than better math scores on their chances of enrolling in and completing a postsecondary program.

​This was not new. Almost 50 years ago, Bowles and Gintis (1976), on the political left, pointed out that an individual’s noncognitive behaviors were perhaps more important than their cognitive skills in determining the kinds of outcomes the middle and upper middle classes expect from their children. Shortly after Bowles and Gintis’s treatise, Jencks and his colleagues (1979), closer to the political right, found little evidence that cognitive skills, such as those taught in school, played a big role in occupational success.

Employment usually depends on certificates or licenses—a high school degree, an Associate’s degree, a 4-year college degree or perhaps an advanced degree. Social class certainly affects those achievements. But Jenks and his colleagues also found that industriousness, leadership, and good study habits in high school were positively associated with higher occupational attainment and earnings, even after controlling for social class. It’s not all about grades, test scores, and social class background: Soft skills matter a lot!

Lleras (2008), 10 years after she studied a group of 10th grade students, found that those students with better social skills, work habits, and who also participated in extracurricular activities in high school had higher educational attainment and earnings, even after controlling for cognitive skills! Student work habits and conscientiousness were positively related to educational attainment and this in turn, results in higher earnings.

It is pretty simple: students who have better work habits have higher earnings in the labor market because they are able to complete more years of schooling and their bosses like them. In addition, Lleras’s study and others point to the persistent importance of motivation in predicting earnings, even after taking into account education. The Lleras study supports the conclusions reached by Jencks and his colleagues (1979), that noncognitive behaviors of secondary students were as important as cognitive skills in predicting later earnings.
So, what shall we make of all this? I think poor and wealthy parents, educated and uneducated parents, immigrant or native-born parents, all have the skills to help their children succeed in life. They just need to worry less about their child’s test scores and more about promoting reading and stimulating their children’s minds through interesting games – something more than killing monsters and bad guys. Parents who promote hobbies and building projects are doing the right thing. So are parents who have their kids tell them what they learned from watching a PBS nature special or from watching a video tour of a museum. Parents also do the right thing when they ask, after their child helps a neighbor, how the doing of kind acts makes their child feel. This is the “stuff” in early life that influences a child’s success later in life even more powerfully than do their test scores.

So, repeat after me all you test concerned parents: non-academic skills are more powerful than academic skills in life outcomes. This is not to gainsay for a minute the power of instruction in literacy and numeracy at our schools, nor the need for history and science courses. Intelligent citizenship and the world of work require subject matter knowledge. But I hasten to remind us all that success in many areas of life is not going to depend on a few points lost on state tests that predict so little. If a child’s stay at home during this pandemic is met with love and a chance to do something interesting, I have little concern about that child’s, or our nation’s, future.

Bowles, S., & Gintis, H. (1976). Schooling in Capitalist America. New York: Basic Books.

Deke, J. & Haimson, J. (2006, September). Expanding beyond academics: Who benefits and how? Princeton NJ: Issue briefs #2, Mathematica Policy Research, Inc. Retrieved May 20, 2009 from:http://www.eric.ed.gov:80/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/28/09/9f.pdfMatematicapolicy research Inc.

Lleras, C. (2008). Do skills and behaviors in high school matter? The contribution of noncognitive factors in explaining differences in educational attainment and earnings. Social Science Research, 37, 888–902.

Jencks, C., Bartlett, S., Corcoran, M., Crouse, J., Eaglesfield, D., Jackson, G., McCelland, K., Mueser, P., Olneck, M., Schwartz, J., Ward, S., and Williams, J. (1979). Who Gets Ahead?: The Determinants of Economic Success in America. New York: Basic Books.

 

The Pandemic Is Far From Over

While the rate of increase per day in the number of deaths is generally down, the COVID-19 pandemic is far from over. In general, more people are still dying each day in the US from this disease than the day before, as you can see from this data, which is taken from the CDC. The very tall bar on day 27 is when New York City finally added thousands of poor souls who had in fact died from this virus. (Day 27 means April 9, and Day 41 means April 30, which is today.)

