Metodoloˇ ski zvezki, V ol. 16, No. 2, 2019, 57–69
The Bad Mathematics of the Bad Luck Theory
Mariia Beliaeva
1
Abstract
The mathematics of the Bad Luck theory of carcinogenesis by Tomasetti and
V ogelstein generated a great deal of controversy among cancer specialists but did
not draw the mathematicians’ attention. Thus the gross mathematical mistakes of
the theory foundation did not get a proper critique and remained unnoticed. As
a result, the sensational quantitative estimates of the role of Bad Luck in cancer
occurrence, though being erroneous, have spread widely among researchers and the
general public and got the unfair popularity. The present paper reviews the actual
mathematical mistakes of Bad Luck theory.
1 Introduction
Almost ﬁve years have passed since Tomasetti and V ogelstein (hereafter T&V) attempted
to explain the variation of cancer risk by the number of stem cell divisions [22]. Their
idea, referred to as Bad Luck Theory, further elaborated in [25, 26, 27], generated a great
deal of controversy. The discussion is still alive [17, 31, 35, 41], for more bibliography
see [19] and [37].
T&V argue that intrinsic stochastic effects play the main role in tumor initiation, and
speciﬁcally that the majority of cancers (22 of 31 considered) “appear to be mainly due to
stochastic effects associated with DNA replication of the tissues’ stem cells” [22]. In their
opinion, it follows that the primary prevention measures—vaccination, altering lifestyle,
environmental control—are not likely to be as effective as the secondary prevention that is
early detection and treatment. The authors believe that their “results could have important
public health implication” and conclude that “secondary prevention should be the major
focus” [22]. If this result had been obtained in a correct way, it would be really signiﬁcant
as making it possible to optimize the research direction and funding.
Unfortunately for the authors and fortunately for all of us they were wrong. T&V
claimed that their theory was based on a mathematical model—one of many cancer math-
ematical models which “are popping up everywhere in cancer research” [15]. The gross
mathematical mistakes however make the Bad Luck model erroneous and its conclusions
unfounded. Surprisingly, these mistakes were neither detected at the peer-review stage nor
noticed after publication, maybe because professional mathematicians rarely read medical
papers.
On the other hand, the statement that some conclusion is based on a mathematical
model usually produces a magical effect on the general public. There are a great many
1
RHYTHM Engineering, Soﬁa, Bulgaria; maria.beljaeva29@gmail.com
58 Beliaeva
ways though for a mathematical model to be incorrect - wrong assumptions, errorneous
formulas, senseless criteria, or arithmetic mistakes. Thus each model has to pass a thor-
ough examination until one can decide to trust it. Nevertheless some cancer specialists
([1, 6, 14, 21]) just believe that the Bad Luck theory is right and use it as a basis for
the further research. The purpose of this paper is to stop that common misconception by
demonstrating the crucial mathematical mistakes in the theory.
The rest of the paper is organised as follows. The three main mathematical mistakes
of Bad Luck theory are described in Sections 2–4, Section 5 draws the reader’s atten-
tion to the unreasonable using of one fashionable method in the theory, in Section 6 we
explain what is wrong with some reasoning of the theory fallacy, Section 7 contains the
conclusions.
2 The First Mistake: Correlation between Logarithms
treatedasCorrelationbetweentheirArguments
T&V tried to calculate the coefﬁcientR of correlation between the human lifetime num-
berlscd of stem cell divisions within different tissues and the lifetime cancer riskr for
them [22, 23]. Having obtainedR = 0:804, they argue that as the coefﬁcient of determi-
nationR
2
  2=3, then approximately 2=3 of differences in cancer risk can be explained
by the number of stem cells random divisions i.e., by bad luck. T&V concluded that “The
stochastic effects of DNA replication appear to be the major contributor to cancer in hu-
mans” [22]. Actually, what T&V have really calculated was not the correlation coefﬁcient
betweenlscd andr themselves but that between their logarithms. These two coefﬁcients
are not equal.
To know it immediately one only has to google logarithms change correlation and ﬁnd
out questions like “I don’t understand why researchers sometimes use logged versions of
their variables and why correlation seems so much higher if you do so” [42] or “Before
logs the correlation is 0.49 and after logs it is 0.9. I thought the logs only change the scale.
How is this possible?” [43]. I failed to ﬁnd any mathematical explanation why logarithms
change correlation in the literature. The proof below ﬁlls the gap.
