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Benjamin Peirce and the Howland WillAuthor(s): Paul Meier and Sandy ZabellSource: Journal of the American Statistical Association, Vol. 75, No. 371 (Sep., 1980), pp. 497-506

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Benamin

Peirce

nd

the

HowlandWil

PAUL

MEIER nd

SANDY ZABELL*

The Howlandwill ase is possibly he earliest nstance nAmerican

law of the use of probabilisticnd statistical vidence. dentifying

30 downstrokes

n

thesignaturefSylviaAnnHowland,

Benjamin

Peirce ttempted o showthat a contested ignature n a will had

been

tracedfrom nother nd genuine ignature. e argued hat

their greement

n

all 30 downstrokes as improbablen the ex-

treme under a binomialmodel. Peirce supportedhis model by

providing graphical est of goodness f fit.We give a critique

of Peirce's model and discussthe use and abuse of the product

rule formultiplyingrobabilitiesf ndependent

vents.

KEY

WORDS: Charles S. Peirce; 19th-century athematical

statistics; aw and statistics; andwritingnalysis; ndependence;

Product ule.

1. INTRODUCTION

On

July , 1865, Sylvia Ann Howland of New Bedford,

Massachusetts, died, leaving an estate of $2,025,000,

and

a single heir-her niece Hetty H. Robinson,

who

had lived with her for ome years. Two milliondollars s

a

respectable sum, even by today's inflated tandards,

and

in

1865 it was certainly prize worth ontesting.

The Howland will, dated September 1, 1863,

and a

codicil of November 18, 1864, lefthalf the estate to

a

number of individuals and institutions. t provided

that the residuewas to be held

in

trustfor he benefit

f

Hetty Robinson and, at the time

of

Hetty's death, to

be

distributed o

the

lineal descendents f Hetty's paternal

grandfather,Gideon Howland, Sr. Hetty's father had

died

a month arlier, eaving her $910,000 outright

nd

the income from

trust of $5,000,000. AlthoughHetty

Robinson was thus already a wealthy woman,

she

claimed

a

right to inheritoutright

he

entire

estate

of

her

aunt, providing

s

evidence forthis claim an earlier

will,dated January11, 1862,which eft he entire state

to

Hetty with nstructions hat

no

later

will was

to

be

honored.

Family feuds are relevanthere,

as is

a

claimed

agree-

ment

between Hetty

and her

aunt to leave mutual

wills

excludingHetty's

father.

For

our

purposes,

t

is

enough

to observe that Hetty Robinsonhad an arguable claim

*

Paul Meier

s

ProfessorfStatistics, he University

f

Chicago,

Chicago,

L

60637,

and

Sandy

Zabell

is

Associate

Professor f

Mathematics,

Northwestern

niversity, vanston,

IL

60201.

Support

for this

research

was

provided

n

part by

NSF Grant

SOC

76-80389.

The

authors re grateful

o

a

number

f friends

and

colleagues

with

whom

hey

discussed

spects

of

this

case-

P.

Diaconis, S. Fienberg, . Langbein,

P.

McCullagh,

J.

Tukey,

and H.

Zeisel, among

others.

Special

thanks re due

to David

Wallace

for

xtensive iscussion

oncerning

he inference

roblem

that

arose, nd Stephen tigler orhelpful orrespondence

n

the

historical ackground.he staff

f

the

New Bedford

ublic

Library

were

especially ccommodating

n

locating

nd

arranging

orthe

copying

f

unique source documents nd exhibits,

nd

to

them

the authors

xpress heir pecialgratitude.

that the earlierwill should be recognized.But this the

Executor,

Thomas Mandell, declined to do. Not only

did he claim

that the later will governed,but he

also

contended

hat two of the three ignatures n the

earlier

will were tracings from the

third, thus providingan

independent ground for

invalidating that will (see

Appendix 1).

Hetty Robinson

retained

distinguished ounsel and

sued.'

Thomas Mandel retained equally distinguished

counsel and defended.