Opening up the economy and encouraging everybody to go back to work, play, and school will mean a rebirth of exponential growth in deaths and in diagnosed cases after about 2 weeks, since this disease takes about that long to be noticed in those who have been exposed. And once everybody is back on the streets and in the stores and schools, the disease WILL spread exponentially. Opening wide right now, when we still can’t test or follow those who may be infected, would be a huge mistake.

us covid deaths per day

Only somebody as clueless as our current Grifter-In-Chief and his brainless acolytes could be recommending something so irresponsible, against the advice of every medical expert. Maybe they think that only the poor, the black, and the brown will get this disease. Wrong.

The shutdown, while painful, appears to have saved a LOT of lives so far

If you recall, the growth of the new corona virus disease in the US (and many other countries) at first looked to be exponential, meaning that the number of cases (and deaths) were rising at an alarming, fixed percent each and every single day.

Even if you slept through your high school or middle school math lessons on exponential growth, the story of the Shah and the chessboard filled with rice may have told you that the equation 2^x gets very, very hairy after a while. Pyramid schemes eventually run out of suckers people. Or perhaps you have seen a relatively modest credit-card bill get way out of hand as the bank applies 8 percent interest PER MONTH, which ends up multiplying your debt by a factor of 6 after just 2 years!

(If the total number of deaths were still increasing by 25 percent per day, as they were during the middle of March, and if that trend somehow continued without slowing down, then every single person residing inside America’s borders would be dead before the end of May. Not kidding! But it’s also not happening.)

However, judging by numbers released by the CDC and reported by my former colleague Ron Jenkins, I am quite confident that THE NUMBER OF CASES AND DEATHS FROM COVID-19 ARE NO LONGER following a fixed exponential curve. Or at least, the daily rate of increase has been going down. Which is good. But it’s still not zero.

Let me show you the data and fitted curves in a number of graphs, which often make complex things easier to visualize and understand.

My first graph is the total reported number of deaths so far in the US, compared to a best-fit exponential graph:

Deaths in US are not growing exponentially

During the first part of this pandemic, during the first 40 or so days, the data actually fit an exponential graph pretty well – that is, the red dotted line (the exponential curve of best fit) fit the actual cumulative number of deaths (in blue). And that’s not good. However, since about day 50 (last week) the data is WAY UNDER the red dots. To give you an idea of how much of a victory that is: find day 70, which is May 9, and follow the vertical line up until it meets the red dotted line. I’ll wait.

Did you find it? If this pandemic were still following exponential growth, now and into the future, at the same rate, we would have roughly a MILLION PEOPLE DEAD BY JUNE 9 in just the US, just from this disease, and 2 million the week after that, and 4 million the next week, then 8 million, then 16 million, and so on.

THAT AIN’T HAPPENIN’! YAY! HUZZAH!

As you can see — the blue and red graphs have diverged. Ignore the relatively high correlation value of 0.935 – it just ain’t so.

But what IS the curve of best fit? I don’t know, so I’ll let you look for yourself.

Is it linear?

Deaths in US are not growing in a linear fashion

This particular line of best doesn’t fit the data very well; however, if we start at day 36 or thereabouts, we could get a line that fits the data from there on pretty well, like so:

maybe this purple line

 

The purple line fits the blue dots quite well after about day 37 (about April 6), and the statistics algorithms quite agree. However, it still calls for over 80,000 Americans dead by May 8. I do not want the slope of that line to be positive! I want it to turn to the right and remain horizontal – meaning NOBODY ELSE DIES ANY MORE FROM THIS DISEASE.

Perhaps it’s not linear? Perhaps it’s one of those other types of equations you might remember from some algebra class, like a parabola, a cubic, or a quartic? Let’s take a look:

Deaths might be growing at a 2nd degree polynomial rate - still not good

This is a parabolic function, or a quadratic. The red dots do fit the data pretty well. Unfortunately, we want the blue dots NOT to fit that graph, because that would, once again, mean about a hundred thousand people dead by May 8. That’s better than a million, but I want the deaths to stop increasing at all. Like this piecewise function (which some of you studied). Note that the purple line cannot go back downwards, because generally speaking, dead people cannot be brought back to life.

maybe this purple line - nah, prefer horizontal

Well, does the data fit a cubic?

deaths fit a cubic very well

Unfortunately, this also fits pretty well. If it continues, we would still have about a hundred thousand dead by May 8, and the number would increase without limit (which, fortunately, is impossible).

