Statistical Problems in a Paper on Variation In Cancer Risk Among Tissues, and New Discoveries

0.71 MB·English

by  Lee Altenberg

Statistical Problems in a Paper on
Variation In Cancer Risk Among Tissues,
and New Discoveries
LEEALTENBERG
TheKLIInstitute,[email protected].
5
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0 January20,2015
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n
a Abstract into“R-tumors”,where“stochasticfactors,presumablyrelated
J
to errors during DNA replication, most strongly appear to af-
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TomasettiandVogelstein(2015)collecteddataon31different fect their risk,” and “D-tumors”, where “deterministic factors
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tissue types and found a correlation of 0.8 between the log- such as environmental mutagens or hereditary predispositions

] arithms of the incidence of cancer (LCI), and the estimated stronglyaffecttheirrisk”. Onthisbasis,theygiveaprognosis
M number of stem cell divisions in those tissues (LSCD). Some for the likely success of interventions to prevent these tumor
Q oftheirconclusionshoweverarestatisticallyerroneous. Their types:
excess risk score, “ERS” (log LCI log LSCD), is non-
. 10 × 10
o monotonic under a change of time units for the rates, which Theseresultscouldhaveimportantpublichealthim-
i
b renders meaningless the results derived from it, including a plications. One of the most promising avenues for
- cluster of 22 “R-tumor” types for which they conclude, “pri- reducing cancer deaths is through prevention. How
q
mary prevention measures are not likely to be very effective”. successful can such approaches be? The maximum
[
Further, r = 0.8 is consistent with the three orders of magni- fraction of tumors that are preventable through pri-

1 tudevariationinotherunmeasuredfactors,leavingroomforthe maryprevention(suchasvaccinesagainstinfectious
v
possibility of primary prevention if such factors can be inter- agents or altered lifestyles) may be evaluated from
5
venedupon. Furtherexplorationofthedatarevealsadditional theirERS.FornonhereditaryD-tumors,thisfraction
0
6 findings: (1) that LCI grows at approximately the square root is high and primary prevention could make a ma-
4 ofLSCD,whichmayprovideacluetothebiology;(2)among jor impact (31). Secondary prevention, obtainable
0 different possible combinations of the primary data, the one in principle through early detection, could further
.
1 maximizing the correlations with LCI is almost precisely the reduce nonhereditary D-tumor-related deaths and is
0 formula used by Tomasetti and Vogelstein to estimate LSCD, also instrumental for reducing hereditary D-tumor-
5 giving support to stem cell divisions as an independent factor related deaths. For R-tumors, primary prevention
1
incarcinogenesis,whilenotexcludingothersuchfactors. measuresarenotlikelytobeveryeffective,andsec-
:
v ondarypreventionshouldbethemajorfocus.
i
X Keywords:cancerincidence,stemcells,correlation,tumortypes,lin-
earregression,somaticmutation,fallacy,symbolicregression,preven- Here I describe statistical problems with the paper that un-
r
a tion dermine these conclusions. These problems are so basic that
theyoughttohavebeencaughtinreview,butapparentlywere
not.
Introduction First, any statistic on “extra risk” should be invariant under
a change in the units of time used to measure the rates. Their
ApaperrecentlypublishedinScience,“Variationincancerrisk ERSstatisticnotonlyfailstobeinvariant,butisnon-monotonic
amongtissuescanbeexplainedbythenumberofstemcelldi- under a simple change of time units in the data. This renders
visions” by Tomasetti and Vogelstein (2015), investigates the meaninglessalltheconclusionsbasedonit.
