NedNotes (not blog): Appendix-l to weekly COVIData sweeps

Appendix-I to Letter #158 to Friends & Familiares
https://nedmcdletters.blogspot.com/2020/04/letter-158b-covid-numbers-for-family.html
Appendix-l to the COVIData Sweeps; added 5th December 2020

INTRODUCTION
This appendix clarifies the meaning of symbols and calculations on the weekly statistical summaries contained in the bellwether states and the table of thirty-four states and five territories. 

The bellwether states consist of a sample of ten of thirty-eight states and one territory that faced the epidemic after the hard Spring for the industrial states, primarily in the Northeast. These ten states comprise half of the 264 million people living in across the South; the Southwest; the Midwest; and the Pacific coast. 

The '38+' table covers compares the magnitude of mortality among thirty-four states and five territories. This group integrates the bellwether states and other states around the country and represents approximately 95% of the 330 million population of the United States. Thes tables provide adequate coverage of the wider country.

Specifically, this appendix delivers the following improvements to the COVIData sweeps:

  • clarification and explanation of the growth rates used in the bellwether states table so the reader can understand the implications of the time weighted versus compounded weekly growth rates;
  • revision of the risk parameters in the 38+ table to reflect actual experience and revised expectations laid out by experts far better informed than l;
  • clean up and explanation of the 'tri-colored' positivity rates; as well as,
  • systematization of the symbology expressing a state's testing posture.

The first section on the time weighted versus geometric growth rates takes up three-quarters of the appendix as it explains the arithmetic and logic underlying the calculations. One familiar with these concepts can punt the screaming yadas. 

GEOMETRIC or COMPOUND versus TIME WEIGHTED GROWTH RATES
The bellwether states, twelve in all, calculate weekly growth rates. They do so for the ten states, not regions within them. One can confuse these data with information in the family cluster muck since the latter refers to monthly growth rates and involves metropolitan areas within nine states; only Colorado is considered on a statewide basis. The two aggregate 'black-&-blue’ growth rates -- one in black and one in blue -- measure the same phenomena differently.

The growth rates in black font represent a geometric, or compound, weekly growth rate. In concept, this calculation is similar to that of compound interest. The calculation basically determines a constant growth rate for each period, assumed to be the same in duration, from the ‘initial (or beginning) value’ of the first period on through to ‘terminal (or final / ending) value’ of the last period.

Arithmetic
When simple is too simple. For example, the initial value of 100 becomes 120 in period-1 and then increases further to 150 during the second period. Using an 'arithmetic' average, one would understand that the value increased by twenty in period-1, or 20% (i.e., 20 / 100 = 0.20 = 20%). Subsequently, the value increased by thirty in period-2 for a periodic increase of 25% (i.e., 30 / 120 = 0.25 = 25%). Taking the arithmetic average simple periodic growth rates of 20% and 25%, one might say that the value increased by 22.5% each year (i.e., [20% + 25%] / 2 = 45% / 2 = 22.5%).

Geometric or Compound
Time for a calculating S.O.B. The geometric, or compound, rate takes the two identical periods of growth and calculates the implied growth rate were the individual periodic growth rates of each the two periods equal to each other. This calculation is the compounded growth rate, factoring the 'interest-on-interest' concept one sees in banking into the growth rate. One calculates this geometric growth rate by taking the terminal value against the initial value and spreading that cumulative growth rate of, in this case, the two periods over each equal period.

This spreading function uses exponents; specifically the 'negative' exponent equal to the number of periods elapsed; in this case, two. Thus, one has 150 at the end of two equal periods for a cumulative growth of 50% (i.e., 150 / 100 = 1.50 = 150% - 100% = 50%).

To calculate the uniform growth rate for each of the two periods, ones takes the 150% end value and calculates for the negative exponent of two periods; this negative exponent, in this case, is simply the square root. One must calculate the ratio of the final to initial value before subtracting the 100% for the initial value lest that person end up with a non-sensical uniform growth rate.

