Metodološkizvezki,Vol. 14,No. 2,2017,75–81
BadLuckofCancer–orMisinterpretedStatistics?
JanezStare
1
Abstract
A paper in Science (January 2015) claimed that the majority, 65% to be precise,
of cancers is due to bad luck, so non-preventable. In this paper we show that the
analyses, presented in the paper, give absolutely no grounds to make such a claim.
Someoftheargumentshaveinthemeantimeappearedelsewhere,butsomehavenot.
We also show that the authors’ assumptions and their data can only support a claim
ofnomorethan5%ofcancersbeingrandom.
1 Introduction
In2015TomasettiandVogelstein(2015)publishedapaperinScienceinwhichtheyana-
lyzed associationbetween thenumber of stemcell divisions ofgiven tissues ina lifetime
and probability of cancers of those tissues. They found a strong correlation of more than
0.8(orR
2
= 0.65)betweenthelogarithmsofthesevariablesandbasedonthisconcluded
that a great majority of cancers, approximately two thirds of them, occurs randomly due
tostemcelldivisions,andthattheotherfactorscontributeonlytotheresidualthirdofall
cancers. Their work immediately met with some negative reactions, published mainly as
letters in Science, but their results were essentially not disputed. Only a year later, Wu,
Powers,Zhu,andHannun(2016),inapaperpublishedinNature,showedthatcorrelation
cannotsayanythingabouttheproportionofrandomcancers.
InthispaperwegiveamoredetailedandmoreversatilecriticismoftheTomasettiand
Vogelstein analysis, and also show that using their assumptions one cannot claim more
than 5%ofallcancers,usedintheiranalysis,beingrandom.
There are different ways to show that correlation, or R
2
, cannot come close to esti-
matingtheproportionofrandomcancers. Inthefirstsectionweshowinadirectwaythat
such an estimation is impossible, and give an illustration which completely mimics the
analysisbyTomasettiandVogelstein,butonadifferentdatasetwhichmakesthemistake
more obvious. In the second section we show how understanding the properties of R
2
makes it obvious that something is wrong with the conclusion about the proportion of
random cancers, and, finally, we show that Tomasetti and Vogelstein’s assumptions and
theirdatacanonlysupportaclaimofnomorethan5%ofcancersbeingrandom.
1
Institute for Biostatistics and Medical Informatics, Faculty of Medicine, University of Ljubljana,
Ljubljana,Slovenia;janez.stare@mf.uni-lj.si

76 Stare
2 ADirectWayofShowingthatR
2
cannotMeasureRan-
domnessofCancer
ThemodelthatTomasettiandVogelsteinassumeis
p
i
= 10
a
d
b
i
wherep
i
istheprobabilityofcancerfortissuei,d
i
isthenumberofstemcelldivisionsfor
thattissue,andaandbarecoefficientstobedeterminedfromthedata. Takinglogarithms
weget
logp
i
= a+blogd
i
and by fitting a regression line to the points (logp
i
, logd
i
) we get the estimates of the
coefficients and an R
2
equal to 0.65. If we say that cancers which occur regardless of
people’s life style, environment or similar, are random, and other cancers non random,
then we can write every probability p
i
as a sum of two probabilities, the probability of
a cancer being random (or stochastic, denoted by s
i
) and a probability of a cancer being
non-random(ornon-stochastic,denotedbyns
i
). Solet’swritep
i
= s
i
+ns
i
and
log(s
i
+ns
i
) = a+blogd
i
If we knew all s
i
then proportion of random cancers (PRC) among all cancers would be
simply
PRC =
P
s
i
P
p
i
The paper by Tomasetti and Vogelstein hints that this proportion is estimated by R
2
thattheyobtained,so
R
2
=

X
s
i

/

X
p
i

= 0.65.
It should be obvious that regression analysis will give the same results, and so the same
R
2
, regardless of how each p
i
is decomposed into s
i
and ns
i
. This means that R
2
has
absolutely nothingtodowithPRC.ThissimplefactisillustratedinFigure1forsubsetof
valuesinTomasettiandVogelsteinpaper. Proportionofrandomcancersisvaried,butthe
total probabilities remain the same. The actual situation should, under the Tomasetti and
Vogelstein assumption, look something like subfigure (c), but the values for randomness
(darkareas)canbealmostanything.
We illustrate this simple fact using another example, completely analogous to the
above,butmuchmoreobvious.
LetuslookattheEuropeancountriesandrecordtheirareasandpopulationsizes. Data
canbe,forexample,foundherehttp://bit.ly/1K2oosV.Ofcourse,anysetofcountrieswill
do. So d
i
now represent areas, and p
i
are proportions of each country’s population in the
total population of Europe. We use the same model as Tomasetti and Vogelstein (so logs
of proportions and areas)and fit a regression line (Figure 2). We get an R
2
of 0.86. If, as
anexample,s
i
andns
i
representproportionsofsmokersandnon-smokersofcountryiin
the total of European population, can we say that there are 86% of smokers in Europe?
Obviouslynot. ButthisisexactlytheargumentthatTomasettiandVogelsteinmake. The
pointis,again,thats
i
andns
i
canbeanytwonumbersaddinguptop
i
,andwewillalways
havethesameR
2
.

