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CHAPTER 5
Dose-Response, Linear Regression,
and Some Other Key Concepts in Our Analyses
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Part 1. Radiation-Induced Cancer: The "Build-Up" Phase
Part 2. Equilibrium Phase: Flat Rates of Radiation-Induced Cancer
Part 3. The "Build-Down" Phase: An Exceedingly Gradual Phenomenon
Part 4. Is There Really Any "Minimum Latency Period" ?
Part 5. Dose-Response: Linearity and Regression Analysis
Part 6. Dose-Response: Perfect Correlation without Perfect
Proportionality
Part 7. Dose-Response: Effects of Imperfect Matching across
Dose-Groups
Part 8. Real-World "Entropic Circumstances" Which Reduce Observed
Correlations
Part 9. Estimating the Impact of Medical Radiation on Cancer
MortRates
Figure 5-A. Annual Delivery-Rates of Radiation-Induced Cancer.
Figure 5-B. The MX Model of Dose-Response.
Figure 5-C. The MX+C Model of Dose-Response.
Figure 5-D. Effect of Imperfect Matching of Dose-Groups.
Figure 5-E. Effect of an Inverse Relationship between Dose and a
Co-Actor.
* Part 1. Radiation-Induced Cancer: The "Build-Up" Years
==========================================================
Because ionizing radiation is a carcinogen (Chapter 2, Part
4), its introduction into medicine, in 1896, had to cause
radiation-induced Cancers. The Cancers, caused by medical
radiation received during 1896, did not all appear at once. Like
products dispensed from an inventory, the Cancers were delivered
gradually (Chapter 2, Part 8). And the Cancers caused by medical
radiation received during 1897 were also delivered gradually. And
the Cancers caused by medical radiation received during 1898 were
gradually delivered, too. We need not name every year for a
century.
1a. Figure 5-A: The "Build-Up" Years
-----------------------------------------
Figure 5-A depicts the effect of gradual delivery: A period of
"build-up" in the annual delivery of radiation-induced Cancer.
Figure 5-A refers to cancer incidence, with no arbitrary interval
between radiation exposure and diagnosis of radiation-induced
cancer (discussion in Part 4, below). We emphasize that Figure 5-A
is a diagram in which we arbitrarily:
(a) show years of annual irradiation only through 1951;
(b) make 120 cases (per 100,000 population) the total
cancer-consequence from each year's irradiation;
(c) make 40 years the maximum delivery-time for the 120 cases
produced by a single year's irradiation;
(d) make delivery of the 120 cases occur at a constant annual
rate: 3 cases per 100,000 population, annually, for 40 years. This
is equivalent to having 40 different latency periods before
diagnosis occurs.
As a result of these choices, the annual deliveries of Cancer
induced by medical irradiation show a build-up in Figure 5-A as
follows:
During 1896: 3 cases delivered per 100,000 population.
During 1897: 6 cases delivered per 100,000 population.
During 1898: 9 cases delivered per 100,000 population.
During 1899: 12 cases delivered per 100,000 population.
During 1900: 15 cases delivered per 100,000 population.
..... The 40th year is 1935.
During 1935: 120 cases delivered per 100,000 population.
Although the numbers in Figure 5-A are merely illustrative,
they make a key point: The introduction and maintenance of medical
radiation necessarily caused a gradual build-up in the number of
RADIATION-induced cases of Cancer per 100,000 population.
1b. Equality of Response:
A Simplifying Assumption in Figure 5-A
-----------------------------------------------
Figure 5-A depicts an "ideal" situation in which the magnitude
of the average radiation dose is the same every year, and the
magnitude of the response is the same (120 radiation-induced
Cancers per 100,000 population).
Equality of response over decades is a condition invoked for
simplification. In reality, several lines of evidence indicate
that the magnitude of carcinogenic response, per rad of radiation
exposure, can be modulated (altered) by the intensity of exposure
to nonradiation co-actors and by the absence of such co-actors.
Therefore, the magnitude of cancer-response, per unit of radiation
exposure, can vary over time according to the abundance or paucity
of co-actors.
The Introduction has already discussed the widely accepted
concept that more than one cause is necessary to produce a case of
fatal Cancer. Ways in which carcinogenic co-actors can multiply
each other's potency is a topic deferred to Chapters 49, 67, and
Appendix-M. Here, we simply point out that --- when Figure 5-A
depicts a response of constant size to a radiation dose of
constant size, decade after decade --- we have invoked the "ideal"
assumption that exposure to co-actors is also constant decade
after decade.
* Part 2. Equilibrium Years:
Flat Rates of Radiation-Induced Cancer
==================================================
When the production-rate and the delivery-rate of Cancer are
equal in the SAME CALENDAR YEAR --- despite the variable and
extended latency periods --- it is because equilibrium has
occurred between two opposite drives. During the equilibrium
years, successive columns add one box at the top --- and subtract
one box at the bottom.
Equilibrium is first reached in Figure 5-A in the year 1935.
That is the first year in which 120 new Cancers/100,000 population
are PRODUCED (for gradual delivery) and also 120 radiation-induced
Cancers are DELIVERED (from earlier years of production plus 1935
production).
Since nothing changes in our ideal model, Figure 5-A shows
that equilibrium continues through 1951. Equilibrium would
continue INDEFINITELY if the average radiation dose were
maintained at a constant level "forever" --- but due to the size
of our page, Figure 5-A completely terminates medical irradiation
after 1951.
Flat Cancer-Rates and the "Law of Equality"
-------------------------------------------
The "Law of Equality" states: If an age-matched population
receives the same level of irradiation and same exposure to
co-actors year after year, ultimately a state of equilibrium will
be reached when the annual delivery-rate of radiation-induced
Cancer per 100,000 population is equal to the annual
production-rate of radiation-induced Cancer per 100,000
population, and the same annual delivery-rate will endure
indefinitely, if the same annual production-rate is maintained.
In other words, the "Law of Equality" leads toward FLAT rates
of radiation-induced Cancer. In Figure 5-A, the rate in every year
of equilibrium is 120 cases/100,000 population. The equilibrium
years, in Figure 5-A, are limited to 1935 through 1951 --- simply
because of the size of the page.
Does the "Law of Equality" depend on assuming that the
delivery rate of a single year's production occurs in equal parts
--- such as 3 cases each year as shown in Figure 5-A? No. The law
is also valid for delivery in unequal parts over the specified
timespan. This is demonstrated in Gofman 1995/96, where all of
Chapter 4 is devoted to the "Law of Equality."