2.1 APieceofTheory
The well-known formula for the varianceD
y
of a non-random functiony =’(x
1
;x
2
;:::x
n
)
is [12, 39]
D
y
  n
X
i=1
  @’
@x
i
  2
m
D
i
+2
X
i<j
  @’
@x
i
  m
  @’
@x
j
  m
K
ij
; (2.1)
where x
1
;x
2
;:::;x
n
are random variables with means m
1
;m
2
;:::;m
n
and variances D
1
,
D
2
;:::;D
n
correspondingly,
  @’
@x
i
  m
=’
0
x
i
(m
1
;m
2
;:::;m
n
); (2.2)
indexm means that the derivatives are taken at the means of the argumens. Note that (2.1)
is [12, 32]
The bad mathematics of the Bad Luck theory 59
• exact when function’ is linear,
• approximate when ’ is non-linear but the range of the values of its arguments is
small enough to make possible the linear approximation of’,
• unreliable otherwise.
Equation 2.1 was derived by means of the error propagation method [32]. Below we
use the same method in order to get the approximate formula for the correlation coefﬁcient
between two nonlinear functions of random arguments and to show, as a consequence, that
correlation between logarithms is not equal to that between the arguments. Lety
1
andy
2
be two functions ofn andk random variables correspondingly:
y
1
=’
1
(x
1
;x
2
;:::x
n
);
y
2
=’
2
(z
1
;z
2
;:::;z
k
):
(2.3)
Their ﬁrst-order Tailor approximations about the mean points (m
1
;m
2
;:::;m
n
) and
(  1
;  2
;:::;  k
) of arguments are
’
1
(x
1
;x
2
;:::;x
n
)
  =A
1
+
n
X
i=1
B
1;i
x
i
;
’
2
(z
1
;z
2
;:::;z
k
)
  =A
2
+
j
X
k=1
B
2;j
z
j
;
(2.4)
where
A
1
=’
1
(m
1
;m
2
;:::;m
n
)  n
X
i=1
  @’
1
@x
i
  m
m
i
;
A
2
=’
2
(  1
;  2
;:::;  k
)  k
X
j=1
  @’
2
@z
j
      j
;
B
1;i
=
  @’
1
@x
i
  m
;i = 1;:::;n;
B
2;j
=
  @’
2
@z
j
    ;j = 1;:::;k:
(2.5)
The covariance betweeny
1
andy
2
is
K(y
1
;y
2
) =M(y
1
y
2
)  M(y
1
)M(y
2
): (2.6)
Using (2.4) we get
K(y
1
;y
2
)
  =M
h 
A
1
+
n
X
i=1
B
1;i
x
i
   A
2
+
k
X
j=1
B
2;j
z
j
 i
  M
  A
1
+
n
X
i=1
B
1;i
x
i
  M
  A
2
+
k
X
j=1
B
2;j
z
j
  : (2.7)
60 Beliaeva
After opening parenthesis and substitutingA
1
;A
2
;B
1;i
;B
2;i
from (2.5) into (2.7) we ob-
tain
K(y
1
;y
2
)
  =
X
i=1;:::;n
j=1;:::;k
  @’
1
@x
i
  m
  @’
2
@z
j
    h
M(x
i
z
j
)  M(x
i
)M(z
j
)
i
; (2.8)
which is equivalent to
K(y
1
;y
2
)
  =
X
i=1;:::;n
j=1;:::;k
  @’
1
@x
i
  m
  @’
2
@z
j
    K(x
i
;z
j
); (2.9)
whereK(x
i
;z
i
) is the covariance betweenx
i
andz
i
.
As well as (2.1) the formula (2.9) is
• exact when equalities (2.4) are exact that is when functions’
1
;’
2
are linear,
• approximate when at least one of those functions is non-linear but the range of the
values of its arguments is small enough to make possible the linear approxima-
tion 2.4,
• unreliable otherwise.