relationshipbetweenthenumberofstemcelldivisionsinvari- Second, there is a fundamental misunderstanding of what
oustissuetypesandtheincidenceoftumorsinthem. Theyfind high correlations imply. The argument that “primary preven-
a strong correlation, 0.8, between the logarithms of these two tion measures are not likely to be very effective” rests on the
values(bothSpearman’srankcorrelationρandPearson’slinear ideathathighcorrelationsbetweenavariablenotsubjecttoin-
correlationr). Theyusetheproductoftheselogarithms,which tervention(numberofstemcelldivisions)andatargetvariable
they call the “extra risk score” (ERS) to classify tumor types (cancerincidence)meansthatthetargetvariableismostlynon-
1
2 LEEALTENBERG
susceptibletointervention. LCI/LSCD.Onthelogarithmicscalethiswouldbelog LCI
10 −
This is wrong on two counts: first, because the correlation log LSCD. Forunknownreasons, theauthorsinsteaddevise
10
isbetweenlogarithms,itispossibleforasecond,unmeasured astatisticERS(LCI,LSCD) := log LCI log LSCD. They
10 × 10
factor to vary, in this case over four orders of magnitude, and alsoconsideredusinglog LCI/log LSCD(seetheirSupple-
10 10
still maintain the correlation of 0.8 for the data. Second, cor- ment)butrejectit,notonfirstprinciples,butbecauseitis“sub-
relations only put limits on the existing variation of unknown optimal”:
factors;theyhavenothingtosayaboutnovelinterventionsthat
“Note that using the ratio between the log values
maybedevelopedwhichchangethatvariation.Thislatterpoint 10
of r [LCI] and lscd, instead of the product, would
is well developed in Feldman and Lewontin (1975), an article
be sub-optimal to estimate the extra risk. ...When
promptedbythemisuseofheritabilitymeasuresofhumanin-
ERS is defined as the product rather than the ratio,
telligence. Thepotentialmisuseofcorrelationmeasuresinde-
theexpectedrelationshipisevident”.
cisionsaboutcancerresearchpromptstheworkhere.
Putting aside the statistical problems in Tomasetti and Vo- Oneexpectedrelationshipthatshouldbeevidentforamea-
gelstein (2015), further exploration of their data reveal some surementofextrariskisthatitbeinvariantunderachangeof
tantalizingcluestothebiologyofcancer. timeunits. Ifthetimeunitswerechangedfromperlifetimeto
First, the best fit to the data indicates that cancer incidence perlifetime/T,thisshouldbeirrelevanttoameasureof“can-
grows not in proportion to the number of stem cell divisions cer risk of that tissue type relative to its number of stem cell
in a tissue, but in proportion close to the square root. Sec- divisions”.
ond,anexplorationofdifferentcombinationsofprimarydata, However, wefindthatERSisextremelyill-behavedinthis
s and d, they use to estimate the lifetime number of stem cell regard: itisnotinvariant,andevenworse,itisnon-monotonic.
divisions shows the correlations with the lifetime cancer inci- Thiscanbeseenfromitsexpansion:
dencearemaximizedalmostpreciselybytheformulatheyuse,
LSCD = s(2+d) 2. This suggests that their estimate for ERS(LSCD/T, LCI/T)
−
the number of stem cell divisions or a closely related formula =ERS(LSCD, LCI)
isarealbiologicalfactorintheincidenceofcancer,whilenot
T(log LSCD+log LCI)+(log T)2.
rulingoutthepossibilityofothercentralfactors. − 10 10 10
TherelationshipbetweenERSforthedatasetusingtimeunits
The Ill-Behaved “Extra Risk Score” T = 1 and T = 1000 is plotted in Figure 1. We see that a
simplechangeoftimemeasurementunitsessentiallyscrambles
(ERS)
theERSscores.
TomasettiandVogelstein(2015)introducetheirERSstatistic:
We next attempted to distinguish the effects of ￿30
this stochastic, replicative component from other log (0.001 LSCD)
10
causative factors—that is, those due to the external ￿35
log (0.001 LCI)
environment and inherited mutations. For this pur- × 10
pose, we defined an extra risk score (ERS) as the ￿40
product of the lifetime risk and the total number of
stemcelldivisions(log values). ...