* Part 3. The "Build-Down" Phase:
An Exceedingly Gradual Phenomenon
=============================================
Due to the size of our page, Figure 5-A COMPLETELY terminates
irradiation of the population after 1951. But deliveries of
radiation-induced Cancer continue, because in 1951, deliveries
from irradiation received during the 1940s and 1930s and 1920s and
even earlier, were not yet complete. The total annual deliveries
can decline only GRADUALLY, even when there is NO additional
production.
The build-down of deliveries can be quantified by counting the
vertical boxes in the post-1951 columns. Each year, just one box
is coming off at the bottom of successive columns. So total
delivery declines to 117 cases in 1952, 114 cases in 1953, 111
cases in 1952, and so forth.
3a. How Does the "Termination" Model Relate to Reality?
------------------------------------------------------------
Depiction of this build-down phase should drive home an
important point. If all medical radiation were abruptly and
permanently terminated (which we certainly do not advocate), and
if exposure to co-carcinogens were held constant, the resulting
reduction in cancer mortality rates would happen gradually over
about 50 years --- due to delivery of radiation-induced cases
already "in the pipeline." The gradual build-down depicted in
Figure 5-A is an important reminder, that uselessly high doses of
xrays administered TODAY will still be causing Cancers 10, 20, 30,
40 (and more) years from now.
3b. Real-World Status of Delivery-Schedules
------------------------------------------------
For radiation-induced cancer cases, delivery-intervals after
irradiation are necessarily much clearer in studies of excess
cases due to exposure at a SINGLE time (such as radio-iodine
exposure from the Chernobyl accident, or gamma-ray exposure from
the Hiroshima-Nagasaki bombs), than in studies of excess Cancer
due to chronic exposure (such as occupational exposures spread
over years or decades). In the latter, it is impossible to know in
WHICH years the radiation produced the carcinogenic lesions. Thus,
analysts rely heavily on the Atomic-Bomb Study for our knowledge
of delivery-intervals and duration (Chapter 2, Part 8).
The observation, of excess Cancers (meaning radiation-induced
cases) from a particular radiation event, refers to the excess
number compared with the number occurring in comparable "control"
groups NOT exposed to extra radiation from the particular
radiation event. Of course, part of the "background" cancer-rate
in the control groups is radiation-induced too --- by OTHER
sources of radiation exposure.
The total cancer-rate (radiation-induced cases plus cases
which would occur anyway) climbs with advancing age --- as
illustrated for 1940 in Chapter 4 (Box 2), and for 1990 in Chapter
4 (Box 4). To the extent that radiation-induced cases occur at
approximately the same ages as radiation-unaided cases (Gofman
1971, p.244; BEIR 1990, p.5), then the AVERAGE interval between
irradiation and delivery of radiation-induced cases will be longer
for cases induced during childhood than for cases induced at ages
near or beyond age 55.
Even though delivery-schedules vary by age at irradiation,
delivery-schedules for radiation-induced Cancers will be the same,
per 100,000 population, in all Nine Census Divisions --- because
the Divisions have been "matched" for age by use of age-adjusted
cancer MortRates.
* Part 4. Is There Really Any "Minimum Latency Period" ?
==========================================================
The notion that there exists a "minimum latency period" of 5
to 20 years after radiation exposure, before any radiation-induced
Cancer is manifest, is almost certainly mistaken (Gofman 1981,
Gofman 1994, Gofman 1995/96). The limited evidence at hand shows
that atomic-bomb-induced Leukemia showed up before five years
(BEIR 1972, p.101; and UNSCEAR 1986, p.222, Fig. 24), and that
Chernobyl-induced Thyroid Cancer also showed up within five years
(Kazakov 1992; Baverstock 1992; WHO 1995).
Indeed, in a non-Chernobyl radio-iodine study (of 38,000
medical patients who received diagnostic doses of iodine-131),
Holm et al mention a large excess of Thyroid Cancer observed
during the first five years after administration of the iodine-131
(Holm 1988). However, Holm et al count NONE of these Cancers as
caused by the radio-iodine. With a single sentence, the cases are
simply discarded from the study --- a decision which appears to be
a highly questionable prejudgment (full analysis in Gofman 1990,
Chapter 22, Part 5).
After the atomic bombings in August 1945, the A-Bomb Survivors
Study is silent about solid Cancers until 1950. The study's first
report on solid Cancers covers the period 1950 through 1954. It
shows that, by then, the 25,203 exposed survivors (all ages and
all doses, combined) had a solid cancer MortRate which was already
11% higher than the rate among the 66,028 participants in the
reference-group (details in Gofman 1990, Table 17-A). If the study
had involved 130,000,000 to 250,000,000 participants --- as our
study of the entire U.S. population does --- then a
radiation-induced excess MortRate of fatal solid Cancers might
have been detectable within 1 to 2 years after the bombings.
We know of no studies which are capable of ESTABLISHING that a
five-year "minimum latency period" truly occurs, between exposure
of a mixed-age population to extra ionizing radiation and delivery
of fatal cases of radiation-induced solid Cancers.
Some Biology-Based Logic for Expecting No Minimum
-------------------------------------------------
We know of no biological basis for EXPECTING any minimum
latency period for Cancer in a large mixed-age population. By
contrast, we know of some reasons for expecting NO such minimum.
In molecular biology, evidence is accumulating that a cell
becomes malignant only after its chromosomes have accumulated
SEVERAL carcinogenic abnormalities (see Appendix D, for instance).
Some of these genetic abnormalities may be inherited, and others
may be acquired at any age after conception. If a cell, which has
already accumulated a full set of carcinogenic lesions except for
one, receives the final necessary lesion from a
radiation-exposure, the delivery-time for that particular case of
radiation-induced Cancer could be extremely short.
Almost certainly, carcinogenic genetic lesions have a range of
effects, from a mild predisposition to Cancer, to a virtual
guarantee of a rapidly lethal malignancy. It is very reasonable to
expect that the speed of cancer development varies with the
particular areas of chromosomal damage or chromosomal deletion
which are present in a cell. For this reason, too, it is very
reasonable to expect that some radiation-induced Cancers will be
delivered almost immediately as overt, clinical cases.
Unless strong epidemiologic evidence develops someday in favor
of a minimum latency period for radiation-induced Cancers, we
think the most reasonable assumption is NO minimum latency period
in populations of mixed ages.