In this case, one calculates the square root of 1.5 (or raises the decimal ratio to negative second power, or (1.5)-2, followed by subtracting one (i.e., the component expressing the initial value of 100 in this case). Thus, one calculates the two year geometric growth rate of

(1.5)-2 - 1 = 1.2247 - 1 = .2247 or 22.47%.

This number is slightly less than 22.5% calculated by average the two periodic rates. The reason for these close values lies in the simplicity of the data and the few number of periods for calculation. It becomes pronounced over time.

Born in Babble-ona, moved to Arizona. (oops, I mean "born in Arizona, moved to Babaloney".) Currently we are on the twenty-ninth week of calculation for the original five bellwether ‘Summertime Blues’ states and the tenth week for the five newbies. Consequently, to calculate the uniform, or smoothed, growth rate, one takes the most recent value relative to the initial value and raises it to the negative twenty-ninth power or the twenty-ninth root. Arizona, for example, has a current value for confirmed cases of 354,050 infections (at 04dec20) versus just 13,631 twenty-nine weeks ago (at 16may20).

  1. The cumulative growth of cases has been 2497% or 25x:
    or, 354,050 / 13,631 - 1 = 25.97 - 1 = 24.97 = 2497%).
  2. A simple arithmetic calculation would yield 86% growth each week:
    or, 24.97 / 29 = 0.86 = 86%).
  3. This approach overlooks the compounding akin to the interest-on-interest component in a savings account or a mortgage loan to be re-paid. The geometric average calculates out to an 11.9% uniform growth for each of the twenty-nine weeks.
  4. Therefore: [354,050 / 13,631]-29 = 25.97535)-29
    = 1.118866 -1 = 0.119
    = 11.9% per week.

Time Weighted
Hey, punk, ¡timing is everything! The blue-italic font is a time weighted, essentially arithmetic, periodic growth rate to account trends or sharp changes, or swings, in periodic growth rates over time. It is a simpler calculation. 

One takes each periodic growth rate and multiplies that particular percentage by a coefficient representing the number of periods that have elapsed since the initial period. Then one sums the product of each calculation and divides that result by a sum of the coefficients (think sum-of-the-digits accelerated depreciation in the denominator).

Speaking of time, could ya, like, move it, ¿s.v.p? In the previous example, one had an initial value of 100, followed by a value of 120 after the first period and ended by a value of 150 after the second equally long period. One has seen that the simple arithmetic rate is 22.5% for each period versus the slightly lower geometric average of 22.47%.

  1. The time-weighted average takes the most recent, second year, 25% growth rate. 
    So, 150 / 120 = 1.25 -1 = 0.25 = 25%
    and in turn multiplies that value by the number of periods since the beginning, or two:
    25% x 2 = 50%.
  2. Likewise, one takes the first year growth rate of 20%.
    Hence, 120 / 100 = 1.2 -1 = 0.20 = 20%
    and then multiplies that percentage by the number of periods lapsed from the beginning, or one:
    20% x 1 = 20%.
  3. Next, one adds up the two products and divides by the sum of the coefficients to calculate a time-weighted periodic growth of 23.3%.
    Stay with me: [50% + 20%] / [2 + 1] = 70% / 3 = 23.3%

This value of 23.33% is visibly above the smoothed compound growth rate of 22.47%.

Why?

Because the more recent growth rate of 25% is higher than the previous growth rate of 20%.

Reversal of fortune, ¿or torture? Had the increments of growth been reversed (i.e., 100 for period-1; 130 for period-2; and, 150 for period-3), the geometric growth rate would remain the same: [150 / 100]-2 = 1.2497 -1 = .2247 = 22.47%.

Would the time weighted periodic rate be same as the first calculation of 23.33%?

NOPE.

In fact, the time weighted growth rate would be noticeably lower at 20.3%

  • [150 / 130] = 1.15385 - 1 = 0.15385 = 15.4% x 2 = 30.8%
  • [130 / 100] =1.30 - 1 = .30 = 30% x 1 = 30%.
  • Accordingly, (30.8% + 30% = 60.8% / 3 = 20.26% = 20.3%).