10 ￿45
TheERSprovidesatestoftheapproachdescribedin
thiswork. IftheERSforatissuetypeishigh—that ￿50
is,ifthereisahighcancerriskofthattissuetyperel-
￿30 ￿25 ￿20 ￿15 ￿10 ￿5 0
ative to its number of stem cell divisions—then one
ERS = log LSCD log LCI
wouldexpectthatenvironmentalorinheritedfactors 10 × 10
would play a relatively more important role in that
cancers risk (see the supplementary materials for a Figure1: Non-monotonicityofthe“ExtraRiskScore”(ERS)under
change of time units for the cancer incidence data and the number
detailed explanation). It was therefore notable that
of stem cell divisions. Here the time unit is rescaled by a factor of
thetumorswithrelativelyhighERSwerethosewith
1/T =0.001.
known links to specific environmental or hereditary
riskfactors(Fig. 2,bluecluster).
Thenon-monotonicityoftheERSstatisticinTomasettiand
The most straightforward statistic to measure the “cancer Vogelstein (2015) under a simple change of time units ren-
riskofthattissuetyperelativetoitsnumberofstemcelldivi- ders any conclusions based on the ERS meaningless. This
sions”wouldbetheratioofrisktothenumberofcelldivisions, includes their Fig. 2, their categorization of “R-tumors” and
STATISTICALPROBLEMSINAPAPERONVARIATIONINCANCERRISKAMONGTISSUES 3
“D-tumors”, their K-means cluster analysis (Supplement Fig.
S2),andthereforetheconclusionoftheirpaper(repeatedfrom 0 CCAAFF
above): 0.533log LSCD 7.61 CCAALL
10 − BBCC
￿1
Themaximumfractionoftumorsthatarepreventable LLAASSHHNNVV HHCCVV
CCAA
throughprimaryprevention(suchasvaccinesagainst DDAAFF
MMEELL
infectious agents or altered lifestyles) may be eval- ￿2 TTPP HHNNSS PPDD
uated from their ERS. For nonhereditary D-tumors, log10 LCI GGNNPP TTEESSLLAA AACCMMLLLLLLHHCC
this fraction is high and primary prevention could GGLLII EESSCC
￿3
makeamajorimpact... ForR-tumors,primarypre- SSII
vention measures are not likely to be very effective, OOSSLLOOOOVVSSTTMM PPDDEEAA
andsecondarypreventionshouldbethemajorfocus. ￿4 MMBB
OOSSAA
OOSSOOPPSSHH
ResidualRisk ￿5
6 7 8 9 10 11 12 13
log LSCD
Clearly, Tomasetti and Vogelstein’s ERS = log10LCI × 10
log LSCD is so ill-behaved that it has no meaning. The two
10
naturalchoicesformeasuringextrariskare:
Figure2: The31tissuestypes(abbreviations)withthelinear
1. Normalized Incidence: This is simply the lifetime can- predictionfunction.
cerincidencedividedbythelifetimenumberofstemcell
divisions. Inlogscale:
Thatthisisfortuitouscanbeseeninthecompletelydifferent
NI:=log LCI log LSCD. mathematicalstructuresofERSandRR:
10 − 10
2. Residual Risk: This is the lifetime cancer incidence di- ERS=log LCI log LSCD,
vided by the incidence predicted from the linear regres- 10 × 10
RR=log LCI (0.533log LSCD 7.61).
sion. Inlogscale: 10 − 10 −
For comparison, the rank correlation between ERS and ERS
RR:=log LCI P[log LCI],
10 − 10 with time units rescaled by 0.001 is only 0.24 (Figure 1), and
between ERS and NI it is only 0.09. Note that ρ(NI,RR) =
where
0.618.
P[log LCI]=0.533log LSCD 7.61 (1) Thiscoincidentalcorrelationmayhavebeenthereasonthat
10 10 −
ERS was retained by the authors, because it sorted the tumor
isthelinearpredictorfunctionfromregressionoflog10LCIon types in an order similar to the vertical position of the points
log10LSCD(plottedwiththedatainFigure2): relativetothepredictorlineinFigure2.
InTable1the31tumortypesaresortedbytheirresidualrisk
If one wanted to distinguish “R-tumors” from “D-tumors”
values,
based on Table 1, one certainly could, since there are several
large gaps between RR values, notably between Gallbladder
RR=log LCI (0.533log LSCD 7.61).