* Part 5. Dose-Response: Linearity and Regression Analysis
============================================================
In Figures 1-A and 1-B of Chapter 1, the boxy symbols show the
nine pairs of PhysPop values and MortRates (one pair for each of
the Nine Census Divisions). Those PhysPop values and MortRates
have a strong linear relationship with each other --- which is
clear because the boxy symbols cluster so closely around a
straight line. If this were a PERFECT linear correlation, the boxy
symbols would fall directly upon a single straight line, with no
scatter at all.
5a. The Linear Dose-Response: Meaning and Expression
---------------------------------------------------------
In a perfect linear relationship, one additional unit of dose
adds exactly the same number of fatal cases to the MortRate, no
matter whether the total dose is low or high. Suppose that each
dose-unit adds 6 fatal cancers to the cancer MortRate. Then 3
additional units of dose will add 18 fatal Cancers to the
MortRate. Thus, increment in MortRate (18 cases) is proportional
to the increment in dose (3 units), and the constant of
proportionality (which relates dose to MortRate) is 6 cases per
unit of dose. 18 additional fatal cases = (6 additional cases /
dose-unit) times (3 additional dose-units). The dose-units cancel
out in this equation, so that additional cases = additional cases.
When dose-response is linear, each MortRate is related to its
corresponding dose by the equation for a straight line: y = mx +
c.
o The y-variable is the MortRate, expressed in "cases per
100,000 population," for example.
o The x-variable is the corresponding dose, expressed in
dose-units (for example, in PhysPop values in this book).
o "m" is the coefficient of proportionality (also called the
X-Coefficient), expressed as "fatal cases per dose-unit." Thus,
potency per dose-unit is the SAME at all dose-levels.
o "c" is a Constant, expressed in the same MortRate units as
"y." The Constant quantifies the number of cases in the total
MortRate which are NOT related to dose.
When the value of the Constant is greater than zero, then each
MortRate is proportional to dose only after the Constant is
SUBTRACTED from the total MortRate: (y - c) = mx. If the value of
the constant is zero, there is nothing to subtract, and the ENTIRE
MortRate is proportional to dose. In that case, y = mx.
5b. Figure 5-B: Perfect Proportionality (MX Model)
-------------------------------------------------------
Figure 5-B, which is located at the end of this chapter,
illustrates what we call the MX model of dose-response --- an
abbreviation of the equation y = mx. This is the model in which
the ENTIRE MortRate (y) is directly proportional to PhysPop (x).
In other words, the MX model reflects the concept that medical
radiation became a contributing cause to nearly all cases of fatal
Cancer, with nearly no cases unaided by medical radiation.
5c. The X-Values and Y-Values for Figure 5-B (MX Model)
------------------------------------------------------------
In Figure 5-B, the x-values are the nine real PhysPops of 1940
(from the Universal PhysPop Table 3-A). The y-values are an unreal
set of MortRates. We have arbitrarily made the highest MortRate
equal to 120 radiation-induced cancers per 100,000 population.
Readers have seen that rate before. It is the annual delivery-rate
of cancer depicted in Figure 5-A during the equilibrium years. In
Figure 5-B, we pair it with the highest 1940 PhysPop value, which
is 169.76 in the Mid-Atlantic Division.
To obtain eight other illustrative MortRate values, which must
be perfectly proportional in the MX model to the eight other
PhysPops of 1940, we do exactly what we did in Chapter 3, Part 6b.
We take the ratio of the y-variable over the corresponding
x-variable: (120 / 169.76) = 0.7068803. Then we multiply each of
the eight PhysPops by 0.7068803 to obtain their matching MortRates
--- thus making the pairs of x,y values perfectly proportional to
each other (y = mx). The value of m (the X-Coefficient) is
0.7068803.
Some Ratios Resulting from Perfect Proportionality
--------------------------------------------------
As a result, the proportionalities demonstrated in Chapter 3,
Part 6b, apply here too. We already know that the ratios of the
MortRates over the PhysPops are 0.7068803 in every Census
Division, because we just made them so. In addition, any two
MortRates will have the same ratio as their corresponding
PhysPops. For example, we can compare the New England Division
with the Mountain Division (data in Figure 5-B). The MortRate
ratio is (114.1969 / 84.74820), or 1.347. The corresponding
PhysPop ratio is (161.55 / 119.89), or 1.347. The same.
It follows, IN THE MX MODEL, that the ratio of PhysPop
Hi5/Lo4, and the ratio of MortRate Hi5/Lo4, must be the same. (The
Hi5/Lo4 ratio was introduced in Chapter 3, Part 4.) From the
Universal PhysPop Table 3-A, we find that the Hi5/Lo4 PhysPop
ratio for 1940 is 1.46. When we calculate the Hi5 average and the
Lo4 average for the synthetic MortRates in Figure 5-B, we obtain
105.68 and 72.53, respectively. Their ratio is also 1.46.
5d. Linearity: Interpreting the Absence of Curvature
---------------------------------------------------------
Before proceeding to linear regression analysis, we want to
comment on the strong linear relationships between MortRates and
PhysPop values, already depicted in Figures 1-A and 1-B of Chapter
1. In view of the data discussed in Chapter 2, Part 5b, how do we
interpret the observation that these correlations are linear
rather than curved?
We refer to the nature of PhysPop itself:
PhysPop is proportional to average accumulated per capita
population dose from medical radiation because the more physicians
there are per 100,000 population, the more radiation procedures
are done per 100,000 persons. The increase in procedures occurs
chiefly because MORE persons per 100,000 receive such attention
--- not because the SAME persons get irradiated more often. In
other words, the average per-PATIENT dose is about the same in the
Census Divisions with low PhysPop values as in Divisions with high
PhysPop values, but the average per-CAPITA dose is higher in
high-PhysPop Census Divisions than in low PhysPop Divisions
because there are more PATIENTS per 100,000 population in
high-PhysPop Divisions.
At the cellular level where xray-induced mutations occur, the
average per-patient dose-level is likely to be very similar in all
Nine Census Divisions. Therefore, the observed absence of
curvature (e.g., the absence of supra-linearity) matches
expectation, in dose-responses between PhysPop and MortRates.
5e. Linear Regression Analysis:
Best-Fit Equation, Best-Fit Line (MX Model)
----------------------------------------------------
Regression analysis is a branch of mathematics which can
evaluate the correlation between sets of x,y pairs. Part 5a has
already emphasized that, in a linear dose-response, each MortRate
(y) is related to its corresponding PhysPop (x) by the equation
for a straight line: y = mx + b.