How about a quartic (fourth-degree polynomial)? Let’s see:

4th degree polynomial is impossible - people do NOT come back to life

I admit that the actual data, in blue, fit the red calculated quartic red curve quite well, in fact, the best so far, and the number of deaths by Day 70 is the lowest so far. But it’s impossible: for the curve to go downwards like that would mean that you had ten thousand people who died, and who later came back to life. Nah, not happening.

What about logarithmic growth? That would actually be sweet – it’s a situation where a number rises quickly at first, but over time rises more and more slowly. Like this, in red:

logarithmic growth

I wish this described the real situation, but clearly, it does not.

One last option – a ‘power law’ where there is some fixed power of the date (in this case, the computer calculated it to be the date raised to the 5.377 power) which explains all of the deaths, like so:

no sign of a power law

I don’t think this fits the data very well, either. Fortunately. It’s too low from about day 38 to day 29, and is much too high from day 50 onwards. Otherwise we would be looking at about 230,000 dead by day 70 (May 8).

But saying that the entire number of deaths in the US is no longer following a single exponential curve doesn’t quite do the subject justice. Exponential growth (or decay) simply means that in any given time period, the quantity you are measuring is increasing (or decreasing) by a fixed percentage (or fraction). That’s all. And, as you can see, for the past week, the daily percentage of increase in the total number of deaths has been in the range of three to seven percent. However, during the first part of March, the rate of increase in deaths was enormous: 20 to 40 percent PER DAY. And the daily percent of increase in the number of cases was at times over A HUNDRED PERCENT!!! – which is off the chart below.

daily percentages of increases in covid 19 cases and deaths, USA, thru April 25

The situation is still not good! If we are stuck at a daily increase in the number of deaths as low as a 3%/day increase, then we are all dead within a year. Obviously, and fortunately, that’s probably not going to happen, but it’s a bit difficult to believe that the math works out that way.

But it does. Let me show you, using logs.

For simple round numbers, let’s say we have 50,000 poor souls who have died so far from this coronavirus in the USA right now, and that number of deaths is increasing at a rate of 3 percent per day. Let’s also say that the US has a population of about 330 million. The question is, when will we all be dead if that exponential growth keeps going on somehow? (Fortunately, it won’t.*) Here is the first equation, and then the steps I went through. Keep in mind that a growth of 3% per day means that you can multiply any day’s value by 1.03, or 103%, to get the next day’s value. Here goes:

in 10 months we are all dead

Sound unbelievable? To check that, let us take almost any calculator and try raising the expression 1.03 to the 300th power. I think you’ll get about 7098. Now take that and multiply it by the approximate number of people dead so far in the US, namely 50,000. You’ll get about 355,000,000 – well more than the total number of Americans.

So we still need to get that rate of increase in fatalities down, to basically zero. We are not there yet. With our current highly-incompetent national leadership, we might not.

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* what happens in cases like this is you get sort of an s-shaped curve, called the Logistic or logit curve, in which the total number levels off after a while. That’s shown below. Still not pleasant.

I have no idea how to model this sort of problem with a logistic curve; for one thing, one would need to know what the total ‘carrying capacity’ – or total number of dead — would be if current trends continue and we are unsuccessful at stopping this virus. The epidemiologists and statisticians who make models for this sort of thing know a lot more math, stats, biology, and so on than I do, but even they are working with a whole lot of unknowns, including the rate of infectiousness, what fraction of the people feel really sick, what fraction die, whether you get immunity if you are exposed, what is the effect of different viral loads, and much more. This virus has only been out for a few months…

logistic curve again

 

What’s the best approach – should we lock down harder, or let people start to go back to work? Some countries have had lockdowns, others have not. How will the future play out? I don’t know. I do know that before we can decide, we need to have fast, plentiful, and accurate tests, so we can quarantine just the people who are infected or are carriers, and let everybody else get back on with their lives. We are doing this lockdown simply because we have no other choice.

How do we fix the CV19 testing problem? By re-testing everybody who tested positive!

