XVII]

(3)

Houses

in course of^{erection in}

Urban and Rural

Districts.

Introduction xxxv

(4) Imbecility

and

Deaf-Mutism.

xxxvi Tables for Statisticians and Biometricians [XVII — _XX

and

therefore

log

P ⁼ _1739602,

and P ⁼

_3-179/10™.

In thefirst

and

third casesadifferenttreatment

must

be used.

For ^ ^{= 14393}

we

use Table XII.

We

have forn

=

4^:

P ⁼

^-801

253 +

-4393 [-

228846] -

$ (-4393) (-5607)

[+

48064]

=

-6948.

Had we worked

from Table

XVII _{by Formula}

(i),

we

should have had

P ⁼

^-6950.

For x

²

=

_-7080,

_we

can use Table XII,

remembering

^{that for}

^

²

=

_0,

P ⁼

^l.

We have

P ⁼

1-000,000

+

-708

[-

198,747]

-

} ('708) (-292)[-30,099]

=

-8624.

Had we worked

from Table

XVII _{by Formula}

_(i),

^we

should have had

P ⁼

*865, close

enough

^for practical purposes.

The

true value of

P worked

from

IV27Tiv

•~ W * ⁺ _?S?

^#

_~***)

by

using Table ^IIis

P ⁼

"8713.

See

_p. xxxviii.

Examining

^{the values of}

P we

see that having regard^to the errors of

random sampling we

can only say ^that there is

no

relation

between

^rural

and

urban districts

and

houses building ^{or built; there is} clearly no ^'distinct association,' forin

69

out of

100

cases in

sampling

from independent ^material

we

should _get

more

_highly associated results.

There

is likewise no association on the given materialinthe

Datura

characters.

The

other three cases have _clearly very

marked

association, quite independent ^of

any

influence of

random

_sampling. If

we

_regard

these three tables the order of ascending association

judged by

^either<f> or

C

2is (4), (5), (2), as against

Mr

Yule's(2), (4), (5). If

we

_disregard the non-significance

and

take merely intensity of association,without regard ^to

random

sampling, the orderis (3), (4), (1), (5), (2), as against

Mr

Yule's order(1), (2), (3), (4), (5).

The

best

method

of inquiry at present ^{for relative} association in the case of four-fold tables _is, I hold, ^first to investigate

P and throw

out as not associated those cases like the ^'Houses, ^built

and

_building^' above.

Then

to use either

"

tetrachoric r_t^" or

C

2 according ^as

we

are _justified in considering ^{the variates} as continuous or not. r_{P (see p. xxxvii)}

may

^{be used as control.}

Tables XVIII— XX

_(pp.

_31—32)

Tables

for

determining ^the Equiprobable Tetrachoric Correlationr_P. (Pearson

and

Bell:

On

a

Novel Method

ofregarding^the Association oftwo Variates classed

XVIII— _XX] Introduction xxxvu

solely in Alternate Categories. Draper's'

Company

Research Memoirs, Biometric Series,

vm. Dulau &

_Co.)

We have

seen

under

the discussion of theprevious Table

how

to find a

measure

of the improbability^of

two

variates being independent,

when

_theyare classed in alternate categories.

The

_difficulty in such cases is to appreciate the relative importance ^ofvery large inverse powers ^of10.

The

object of thepresent tables ^is to enable us to deduce a tetrachoric correlation, r

t, of which the improbability

is the

same

as that of the given system supposing ^{it to arise,}

when

the

two

variates have the

same

_marginal frequencies but are _reallyindependent. In order to do this

we

have to determine <rr for the given marginal frequencies, ^i.e. the standard deviation ofrt on theassumption ^{that ris}really zero. This

may

^be easily found from

Abac _Diagram XXI

or from Table

XXIV

_{(see below).} Table

XVIII

then _{gives us} the value of

(—

log

P)

^{foreach value of rtand a>. If}

we now

turn to our _original table

and

calculate its rf, this as

we

have seen will correspond to a given

(—

log P).

We now make

the

(—

log

P)

^{from our}

^

²correspond ^to the

(— logP)

^{from ourrt}

and

„a,.,this givesus a value of rt

which

has the

same

_degree of improbability ^as our observed table. In other _words, instead of trying ^to appreciate the

meaning

^{of inverse} high powers ^of 10,

we

say that a table of the

same

marginal frequency

would

be as improbable ^ifit

had

a tetrachoric correlation r_t arising from

random

_sampling of independent ^variates.

Thus we

read our improbability on a scale of tetrachoric correlation.

We

useourcorrelation merely as a scale to

measure

probability ^on.

As _log%

² _{provides a}

more

satisfactory ^{basis for} interpolation,

and

as

many

readers use logarithm ^tables

and

not calculators,

log^

^{2will be the form in}

which

X

^{1will be} often presented. Table

XX

provides the value of r_t corresponding ^to given

a

r

and

_{given log}

%

^2.

We

^will

assume

forthe present that „cr,.can be _readilyfound from the marginal totals: see _p. xli below.

Illustration. Obtain the values of r_P for the five tables given above on pp. xxxiv

—

v.