In earlier decades, we had to do the calculations for
regression analysis by hand. Now, we can just enter the two
columns of data (the x-values and the corresponding y-values) into
the proper location of a computer spreadsheet, and use a
regression-analysis program to do the calculations for us. The
program which we use, in the Lotus 123 spreadsheet, produces
standard output from the method of least squares. The program is
described in the Lotus Journal by Chuck Sullivan, a systems
engineer for the Lotus Development Corporation (Sullivan 1986).
Every regression analysis has input and output.
Obtaining the Equation of Best Fit, from Figure 5-B
---------------------------------------------------
The input-data for the regression analysis of Figure 5-B: The
x-values are the nine real 1940 PhysPops, and the y-values are
nine corresponding MortRates, calculated in Part 5c in order to
illustrate a PERFECT linear correlation. The additional x-entries
and "Best-Fit Calculated MortRates" in Figure 5-B are needed for
graphing, as explained below.
The regression output: The output is located at the top-right
of Figure 5-B. From it, we obtain the values of the X-Coefficient
and the Constant (discussed in Part 5a) which are required in
order to write the best-fit equation for this set of data.
Patterned on the straight-line equation, y = mx + c, the equation
of best fit for Figure 5-B is:
MortRate = (0.7068803 * PhysPop) + Zero. [ * denotes
multiplication.]
Generating the LINE of Best-Fit from the Best-Fit Equation
----------------------------------------------------------
Using this best-fit equation, we can "plug in" any value for
PhysPop, and calculate a corresponding MortRate. Each MortRate
requires a separate calculation. To distinguish such MortRates
from real-world observations ("observed MortRates"), it is
customary to call them "calculated" or "estimated" or "best-fit"
MortRates.
By using such calculations, we obtained the column of best-fit
MortRates in Figure 5-B --- including MortRates when PhysPop = 90,
when PhysPop = 80, when PhysPop = 70 ... right down to PhysPop =
0. The LINE OF BEST FIT, which is graphed in Figure 5-B, connects
these pairs of x,y values (various PhysPops, best-fit MortRates).
5f. The X-Coefficient and the Constant (MX Model)
------------------------------------------------------
X-Coefficient: Because in Part 5c, we made the pairs of x,y
values perfectly proportional to each other (Mort Rate = 0.7068803
times PhysPop, where PhysPop is the x-variable), the regression
output had to produce 0.7068803 as the "X-Coefficient." The
X-Coefficient is simply "m" in the equation, y = mx. Re-arranged:
m = y/x. So "m" evaluates how many units of y (the MortRate) occur
per unit of x (dose). In short, the X-Coefficient describes how
steep the slope is, of the best-fit line.
Constant: The Constant is "c" in the straight-line equation, y
= mx + c. The Constant is the value of y, when x = zero. The value
of the Constant (the c-value) never changes --- which gives it the
name "Constant." When the x-value changes to a new value, the
X-Coefficient ("m") determines the new value of the product, "mx",
which gets added to the c-value to produce the new and
corresponding best-fit y-value.
In Part 5c, we made every MortRate = (0.7068803 times
PhysPop), a procedure for which the equation is y = mx. So, when
the resulting pairs of x,y values were fed into the linear
regression analysis, there was no "room" for any c-value other
than zero in the regression's straight-line equation (y = mx + c).
Quite predictably, the regression output in Figure 5-B shows the
value of the Constant to be zero.
In the MX model, the Constant has a value of zero, and so the
ENTIRE value of the MortRate is directly proportional to every
corresponding value of PhysPop (Part 5b).
The Y-Axis Intercept and the "Origin"
-------------------------------------
Because the Constant is the value of y, when x = zero, the
Constant is the value of y wherever the best-fit line intercepts
the vertical y-axis. Thus, in our graphs, the Constant (also
called "the y-intercept") is the value of the MORTRATE, when the
value of PHYSPOP is zero. The spot where both y = 0 and x = 0 is
called the "origin" in such graphs.
5g. The R-Squared Value and the "Std Err of Coef" (MX Model)
-----------------------------------------------------------------
The regression output at top-right of Figure 5-B provides some
measures of how good (how strong) the x,y correlation is.
R-Squared Value: The R-squared value measures the "goodness of
fit" between the line of best fit and the pairs of input-data. The
input-pairs are depicted by the boxy symbols in our graphs. Only a
PERFECT correlation produces an R-squared value of 1.00 from
regression analysis, as we emphasized in Chapter 3, Parts 6 and 7.
Imperfect correlations generate R-squared values less than 1.00. A
rule of thumb is that R-squared values below 0.3 are not
considered to be statistically significant (at about the 90%
confidence level). As readers study the chapters on
non-malignancies in this book, they will see some R-squared values
quite a bit lower than 0.3 --- meaning no detectable correlation
whatsoever between the x,y pairs.
Standard Error of the X-Coefficient: The Standard Error (SE)
of the X-Coefficient is an indicator of how reliable is the SLOPE
of the best-fit line. The certainty of a slope and the strength of
a correlation diminish as the distance grows between the best-fit
line and some of the boxy symbols, of course.
"The smaller, the better," is the rule for the size of the
Standard Error (SE) of the X-Coefficient, relative to the size of
the X-Coefficient itself. In Figure 5-B, the MX model produces
"zero" as the SE of the X-Coefficient, because the slope of the
best-fit line is not in any doubt when there is a perfect
correlation (R-squared = 1.00).
90% Confidence Limits on the X-Coefficient
------------------------------------------
The 90% confidence-limits (CLs) on the X-Coefficient are
calculated from the SE. The upper limit is (X-Coef) + (1.645 times
SE) and the lower limit is (X-Coef) - (1.645 times SE).
For example, if the X-Coefficient from regression output is
(0.203) and its Standard Error is (0.045), then (at the 90% CL)
the upper limit on the X-Coefficient is (0.203) + (1.645 times
0.045) = (0.203 + 0.074) = (0.277). The lower CL is (0.203) -
(1.645 times 0.045) = (0.203 - 0.074) = (0.129). In other words,
if a great number of samples were measured and regressed, 90% of
the X-Coefficients would fall in the range of 0.129 through 0.277.
However, the central value (provided by the regression output) is
the most likely value --- and therefore, the central value is
often called "the best value."
