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Accident Analysis and Prevention 60 (2013) 5056

Contents lists available at ScienceDirect

Accident Analysis and Prevention

journal homepage: www.elsevier .com/ locate /aap

The killing zone revisited: Serial nonlinearities predict general

aviation accident rates from pilot total flight hours

William R. Knecht

Civil AerospaceMedical Institute, Federal Aviation Administration,MMAC/AAM-510, 6500S.MacArthurBoulevard,OklahomaCity,OK 73079, UnitedStates

a r t i c l e i n f o

Article history:

Received 17 January 2013

Received in revised form 24 July 2013Accepted14 August 2013

Keywords:

General aviation

Accidents

Risk

Flight hours

Modeling

Nonlinear

a b s t r a c t

Background: Is there a killing zone (Craig, 2001)arange ofpilot flight time overwhichgeneral aviation

(GA)pilots areatgreatest risk?Morebroadly,canwepredictaccidentrates, givenapilots totalflighthours

(TFH)? These questions interest pilots, aviation policy makers, insurance underwriters, and researchers

alike.Most GA research studies implicitly assume that accident rates are linearly related to TFH, but that

relation may actually be multiply nonlinear. This work explores the ability ofserial nonlinearmodeling

functions to predict GA accident rates from noisy rate data binned by TFH.

Method: Two sets ofNational Transportation Safety Board (NTSB)/FederalAviation Administration (FAA)

data were log-transformed, then curve-fit to a gamma-pdf-based function. Despite high rate-noise, this

produced weighted goodness-of-fit (R2w) estimates of .654 and .775 for non-instrument-rated (non-IR)and instrument-rated pilots (IR) respectively.

Conclusion: Serial-nonlinearmodels could be useful to directly predict GA accident rates from TFH, and

as an independent variable or covariate to control for flight risk during data analysis. Applied to FAA

data, these models imply that the killing zone may be broader than imagined. Relatively high risk for

an individual pilot may extendwell beyond the 2000-hmark before leveling offto a baseline rate.

Published by Elsevier Ltd.

1. Introduction

Totalflighthours isoftenused torepresentpilot flightexperience

and/or risk exposure (e.g., Dionne et al., 1992; Guohua et al., 2003;

Mills, 2005;Sherman,1997). TFHhasoftenservedas eitheran inde-

pendent variable (IV) in itsownright or as a statistical covariate, to

control for the effects of experience or risk.

However, many investigators have found no reliable relations

between TFH and their dependent variables-of-interest (DV). This

may be because most typically assume an underlying linear rela-

tion( y = a+ bx) between IVandDVfor instancebetweenTFHand

accident frequency and/or rate.

Strong evidence is beginning to emerge that some TFH rela-

tionsmaybe nonlinear. For instance, Bazargan andGuzhva (2007)

reported that the nonlinear logarithmic transform log(TFH) signif-

icantly predictedGA fatalities in a logistic regression model.

In his popular 2001 book, TheKilling Zone, Paul Craig presented

extensive evidence that GA pilot fatalities and accident frequen-

cies relate nonlinearly to TFH. Fatalities, for instance, occur most

frequently at middle ranges of TFH (50350), perhaps because

pilots are overconfident yet lack actual experience and skill with

Tel.: +1 405 954 6848.

E-mail address:[email protected]

rare, challenging events. Craig supported this hypothesis with his-

tograms of GA accident frequencies, although not with a formal

model. In group after group, a similar pattern emerged, one of a

skewed camel hump with a long tail at higher TFH.

Following Craig, I investigated a nonlinear model for accident

frequencycounts (Knecht, 2012). Derived fromNTSB andFAAdata,

Fig. 1 typifies thelookandfeelof TFHversusfatal accident counts

for a group of 831 U.S. instrument-rated (IR) GA pilots from 1983

to 2011.

This skewed hump is quite durable, persisting whether we

break out different data categories (e.g., IR vs. non-IR or serious

vs. fatal accidents) or even combine them. For example, Fig. 2

shows GA accident data for 832 pilots from2003 to 2007 for NTSB

SERIOUS+ FATAL accident categories that were combined to boost

frequency counts.

These data were successfully modeled by a gamma probabil-

ity density function (pdf) operating on log-transformed TFH. We

conclude that frequency count histograms support the notion of a

nonlinear killing zone somewhere in the lower range of TFH.