Between them, both sides called

as

witnesses ome

of

the most

prominent cademicians

of

the day. Oliver

Wendell

Holmes, Sr.,

Parkman

Pro-

fessor fAnatomy nd Physiologyn theMedical School

of

Harvard

University,

xamined

the

contested signa-

tures

under

a

microscope,

s did Louis

Agassiz, and

testified for Robinson) that

he could find no

evidence

of

pencil

marks

uch

as

mighthave

appeared

in

tracing.

(The

cross-examination f ProfessorHolmes was

both

brief nd

painful;

ee

Appendix

2.) Among

those

testify-

ing for he Executor were

Benjamin Peirce, Professor f

Mathematics at Harvard, and his

son,

Charles

Sanders

Peirce, who

was later to do importantwork in several

branches

of

philosophy.

The Peirces were

among the

first o contribute o the development f mathematical

statistics

n

the

United States;

a

recent urvey of their

work n thisarea is given n Stigler 1978, pp.

244-251).

Professor Peirce

undertook

to

demonstrate by

sta-

tistical

means

that the disputed ignatures n the second

page of the

will

were indeed forgeries.

is

method

was

to

contrast he similarities etween

one of the

disputed

signatures

referred

o

in

the trial record

as

signature

#10) and

the unquestioned original (referredto

as

signature# 1) with the lesser

degree

of

similarity

o be

found

n

pairing

42

other

signaturespenned by Sylvia

Ann

Howland

in

her

later years

on

other

documents.

(See Figures

A

and

B.)

It

is

this analysis

with

which

we shall be concerned.

The Peirceanalysis, s weshallsee, sat once ngenious

and in

some

respects naive. It

leads

to

a

distributional

problem, discussed

in

Appendix

4,

which is

nontrivial

and has some

nterest

n

its

own

right.

he data

available

1December, 865. Testimonywas taken n 1867, decision

ren-

dered October 1868 (Robinson . Mandell, 20 Fed.

Cas. 1027).

A copyofthe trial ranscripts available n theNew Bedford ublic

Library nd the testimonyn the text and Appendixes and 3

is

extracted rom hat source.

? Journal

f the

American

Statistical

Association

September

1980,Volume

75, Number

371

Applications

Section

497

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7/25/2019 2287637

3/11

498

Journal

f the American

tatistical

ssociation,

eptember

980

A.

The

Unquestioned

riginal

ignature

no.

1)

and

the

Two

Disputed

Signatures

nos.

10

and

15)

in

the record

is

sufficient o allow

some

interesting

analyses,

but

tantalizing

or

ts failure o include

details

of

major

relevance for the

problem

at issue.

However,

as weproceed, tis welltokeep nmind hat no organized

theory

of statistical nference

xisted at the

time,

and

that,

for

example,

casual

imputations

of

independence

weremade almost

utomatically y

those

eeking

o

bring

considerationsof

probability

to bear on

problems

of

human

affairs.

We

should

not

be

surprised,

herefore,

o

find

uch

mputations

s a

matter

f

course

n

the

present

problem.

2.

THE

PROBLEM

Let

us

start

with

the

evidence

as

given,

making

assumptions s

needed, nd returnater to judge which

seem

plausible and

which

not. The

evidence

consisted

of

photographic

opies

of 42

signatures,

n

addition

to

the

original

#1)

and one

of the

purported

opies

(#

10).

Suppose

we

are

given

some

index of

agreement be-

tween

signatures. n the

present

case

Professor

Peirce

chose the

numberof

coincidences-in

length and

posi-

tion-among

the

30

downstrokes

n each

of

a

pair

of

signatures,

when

the two

were

uperimposed.We

might

then et

Xi,

.. .,

X42

be the

values of

that

ndex

obtained

in

matching

ach

of

the

42

uncontested

nd

uninvolved

signatures

with

#1,

and

let

Y

be

the

value

of

that

index

in

matching#10 with

#

1.

We

could

then

pose the

prob-

lem as one of udgingwhetherY could reasonably have

been drawn

at random from

he

same

population

that

gave

rise

to

the

sample

of

size 42.