I guess I’ve re-discovered a form of Bayes’ Theorem  regarding the problem that is posed by the high numbers of false negatives and false positives when testing for the feared coronavirus.  What I found is that it doesn’t really even matter whether our tests are super-accurate or not. The solution is to assume that all those who test negative, really are negative, and then to give a second test to all those who tested positive the first time. Out of this group, a larger fraction will test positive. You can again forget about those who test negative. But re-test again, and if you like, test again. By the end of this process, where each time you are testing fewer people, then you will be over 99% certain that all those who test positive, really have been exposed.

Let me show you why.

Have no fear, what I’m gonna do is just spreadsheets. No fancy math, just percents. And it won’t really matter what the starting assumptions are! The results converge to almost perfect accuracy, if repeated!

To start my explanation, let’s start by assuming that 3% of a population (say of the US) has antibodies to CV19, which means that they have definitely been exposed. How they got exposed is not important for this discussion. Whether they felt anything from their exposure or not is not important in this discussion. Whether they got sick and died or recovered, is not going to be covered here. I will also assume that this test has a 7% false positive rate and a 10% false negative rate, and I’m going to assume that we give tests AT RANDOM to a hundred thousand people (not people who we already think are sick!) I’m also assuming that once you have the antibodies, you keep them for the duration.

This table represents that situation:

math of CV19 testing

If you do the simple arithmetic, using those assumptions, then of the 100,000 people we tested, 3%, or three thousand, actually do have those antibodies, but 97%, or ninety-seven thousand, do not (white boxes, first column with data in it).

Of the 3,000 folks who really do have the antibodies – first line of data – we have a false  negative rate of 10%, so three hundred of these poor folks are given the false good tidings that they have never been exposed (that’s the upper orange box). The other 90% of them, or two thousand seven hundred, are told, correctly, that they have been exposed (that’s the upper green box).

Now of the 97,000 people who really do NOT have any antibodies – the second line of data – we have a false positive rate of 7%, so you multiply 0.07 times 97000 to get six thousand, seven hundred ninety of them who would be told, incorrectly, that they DID test positive for Covid-19 – in the lower orange box. (Remember, positive is bad here, and negative is good.) However, 90,210 would be told, correctly, that they did not have those antibodies. (That’s in the lower green box.)

Now let’s add up the folks who got the positive test results, which is the third data column. We had 2,700 who correctly tested positive and 6,790 who wrongly tested positive. That’s a total of 9,490 people with a positive CV19 antibody test, which means that of that group of people, only 28.5% were correctly so informed!! That’s between a third and a fourth! Unacceptable!

However, if we look at the last column, notice that almost every single person who was told that they were negative, really was negative. (Donno about you, but I think that 99.7% accuracy is pretty darned good!)

However, that 28.5% accuracy among the ‘positives’ (in the left-hand blue box) is really worrisome. What to do?

Simple! Test those folks again! Right away! Let’s do it, and then let’s look at the results:

math of CV19 testing - round 2

Wowser! We took the 9490 people who tested positive and gave them another round of tests, using the exact same equipment and protocols and error rates as the first one. The spreadsheet is set up the same; the only thing I changed is the bottom two numbers in the first data column. I’m not going to go through all the steps, but feel free to check my arithmetic. Actually, check my logic. Excel doesn’t really make arithmetic errors, but if I set up the spreadsheet incorrectly, it will spit out incorrect results.

Notice that our error rate (in blue) is much lower in terms of those who tested positive. In fact, of those who test positive, 83.7% really ARE positive this time around, and of those who test negative, 95.9% really ARE negative.

But 84% isn’t accurate enough for me (it’s either a B or a C in most American schools). So what do we do? Test again – all of the nearly three thousand who tested positive the first time. Ignore the rest.

Let’s do it:

math of CV19 testing - round 3

At this point, we have much higher confidence, 98.5% (in blue), that the people who tested ‘positive’, really are ‘positive’. Unfortunately, at this point, of the people who tested negative, only about 64% of the time is that correct. 243 people who really have the antibodies tested negative. So perhaps one should test that subgroup again.

The beautiful thing about this method is that it doesn’t even require a terribly exact test! But it does require that you do it repeatedly, and quickly.