This is useful information. But, it now raises a new question:

How can webe certain that this relation is not an artifact,merely

a side-effect of the fact that there are just fewer pilots at higher

TFH, hence fewer accidents? Fig. 3 shows how frequency counts

for amatched sampleofnon-accidentpilotsalso drop offmarkedly

as TFH increase.

0001-4575/$ seefrontmatter.Published by Elsevier Ltd.

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W.R. Knecht/ Accident Analysis andPrevention 60 (2013) 5056 51

Fig. 1. Frequencyhistogram ofU.S. GA IR fatal accident counts, 19832011 (y-axis)

as a function of total flight hours (x-axis, binwidth= 100TFH).

Fig.2. Frequency histogramof GA IRSERIOUS+ FATAL accident counts as a function

ofTFH (bin width= 100TFH).

Figs 2 and 3 look suspiciously similar. In fact, the Pearson cor-

relation between them is .845 (p< .001). This implies that most

(r2 .71) of the killing zone seen in accident frequency counts

can beexplained simply by the fact that the numbers of pilots tend

to decrease as TFH increases.

With that inmind,whatsupportnow remains forthehypothesis

of an actual killing zone? And, if it exists, what form does it likely

Fig. 3. A histogramof GA IRnon-accident pilot countsfromthe same data cohortas

Fig. 2, showinghow numbers of pilotsdecrease rapidly as TFHincrease.

Fig. 4. Anoisyhistogramshowingcombined serious+ fatal accident ratesfor all

GA pilots as a function of TFH (binwidth= 100).

Data fromKnecht andSmith (2013).

take? We can address these questions by analyzing accident rate

histograms, in which the ith data bins rate is

ratei = accidentsi

accidentsi + nonaccidentsi(1)

Rateswill control forthe fact that ourfrequency countsdecrease

as TFH increases. If a killing zone continues to persist with rates,

thenwe have gained new insight intoGA flight risk.

1.1. The problem with rates

Unfortunately, rate data can be quite noisy, as Fig. 4 depicts.

Rate-noise results fromsamplingerrorrandomfluctuations in

rate from one bin to the next. It is a vexing problem at higher TFH

where Eq. (1)s denominator is small compared to its numerator,

and even small changes in frequency count can cause substantial

rate variations.It is nothard toappreciate theusefulness of amodeling function

here. A good model would smooth the noise in the data, allow-

ing investigators to better predict rates, given TFH. This would

be directly useful in areas such as determination of insurance

premiums,allocation of resources forpilottraining,accident inves-

tigation staffing, and public relations. To safety researchers, it

would be useful as an IV or statistical covariate, to either factor

in or out risk as a function of flight experience. This would have

broad application in numerous types of aviation research.

With this inmind, the present workmoves fromfittingGAacci-

dent frequency countdata to fitting accident rates as a function of

GA pilot TFH. The calculation of rates is a key step in risk analy-

sis because rates will control for the actual number of individuals

within a given data bin.

2. Method

2.1. The modeling function

Ideally, modeling functions should be motivated by theory

involving causal processes inherent to the data. Unfortunately, in

aviation accidents, causes tend to be numerous and hard to disen-

tangle, making theory-basedmodeling difficult.

We therefore proceed with a less ambitious goal of simply try-

ing to fit a well-behaved, continuous mathematical function to

empirical GA accident rate data. Standard techniques are used,

namely minimization of least-squares residuals between a model

and empirical data.

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52 W.R. Knecht/ Accident Analysis andPrevention60 (2013) 5056

Fig. 5. pdfwith various values of and .

The model itself begins with the versatile probability density

function pdf (Spanier and Oldham, 1987). The x-axis representsTFH. They-axiscanrepresenteither frequencycountor proportion.It contains a shape parameter, >0,and a scale parameter, >0.

pdf.c(x;,) =ex/ x1

() (2)

Thegammafunction() itselfis describedas theEuler integral,defined for >0

() =

0

t1etdt (3)

Shown in Fig. 5, pdf has the attractive feature of being able

to represent overdispersion (variance> mean) or underdispersion

(variance1 =100i50.Note that, in calculating a series of binwise accident rates, we

are not technically constructing a pdf. Rather, we are constructing

a rate histogram and using Eq. (4) (which is based on a pdf) to fit

that histogram.