The

valuesof

log^

²

and

cr_rare as follows:

xxxviii Tables for Statisticians and Biometriciaus [XVIII — _XX

regard ^to the spacings ^{of the} correlation curves, the value of the equiprobable correlation is under "0:3, say ^'027. In other words no significant association can be asserted.

In the case of _Oov

=

1941

we

are

thrown

back on the _original formulae*. In the first

place

we must

find

P

^for _{the given} ^value of

^

^{2, i.e. -}

⁷⁰⁸⁰

^(see p. xxxv).

But

for

w'=

4 from formula _(xxix),

=

2 (-200,0578

+

-280,0088x "84142}

=

_-871,3256.

To

obtain r

we have

to use the formula _below,

where

O

ov=

"1941,

and

m—

^A (

•

—

8), ^the /*„, fit, /is being ^the

normal moment

functions of Table IX.

W2irJ;

e X

P ⁼

-^5=

[j

Mo

(V2«0 - &

^(V2//H-)}

^- _lj ^ ^{(V2m) -}

^to

^{(^2mr)j}

-

sss-

^!/"c

(V ^"

_}

^" ^

(V2Vir)1

+ m&

^!ms

^(V2

^~'7t)

^{~ *}

⁽

^{vi ™}

^r>5j (xxxii).

Substituting ^{the valuesof na,.}

=

-1941

and

V2wt

=

_4-852,107,

_we

have for r

=03, P=

-90550,

r

=

_-04,

P=

-86501.

Whence

for

P ⁼

-87133,

we have

r

=

-038.

We now

turn to the three cases which fall inside Table

XX.

(2) Eye-colour, Father

and

Son.

log

x

²

=

2-1249 „<7,

=

_-0514,

r

=

_0-5

log x"

-

2-0942

n0-,

=

0o

^

=

of)

j^ ^ ⁼

²

²⁷⁴⁸

r-06

_log

_x

²

⁼

^2-1239

^,-•06

5=0

^{.7 i}

_og%2=

^2-2935.

Linear differences will suffice

„<r,

=

-05 r

=

_0-5

+.^[1] ⁼

^0-517,

o£r)^.

=

^.OG r

=

0-G+;^[-l] ^{= 0G01.}

Hence

Oo-r

=

_{-0514 gives}

r

=

-517

+

^x-084

=

•517

+

^-012

=

-529.

* Drapers' Company ^{Research Memoirs. Biometrie Series VII.}

"A

Novel Method," ^{etc.: see} pp. 12, 13.

XVIII— XX] Introduction XXXIX

Interpolating ^{for <7,.} first,

,•

=

•5 „<r,.=

-05U

_logx"

=

_2-0737,

r

=

-6 _O0V

=

-051-1- logx'

=

2-2537.

Hence

for log x~

=

_2-1249:

•0512 ri^, .__

r_"

=

^'O

+

-1800 ^[

'

1]

=

"°28-

We

conclude that the equiprobable correlation^{is -}53.

(4) Imbecility

and

Deaf-mutism.

logx"

=

_{3-9039 cr,.}

=

_-Ol75,

r

=

_0-95, Oo-,

=

-01_, log

x

²

= _*3673

; „°v

=

_"02,

log

x

²

= _37660.

Hence:

r

=

_{0'J5, Oo-,}

=

_-0175,

log

x

²

= _39163.

Again

^:

r

=

_{090, „ct,.}

=

_-01,

log

x

²

=

-i-2207 _; „o-,.

=

_{-02, log}

_%

²

=

3-6197.

Hence: r-090,

^<r,.

⁼

^-0175, log

X

^*

=

3-7699.

Interpolatinglogx'-

= _{39039 between 39163 and 37699,} we

find r_P

=

0-946.

(5) Developmental Defects

and

Dullness.

log

x

^*

=

_{3-5128, o-,= 0201.}

r

=

_0-8,

<r,.

-

_-02,

log

X

²

=

_3'4097

; <x,.

=

_03,

logtf

=

3-0598.

Hence

: „o-,

=

_0201,

log

X

²

=

3"4062.

r

=

_0<), ,«r-"02, log

x

²

=

_3-6197; O o-,.

=

_-03,

log %=

=

3-2690.

Hence

^: _logtf

=

_3-6162, for_„cr,.

=

-0201.

Thus,

by

interpolating

logx

^s

=

3-5128

between

3'4062

and

3-6162,

we

find

r>=-851.

We

have accordingly the _followingresults^:

xl

Tables for Statisticians and Biometrieians [XXIII — _XXIV

p.xxxiv.

Both

_rP

and Q

give veryhigh results for (4)

and

_(5),and thisis inaccord- ance with the view elsewhere expressed ^{that for}

extreme

dichotomies

Q

^{is not to}

be trusted. It

may

^further be doubted,

whether

for such dichotomies the theory of the distribution of deviations on

which

_rP ^is based can in its turn be _accepted.

On

the whole r_t

seems

to

me

the

most

satisfactory coefficient of association, to be controlled

by

results forrP ⁱⁿ the cases

where

neither the dichotomiesare extreme, nor the

numbers

so large or so small as to fall outside the

moderate

_range of Tables

XVIII— XX

^or

Abacs XXI and XXII.