Let me assure you that the exact level of accuracy, and the exact number of exposed people, doesn’t matter: If you test and re-test, you can find those who are infected with almost 100% accuracy. With that information you can then discover what the best approaches are to solving this pandemic, what the morbidity and mortality rates are, and eventually to stop it completely.

Why we don’t have enough tests to do this quickly and accurately and repeatedly is a question that I will leave to my readers.

Addendum:

Note that I made some starting assumptions. Let us change them and see what happens. Let’s suppose that the correct percentage of people with COVID-19 antibodies is not 3%, but 8%. Or maybe only 1%. Let’s also assume a 7% false positive and a 10% false negative rate. How would these results change? With a spreadsheet, that’s easy. First, let me start with an 8% infection rate and keep testing repeatedly. Here are the final results:

Round Positive accuracy rating Negative accuracy rating
1 52.8% 99.1%
2 93.5% 89.3%
3 99.5% 39.3%

So after 3 rounds, we have 99.5% accuracy.

Let’s start over with a population where only 1% has the antibodies, and the false positive rate is 7% and the false negative rate is 10%.

Round Positive accuracy rating Negative accuracy rating
1 11.5% 99.9%
2 62.6% 98.6%
3 95.6% 84.7%
4 99.6% 30.0%

This time, it took four rounds, but we still got to over 99.6% accuracy at distinguishing those who really had been exposed to this virus. Yes, towards the end our false negative rate rises, but I submit that doesn’t matter that much.

So Parson Tommy Bayes was right.

More on the “false positive” COVID-19 testing problem

I used my cell phone last night to go into the problem of faulty testing for COVID-19, based on a NYT article. As a result, I couldn’t make any nice tables. Let me remedy that and also look at a few more assumptions.

This table summarizes the testing results on a theoretical group of a million Americans tested, assuming that 5% of the population actually has coronavirus antibodies, and that the tests being given have a false negative rate of 10% and a false positive rate of 3%. Reminder: a ‘false negative’ result means that you are told that you don’t have any coronavirus antibodies but you actually do have them, and a ‘false positive’ result means that you are told that you DO have those antibodies, but you really do NOT. I have tried to highlight the numbers of people who get incorrect results in the color red.

Table A

Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 50,000 10% 45,000 5,000
Actually Negative 950,000 3% 28,500 921,500
Totals 1,000,000 73,500 926,500
Percent we assume are actually positive 5% Accuracy Rating 61.2% 99.5%

As you can see, using those assumptions, if you get a lab test result that says you are positive, that will only be correct in about 61% of the time. Which means that you need to take another test, or perhaps two more tests, to see whether they agree.

The next table assumes again a true 5% positive result for the population and a false negative rate of 10%, but a false positive rate of 14%.

Table B

Assume 5% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 50,000 10% 45,000 5,000
Actually Negative 950,000 14% 133,000 817,000
Totals 1,000,000 178,000 822,000
Percent we assume are actually positive 5% Accuracy Rating 25.3% 99.4%

Note that in this scenario, if you get a test result that says you are positive, that is only going to be correct one-quarter of the time (25.3%)! That is useless!

Now, let’s assume a lower percentage of the population actually has the COVID-19 antibodies, say, two percent. Here are the results if we assume a 3% false positive rate:

Table C

Assume 2% really exposed, 3% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 20,000 10% 18,000 2,000
Actually Negative 980,000 3% 29,400 950,600
Totals 1,000,000 47,400 952,600
Percent we assume are actually positive 2% Accuracy Rating 38.0% 99.8%

Notice that in this scenario, if you get a ‘positive’ result, it is likely to be correct only a little better than one-third of the time (38.0%).

And now let’s assume 2% actual exposure, 14% false positive, 10% false negative:

Table D

Assume 2% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 20,000 10% 45,000 2,000
Actually Negative 980,000 14% 137,200 842,800
Totals 1,000,000 182,200 844,800
Percent we assume are actually positive 2% Accuracy Rating 24.7% 99.8%

Once again, the chances of a ‘positive’ test result being accurate is only about one in four (24.7%), which means that this level of accuracy is not going to be useful to the public at large.