Note also that we must openly address the issue of whether

these data, being a sample of convenience, may contain hidden

biases that could affect parameterization of Eq. (4). That is not

impossible. By definition, hiddenbiases areunknown andtypically

revealed only after more comprehensive data become available.

Until then, however, we argue there is substantial value in dis-

cussing serial nonlinearities, and layingoutamethodology for their

future use.

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2.3. Parameter estimation

The NonlinearModelFit function ofMathematica 7.0 (Wolfram,

2008) is used for parameter estimation. Specifically, since param-

eters b, , and of Eq. (4) are constrained to remain >0, theKarushKuhnTucker (KKT) method is used.

2.4. The problem of noise

NoisydatasuchasFig.4s posegreatdifficulty forstandardleast-

squares minimization. Parameters often fail to converge in rugged

errorscapes, oscillating around a saddlepoint, andmay converge to

a local minimum rather than a global one.

To confront the problem of noise, we first tried the simplest

approach of usingwider data bins. Surprisingly, this hadalmostno

effect. Accidents within these data tended to bunch up in several

adjacent bins, leaving noway to smooththemwithout resorting to

arbitrary bin widths of convenience.

We next tried dropping outliers greater than 2.53 s (standard

deviations) from thegroupsamplemeany. Thismethodwas finallyabandoned because it relied on an invalid assumption of distri-

butional normality, destroyeduseful information, andbecause the

extremely long tail of the fitting function resulted in an s so small

that very few of the very noisy values in the long tail were being

dropped anyway.

A thirdapproachwasnexttried, involvingdata-smoothingusing

moving averages.Unfortunately, this also hadtheeffectof destroy-

ing information by smearing.

Simulatedannealingwasthe fourthapproach(Kirkpatrick et al.,

1983)adding noise to each parameter start value before engag-

ing in error-gradient descent, then cooling the noise, allowing

parameter estimates to bounce around and out of local minima

until a global near-minimum could be found. Nonetheless, even

this approach would unfortunately still leave one major logical

problem:Allourdata binsincluding thenoisywould have equal

opportunity to influence the residual sum-of-squares during the

data-fit. In fact, any one bin, one vote approach would implic-

itly allowall bins equal influence,nomatter howmany individualseach binrepresentedor what that binsexpectedvariancewas. This

would confer inordinately large influence to sparsely populated

bins and those with inherently low reliability.

After weighing the above considerations, we finally settle on

the procedure ofweighting empty bins by 0 while weighting non-

empty bins accident rate ri by the inverse of its sample variance

(1/s2i). This is easily done in Mathematica by supplying a vector

Weightsw to NonlinearModelFit, wherew= {w1,w2, . . .,wi}.Since our data are rates are proportions, 0 ri1, the ith bin

weight wibecomes

wi =1

s2i

=ni 1

ri(1 ri)(1 ni/Ni) (5)

ni being the ith bins total frequency count (accidents+ non-accidents) used to derive ri, and Ni being the number of pilots in

the general populationwith that rangeof TFH.

Inthisparticularinstance,wedo notknowNi,soweshallassume

a constant sampling ratio ni/Ni, making the term 1 (ni/Ni) con-

stant for all bins. Since weights only need to be expressed relative

to each other during curve-fitting, we can eliminate this constant

term, leading to a functional weight of

wi =ni 1

ri(1 ri) (6)

Unfortunately, Eq. (6) is ill-behaved when ri =0 or 1, leading to

divisionbyzero. This canarisewhen eitherTFHbins aretoo narrow

orwhenTFH is large,making the bin sample sizenitoo small, since

few pilots have high TFH. Fig. 6 (top) illustrates the problem.

Fig. 6. (red, solid lines) With Eq. (6), as r0 or 1, 1/(r(1 r)); (top, dashed,

blue line) With =001, Eq. (7) gives a good approximation across most ofr, from

around .006< r< .994, but gracefully self-limits at r=0 and 1. (For interpretation of

the references tocolorin this figurelegend,the readeris referredto thewebversion

ofthe article.)

To address this issue, Eq. (6) is modified slightly by adding a

term designed to constrain behavior at 0 and 1:

wi =ni 1

(ri + )(1 ri + ) (7)

As Fig. 6 shows,when is small, Eq. (7) closelyapproximatesEq.(6) over most of its range but self-limits gracefully at ri =0 and 1.