10 − 10 − and Glioblastoma. But an goodness-of-fit test (Anderson and
(this has been posted on at least one online blog, “Peer 3” Darling,1952)showsthedistributionofRRvaluestobeindis-
(2015)). tinguishablefromaGaussiannormaldistribution(P =0.977),
Topping the list is lung cancer in smokers, followed princi- andthegapsizesnotsignificantlydifferentfromanexponential
pallybythetumorslabeledwithinheritedorviralriskfactors. distribution (P = 0.51). Caution should therefore be used in
ThissupportstheargumentofTomasettiandVogelstein(2015) makinganyclaimsbasedonthesegaps.
for“incorporationofareplicativecomponentasathird,quan- We also see in Table 1 that the residual risk spans 2.862 =
titativedeterminantofcancerrisk”inaddition“environmental 1.211+1.651 orders of magnitude. A correlation of 0.8 be-
orinheritedfactors.” tweenthelogarithmsofLSCDandLCIthereforedoesnotpre-
Wealsoseesomethingremarkable. Thetypesidentifiedby clude three orders-of-magnitude variation in LCI due to other
Tomasetti and Vogelstein as “excess risk” based on the spu- factors. Thisiselaborateduponinthenextsection. Ifwecon-
rious ERS statistic largely maintain their positions in the top sider only those tumor types with negative RR, there are still
ofthelistbasedontheRRstatistic. Thereasonforthisispure almost2ordersofmagnitudevariationduetounknownfactors.
coincidence:ithappensthat,onthisdataset,thecorrelationbe- Ifnotfromerrorsinthedata,somethingissuppressingthein-
tweenRRandERSisfortuitouslyhigh: r(ERS,RR) = 0.83, cidence of certain tissue tumors to only 2% of that predicted
ρ(ERS,RR)=0.80. fromthelogLSCDregression.
4 LEEALTENBERG
tionsbetweenLCIandLSCD+ ( 2,2),where ( 2,2)isa
Table 1: Excess of log LCI above the best fit linear regres- U − U −
10 uniformly distributed random variable on the interval [ 2,2].
sion: RR = log LCI (0.533log LSCD 7.61). In bold −
10 − 10 − Under 10,000 replicates they find the addition of this four
face are the types called “D-tumors” in Tomasetti and Vogel-
orders-of-magnitude noise only reduces the median value of
stein (2015). Their position near the top of the list is due to
Spearman’srankcorrelationρ(Spearman,1904)from0.81 =
the coincidentally high rank correlation between log LCI
10 × ρ(LCI,LSCD) to 0.67 = ρ(LCI,LSCD+ ( 2,2)), which is
log LSCDandlog LCI (0.533log LSCD 7.61)onthe U −
10 10 − 10 − stillsignificantlydifferentfromzero:
dataset.
Thus,thoughthetotalrangeforLSCDis 6orders
∼
RR TumorType of magnitude and we allowed four 4 [sic] orders of
1.211 Lungadenocarcinoma(smokers) magnitudevariationforeachdatapoint, thecorrela-
1.183 ColorectaladenocarcinomawithFAP
tions generated were always statistically significant.
0.952 Thyroidpapillary/follicularcarcinoma
0.916 Head&necksquamouscellcarcinomawithHPV-16 Thisprovidesstrongevidencethatourresultsarero-
0.886 DuodenumadenocarcinomawithFAP bust. [Supplementp. 11]
0.882 ColorectaladenocarcinomawithLynchsyndrome
0.853 Gallbladdernonpapillaryadenocarcinoma
0.460 Glioblastoma But the robustness of the high correlation to order-of-
0.403 Basalcellcarcinoma magnitudevariationsinthedatahasitsconverseimplication:it
0.373 HepatocellularcarcinomawithHCV meansthathighcorrelationcannotruleoutorder-of-magnitude
0.314 Ovariangermcell
variationinotherfactorsthatmaydeterminetheratesofcancer.
0.201 Osteosarcomaofthelegs
0.178 Osteosarcoma WeseethisinTable1.
0.156 Head&necksquamouscellcarcinoma Toillustratethisconverseimplication,supposethatLCIwere
0.106 Testiculargermcellcancer
determinedbytwofactors,LSCD,andanother,unknownpre-
0.062 Esophagealsquamouscellcarcinoma
−0.016 Thyroidmedullarycarcinoma ventativefactor“X”thatreducestherateofcancerinproportion
−0.045 Lungadenocarcinoma(nonsmokers) toitsvalue. HowmuchvariationinXcouldtherebeandstill
−0.136 Colorectaladenocarcinoma obtaincorrelationsof0.8betweenlog LSCDandlog LCI?