Final set of assumptions: 3% actual positive rate, and excellent tests with only 3% false positive and false negative rates:

Table E

Assume 3% really exposed, 3% false positive rate, 3% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 30,000 3% 45,000 900
Actually Negative 970,000 3% 29,100 940,900
Totals 1,000,000 74,100 941,800
Percent we assume are actually positive 3% Accuracy Rating 60.7% 99.9%

Once again, if you test positive in this scenario, that result is only going to be correct about 3/5 of the time (60.7%).

All is not lost, however. Suppose we re-test all the people who tested positive in this last group (that’s a bit over seventy-four thousand people, in Table E). Here are the results:

Table F

Assume 60.7% really exposed, 3% false positive rate, 3% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 45,000 3% 43,650 1,350
Actually Negative 29,100 3% 873 28,227
Totals 74,100 44,523 29,577
Percent we assume are actually positive 60.7% Accuracy Rating 98.0% 95.4%

Notice that 98% accuracy rating for positive results! Much better!

What about our earlier scenario, in table B, with a 5% overall exposure rating, 14% false positives, and 10% false negatives — what if we re-test all the folks who tested positive? Here are the results:

Table G

Assume 25.3% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 45,000 14% 38,700 6,300
Actually Negative 133,000 10% 13,300 119,700
Totals 178,000 52,000 126,000
Percent we assume are really positive 25.3% Accuracy Rating 74.4% 95.0%

This is still not very good: the re-test is going to be accurate only about three-quarters of the time (74.4%) that it says you really have been exposed, and would only clear you 95% of the time. So we would need to run yet another test on those who again tested positive in Table G. If we do it, the results are here:

Table H

Assume 74.4% really exposed, 14% false positive rate, 10% false negative
Group Total Error rate Test says they are Positive Test says they are Negative
Actually Positive 38,700 14% 33,282 5,418
Actually Negative 13,300 10% 1,330 11,970
Totals 52,000 34,612 17,388
Percent we assume are really positive 74.4% Accuracy Rating 96.2% 68.8%

This result is much better, but note that this requires THREE TESTS on each of these supposedly positive people to see if they are in fact positive. It also means that if they get a ‘negative’ result, that’s likely to be correct only about 2/3 of the time (68.8%).

So, no wonder that a lot of the testing results we are seeing are difficult to interpret! This is why science requires repeated measurements to separate the truth from fiction! And it also explains some of the snafus committed by our current federal leadership in insisting on not using tests offered from abroad.

 

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EDIT at 10:30 pm on 4/25/2020: I found a few minor mistakes and corrected them, and tried to format things more clearly.

COVID-19 Numbers in the US do not seem to be growing exponentially

Looking at the past month of CDC-reported infections and deaths from the new corona virus, I conclude that there has been some good news: the total number of infections and deaths are no longer following an exponential growth curve.

The numbers are indeed growing, by either a quadratic (that is, x^2) or a quartic (x^4) curve, which is not good, and there is no sign of numbers decreasing.

BUT it looks as though the physical-social distancing and self-quarantining that I see going on around me is actually having an effect.

Yippee!

Here is my evidence: the actual numbers of infected people are in blue, and the best-fit exponential-growth equation is in red. You can see that they do not match well at all. 

total cases US not looking exponential

If they did match, and if this were in fact exponential growth, we would have just about the entire US population infected by the end of just this month of April – over 300 million! That no longer seems likely. Take a look at the next graph instead, which uses the same data, but polynomial growth:

total cases US looking second power

Just by eyeballing this, you can see that the red dots and blue dots match really, really well. When I extend the graph until the end of April, I get a predicted number of ‘only’ 1.5 million infected. Not good, but a whole lot better than the entire US population!

Also, let’s look at total cumulative reported deaths so far. Here are the CDC-reported numbers plotted against a best-fit exponential curve:

deaths do not seem to be exponential

Up until just a few days ago, this graph was conforming pretty well to exponential growth. However, since about April 8, that seems to be no longer the case. If the total numbers of deaths were in fact growing at the same percentage rate each day, which is the definition of exponential growth, then by the end of April we would have 1.5 million DEAD. That’s THIS MONTH. Continued exponential growth would have 1.2 BILLION dead in this country alone by the end of May.

Fortunately, that is of course impossible.