2.5. Parameter start values and evolution

Inmultidimensionalparameterspaceswithunknownlocalmin-

ima, the choice of start values can be critical. A hybrid approach is

used here. Start values are first selected that roughly approximate

the shape of the binned data, namelyA=1, =6, =0.5, =b=0.Toallowparametersto evolveinamannerthatemulatesanneal-

ing, NonlinearModelFitis setup touse weighted binneddata and to

runmultiple times insidea For[i=1, >1.0, i= i+1] loop. Inside theFor[] loop, to emulate noise injected at each iteration, the value

of each parameterpjismultiplied by a randomreal numberdrawn

fromtherange1.0withstartingat0.02.Duringeach (ith)iter-

ation of the For[] loop,NonlinearModelFitis itself allowed to run for

i iterations, afterwhich i is increased by1while is decreasedby asmall fixed amount (.0002).thus emulates thecooling rate. Even-tually,fallsbelow1.0, halting theinjectionofparameternoiseand

terminating the For[] loop after one final gradient descent.

Running this method repeatedly on a given dataset, we can

broadly sample the parameter space. After inspection of the

graphic data-fit plots verifies a good fit, and multiple runs shows

parameters similar to at least three significant figures, we can be

confident that those parameter values represent a near-optimal

data-fit.

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Fig. 7. Annualized non-IR GA accident rates (y=predicted rate for x50 TFH,

medianTFH=250.5).

2.6. Model evaluation

Goodness-of-fit can bemeasured by the coefficient of determi-

nation, R2. The weighted form is usually taken as

R2

w = 1

SSw,error

SSw,total = 1ni=1wi(yi fi)2

ni=1

wi(yi yw)2 (8)

with weightedmeans, for instanceYw , oftheform

yw =

niwiyiniwi

(9)

Inour case, since Eq. (8) sometimesthrows values

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4.2. A point-estimate of relative GA flight risk

To estimate an annualized accident rate for a given value of

TFH=x50, simply populate Eq. (4)s parameters and insert x.2

To illustrate, for non-IR pilots at the median value of TFH =xNIR =250.5, Eq. (4) becomes

.00252081+ .0130177

e(ln(250.5).364805)/.0923902(ln(250.5) .364805)64.37621.092390264.3762

(64.3762)

Theresult is approximately.0068(1 in150)expectedaccidents

per 100 TFH,which we can see is correct fromFig. 7.

This point-estimatecanbeusedaseitheranIV orcovariatewhen

our data contain only a single value of TFH for each pilot. Keep in

mind that, in using such a relative estimate, we implicitly assume

that (a) all pilots fly exactly 100 TFH per year, and (b) their acci-

dent rate itself does not change during that time. Naturally, both

these assumptions are false for most pilots and can only be safely

ignored where large sample sizes dampen random fluctuations.

Thismay seem like a grave restriction.But, theexact same assump-

tionsusuallyunderliethelinearmodel,and aremerely rarely stated

so explicitly.

4.3. A definite-product estimate of annual GA flight risk

In thefortunatecircumstancewhere ourdata contain twoaccu-

rately time-stamped values of TFH for each pilot,3 we can better

estimate each pilots absolute predicted accident rate based on

the general form:

risk = 1

TFH2x=TFH1

(1hourly) (11)

where risk estimates the chance of having an accident over thetimespan TFH

2TFH

1. We calculate the definite product (),

which simply subtracts from 1.0 the multiplied individual esti-

mated hourly probabilities ofnothaving an accident (1hourly),with hourlydefined as

hourly =1

100

b+A

e(ln(x))/(ln(x) )1

()

(12)

Here, the constant 100normalizes for each of our accident-rate

bins having been 100TFHwide when parameterized.

riskavoidsamajor deficiencyof thepoint-estimatebypredict-ing cumulative accident probability over any desired number of

TFH rather than assuming risk over an arbitrary span of 100 TFH.

To illustrate, suppose that an IR pilot reports having flown 310

TFH on6 September,2012, and 433 TFH on29August, 2013(a span

of 357 days, or .978 years). Eqs. (11) and (12) become

risk = 1

433x=310

1

1

100

.00112073+ .0173821

e(ln(x)2.24346)/.0890114(ln(x) 2.24346)51.34721.089011451.3472

(51.3472)

= 1

433x=310

.9999887927

.0001132895

(ln(x) 2.24346)50.3472

x11.23451603

2 (64.3762)9.460741087. (51.3472)1.188441065. Those using SPSS

may find only the natural log of, so re-transformas eln(()) .3 Alldatashould befilteredto ensurethatt2> t1and TFH2> TFH1 . Experiencewith

FAA data provesthat this is usually, butnot always, thecase.