Why the difference? Because the lower growth year came later, emphasizing the slowing periodic growth.

CCC = compare, contrast, confuse
So, which rate is the better to use? The geometric rate has the advantage of giving a smooth rate that accounts for the accumulation of value, on which to grow, over time. But it fails to account for swings in individual periodic growth rates.

For example, say the initial value is 100 but falls to 75 after period-1 and rebounds to 150 in period-2. The geometric or compound growth rate is straight-forward at the previous value of 22.47%.

  1. Here we go again: [150 / 100]-2 = [1.50]-2 = 1.2247 - 1 = .2247 = 22.47%

But does this uniform rate truly capture the growth pattern? The time weighted value calculates to 58.3%.

  1. Ee-gadds, again; for period-1: 75 / 100 = .75 - 1 = -.25 = -25.0% x 1 = -25.0%
  2. for period-2: 150 / 75 = 2.0 - 1 = 1.0 = 100% x 2 = 200%
  3. therefore, [-25.0% + 200%] / [2 + 1] = 175 / 3 = 175% / 3 = 58.3% (¡or 2.6x the compound rate!).

On the other hand, the order of swings reverses, and one sees the following periodic values over equally long periods:
100 initial value; 175 after period-1; and, 150 after period-2.
Again, the geometric rate is transparent.

  1. ¡One more time! 22.47% (i.e., [150 / 100]-2 - 100 = [1.50]-2 -1 = 0.2247 = 22.47%.

Yes, this geometric growth rate is equal to the first example but is it really the same? The time weighted periodic growth comes out to 15.5%

  1. ¡Oy vey! First period: 175 / 100 = 1.75 -1 = 0.75 = 75.0% x 1
  2. For the second period, 150 / 175 = 0.8571 - 1 = .1429 -1 = -14.3% x 2 = -28.6%
  3. Consequently, [75.0% - 28.6%] / [2 + 1] = 46.4% / 3 = 15.5%, (roughly two-thirds of the compound periodic growth rate of 22.47%).

History repeats itself. The time weighted growth accents the lower value in the second period. So, the time weighted trend obviously delivers a more useful calculation because it weights the recent trend, ¿right?

Easy, sailor; not quite that simple.

In the example above, one sees a symmetrical set of changes in values: one goes down by twenty-five and then up by seventy-five; the other goes up by seventy-five then down by twenty-five. So why are the two time weighted averages – 58.3% (down 25; up 75) versus 15.5% (up 75; down 25) – so far apart?

Because the timing of the swings makes them sharper with respect to the preceding values used as sequential reference points in the calculations.

  • 100 => 75 => 150:  -25.0% then +100% versus
  • 100 => 175 => 150: +75.0% then -14.3%

So, time weighted averages are helpful but not sufficient since they vary depending upon the timing and magnitude of the swings in particular value. Additionally, time weighted rates are arithmetic calculations with a trend-weighting integrated into the calculation. Arithmetic calculations tend to overstate growth rates relative to their compounding geometric counterparts.

My recommendation? Use both periodic rates and compare them with following ideas in mind.

  1. When the time weighted growth rate runs notably higher than the compound growth rate, chances are that growth has been accelerating or gains have been increasingly greater than losses over time.
  2. When the two periodic growth rates are similar, one can infer that the growth rates are consistent over time.
  3. When the geometric growth rate is noticeably higher than the time weighted rate, growth is either decelerating or recent losses have been higher relative to recent gains.

Conclusion on bellwether bloviation
Are these conclusions axiomatic? No way. But they can give one a quick sense of what is happening. Of course, further investigation is required. Looking again at Arizona's case load for twenty-nine weeks, ending at 354,070 confirmed cases on 04dec20, ones sees a compound growth rate of 11.9% and a time weighted periodic rate of 7.6%.

What can a casual observer surmise with this difference? There is a strong likelihood that weekly growth rates have slowed over time. Yet, that onlooker sees that, over the last nine weeks, the weekly growth rates have started at under 2% and now exceed 11.1%.