Finally, let us consider the particular casen =k = 1. Suppose’
1
and’
2
are univari-
ate functions:
y
1
=’
1
(x);
y
2
=’
2
(z):
(2.10)
Now (2.9) takes the form
K(y
1
;y
2
)
  =
  @’
1
@x
  m
  @’
2
@z
    K(x;z): (2.11)
Passing to correlation coefﬁcients we get
r
y
1
;y
2
  y
1
  y
2
  =
  @’
1
@x
  m
  @’
2
@z
    r
x;z
  x
  z
; (2.12)
where  y
1
=
p
D
y
1
,  y
2
=
p
D
y
2
, and variancesD
y
1
;D
y
2
are evaluated using (2.1):
D
y
1
  =
  @’
1
@x
  2
m
D
x
D
y
2
  =
  @’
2
@z
  2
  D
z
;
hence
  y
1
  =
  @’
1
@x
  m
  x
  y
2
  =
  @’
2
@z
      z
:
(2.13)
The bad mathematics of the Bad Luck theory 61
After substituing (2.13) into (2.12) we get
r
y
1
;y
2
  =r
x;z
Being a special case of (2.9), the last equation is reliable if either both functions’
1
;’
2
are linear, or if the scatter of arguments of both of them is small enough to make their
linear approximation reasonable. None of these conditions is met in the considered case:
the function y = log
10
x is not linear while the interval from 10
5
to 10
13
in which lscd
varies [22] cannot be considered as small.
Therefore the coefﬁcient of correlation between the number of stem cells divisions
and cancer risk cannot be calculated as that of their logarithms, hence the estimate 2=3 of
the part of cancers occuring because of bad luck, is unfounded.
Indeed, having calculated the true value of the correlation coefﬁcient betweenlscd and
r for the data set from [22] we getR = 0:96 thusR
2
= 0:93 which differs substantially
from the value obtained by T&V . Interestingly, the authors did calculate it in [23] but
the 93% fraction of unpreventable cancers seemed too high to them: “The distribution of
these cancer risks is extremely skewed in the original scale . . . ” That’s why they “purposly
chose the more conservative estimate, based on log-log transformation of the data” and
claimed that “This is standard practice in statistics” [23].
It is not. Correlation coefﬁcient is a meaningful characteristic and one cannot use non-
linear functions of the original variables instead of variables themselves in order to get its
desired value. That is valid not only for correlation but for other mathematical operations
as well. When planning a car trip, we calculate its duration by dividing the distance S by
the average speed v and get 10 hours forS = 600 km andv = 60 km/h. Using logarithms
ofS andv instead results inlog
10
600=log
10
60  1:56 but that does not mean that the trip
will take less than two hours.
3 The Second Mistake: Causation treated as a Conse-
quenceofCorrelations
Knowing now the trueR value and following the logic of Bad Luck theory one can con-
clude that not 2=3 but 93% of cancers should occur due to bad luck. This result however
is not consistent with statistical data which show the large inﬂuence of environmental,
occupational, and lifestyle factors on cancer emergence [4, 9, 18].
The paradox is resolved by means of the well-known fact that even high correlation
does not imply causation. If you don’t know the nature of real interdependence, neither
correlation coefﬁcient nor coefﬁcient of determination is important. The famous statis-
tician Sir Ronald Fisher wrote [7]: “. . . if two factors, A and B, are associated – clearly,
positively, with statistical signiﬁcance, as I say – it may be that A is an important cause
of B, it may be that B is an important cause if A, it may be that something else, let us
say X, is an important cause of both”. Moreover, in the same article he gives an example
of strong correlation, statistically signiﬁcant at a high level of signiﬁcance, between the
variables which are obviously unrelated with each other: “. . . in the years in which a large
number of apples were imported into Green Britain, there were also a large number of di-
vorces . . . But no one, fortunately, drew the conclusion that the apples caused the divorces
62 Beliaeva
or that the divorces caused the apples to be imported”. To know more examples of high
correlations of that sort, known today as spurious, one can visit [36] or listen to Harriett
Hall aka Scepdoc [11].
“The value of a correlationr
XY
shows only the extent to whichX andY are linearly
associated. It does not by itself imply that any sort of causal relationship exists between X
and Y . Such a false assumption has lead to erroneous conclusions in many occasions” [5].
And of course, if R knows nothing about the cause, then why R
2
would? In general,
R
2
contains less information thanR since it loses the sign. Obviously, neitherR norR
2
say anything about comparative signiﬁcance of two or more factors, only their regression
coefﬁcients (not to be confused with coefﬁcient of correlation) can. For example, if one
has found thatr =  +100  lscd+2  h+", wherer is risk,  is a constant,h is some factor
characterizing heredity," is noise, then one could claim that one unit increase inlscd is
associated with a 100 unit increase inr (holdingh constant), while a one unit increase in
h is associated only with a 2 unit increase inr (holdinglscd constant). With a suitable
choice of units one could claim thatlscd is 50-fold more signiﬁcant than heredity.