Since,

n

this

case,

Y

=

30,

that

s, #

1

and

#10

coin-

cide

in

all

30

downstrokes,

nd

the

Xi

are

presumably

B. Some of

the

42

Comparison ignatures

J

,

49

od- Zn9Sy0,f Z.o{e

4

O

t.^:Fg.v..*~~~~4-

,-.fA

w

,72

?o

At KX

'

r4

tjtt..

o.

4s

r

'7

4.

4/

7/

i14-L

p0r.zzo.

jy?

2/6to

. /is

7$6./CA

r%,r

? *; ;- ie1*

;

f

,

4

0

0

.,,,ta'47'

/

LW4zEtv;

A1

~ ~ ~ ~ ~ ~ ~ ~ ~

A

.

/K.

*rt

7/25/2019 2287637

4/11

Meier

nd Zabell:

Benjamin

eirce

nd

the Howland

Will

499

all smaller

han 30,

we might ssign

a significance

rob-

ability

of

1/43

=

2.3%

to

this outcome.

This

is

a small

enoughprobability o

raise suspicion,

but far

from ver-

whelming.2

If we had

available the individual

Xi

values

(which

we do not)

we might

e willing o

proceed little

more parametrically.

We

can determine see

Table 1)

that

the

average

index in

random pairings

of the 42

signatures s near 6 with a standarddeviationof about

2.5.

Thus

Y is nearly 10 sample

standard

deviations

away from

X

and,

if

we believe

that the

population

sampled

is

at least approximately

normal, the

dis-

crepancy

would

be

udged

significantt

an extreme evel.)

Professor eirce

proceeded

n a different

ay and came

to

the conclusion hat

the odds against

such

a coincidence

as that

between ignatures

#

1

and #

10

were stronomical

(of

order

1020). To see how he

arrived at this

striking

result,

et

us

follow

the

order of

testimony

n

the

case.

2.1 Testimony

f Charles

anders

eirce

The son, Charles Sanders Peirce-then a memberof

the

staffof the

United States Coast

Survey-testified

on

the work he carried out under

the directionof

his

father.

Charles

was provided with

photographic opies

of

signatures

#1

and

#10 (the

third

signature

s

not

mentioned

n

the testimony)

and with copies of 42

signatures

of

Sylvia

Ann Howland on leases

and

other

documents.Following

nstructions

iven

to

him

by

his

father,

Charles undertook

o

match every possible

pair

of

the

42

uncontested

signatures

and

to

observe

the

agreement

etween

ach

pair

n

respect

o

the

number ut

of

30

identified downstrokes

n which

they

were

in

agreement.There being ( ) = 861 possible pairs, and

30 downstrokes

o

be checked for each,

this meant

a

total of

25,830 comparisons,

or

which Charles

reported

agreement

in

5,325

or

20.6

percent

of them-very

nearly

1

in 5. More particularly,

e

gave

the

distribu-

tion

of number

f

agreements

s

shown n

Table

1.

Thus,

for

xample,

15

pairs

each had twocoincidences ut

of

30,

for total

of

2

X

15

=

30 strokes

oinciding.

The

precise

details

of the

matching rocess

nd of

the

criteria for

judgment

of

agreement

were

only briefly

touched

on

in

C.S.

Peirce's

testimony:

Q.

Whatwas

the standard

..

of coincidence?

A. I

considered

hat

two

ines

coincidedwhenever

hey

coin-

cided as well as .

.

. nos. 1 and 10, in those same lines.

Later,

under

cross-examination,

harles

added

some

further etails:

I

made

it

a

condition that

in

shifting

one

photograph

over

another

n

order

o

make as

many

ines s

possible

oincide,-

2

One

might ry

to

improve

n this a

little

by

considering

he

probability

hat

out of 44 signatures, given

pair

will have

the

greatest

ndex

of coincidences,

hat is,

(4Y4)1

=

.0011. (Strictly

speaking,

we

are

not

given

the

number

f coincidences

etween

either

ignature

#1

or

#10

on the

one

hand,

and

any

of the 42

other ignatures

n the other, ut

presumably one

of these

even

approaches

he 30

of #1 and #10.) The appropriateness

f

such

a significanceest is an interestingssue, but one that we will

not touch

on.