Unfortunately all that means is that the virus would run out of people to infect and kill, and we would get logistic growth (which is the very last graph, at the bottom).

death seem to be 4th power polynomial

This fourth-degree mathematical model seems to me to work much better at describing the numbers of deaths so far, and has a fairly good chance of predicting what may be coming up in the near future. It’s still not a good situation, but it shows to me that the social and physical distancing we are doing is having a positive effect.

But let’s not get complacent: if this model correctly predicts the next month or two, then by the end of April, we would have about 60 thousand dead, and by the end of May we would have 180 thousand dead.

But both of those grim numbers are much, much lower than we would have if we were not doing this self-isolation, and if the numbers continued to grow exponentially.

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FYI, a logistic curve is shown below. Bacteria or fungi growing in a broth will grow exponentially at first, but after a while, they not only run out of fresh broth to eat, but they also start fouling their own environment with their own wastes. WE DO NOT WANT THIS SITUATION TO HAPPEN WITH US, NAMELY, THAT WE ALL GET INFECTED!!!

logistic curve again

Various graphs for deaths from COVID-19, so far

I wrote that I would show you what various graphs of various types of simple models look like for deaths so far due to the current corona virus: linear, exponential, polynomial, and so on. I think that a fourth-degree (not third-degree, like I wrote earlier) seems to fit the data best so far, and that’s better than exponential growth.

First, let’s look at a straight-line best-fit model, superimposed by Excel on the data. (Note: deaths are on the Y, or vertical, axis; the X-axis represents days since the beginning of March. So today, the 6th of April, is day 37 (31 + 6). The dotted red line represents the line of best fit, and the blue dots are the CDC-announced numbers of deaths so far.

is it linear

As you can see, the straight dotted line doesn’t fit the data very well at all. R-squared, known as the correlation coefficient, tells us numerically how well it fits. If R or R^2 equals 1.000, then you have absolutely perfect correlation of the data to your model. Which we do NOT have here. By the way, in that model, then by mid-June we would have about 22,000 dead from this disease.

OK, let’s look at an exponential curve-of-best fit next:

is it exponential

As you can see, this red curve fits the data a LOT better, and R-squared is a lot higher.

Unfortunately.

We do NOT WANT EXPONENTIAL GROWTH OF THIS OR ANY OTHER DISEASE, BECAUSE IT MEANS WE ALL GET IT! In fact, if this model is accurate and isn’t slowed down, then by mid-June, just plugging in the numbers, we would have 3.3 BILLION (not million) people dead in the US alone. Fortunately, that won’t happen.

BUT there are some parts of the data where the curve doesn’t fit perfectly — let me point them out:

is it exponential -2

At the upper right-hand end, the red dotted line is quite a bit higher than the blue dots. Fortunately. And near the middle of the graph, the blue dots of death are higher than the red line.

OK, let’s look at some polynomial models instead:

is it a second degree polynomial

This is a fancy version of the simple y=x^2 parabolas you may have graphed in Algebra 1. Once again, this doesn’t do a terrific job of conforming to the actual data. At the right-hand end, the blue dots of death are higher than the curve. In addition, if we continued the red curve to the left, we would find that something like two thousand people had already died in the US, and presumably came back to life. Which is ridiculous.

However, if this model were to hold true until mid-June, we would have 127 thousand dead. Not good.

Let’s try a third-degree polynomial (a cubic):

is it a third degree polynomial

That’s pretty remarkable agreement between the data and the equation! That’s the equation I was using in my earlier post. The R-squared correlation is amazing. Unfortunately, if this continues to hold, then we would have about 468 thousand dead in the US.

Let’s continue by looking at a fourth-degree polynomial curve fitted to the data:

is it a fourth degree polynomial

That is an amazingly good fit to the data! Unfortunately, let’s hope that it won’t continue to fit the data, because if it does, then we are looking at a little over a MILLION dead.

Let’s hope we can get these totals to level off by physically distancing ourselves from other households, washing our hands, and getting proper protective garments and testing technology to our medical personnel.

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Here’s another model that unfortunately does NOT work: logarithmic growth. If it were the case, then we would have about 10,700 deaths by mid-June.

is it logarithmic

 

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