Sooo, ¡what gives?

If one goes further back, (s)he will find that weekly growth rates, during the short-&-sweat Spring of 2020, weekly growth rates trended up rapidly from 18% to 42% between late May and late June. The State's restrictions kicked in and the fever broke as weekly growth rates slowly headed back down to 1.5-2.0% range during the Summer months. The growth rate is rising again but not as abruptly, yet at least, as in the Spring.

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REVISION of RISK LEVELS in the TABLE of 34+ STATES
The risk levels assigned to the diverging fatalities of the several states have changed to reflect nine months of experience with the epidemic in the United States. The basis for the classification is the common-sizing of each state’s mortality experience to make it comparable to the national level of deaths for that week and to the ‘normalized’ values of other states. Rather than calculating deaths per million or hundred thousand, I have opted to gross the state population up to the national population of 332,609,102.

Accordingly, I apply that multiple to the actual number of the state’s deaths to yield common-sized data directly comparable. The improved risk parameters center upon the first year (through 05mar21) base case fatality level of 335,301 souls that I forecast eight months ago, together with the updated projection from the University of Washington of 470,974 deaths. The former datum serves as a benchmark for assessing the Trump Admin. performance, particularly over the last six weeks, end 04dec20. The classifications remain the same as before with new values assigned:

  • “very low” changed from below 40,000 confirmed cases to below 100,000;
  • “low” ranging 100-225,000 cases rather than 40-100,000;
  • “moderate” ranging 225-375,000 versus 100-175,000 cases;
  • “high” lying between 375-500,000 cases, not 175-350,000; and,
  • “very high” changed from above 350,000 to above 500,000 cases.

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REFORM / CORRECTION of POSITIVITY RATES
Changes in positivity rates now follow these revised parameters due to my previous and unannounced shift in practice conveying a mixed message or metaphor in past weeks. My apologies for the confusion. Please note that these color-coded changes of positivity rates are the percentage changes of percentages; that can render interpretation of such data rather tenuous. The new parameters are:

  • italics red font for increases (i.e., deteriorations) of the positivity rates between 10-20% over the prior week’s percentage positivity rate;
  • italics blue font for decreases (i.e., improvements) of positivity percentages between 10-20% week-over-week.
  • plain black font for changes up (for the worse) or down (for the better) from the percentage of the previous week within 10%;
  • bold italics blue font for decreases (i.e., improvements) of positivity percentages in excess of 20% over those of the preceding week; as well as,
  • bold italics red font for increases (i.e., deteriorations) of the positivity rates more than 20% week-over-week.

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ASSESSING a STATE’s COMMITMENT TO COVID TESTING
A refresher on the assessment of a state’s testing capacity follows. Please note that “the percentage of population tested” is misleading. The datum implicitly assumes that each person tested is examined only once; not the case. The calculation is simply the number of tests per million restated as a percentage. Refer to the revised parameters below, s.v.p.:

  • two thumbs down for states with a testing ratio more than fifteen percentage points below the national level, expressed as a percentage, for the same week;
  • one thumb down for states between five and fifteen percentage points below the national benchmark for the week;
  • an index finger pointed left for testing levels up to five percentage points below the concurrent national average;
  • two index fingers pointing at each other for a testing level immaterially different from the weekly benchmark;
  • an index finger pointing right to indicate testing levels up to five percentage points higher than the concurrent benchmark;
  • one thumb up for testing ranging between five and fifteen percentage points higher than weekly average;
  • two thumbs up for testing penetration more than fifteen percentage points higher than the relevant national level;
  • symbols in red font indicating a decline in testing capacity or activity (e.g., changing from four percentage points below the weekly national benchmark to nine points below, or from a left-pointing index finger to one thumb down);
  • symbols in black font denoting no change of classification; as well as,
  • symbols in blue font indicating an up-grade of standing vis à vis the concurrent average (e.g., rising from a level of seven percentage points above the national level, expressed as a percentage, to seventeen points above, or from one thumb up to two thumbs up).

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