1. C. S. Peirce's Table of Coincidences

Coincidences Number

of Pairs Number of

Strokes

?l0

0

2

15

30

3

97 291

4

131

524

5 147 735

6 143 858

7 99

693

8 88

704

9

55

495

10

34

340

11

17 187

12 15 180

13

20 288

30_

861 5325

that the line of writinghould not be materially hanged.

By materially

hanged, mean so much

changed hat there

couldbe

no

question hattherewas a difference

n the general

direction f he two

ignatures.

2.2 Testimony

f Benjamin eirce

C.S.

Peirce

was

followed o the stand

by his father,

who testified s to his own

examination nd comparison

of

#

1

and

#10.

The coincidences extraordinary

nd of such kind s

ir-

resistably

o suggest esign,nd especially

hetracingf

10

over

1.

There re smalldifferences

. . butthe differences

re

such

as

to

strengthen he argument fordesign.

.

.

.

The

co-

incidences

re manifestly

hose of position; while the

dif-

ferences re thoseof form.Coincidence f position s easily

effected y design, nd can be tolerably

well accomplished

by

an unskilful

and.

t

especiallybelongs

o

the

downward

stroke.

. .

. It involves a

close agreement n the average posi-

tion of the

stroke,

n

agreement

n its slant,

nd no extrava-

gant

diversity

f

curvature.

ut

coincidence

f form s ex-

ceedingly

ifficult....

In

a

long ignature. omplete oincidence

fposition

s

an

infallible

vidence

f

design.

The mathematical

iscussion

of

this

subject has never, o my knowledge,

een proposed,

but it is

not

difficult;

nd a numerical xpression pplicable

to

this

problem, he correctness

f which

would

be instantly

recognized

y all the mathematiciansf

the world, an be

readily btained.

ProfessorPeirce

went on to describe

his

son's

work

and, without ualification, sserted hatsince theoverall

proportion

of matches was very

nearly one-fifth, he

probability

of finding

30 matches

in

a

given pair

of

signatures

was

one

in 530

or

once

in

2,666

millions

of

millionsof

millions.

(Note:

There

is some error

here,

since 530

=

9.31

X 1020.)

Lest the

point

be

missed,

Peirce

added:

This

number

ar

transcends uman

experience.

o

vast an

improbability

s

practically

n

impossibility.

uch evanescent

shadows

f

probability

annot

belong

o actual ife.

They

are

unimaginablyess

han hose east

hings

hich he

aw cares

notfor.

When asked

the purpose of his son's display of

the

distribution f the number of coincidences,Peirce ex-


7/25/2019 2287637

5/11

500 Journal

f

the Americantatistical

ssociation,eptember980

2. BenjaminPeirce's Table (and

chi-squared omputations)

Number of

Agreements Obs. Exp. (O

-

E)

(O

-

E)2IE

? l?0 9.1 -9.1 9.1

2 15 29.0 -14.0 6.7

3

97

67.6

29.4

12.8

4 131 114.1 16.9 2.5

5

147

148.3 -1.3 0.0

6 143 154.5 -11.5 0.9

7

99 132.4 -33.4 8.4

8 88 95.2 -7.2 0.5

9

55

58.2 -3.2 0.2

10 34 30.5 3.5 0.4

11 17 13.9 3.1 0.7

12

15 5.5 9.5 16.4

13

20 2.7 17.3

112.1

30J

861 861 0.0 170.7

plained that

It was orderedby

me

in

order

to

obtain

as severe

a

test

as

possible

from

bservation

or

the

criticism f

my

mathe-

maticaldeductions.

To facilitate comparison,

Peirce

provided

a

table,

recapitulatinghis son's

counts,

together

with the

ex-

pected number f counts

under

the

binomial model

/30\ 1

i4 30-j

El

Yj}

=

861

(

3)(1)()

(See Table 2. The table given by Peirce omitsthe 15

cases with two or fewer

coincidences,

nd

the

20 with

more

than

12.)3

Peirce

commented:

This is

a species of comparison amiliar

o mathematicians

and such

a

coincidence

s above

would

be

regarded

s

quite

extraordinary,

nd a

sufficient

emonstrationhat the true

law

of

the case was embodied

n

the

formula.