−0.334 Osteosarcomaofthearms 10 10
−0.373 Osteosarcomaofthepelvis Thegeneralproblemistocomputethecorrelation, r (Pear-
−0.400 Pancreaticductaladenocarcinoma son,1901),betweenY,andY plusuncorrelatednoise,Z:
−0.411 Melanoma
−0.520 Osteosarcomaofthehead
−0.593 Chroniclymphocyticleukemia Cov(Y,Y +Z)
r(Y,Y +Z)=
−0.627 Hepatocellularcarcinoma (cid:112)Var(Y)(cid:112)Var(Y +Z)
−0.696 Acutemyeloidleukemia
−0.840 Medulloblastoma 1
= .
−1.181 Duodenumadenocarcinoma (cid:114) Var(Z)
−1.312 Pancreaticendocrine(isletcell)carcinoma 1+
Var(Y)
−1.651 Smallintestineadenocarcinoma
0.000 Total
Thesolutionofr(Y,Y+Z)=0.8isVar(Z)=0.5625Var(Y).
Tobeconcrete,letLCI=c LSCD X,wheretheconstantc
∗ ∗
The fallacy that high correlations pre- doesnotenterintothecorrelation. Thenlog10LCI=log10c+
log LSCD+log X. We let log LSCD be distributed as a
10 10 10
clude intervention uniform random variable on the interval [6.50,12.55], the ac-
tual range from the data. We let log X be distributed as an
10
TomasettiandVogelsteinpointoutthatthecorrelationbetween independentrandomvariableuniformon[ w,0],andaskhow
−
log LSCD and log LCI is extremely robust to noise, and bigwcanbeandyetmaintain0.8=r(log LCI,log LSCD).
10 10 10 10
therefore makes their results robust. What they do not realize Thesolutionisw =4.5—fourordersofmagnitude.
isthatthisveryrobustnessarguesagainsttheirconclusionthat It may be helpful to see what the distribution of this hypo-
“for R-tumors, primary prevention measures are not likely to thetical preventative factor X looks like, in Figure 3. We see
beveryeffective”,becauseitallowslargevariationinunknown hugevariationinthepopulationincidenceofcancerduetohy-
factorsthatcouldalsocontrolcancerincidencerateswhilehav- pothetical factor X, even though there is still a correlation of
inglittleeffectonthecorrelation. 0.8 between log LSCD and log LCI. Therefore, the corre-
10 10
TomasettiandVogelsteinmeasurethisrobustnessbyadding lation of 0.8 does not rule out the presence of other factors—
noise to the data to see how much it changes the correla- genetic,environmental,physiological,ormeasurementerror—
tion. They add both Gaussian and uniformly distributed ran- that can produce over four orders-of-magnitude variation in
dom noise to their estimates of lifetime number of stem cell cancer rates. Such factors could potentially include environ-
divisions. Fortheuniformvariation,theyexaminethecorrela- mentalvariationthatissubjecttointervention.
STATISTICALPROBLEMSINAPAPERONVARIATIONINCANCERRISKAMONGTISSUES 5
1.0 the mean squared displacement grows in proportion to time
or the number of steps taken. In such cases we would need
0.8
to ask what properties of cells follow a random walk under
PREVENTATIVE
0.6 cell division. Models of telomere length come to mind here
FACTOR
(Blythe and MacPhee, 2013; Duc and Holcman, 2013). An-
"X" 0.4
other source could be geometry—boundaries of regions that
0.2 grow along two-dimensions (e.g. the perimeter of a growing
circle,orthesurfaceofanelongatedcylindergrowingindiam-
2000 4000 6000 8000 10000 eter)canhaveclosetoasquare-rootrelationshiptothatgrowth.