Ratio of the X-Coefficient over Its Standard Error (SE)
-------------------------------------------------------
In the example above, the ratio of the X-Coefficient over its
SE is (0.203 / 0.045), or 4.51. A rule of thumb is that the value
of the X-Coefficient over its SE needs to be at least 2.0 before
the X-Coefficient is regarded as reasonably reliable. In our
dose-response studies, we will calculate the ratio for each
regression. Readers will see see some ratios as high as 5 and
higher --- which means that those slopes are highly reliable.
5h. Effect of a Single Deviant Datapoint upon the Constant
---------------------------------------------------------------
Whenever real-world data fit the MX model of dose-response
rather closely, but not perfectly, the effect of a single deviant
datapoint (boxy symbol) upon the Constant deserves appreciation.
For example, if we move only a high datapoint in Figure 5-B
far above the best-fit line, we would need a new regression
analysis using the altered input-data. The new regression analysis
would produce a steeper slope and a NEGATIVE Constant, instead of
a Constant of zero. The new best-fit line would intersect the
vertical y-axis BELOW the origin. We would see a similar result if
we had moved a low datapoint to a new location far BELOW the
best-fit line. In similar fashion, different "moves" of a single
datapoint could tip the slope to be LESS steep, in which case the
Constant of zero (which characterizes perfect proportionality)
would rise to a positive value. In real-world data, single
"out-lying" datapoints can have such effects.
5i. What Results Would We Expect
from Ideal Research Circumstances?
-------------------------------------------
To obtain an overview of the architecture of our dose-response
studies, between PhysPop and cancer MortRates, it is useful to
imagine that real-world conditions will be "ideal" for such
studies.
"Ideal" conditions would resemble the conditions described for
Figure 5-A. However, Figure 5-A refers only to ONE population. Our
studies compare the NINE different populations in the Nine Census
Divisions. "Ideally," there would be no migration among the Census
Divisions, and each separate population would receive exposure to
a CONSTANT annual average per capita dose of medical radiation,
decade after decade, with constant levels of co-actors decade
after decade.
Under such conditions, what should we expect to observe with
respect to the dose-response relationship between PhysPop and
cancer MortRates, after the introduction of radiation into
medicine in 1896?
We would expect to observe a positive and linear
dose-response, by Census Divisions, between the nine MortRates and
the nine corresponding PhysPops, decade after decade. If
regression analysis produced a Constant greater than zero, we
would subtract the Constant from each of the nine Observed
MortRates, and we would expect the nine remaining MortRate values
to stay always in the same proportions with each other as the
fixed proportions among the nine PhysPop values. In other words,
we would expect the variation in cause to control the variation in
effect.
The same expectation can be expressed somewhat differently.
Under ideal research conditions, we would have nine separate
populations which never mix from one Census Division to another,
and each population would constantly receive its own, fixed, per
capita average dose of medical radiation, decade after decade.
Each of the nine, different, average per capita doses would
produce its own separate stream of radiation-induced Cancers in
the population of its own Census Division. Under such conditions,
of course we would expect that these nine separate streams of
radiation-induced Cancer (expressed as excess age-adjusted cancer
MortRates per 100,000 population) would have proportions with each
OTHER which mirror the proportions that the nine causal doses of
medical radiation have with each other.
It remained for us to learn, just how severely REAL-world
research conditions might depart from the ideal, as we undertook
to examine much of a century.
* Part 6. Dose-Response:
Perfect Correlation without Perfect Proportionality
===============================================================
In contrast to the MX model of dose-response, the MX+C model
reflects the concept that medical radiation does not contribute to
EVERY case of fatal Cancer. The Constant quantifies the number of
cases which occur without help from medical radiation.
6a. Figure 5-C: One Alteration in the Input Data of Figure 5-B
-------------------------------------------------------------------
Figure 5-C, located at the end of this chapter, depicts the
MX+C model of dose-response. It is designed to be exactly like
Figure 5-B except for ONE type of alteration. Every MortRate in
Figure 5-B has had 20 Cancers (per 100,000 population) added, for
Figure 5-C. In other words, we have given the Constant a value of
20. When PhysPop = zero, the cancer MortRate is 20 cases (per
100,000 population). In Figure 5-C, the input-data for the
x-variable (the nine PhysPops) are the same as in Figure 5-B.
How does the regression output differ in Figure 5-Cfrom the
output in Figure 5-B?
Only the Constant has changed, from zero to 20. But the slope
of the best-fit line is still the same, with the X-Coefficient at
0.7068803, and with the standard error still at zero. And the
correlation between the pairs of x,y variables is still perfect,
with an R-squared value of 1.00.
The equation of best fit is now: MortRate = (0.7068803 times
PhysPop) + 20. And with that equation, we calculated MortRates in
order to graph the line of best fit. The graph shows the
y-intercept at 20, of course. And the nine pairs of actual
input-data (the nine boxy symbols) sit right upon the line of best
fit, with no scatter, because R-squared = 1.00.
6b. Perfect Correlation
without Perfect Proportionality (MX+C Model)
-----------------------------------------------------
In Figure 5-B, we illustrated perfect proportionality between
the entire MortRate and PhysPop (y = mx), as well as perfect
correlation (R-Squared = 1.00).
By contrast, Figure 5-C illustrates perfect correlation
between PhysPops and MortRates (R-squared = 1.00), but not perfect
proportionality between the ENTIRE MortRates and their PhysPops.
In order to see the proportionality between dose and response, one
must first SUBTRACT the Constant from each MortRate, because the
Constant represents a contribution to each MortRate which occurs
"anyway" (even when dose = zero) and such a contribution is NOT
proportional to dose.
6c. Can Perfect Correlation Persist,
If X-Values Rise and Y-Values Fall?
--------------------------------------------
The answer to the question in the subtitle is "Yes." To
illustrate, we will do three linear regressions below. The first
one reproduces the regression in Figure 5-C, so that we begin with
"old" values (for x and y) which already have demonstrated their
perfect correlation. In the second regression, each x-value of the
first regression has been multiplied by 1.4, but the y-values stay
as they are in the first regression. In the third regression, the
x-values stay as they are in the second regression, but each
y-value is multiplied by 0.8. So, the third regression shows a
perfect correlation persisting even after all the x-values rose by
one factor (1.4) and all the y-values fell by another factor
(0.8).
