| 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.
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.
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.
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.
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.
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.
The
y-variable is the MortRate, expressed in "cases per 100,000 population," for example.
The
x-variable is the corresponding dose, expressed in dose-units (for example, in PhysPop
values in this book).
"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.
"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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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).