Provided the right geometry, a square-root relationship could
conceivably emerge if the boundaries of tissues played a role
Figure3: Variationinahypotheticalmultiplicative“preventa-
incarcinogenesis. Theremaybeplausiblesourcesofone-half
tivefactorX”oncancerincidenceconsistentwithacorrelation
powerlawstobeelicitedfromtheworkonscalingrelationship
of 0.8 between log LSCD and log LCI. Sorted along the
10 10
inbiology(c.f.Savageetal.(2013)).
X-axisare10,000samplesofX,wherelog X U[ 4.5,0].
10 ∼ −
TheFormulaforLSCD
Further Explorations
The appearance of the exponent 0.533 in the relationship be-
The problems described above do not diminish the fact that tweenLCIandLSCDpromptsustoexaminemorecloselyhow
Tomasetti and Vogelstein (2015) have produced an important the estimate of LSCD was made. Their Supplement explains
dataset,andthestrongcorrelationtheyfindbetweentheloga- thattheformulaforLSCDis:
rithmsoflifetimenumberofstemcelldivisionsandtheratesof
cancerindifferenttissuesprovidesevidenceforarealbiolog- LSCD=s(2+d) 2, (3)
−
icalphenomenon. Theirdatathereforedeservedeeperinvesti-
gation. where
s is the total number of stem cells found in a fully developed
AOne-HalfPowerLaw? tissue,and
The predictor function (2) translates to an approximate power d is the number of further divisions of each stem cell in the
lawof lifetime of that tissue once the tissue is fully developed,
duetonormaltissueturnover.
7.61 0.533
LCI 10− LSCD . (2)
≈ Suppose instead of this formula, we explored the space of
Apowerlawwithexponent0.533isnotwhatwewouldhave possible combinations of the primary data s and d (i.e we are
expectedfromasimplestemcelldivisionhypothesis. Thesim- pursuing symbolic regression (Koza, 1990)). Could modern
plehypothesiswouldpredictthecancerincidenceshouldgrow data mining software discover the stem cell division hypoth-
inproportiontothenumberofstemcelldivisions, withexpo- esis by finding patterns in the data (c.f. Schmidt and Lipson
nent 1.00. What we find instead is that it grows in proportion (2009))? Could we find combinations of s and d with higher
toapproximatelythesquarerootofthenumberofstemcelldi- correlationstoLCI?
visions(thishasbeennotedincommentsonatleastoneonline Wefirstexploretheeffectofchangingtheadditiveconstants
blog,Kuenzel(2015)). in (3). For Spearman’s ρ, the constant 2 is irrelevant. We
−
Thevalue0.533may,however,reflectregressiondilution—a embeds(2+d) 2inthefamilyofformulae,ψ(c):=s(c+d)
− −
systematicreductionintheslopeofthepredictorfunctiondue 2. Forc [0,20],ρ(ψ(c),LCI)variesonlybetween0.776and
∈
to large amounts of noise in the independent variable. This 0.810,withabroadpeakfor0.8 c 6.1. Therobustnessof
≤ ≤
is likely here because the estimate of the number of stem cell ρtovariationintheconstantsmeansthedatagivenoparticular
divisions,LSCD,combinesmultipleandsometimesuncertain validationtotheirvaluesintheformulaforLSCD(3).
primary data. Improvements in the estimation of LSCD and Sincelog LSCD log s+log d,hence(1)isapproxi-
10 ≈ 10 10
inclusion of more tissue types in the data would therefore be matedby
expectedtoincreasetheslope.
P[log LCI] 0.533(log s+log d) 7.61.
If, however, further data lent support to a power law with 10 ≈ 10 10 −
anexponentofone-half,itwouldinvitebiologicalspeculation.
Suppose instead of equal weights on log s and log d, we
Onewouldwanttoexamineprobabilisticmodelsforthemuta- 10 10
embeds(2+d) 2inthefamilyofformulae,
tionalbasisofcarcinogenesistoseewhereprobabilitiesscaled −
inproportiontothesquarerootofthenumberofcelldivisions.
(2 t) t
Randomwalksanddiffusionsprovideanotherpossibility,since φ(t):=s − (2+d ) 2.