Old-x Old-y #1. Regression Output:
159.72 132.90 Constant 19.9974
161.55 134.20 Std Err of Y Est 0.0029
123.14 107.05 R Squared 1.0000
169.76 140.00 No. of Observations 9
133.36 114.27 Degrees of Freedom 7
119.89 104.75
103.94 93.47 X Coefficient(s) 0.7069
85.83 80.67 Std Err of Coef. 0.0000
100.74 91.21 Except for rounding, input and output
are the same as Figure 5-C.
-----------------------------------------------------------------------------
new-x old-y #2. Regression Output:
223.61 132.90 Constant 19.9942
226.17 134.20 Std Err of Y Est 0.0031
172.40 107.05 R Squared 1.0000
237.66 140.00 No. of Observations 9
186.70 114.27 Degrees of Freedom 7
167.85 104.75
145.52 93.47 X Coefficient(s) 0.5049
120.16 80.67 Std Err of Coef. 0.0000
141.04 91.21 Note: X-values are 1.4 times x-values in #1.
Note: This X-Coef = (0.7069 from #1) divided by 1.4 = 0.5049
-----------------------------------------------------------------------------
new-x new-y #3. Regression Output:
223.61 106.32 Constant 15.9953
226.17 107.36 Std Err of Y Est 0.0025
172.40 85.64 R Squared 1.0000
237.66 112.00 No. of Observations 9
186.70 91.42 Degrees of Freedom 7
167.85 83.80
145.52 74.78 X Coefficient(s) 0.4040
120.16 64.54 Std Err of Coef. 0.0000
141.04 72.97 Note: Y-values are 0.8 times y-values in #2.
Note: This X-Coef = (0.5049 from #2) * 0.8 = 0.4039
Note: This Constant = (19.9942 from #2) * 0.8 = 15.9954
* Part 7. Dose-Response:
Effects of Imperfect Matching across Dose-Groups
============================================================
For multi-cause diseases such as Cancer and Ischemic Heart
Disease, we can define co-actors as necessary co-causes in
producing single cases of those diseases (Introduction, Parts 4
and 5). When analysts want to study the dose-response between ONE
co-actor (for instance, medical radiation) and the mortality rate
from the disease, they hope to compare study-groups which differ
in dosage of the ONE co-actor but which are alike ("matched") with
respect to the other co-actors (for instance, smoking). In this
book, the study-groups (or dose-groups) are the populations of the
Nine Census Divisions.
7a. The Real World: Imperfect Matching across Dose-Groups
--------------------------------------------------------------
In the real world of cancer-studies, perfect matching across
dose-groups is never possible. Practical obstacles are immense. In
addition, all causes of Cancer are probably not even recognized
yet, and it would be impossible to match dose-groups for
unrecognized co-actors. For both reasons, imperfect matching
always occurs.
Imperfect matching for co-actors can interfere with detection
of a positive correlation which is truly present, or can produce
an apparent correlation which is spurious. The power of
"confounding variables" is a major concern for all analysts. In
this book, we need not worry about finding a spurious positive
correlation (between medical radiation and cancer MortRates),
because a causal relationship between ionizing radiation and fatal
Cancer has been well established by a multitude of earlier studies
(Chapter 2, Part 4c). But we need to appreciate the power of
imperfect matching to OBSCURE the correlation in a set of data.
7b. Figure 5-D: Inconsistency with the "Correlation Axiom"
---------------------------------------------------------------
Comparison of Figures 5-B and 5-D, at the end of this chapter,
illustrates how imperfect matching for co-actors can change a
perfect correlation (R-Squared = 1.00) into an imperfect
correlation with an R-Squared value of 0.7112.
Figure 5-D uses the real 1940 PhysPops as the x-values, as did
Figure 5-B. However, Figure 5-D depicts the consequence of Census
Divisions which are imperfectly matched for co-actors. The UNEQUAL
average exposure to nonradiation co-actors, in the Nine Census
Divisions, can degrade the PhysPop-MortRate correlation in two
ways. One: Xray potency per rad is modulated differently in the
various Census Divisions (Chapter 6, Part 6; and Chapter 49, Part
2). Two: The number of cases in which xrays are not a co-actor may
differ across the Census Divisions. As a result of one or both
phenomena, the MortRates from Figure 5-B increase by irregular
numbers (purely illustrative) as follows:
MortRate Increments in MortRate due to
Fig.5-B Imperfect Matching of Co-Actors
"y" in Figure 5-D
Pacific Division: 112.9029 + 25 = 137.9029
New England: 114.1965 + 11 = 125.1965
West North Central: 87.0452 + 20 = 107.0452
Mid-Atlantic: 120.0000 + 17 = 137.0000
East North Central: 94.2696 + 35 = 129.2696
Mountain: 84.7479 + 11 = 95.7479
West South Central: 73.4731 + 21 = 94.4731
East South Central: 60.6715 + 45 = 105.6715
South Atlantic: 71.2111 + 31 = 102.2111
As a result of imperfect matching, the R-squared value of 1.00
in Figure 5-B falls to 0.7112 in Figure 5-D. The true biological
correlation is OBSCURED (but not changed) by imperfect matching of
co-actors across the dose-groups. Imperfect matching is not
consistent with what we can abbreviate as the "Correlation Axiom,"
below.
The Correlation Axiom
---------------------
Correlation Axiom: Increment in cancer MortRate is perfectly
proportional to increment in radiation dose (PhysPop), provided
that co-actors are perfectly matched across the dose-groups. The
Correlation Axiom describes (a) the linear dose-response, and (b)
the matching of dose-groups --- which is a fundamental principle
of dose-response research, even though it is never fully
achievable (Part 7a; also Chapter 3, Part 2d).
7c. Figure 5-E:
A Truly Positive Correlation Which Looks Negative
----------------------------------------------------------
Imperfect matching for co-actors can interfere --- much more
severely than illustrated in Figure 5-D --- with detection of a
positive correlation which is truly present. With Figure 5-E, we
will demonstrate how imperfect matching can even make a truly
positive correlation appear negative.