A

graphical display

of

the comparison

was

also

pro-

vided,

which

s

reconstructed

n

Figure

C

according

to

the

descriptiongiven

in the trial record

(the

actual

display

was

omittedfrom he trial record).

3.

CRITIQUE

OF PEIRCE'SMODEL

3.1 Goodness

f Fit

How

convincing

s

Peirce's

model

and his

evidence

for it? No

formal

theory

of

goodness-of-fitesting

ex-

istedat the time,and Peirce can scarcelybe faultedfor

failing

to

provide

one.

Still,

the quality

of fit

can

rea-

sonably be questioned. Peirce's graph

shows

only

the

middle

range,

and

even

here there

appears

to

be

a

con-

sistent

rend n the

deviations.What the graph

does not

show

s the

substantial bsence of really poormatches-

a The table n the trialrecord ives41 cases as undistributed,

but it is clear from eirce's estimonymmediatelyollowinghat

35 was meant. My son had also 35 undistributedases

...

no cases of 0 or 1

agreement,

with

9 expected-and

the

considerable excess of outstandingly ood

matches-20

(with 13 or more

greements),

with ess than 3 expected.

If

we calculate the

usual

goodness-of-fithi-squared

for this

frequency

able we

get X2

=

170.7 on 12

df.

We can

improve

this a little

by choosingp

to

minimize

chi-squared,

but the minimizing p

=

.211

actually

worsens things n the middle range of the graph and

gives

a xI

=

135.3,

which

remains n

exceedingly oor

fit. (The appropriateness

f the

chi-squared

tatistic s

discussed

n

Sec.

3.2.)

If one

compares

Peirce's

graphical

test of fit with

the

chi-squared

statistic,

t

is

obvious

that

the visual

impact

of the

graphical display

is to

weight

ll

deviations

from he

expected values

equally,

while

chi-squared weights

such

deviations relative to

the

magnitude

of each

expected

value.

For

this

reason,

discrepancies

etween the

observed

and

expected

values

in

the

tails,

even

if

they

had been

plotted,

might not

have appeared

to be more

serious than

those in

the

middleofthe distribution, hile t is precisely he con-

tribution

of the

tails

to

chi-squared

that

makes the

latter ignificant.

Clearly

the most

frustratingspect

of

this

data set is

its

lack

of

detail with

regard to

the 20

good matches.

Were even

one of them to show

30

agreements,

r

even

29,

the persuasiveness

of

the mathematical

evidence

would evaporate.

Indeed,

as

shown

n

the earlier

table,

we

can

calculate

from

the data

given

that

the total

number

of agreements

n

these 20 cases was

288,

or

an

average of

14.4

for

those

signatures

with 13 or more

agreements.

hus,

although

one

or two

very highvalues

remains numerical

possibility,

t seems

likely

that few

ifany values were as high s 20.

Unfortunately

for

the

record,

Professor Peirce's

demeanor

nd

reputation

s

a

mathematician

must have

been

sufficiently

ntimidating

to

deter

any

serious

mathematical

ebuttal.He

was made

to confess lack

of

any general xpertise

n

udginghandwriting,

ut

he was

C.

Distribution

f Coincidences

FREQUE14CY

200

---

OBSERVED

FREQUENCY

EXPECTED

FREQUENCY

150-

00- S 0 1 0 5

I~~~

~NME OF GREMNT


7/25/2019 2287637

6/11

Meier nd

Zabell:

Benjamin eirce

nd

the Howland

Will

501

not cross-examined

t all on the

numerical nd mathe-

matical

parts of

his

testimony,

et alone made to

list the

20 high values

(see Appendix3).

3.2

Model Assumptions

The frequent aralysis

of the law before demonstra-

tion of mathematicshas long been noted. An extreme

view,

expressedby

Laurence

Tribe

in

his 1971 Harvard

Law

Review rticle Trial by Mathematics

(Tribe

1971)

is that laymen

are too readily mpressed nd misled

by

such

presentations

nd that they should

therefore e

excluded-at

least in jury trials.