−
6 LEEALTENBERG
Theparametertvariesφ(t)continuouslyfromφ(0)=3s2 2, Nonetheless,theoptimalityoftheformulaofTomasettiand
−
toφ(1)=LSCD,toφ(2)=d2. Wefindthat Vogelsteinisatleastsuggestive: thatsomethingclosetos d
×
hasarealbiologicalroleintheincidenceofcancer.Itisasepa-
ρ(φ(0),LCI)=ρ(s,LCI)=0.68, rateformofevidencefromthe0.8magnitudeofthecorrelation
ρ(φ(1),LCI)=ρ(LSCD,LCI)=0.81, itself.
ρ(φ(2),LCI)=ρ(d,LCI)=0.60.
Discussion
The publication of Tomasetti and Vogelstein (2015) was an-
0.96 0.99
− nounced by Johns Hopkins University with the press release
ρ=0.819 headline,“BadLuckofRandomMutationsPlaysPredominant
0.80 r=0.810 RoleinCancer,StudyShows.” Probabilitytheorywascreated
to make reasoning about “luck” a rigorous science. People’s
Correlation 0.75 0.98 reasoning about probabilities—luck—is a notoriously error-
ρ
proneactivity,andthecharacterizationoftheseerrorsisnowa
0.70
scientificfieldthathasburgeonedsincethepioneeringworkof
(TverskyandKahneman,1974). Thereisawideintuitionthat
0.65 r
thereisazero-sumtradeoffbetweencontrolandluckindeter-
mining events: the more control we have, the less we depend
0.0 0.5 1.0 1.5 2.0
Exponent t parameter on luck. This intuition of a tradeoff resonates with the math-
ematical structure of the analysis of variance, in which total
varianceispartitionedintoasumofvariancesandcovariances.
Figure 4: Correlations r(log LCI,log φ(t)) and ρ(LCI,φ(t))
10 10 However, when two factors, “luck” (L) and “control” (C) in-
(2−t) t
forφ(t):=s (2+d )−2. Thepeaksareneart=1,where teract multiplicatively, as L C, then there is no necessary
φ(1)=LSCD. tradeoffbetweenthematall.A×ndasdescribedhere,becauseof
thenatureoflogarithms,ahighcorrelationbetweenlogL,and
Fortherangeoft,Figure4plotsthecorrelationsρ(φ(t),LCI) logL+logC,doesnotprecludelargevariationinC.
and r(φ(t),LCI). There peaks are very near the value t = 1 Beyond this issue of correlations between logarithms, and
whereφ(1)=LSCD. theerroneoususeoftheill-behaved“extrariskscore”(ERS),
Therefore, by simply looking for correlations between LCI thereistheadditionalpitfallofcognitiveframing. IfTomasetti
anddifferentcombinationsofsandd,wecomeupwithafor- andVogelsteinhadpresentedan80%correlationbetweenloga-
mulaveryclosetotheformuladevelopedbyTomasettiandVo- rithmsofthenumberofstemcelldivisionsandcancermortality
gelstein from first principles for the total number of stem cell fordifferenttissues,insteadofincidence,wouldtheyhavethen
divisionsinatissue. Thereisnoapriorireasonthatthepeak argued“cancertreatmentmeasuresarenotlikelytobeveryef-
should occur near t = 1.0, since s, d, and LCI are indepen- fective,andpreventionshouldbethemajorfocus”? Ofcourse
dentlyobtaineddatasets(therankcorrelationbetweensandd not,becausenovelcancertreatmentsaresomethingnewunder
isonly0.23). thesun. Thecognitiveframeinthiscasedrawsonourfamiliar-
Could this be a coincidence? We can get some idea of that itywiththehistoryofmedicineinwhichmanynewcureshave
likelihood. Bootstrapresamplingofthe31tissuetypesforthe beenfoundthatobliteratepastcorrelations.
values of t that maximize ρ(φ(t),LCI) gives a central 50% The experience of “dramatic cure” has no parallel in expe-
interval of (0.909,1.009), with a median and sharp mode at riences of “dramatic prevention” (except to a statistician) be-
t=0.971.Subjectingsanddtomultiplicativenoise(replacing cause prevention is a non-event. Therefore, the fallacy that
sanddbys ν andd ν ,wherelnν ,lnν (0,1/82), “high correlation precludes intervention” can find an easier
1 2 1 2
× × ∼N
ν ,ν independent), gives a median of 0.954 and 50% central homewhenreasoningaboutprevention.