Sornette and Favre [34] used the model of populations to demonstrate the fallacy of
Bad Luck theory, Weinberg and Zaykin [38] as well as and Wu et al. [40] invented biolog-
ical thought experiments to demonstrate that coefﬁcients of determination and correlation
between cancer risk and cell divisions number cannot differentiate between the contri-
butions of intrinsic and extrinsic factors. The easier way to this conclusion is just to
remember one of the eternal mathematical maxims which should be well understood by
everybody before attempting to use statistics: correlation does not imply causation or,
simply put, correlation is not causation [7].
4 The Misterious ERS: Calculating the Distance from a
PointtoaLine
The “extra risk score” (ERS) was presented in [22] as a criterion for ranging cancers
according to the role of Bad Luck in their occurence. Without saying a word about ERS
physical or mathematical sense T&V deﬁned it as
ERS =log
10
r  log
10
lscd (4.1)
When Potter and Prentice [20] pointed out the weirdness of ERS, they got even more
weird explanation from its authors: “It may seem intuitive to multiply rather than add
logarithms” [25], though neither Potter and Prentice nor anybody else suggested adding
logarithms.
Only in the unpublished “Musings” [23] T&V revealed the geometrical idea of ERS.
Besides, if a mathematician took part in peer review process, the Bad Luck theory [22]
certainly would not see the light of the day until its authors included therein the decisive
criterion explanation.
According to “Musings”, ERS indicates how far is the point (log
10
lscd;log
10
r) from
the regression line. The authors deﬁned that as the area of the rectangle formed by the
point and the coordinate axes, see Figure 1. They claim that the smaller is this area the
larger is the distance from the point to the regression line hence the greater is the role
The bad mathematics of the Bad Luck theory 63
of extrinsic factors in the occurence of the corresponing cancer. Despite the fact that
again using logarithms instead of the variables themselves, while discussing the latter
correlation, is a mistake itself (see Section 2), the distance between a point and a line
is measured by the length of the perpendicular from the former to the latter, not by any
rectangle area. For example, the area of the rectangle corresponding to cancer F (Figure 1)
is less than that of cancer A. According to T&V , point F is farther from the regression line
than point A. That is obviously wrong.
Figure1: Ranking cancers according to ERS (Figure TR2 [23] with some additions)
To rank objects according to some characteristicI (distance from the regression line)
which depends on an unknown parameterk (the slope of the line), one has to compare the
average valuesI on the whole set ofk values. T&V though simply throw away indicator
I and take instead another one,J (the rectangle area) which does not correspond to the
real relationsheep between the ranked objects.
One cannot form a working criterion by combining variables arbitrarily. One could
multiply tangents of lscd and r or divide their integrals by each other – it would be as
meaningless as multiplying logarithms.
Consequently, dividing all cancers into two subsets with high and low ERS (rectan-
gle areas) and state that to those with low ERS the “contribution of classic determinants
(external environment and heredity) . . . is minimal” [22] does not make any sense.
The mistakes described in Sections 2–4 ruin the mathematical foundation of the Bad
Luck theory completely. All of them would be detected immediately by any mathemati-
cian provided that one had examined the article [22] during the peer-review stage.
64 Beliaeva
5 OnefunnyThing
T&V [22, 24] use no less than machine learning, speciﬁcally hierarchical clustering, in
order to split a single set of thirty numbers into two parts. The method is efﬁcient when
complicated, predominantly multidimensional, sets are to be divided, or when there are
a lot of sets. But to put once 30 ﬁgures in the increasing order and to divide this set into
two clusters of relatevely small and big values one needs no more than one’s own eyes.
Using machine learning in such a case is, putting it mildly, a bit redundant, and looks like
an attempt to decorate medical research with fashionable mathematical terms.
6 CritiquingtheCritique
6.1 TheAssumptionthatwasnotmade
The article by Stare [35] is the ﬁrst one concerning the mathematical errors of Bad Luck
theory, except for the aforementioned [20] and my own preprints [2],[3]. Stare is perfectly
right that Bad Luck theory is erroneous but his reasoning is partly not correct. Speciﬁcally,
he expresses the probabilityp(i) of the particular cancer as the sum of two probabilities:
• the probabilitys(i) of the cancer being random that is occuring due to stem cells
divisions) and
• the probabilityns(i) of it being non-random that is occuring due to life style, envi-
ronment, etc.