Whatever one

thinks

of this

position

as a

generality, eirce's

model, as

well

as his data analysis,

deserves closer

examination han it

receivedduring

he trial.

One might,

for example, question the

assumption-

required

for the binomial model-that

the probability

of match houldbe the

same at each of

the 30 positions.

One can easily imagine that certain strokes-perhaps

those early

in the signature-would

have a

higher

probability

f matching han do

others.The assumption

of

equal probability

could readily

have been

checked

in the course

of C.S. Peirce's work,but

it seems never

to

have

been questioned. (Of course,

the effect f such

inequalitywould

be to reduce he

variance n the

number

of matches, and thus to

make the probability

of 30

matches ven smaller

han that quoted.)

Far

more serious

is

the implicit

ssumption of inde-

pendence.

One would expect

that agreement

n, say,

positions

1

and 3, would

make

far

more ikely

n

agree-

ment

n position

2.

The effect

f such dependence

would

be to increase he variance and thus increase the prob-

ability of 30 matches

over that quoted,

quite possibly

by

orders of

magnitude.

To the extent

that the

data

bears on this

point,

t does

indeed

suggest

he

possibility

of dependence by exhibiting

what appear to

be

far

too

many cases with over

12

matches

to agree

with the

binomial model.

A

final

critique

of Peirce's

discussion

s

the absence

of

any comment

on correlation

between signatures

made

close

in

time. It

is

at

least plausible

that signatures

ate

in

life

would differ ubstantially

from hose

made at

a

younger

ge,

and also

plausible

that

signatures

made at

a

single sittingwould be more similar than those made

days

or weeks

apart. (Similar

problems

arise

in

the

statistical

analysis of

authorship,

where both style

and

vocabulary

may depend

on the writer's

ubject

matter

and

may vary over

his

ifetime;

ee Mosteller

nd

Wallace

1964,

pp. 18-22, 195-199,

265-266.)

Since many

of

the

42

signatures

sed

in

the study

were on leases

and other

dated

documents,

this

question

could

also

have

been

examined,

ut there

s no

evidence

hat t

was considered.

In

part

these challenges o

the fairness f

Peirce's

test

can

be

met by the nonparametric

rocedure

uggested

earlier.The questions

of uniform robability f

matching

in each of the 30 positions, nd

independencebetween

them,

would then be

irrelevrant,.

The

problem of time

dependence,on the other

hand, remains n full

force.)

Thattest,

however, s noted

before, ieldsonly

strongly

suggestive

significance

robability,but nothing

at

all

as

compelling s Peirce's

calculation.

For all that one

can raise questions about

Peirce's

model

and

analysis, t remains

o consider he extent o

which

the evidence available

really contradicts

t. Al-

thoughthe conventional hi-squared est clearlyrejects

the binomial

model, t

is

not

at

all

clear that the chi-

squared

statistic should have

even

approximately he

familiar

distribution.Under

Peirce' model the

prob-

ability that a random

pair of true

signatures should

match in

k

stokes is indeed binomial,

as

claimed,

and

thus

the

expected fraction f

pairs with

k

agreements s

also binomial.

However, the (42)

=

861 pairs are

gener-

ated from nly 42

signatures nd are not

at all the same

as

861 independent rials

from binomial

population. t

is to be

expected, herefore,hat

the relative

frequencies

will

show variability

greater han

multinomial nd that

the chi-squared statistic will also exhibit greater vari-

ability. To

assess more

rigorously he possible

lack of

fit

of

Peirce's

model

we

turn

to

considerationof the

rather

arge excess of

pairs with over

12

matches (20

observed,

only 2.7 expected). The

Markov

inequality

(e.g.,

see

Billingsley1979, p. 65), of

which

the familiar

Chebyshev inequality

is

a direct

corollary, tates

that

for

ny nonnegative andom

variableZ,

one has

Pr{Z

>

t} < t-'E{Z}

.

Taking

Z

=

the number

f pairs

out

of the

861

showing

13

or

more

agreements,

nd

t

=

20, we

find

PrIZ

>

t}

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