Bad Luck of Cancer ... 77
0.00 0.01 0.02 0.03 0.04
0.00 0.01 0.02 0.03 0.04
0.00 0.01 0.02 0.03 0.04
Figure1: Differentproportionsofrandomcancersdonotchangetheoverallproportions
Figure2: Regressionofproportionsofcountries’populationsinthetotalEuropean
populationoncountries’areas(logscales)
3 IndirectWaysofShowingthatR
2
cannotMeasureRan-
domnessofCancer
Tomasetti and Vogelstein calculate the correlation for a certain range of the number of
stemcelldivisions. IfR
2
didindeedestimateaproportionofrandomcancers,forcancers
included in their analysis, then we should get a similar result if we calculate this propor-
tion for a subrange. For example, if we did this separately for the tissues with stem cell
divisions below the median, and above the median, then we should be able to combine
these results into the overall number. For example, if R
2
below the median was a, and

78 Stare
above the median b (of course, any subdivision would do), then a weighted average of
these two numbers should give us 0.65. In fact, the numbers that we get for R
2
are 0.43
and 0.21 (Figure 3). So, parts have less random cancers than the whole? On the other
hand, if, in future, say, other cancers are included which have numbers of stem cell di-
visions lower than the present minimum, or higher than the present maximum, R
2
will
increase.
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
Stem cell divisions in 10^
probablity of cancer in 10^
Data from the paper
6 7 8 9 10 11 12 13
−5
−4
−3
−2
−1
0
overall R
2
= 0.65
R
2
= 0.43 R
2
= 0.21
Figure3: Proportionofexplainedvariationdependsonthechosenintervalofthe
independentvariable
Another way of showing that R
2
cannot estimate PRC is to assume that all probabili-
tiesintheirdataweremultipliedbyacertainfactor. Onecanimagineacountrywhichhas
more (less) risk due to some extra factor (or lack of some factor). Since the probabilities
of random cancers cannot change, their proportion in the overall number of cancers will
nowbedifferent,smallerorlarger,dependingonthefactor. ButR
2
willnotchange! This
isillustratedinFigure4.
4 ANoteonData
Data which Tomasetti and Vogelstein paper analyze contain some points that should not
be there. For example, probabilities for lung cancer are given separately for smokers and
nonsmokers. Theseareconditionalprobabilities,givenvaluesofsomeextravariable,and
are certainly not the probabilities which one would predict based on the number of the

Bad Luck of Cancer ... 79
Figure4: Increaseordecreaseofcancerprobabilitiesbyagivefactordoesnotchangethe
R
2

80 Stare
stem cell divisions. They argue that leaving them like this does not change the results of
theanalysis,butthisisnotavalidargument. Weareinterestedinprobabilitiesofcancers,
given the number of stem cell divisions, nothing more. For our calculations in the next
section we corrected these data, so that every number of stem cell divisions has just one
corresponding probability of a cancer of that tissue. We used data from Tomasetti and
Vogelstein supplementary material to do this. For example: lifetime risk of lung cancer
fornonsmokersis0.0045,andforsmokers0.081. Assuming(asTomasettiandVogelstein
report in their supplementary material) that the proportion of smokers is 0.3, then the
overallprobabilityoflungcanceris 0.0045×0.7+0.081×0.3 = 0.02745.
5 ADifferentWayofEstimatingtheProbabilityofRan-
domCancers
Ifweassume,asTomasettiandVogelsteindointheirmodel,thatprobabilitiesofcancers
onlydependonthenumberofdivisions,thenthecandidatesforrandomcancersarethose
lyinglowonthegraph. InFigure5weconnectedtwolowpointsonthegraphandvalues
on this line could be seen as (log of) probabilities for random cancers. Anything above
them must be non random. Of course, assuming that those two points themselves repre-
sent probabilities of random cancer for those two tissues is probably overestimating the
trueprobabilities. Andstill,thetotalprobabilityofrandomcancer,calculatedinthisway,
isverylow.
Figure5: Regressionlinethroughtwopointsrepresentingpossiblerandomcancers
Another,endevenbetter,wayofcalculatingtheoverallprobabilityofrandomcancers,
istodothefollowing

Bad Luck of Cancer ... 81
1. Findcancerwiththelowestprobabilityperdivision.
2. Take this (or part of this) probability to be the probability of random cancer per
division.
3. Multiplythisprobabilitybythenumberofdivisionsforeachtissue.
4. Thisgivesusprobabilitiesofrandomnesspereachtissue.
WhenweapplyabovetotheTomasettiandVogelsteindata,itturnsoutthatthelowest
probability per division is for small intestine cancer. Assuming (probably unreasonably)
that all small intestine cancers are random, and continuing with points 2. to 4. above,
and then summing up all the obtained probabilities, we get that the overall probability
of random cancers is 1.6%! This translates into no more than 5% of all cancers in their
analysisbeingrandom,dependingonhowmuchindependencewewanttoassumeamong
cancers. Of course we do not believe these numbers, we simply show that the claim of
mostcancersbeingrandomrestsonavaryshakyground.
6 Conclusion
There is no way one can claim any proportion of cancers being random, based on the
analysis of Tomasetti and Vogelstein. In fact, their inherent assumption of all divisions
being equally likely to produce cancer, yields a very low estimate of the proportion of
randomcancers.
References
[1] Tomasetti, C. and Vogelstein, B. (2015): Variation in cancer risk among tissues can
beexplainedbythenumberofstemcelldivisions. Science,347,78–81.
[2] Wu, S., Powers, S., Zhu, W., and Hannun, Y. A. (2016): Substantial contribution of
extrinsicriskfactorstocancerdevelopment. Nature,529,43–47.