Here, risk represents a cumulative .0067 chance of having atleast one accident over the TFH range of 310433, with data accu-

mulated over .978 years. This can be used as-is for some purposes.

For use as an IV or covariate, it can be annualized by multiplying

.0067 (1/.978)= .0069.

5. Discussion

Can total flight hours predict general aviation accident rates? If

so, what does that relation look like? Is there a killing zonea

range of TFH over which GA pilots are at greatest risk? These

questions are of great interest to pilots, aviation policy makers,

researchers, and insurance underwriters alike.

Using accident frequency counts, Craig (2001) proposed that

such a zone spans a range of approximately 50 to 350 total flight

hours. The current work revisits the question, using accident rates

to incorporate the essential additional information of numbers of

pilots at each level of TFH. Rates, of course, introduce their own

difficulties, andourmethodology attempts to address these.

If such a killing zone truly exists, what does it look like? Many

aviation research studies implicitly assume that accident rates are

linearly related to TFH. In fact, that relation appearsmarkedly non-

linear, and may even be a function of serial nonlinearitiesforexample, a log transform of TFH followed by a gamma probabil-

ity density function-like transform incorporating a base rate (see

Eq. (4)).

The present work uses such a serial nonlinear function rateto predict GA accident rates from TFH, given extremely noisy

real-world data. Two sets of 20032007 NTSB/FAA data pro-

duce weighted goodness-of-fit (r2w) estimates of .654 and .775 for

non-instrument-rated and instrument-rated pilots, respectively,

considered moderately good by convention.

Shownpreviously, Figs. 7 and 8 illustrate therawrate data with

the modeling function rate =y=f(x) superimposed. Each model

y-value represents the estimated probability of having a serious-

to-fatal accident for a pilot flying 100 TFH (x50) over the course

of oneyear. Theyellowregionsurrounding themodel curve repre-sents the .95 confidence interval.

Consistent with our intuition and previous frequency count

studies, these models suggest that a killing zone may indeed

exist. Accident rates may increase for GA pilots early in their

post-certification careers, reach a peak, decline with greater flight

experience, then asymptote to some baseline level.

However, thekilling zonemaybe farbroader than earlier imag-

ined. Relatively high risk for an individual pilot may extend well

beyond the 2000-hmark before leveling off to a baseline rate.

For now, we should consider these conclusions tentative. First,

the 67,687 pilots in these datasets constitute only slightly more

than 10% of those currently licensed in the U.S. While this might

represent an excellent dataset size in other venues, the relatively

tiny number of pilots at high TFH leads to considerable noise in

high-TFH rates. Data-weighting can overcome part of the problem

but leads to wide confidence intervals at high TFH, reflecting the

lower statistical reliability of high-TFH data.

The net effect is that, at this point in time, we have high con-

fidence in our data only at relatively low values of TFH. Better

results await better cross-communication between the FAA and

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NTSB databases, which will result inmuch larger sample sizes and

much greater reliability across the full spectrum of TFH.

Weendwithanappeal tothe FAA andNTSB. The dataneeded to

compute accident rates such as these come frommultiple sources

that are hard to access in the U.S. These include the FAA CAIS and

DIWS databases, as well as the NTSB Aviation Accident Database.

These databases do not intercommunicate well, and could be

greatly improved by better record-keeping, andmost particularly

by sharing a common pilot identifier code. The NTSB has previ-

ously made similar recommendations (NTSB, 2005) that remain

outstanding.

This studyhas exploredamethodologythatwill hopefullyprove

useful, both in its own right and as a stimulus to further research.

Many researchers have olddata that canbe easily reanalyzed. And,

the sooner we challenge our nave assumptions that all indepen-

dentvariablesrelate linearly to theirdependentvariables, themore

accurate anduseful aviation risk research will become.

Acknowledgements

Deep appreciation is extended to Joe Mooney, FAA (AVP-210),

for his help reviewing the NTSB database queries that provided

the accident data used here, to David Nelms, FAA (AAM-300), for

providing TFH from the DIWS database, and to Dana Broach, FAA

(AAM-500) forvaluable commentary on this seriesofmanuscripts.

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