We are preparing to study the dose-response between PhysPop
(surrogate for medical radiation) and cancer MortRates. Suppose
that a carcinogenic co-actor, such as smoking, occurs with the
most intensity where PhysPop values are the lowest, and with the
least intensity where PhysPop values are the highest. In other
words, suppose there is an INVERSE relationship between PhysPop
and smoking. In such a situation, smoking will increase the cancer
MortRates more in Census Divisions with low PhysPop values than in
Census Divisions with high PhysPop values. Below, starting with
the values from Figure 5-B, we arrange the Census Divisions in
descending order of their 1940 PhysPop values, and then we add to
the 1940 cancer MortRates from Figure 5-B in a way inverse to the
trend of PhysPop values:
1940 MortRate Increments in MortRate due to
PhysPop Fig.5-B Imperfect Matching of Co-Actors
"y" Fig.5-E Regression Output:
Mid-Atl 169.76 120.0 + 20 = 140.0 Constant 176.7119
New Eng 161.55 114.2 + 30 = 144.2 Std Err of Y Est 4.8615
Pacific 159.72 112.9 + 40 = 152.9 R Squared 0.6322
ENoCen 133.36 94.3 + 50 = 144.3 No. of Observations 9
WNoCen 123.14 87.0 + 60 = 147.0 Degrees of Freedom 7
Mtn 119.89 84.7 + 70 = 154.7
WSoCen 103.94 73.5 + 80 = 153.5 X Coefficient(s) -0.2003
SoAtlan 100.74 71.2 + 90 = 161.2 Std Err of Coef. 0.0577
ESoCen 85.83 60.7 + 100 = 160.7 X-Coef / S.E. = -3.4686
The regression-output in Figure 5-E shows that the sign on the
X-Coefficient has become 0.20 with a NEGATIVE sign, which means
that when PhysPop increases by one unit, cancer MortRate FALLS by
0.2 unit. In other words, the true POSITIVE correlation between
PhysPop and cancer MortRate has been so well concealed by the
non-matched co-factor (smoking), that the OBSERVED correlation
between PhysPop and cancer MortRates will be INVERSE in such a
situation. But imperfect matching of co-actors is just an error,
an inconsistency with the Correlation Axiom. Such errors have no
power to repeal the laws of physics and human biology --- the laws
which established the Correlation Axiom for ionizing radiation
(PhysPop) in the first place.
* Part 8. Real-World "Entropic Circumstances"
Which Reduce Observed Correlations
================================================
The ideal MX model and the ideal MX+C model both reflect
perfect correlation between dose and response. They are very
orderly models. But in the real world, order is opposed by the
tendency toward disorder. Most systems move spontaneously from
states of order toward states of disorder. In chemistry, the
molecular chaos of a substance or a system is measured by a
property called "entropy."
What Do We Mean in This Book by "Entropic Circumstances"?
---------------------------------------------------------
In this book, we need a name for the group of real-world
events which perturb the orderly, ideal models of this chapter.
Our name is "entropic circumstances." Entropic circumstances
operate generally AGAINST order --- they DO NOT CREATE order.
(Weiss 1998 describes some recent insights about entropy.)
8a. Some Specific Entropic Circumstances of Concern
--------------------------------------------------------
For our dose-response studies, we know that two entropic
circumstances of great concern have to be migration of populations
from one Census Division to another (discussion in Chapter 3, Part
2c), and PhysPop deviations from "lockstep" over time (discussion
in Chapter 3, Parts 2c and 8).
Both migration and deviations from PhysPop "lockstep" degrade
PhysPops as surrogates for ACCUMULATED radiation dose-differences
from medical applications. Neither migration nor deviations from
PhysPop "lockstep" would be serious problems in our dose-response
studies if complete delivery of radiation-induced cancers occurred
within 2 or 3 years. They become problems because of the very
gradual delivery-times for radiation-induced cancers --- with such
delivery-times stretching over at least 40 years (or longer) for
mixed-age populations. By comparison, other entropic circumstances
may be less important --- and we emphasize "may."
8b. Finding the Maximum Real-World Correlations (PPs with MRs)
-------------------------------------------------------------------
Because entropic circumstances operate AGAINST orderly
phenomena (such as correlations), entropic circumstances reduce
R-Squared values. Therefore, if we seek the best approximation of
the real dose-response relationship, between PhysPops and cancer
MortRates, we will seek and accept the HIGHEST values of R-squared
which survive erosion by entropic circumstances.
1940 is our first year of MortRate data with all 48 states
represented. And 1921 is the year of our earliest PhysPop data. In
our search for the strongest correlation, we regressed the 1940
MortRates serially on every set of prime (not interpolated)
PhysPop data between 1921 and 1940 --- including the 1940
PhysPops. Although cancer MORTALITY during 1940 can hardly be
influenced by medical radiation received during 1940, the 1940
PhysPops are nearly in "lockstep" with the PhysPops of many
preceding years (Chapter 3, Table 3-C) --- and thus, 1940 PhysPops
reflect the approximate differences in accumulated dose of medical
radiation from many PRIOR years.
* Part 9. Estimating the Impact
of Medical Radiation on Cancer MortRates
====================================================
We undertook this project in order to explore Hypothesis-1,
that medical irradiation is the principal cause of cancer
mortality in the USA during the Twentieth Century. We remind
readers that we are not trying to ESTABLISH the existence of a
positive correlation between ionizing radiation and cancer
mortality. That was proven many years ago. Instead, we are making
use of that knowledge to test Hypothesis-1.
We begin, in Section Two of this book, by looking at what we
can learn about Hypothesis-1 from regressing 1940 cancer MortRates
on earlier PhysPops. In Section Five of this book, we examine the
whole 1940-1990 period. We arrive at estimated Fractional
Causation of cancer mortality by medical radiation. Such results
clearly support Hypothesis-1.
>>>>>>>>>>
------------------------------------------------------------------
Figure 5-A. Annual Delivery-Rates of Radiation-Induced Cancer
Related Text = Parts 1, 2, + 3.
------------------------------------------------------------------
o - Each box in the grid represents 3 cases of
radiation-induced cancer per 100,000 population (mixed ages).
o - Each horizontal row of 40 boxes represents gradual
delivery of 120 cancers per 100,000 population. In this
illustration, 120 is the number of cases produced by the radiation
received during a single calendar-year. These 120 cases are
delivered gradually at the rate of 3 cases per year for 40 years.
o - Each vertical column represents the number of
radiation-induced cancers delivered during a single calendar-year,
per 100,000 population, from all earlier years of irradiation. All
boxes in a column were produced by radiation received in different
calendar-years.
o - Both SHADED columns have 40 vertical boxes
(representing 120 cancers) as do the columns BETWEEN the two
shaded columns. Such columns demonstrate the "Law of Equality":
The annual radiation- induced delivery of 120 - the annual
radiation-induced production of 120.