1 2
rangeof(0.928,0.973). Cures, primary prevention, and secondary prevention are
So,yesitcouldbeacoincidencethattheoptimaltissoclose simplyinterventionsatdifferentstagesofdisease. Correlations
to 1.0. Chance can’t be ruled out as to why the formula of foundinthepresentdonotbearonwhatinterventionsmaybe
TomasettiandVogelsteinseemstoprovidethemaximalcorre- foundinthefutureforanyofthesediseasestages. Ifanything,
lationtotheratesofcancerincidence. Wecannotruleoutthere thetrackrecordofhumandiscoverytodatehasbeenthatcan-
beingotherrelationshipsbetweensanddthatgiveevenhigher cerpreventionismucheasiertodiscoverthancancercure. But
correlationswiththecancerincidencedata. Itwillrequireex- thefuturehasyettobewritten.
pansionofthedatasettomoretissuetypes,morepreciseesti- Leaving aside the erroneous statistics of Tomasetti and Vo-
matesofthenumberofstemcelldivisions,andexplorationof gelstein(2015)andtheconclusionsbasedonthem,asbiology
othermodelsofthebiologytoresolvethisquestion. it is a significant finding that there is a high rank correlation
STATISTICALPROBLEMSINAPAPERONVARIATIONINCANCERRISKAMONGTISSUES 7
between(1)cancerincidence,and(2)theestimatednumberof “Peer 3” 2015. Comments. Pub Peer (blog), January
stemcelldivisionsinatissue.Additionalupportforthatsignif- 3 (3:54 pm UTC). http://pubpeer.com/publications/
icanceispresentedhere:thatamongvariouspossiblecombina- 867D3AFB2EBDF34A37FFCBA075D1BE.
tionsoftheprimarydata,theonethatproducesthehighestcor-
Savage, V.M., Herman, A.B., West, G.B., andLeu, K.2013. Us-
relationtocancerincidenceispreciselytheformulaTomasetti
ingfractalgeometryanduniversalgrowthcurvesasdiagnosticsfor
andVogelsteinobtainfrombiologicalfirstprinciples.Although
comparing tumor vasculature and metabolic rate with healthy tis-
this could be a statistical coincidence and requires additional sue and for predicting responses to drug therapies. Discrete and
data to confirm, it can be seen as a novel form of support for ContinuousDynamicalSystems.SeriesB,18(4).
the hypothesis stem cell divisions are an independent causal
Schmidt, M.andLipson, H.2009. Distillingfree-formnaturallaws
factorforcancer.
fromexperimentaldata. Science,324(5923):81–85.
MaterialsandMethods Spearman, C. 1904. The proof and measurement of association be-
tweentwothings. TheAmericanJournalofPsychology,15(1):72–
101.
DatasetswereobtainedfromtheSupplementtoTomasettiandVogel-
stein (2015). All computations and statistics were carried out using
Tomasetti,C.andVogelstein,B.2015.Variationincancerriskamong
MathematicaTM.Thecodeusedisavailableuponrequest. tissuescanbeexplainedbythenumberofstemcelldivisions. Sci-
ence,347(6217):78–81.
ACKNOWLEDGEMENTS Tversky, A. and Kahneman, D. 1974. Judgment under uncertainty:
Heuristicsandbiases. Science,185(4157):1124–1131.
IthankmycolleaguesattheKLIfortheirspiriteddiscussion—
GerdMu¨ller,IsabellaSarto-Jackson,OlivierMorin,andMath-
ieu Charbonneau, and thank Claus Vogl and Marcus W. Feld-
man for their comments. I gratefully acknowledge support
fromTheKLIInstitute,Klosterneuburg,Austria,andtheMath-
ematical Biosciences Institute at Ohio State University, USA,
throughNationalScienceFoundationAward#DMS0931642.
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