Such a decomposition means that the two causes of cancer are considered as incompatible
random events. Thus the following assumption is made, though not formulated explicitly:
the particular cancer can occur only in one of two alternative ways - either randomly as
a stochastic result of bad luck, or non-randomly (non-stochastically) - because of envi-
ronment, behavior, etc; both events are considered random since this all is about proba-
bilities. In doing so the author withdraws the very idea that cancer can occur as a result
of the combination of bad luck and other factors.
Then, Stare shows that the Bad Luck model does not give plausible results under that
assumption, and claims that thus he proves the incorrectness of the model (Section 2 “A
direct way of showing that R
2
cannot measure the randomness of cancer” [35]). That
conclusion would be right if T&V had made that unnatural assumption. They however
had not. Moreover, they argued that “Every individual cancer is the result of a possible
combination of bad luck, “bad” genes, and “bad” environment”, referring to their favorite
analogy of an auto accident which is a result of many factors sumaltaneously: “a long trip
in a car with a drunk driver, bad tires, and poor brakes who was driving in torrential rain
and hit by an out-of-control tractor-trailer” [23].
One cannot prove the fallacy of the theory by means of making assumptions which
contradict the theory’s foundation. Actually the conclusion that “There is no way one can
claim any proportion of cancers being random, based on the analysis of Tomasetti and
V ogelstein” [35] is absolutely correct, albeit for different reasons listed in the previous
Sections.
The bad mathematics of the Bad Luck theory 65
6.2 OneInterpretation
While critisizing the Bad Luck theory Kelly-Irving et al. [13] stated that “Even when
stem-cell mutations occur as random, the initiation and development of a cancer cannot
be viewed as random process”. The reason is named as the behaviour of the immune
system has not been shown to be random.
Figure2: The intersection points (circles) of immunity (thick line) and number of stem cells
divisions (thin line) are random when the immunity is nonrandom (left) as well as when it is
random (right).
Figure 2 illustrates that this is wrong. If the T&V idea was correct, the number of
stem cell mutations could sometimes exceed some threshold and the resistance (immu-
nity) would fail thus giving start to cancer. If one of two processes is random and the
other is not, then their intersections still form a random sequence (Figure 2a)). And of
course immunity itself is a random process (Figure 2b)) as it is formed in the environment
(the body) which has a lot of random characterics and is inﬂuenced by a lot of random pro-
cesses – heredity, external environment, accompanying deseases, age, the state of mood
etc.
7 Conclusions
Confusing correlation between the variables with that of their non-linear functions, infer-
ring causation just from correlation, using a senseless criterion for dividing cancers into
sorts are mathematical mistakes that make the Bad Luck theory mathematically incorrect
and its conclucions meaningless. That is not to say that the hypothesis about the prevail-
ing role of bad luck in carcinogenesis is wrong. At the moment though, it is no more than
mathematically unfounded assumption.
However, having been shared widely by mass-media ([16, 33] and many others) it
became popular among cancer researchers and general public giving a false impression
of having been mathematically proven. That can have, and maybe already has, serious
consequences, on one hand allowing people to think that they don’t have to worry about
environmental conditions or lifestyle because a tremendous fraction of cancers are all the
same unpreventable, and inﬂuencing the decision makers on the other hand. Meanwhile,
in their later publications [27, 28, 26] T&V extend their erroneous method, formerly
66 Beliaeva
conﬁned to US population only, to the large set of countries, omitting now the very word
logarithm as well as the deﬁnition of extra risk score. Thus those who begin their acquain-
tance with the Bad Luck theory from those later papers don’t have a chance to recognize
the fatal mistakes.
Besides, the authors themselves did a lot to make their theory as popular as possible:
gave interviews, held a brieﬁng [29], recorded a video [30]. Sorry for the pun, but this
was a real error propagation. After getting extensive criticism, based mostly on biological
reasoning, of their ﬁrst Bad Luck article [22] T&V stopped announcing that early detec-
tion should be of major focus as being more effective than altering lifestyle for tumors
with high ERS. Instead they stated that they were misunderstood and that their model is
just explaining the differences between cancer rates in different tissues. This new inter-
pretation certainly reduces the possible harmfulness of the Bad Luck theory making its
results more of academic than of practical interest, but does not make them correct. The
wrong mathematics proves nothing as falsity implies anything [8].
To conclude: the mathematician’s assistance in developing mathematical models would
help cancer specialists to avoid many mistakes and save them time and effort.
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