[Figure 5-A]
------------------------------------------------------------------
Figure 5-B. The MX Model of Dose-Response
Related Text = Part 5.
------------------------------------------------------------------
Census Divisions 1940 1940 Best-Fit
"x" "y" Calc.
PhysPops MortRates MortRates
Pacific 159.72 112.9029 112.9029 Regression Output:
New England 161.55 114.1965 114.1965 Constant 0.00000
West No. Central 123.14 87.0452 87.0452 Std Err of Y Est 0.00000
Mid-Atlantic 169.76 120.0000 120.0000 R Squared 1.00000
East No. Central 133.36 94.2696 94.2696 No. of Observations 9
Mountain 119.89 84.7479 84.7479 Degrees of Freedom 7
West So. Central 103.94 73.4731 73.4731
East So. Central 85.83 60.6715 60.6715 X Coefficient(s) 0.706880
South Atlantic 100.74 71.2111 71.2111 Std Err of Coef. 0.000000
Additional PhysPops 90.00 63.6192
--- not "observed" --- 80.00 56.5504
down to zero PhysPop 70.00 49.4816
(zero medical 60.00 42.4128
radiation). For each, 50.00 35.3440
we calculate a best- 40.00 28.2752
fit MortRate. These 30.00 21.2064
additional x,y pairs 20.00 14.1376
are also part of the 10.00 7.0688
best-fit line. 0 0.0000
[Figure 5-B]
------------------------------------------------------------------
Figure 5-C. The MX+C Model of Dose-Response
Related Text = Part 6a.
------------------------------------------------------------------
Census Divisions 1940 1940 Best-Fit
"x" "y" Calc.
PhysPops MortRates MortRates
Pacific 159.72 132.9029 132.9029 Regression Output:
New England 161.55 134.1965 134.1965 Constant 20.0000
West No. Central 123.14 107.0452 107.0452 Std Err of Y Est 0.0000
Mid-Atlantic 169.76 140.0000 140.0000 R Squared 1.000000
East No. Central 133.36 114.2696 114.2696 No. of Observations 9
Mountain 119.89 104.7479 104.7479 Degrees of Freedom 7
West So. Central 103.94 93.4731 93.4731
East So. Central 85.83 80.6715 80.6715 X Coefficient(s) 0.706880
South Atlantic 100.74 91.2111 91.2111 Std Err of Coef. 0.000000
Additional PhysPops 90.00 83.6192
--- not "observed" --- 80.00 76.5504
down to zero PhysPop 70.00 69.4816
(zero medical 60.00 62.4128
radiation). For each, 50.00 55.3440
we calculate a best- 40.00 48.2752
fit MortRate. These 30.00 41.2064
additional x,y pairs 20.00 34.1376
are also part of the 10.00 27.0688
best-fit line. 0 20.0000
[Figure 5-C]
------------------------------------------------------------------
Figure 5-D. Effect of Imperfect Matching of Dose-Groups
Related Text = Part 7b.
------------------------------------------------------------------
o Regression input for the x-variable (PhysPop) is the same
as in Figure 5-B.
o Regression input for the y-variable (MortRate) comes from
the text of Chapter 5, Part 7b. The MortRates differ from Figure
5-B in a manner which reflects Census Divisions which are
imperfectly matched for radiation's carcinogenic co-actors.
o Each Best-Fit MortRate (to make the graph) is calculated
with the equation of best fit provided by the regression output:
MortRate = (0.4929 * PhysPop) + 51.5299.
Census Divisions 1940 Part 7b Best-Fit
"x" "y" Calc.
PhysPops MortRates MortRates
Pacific 159.72 137.9 130.3 Regression Output:
New England 161.55 125.2 131.2 Constant 51.5299
West No. Central 123.14 107.0 112.2 Std Err of Y Est 9.9955
Mid-Adantic 169.76 137.0 135.2 R Squared 0.7112
East No. Central 133.36 129.3 117.3 No. of Observations 9
Mountain 119.89 95.7 110.6 Degrees of Freedom 7
West So. Central 103.94 94.5 102.8
East So. Central 85.83 105.7 93.8 X Coefficient(s) 0.4929
South Atlantic 100.74 102.2 101.2 Std Err of Coef. 0.1187
Additional PhysPops 70.00 86.0 XCoef/SE 4.1523
--- not "observed" --- 60.00 81.1
down to zero PhysPop 50.00 76.2
(zero medical 40.00 71.2
radiation). For each, 30.00 66.3
we calculate a best- 20.00 61.4
fit MortRate. These 10.00 56.5
additional x,y pairs 0 51.5
are also part of the
best-fit line.
[Figure 5-D]
------------------------------------------------------------------
Figure 5-E.
Effect of an Inverse Relationship between Dose and a Co-Actor
Related Text = Part 7c.
------------------------------------------------------------------
o Regression input for the x-variable, (PhysPop) is the
same as in Figure 5-B. The sequence here is in order of descending
values. (Sequence does not affect regression output.)
o Regression input for the y-variable (MortRate) comes from
the text of Chapter 5, Part 7c. The MortRates differ from Figure
5-B in a manner which reflects an inverse relationship between
PhysPop and intensity of a co-actor across the Census Divisions.
Census Divisions 1940 Part 7b Best-Fit
"x" "y" Calc.
PhysPops MortRates MortRates
Mid-Atlantic 169.76 140.0 142.7 Regression Output:
New England 161.55 144.2 144.4 Constant 176.7119
Pacific 159.72 152.9 144.7 Std Err of Y Est 4.8615
East No. Central 133.36 144.3 150.0 R Squared 0.6322
West No. Central 123.14 147.0 152.0 No. of Observations 9
Mountain 119.89 154.7 152.7 Degrees of Freedom 7
West So. Central 103.94 153.5 155.9
South Atlantic 100.74 161.2 156.5 X Coefficient(s) -0.2003
East So. Central 85.83 160.7 159.5 Std Err of Coef. 0.0577
Additional PhysPops 70.00 162.7 X-Coef S.E. -3.4686
--- not "observed" --- 60.00 164.7
down to zero PhysPop 50.00 166.7
(zero medical 40.00 168.7
radiation). For each, 30.00 170.7
we calculate a best- 20.00 172.7
-fit MortRate. These 10.00 174.7
additional x,y pairs 0 176.7
are also part of the
best-fit line.
[Figure 5-E]