j Pharmacol Exp Ther 1949 Litchfield 99 113

7/26/2019 j Pharmacol Exp Ther 1949 Litchfield 99 113

1/15

A S IM P L IF IE D M E T H O D O F E V A L U A T IN G

D O S E -E F F E C T E X P E R IM E N T S

J . T . L IT C H F IE L D , JR . N F . W IL C O X ON

S tam ford R esearch L ab ora to rie s A m erican C yana m id C om p any

S tamford C onnec t icu t

R ece ived fo r pub licatio n Janua ry 15 ,

1 9 4 8

T he inc re ase d e m ph as is on qu an tita tiv e b io log ica l stud ies in re ce n t ye ars ha s

re su lte d in the w id esp rea d u se o f s ta tis tic a l m eth od s fo r ev a lua tin g b io lo g ic a l

d ata . M uc h o f th is da ta is o f th e a ll-o r-no ne typ e a nd , co nse que n tly , it is n e ce s-

sa ry to so lv e a d ose -pe r c en t effe c t cu rve . B y c on ve rtin g do ses to log arithm s

an d pe r c e n t e ffec ts to p ro b its 1 ), log its 2 ), o r a ng les 3 ), a s tra ig h t l ine m a y b e

fitte d by th e m e th od o f w eigh ted le as t sq ua res . F rom th e v ie w p o in t o f m a ny

b io log is ts , suc h p roc e du re s are n o t p lea san t to co n tem pla te b ec a use th e d a ta

m u st b e c on ve rte d to un its w h ic h a re m e an ing le ss to m a ny a nd th e ca lc u la tio ns

are d iffic u lt, ted iou s an d often qu ite in c om pre he ns ib le . I t is no t su rp ris ing

the re fo re th at th ere is w id esp rea d use o f a va rie ty o f ap p ro x im a te m e tho ds fo r

so lv ing dose -pe r c en t effe ct cu rve s . It m ay be a rgu e d tha t su ch m ethod s a re

u nde s ira b le be ca use the y do no t m ak e use o f a ll o f the in fo rm ation co n ta in ed in

the da ta , a nd are th ere fo re in effic ie n t in a s ta tis tic a l se nse . O n the o the r ha nd ,

th e c om pu tat ion s ne c essa ry in u s ing e ffic ien t m eth od s are o fte n so tim e -co nsu m -

in g a nd lab o rio us th at the b usy e xp erim e n te r is d e terre d fro m u sing th em , a nd

thu s lo ses th e a dv an ta ge o f a s ta tis tic a l ev alu at ion o f h is re su l ts . A n e xa m ina -

t ion o f the va riou s a pp rox im a te m e tho ds , w h ic h h av e be en p ro po sed fo r the

so lu tion o f do se -e ffec t e xp erim e n ts o f the a ll-o r-n on e typ e , lea d s to th e c on clu s ion

tha t no ne o f the m a re e n tire ly satisfa c to ry in co m bin ing e ase o f c om pu tat ion w ith

effic ie nc y an d a c cu rac y . In o rde r to a pp re c ia te th is fac t, i t is h e lp fu l to lis t

the essen tia ls o f a sa tis fac to ry a p p rox im ate m e th od fo r the so lu tio n o f d ose -e ffe ct

e xpe r ime n t s .

1 ) T he m e tho d sh ou ld g ive n o t on ly th e E D 5 0 a nd s lop e o f the c u rve , b u t a lso

th eir c on fide nc e l im its.

2 ) T he m e th od sho u ld use th e d ata in o rig in al u n its th rou gh ou t.

3 ) Z ero a nd 1 00 p er c en t effe c ts sho u ld b e use d e ffe ctiv e ly .

4 ) T h e m e th od sho u ld m a ke it po ss ib le to c a rry ou t th e n ec e ssa ry c a lcu la tion s

w ith in 1 0 -1 5 m in u te s w itho u t a c alc u la ting m a c h in e , a n d w ith ou t reso rt to

loga r i t hms .

5 ) T he m e th od sho u ld re co gn ize he te rog en eity w h en p rese n t a nd g ive c o r-

re c ted c on fide nc e lim its in suc h ca ses .

6 ) T he m e tho d sho u ld fa c ilita te bo th th e co m pa rison o f th e tw o c urv es fo r

pa ra l le l ism a nd the co m p u ta tio n o f re la tiv e po ten c y w ith its c on fide nc e lim its .

7 ) T he m eth od sh ou ld no t un du ly sac rific e a cc u ra c y in fa vo r o f s im p lic ity

an d sp ee d .

T h e v ario us a pp rox im ate m ethod s fo r so lv in g do se -pe r ce n t e ffe ct cu rve s fa il

99

his ne

I II1 1 1 1 0 1 1lll III IINI

J P 6 J Q 3 D W 3 U N


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T . L I T C H F I E L D , J R . A N D F . W . W I L C O X O N

i n vary ing degrees to meet these requi rements. T he double integrati on method

4-7 w hich uses data in original uni ts and the methods of averages 8-10 w hich

requi re logar i thms prov ide an estimate of the ED 50. W i th a restr i cted experi -

mental design, the conf idence l im i ts of the ED 50 can be obtained by the methods

of averages, w i th vary ing amounts of calculati on.

O f the methods w hich may be considered to approx imate that of B l i ss 1 ,

one, using the data in or iginal uni ts gi ves only the conf idence l im i ts of the ED 50

11 ; a second, using logar i thms gives, in addi ti on to the above, the slope con-

stant, but not i ts l im i ts 12 ; and a thi rd, using logar i thms and probi ts gi v es

both parameters and thei r conf idence l im i ts 13 . N one of the approx imate

methods use 0 or 100 per cent observations to best ef f ect, or recognize hetero-

genei ty , i f present.

The method of L i tchf i eld and Ferti g 13 , w hich gives conf idence l im i ts of both

parameters, appeared to of fer the best starti ng point f or developing a rev i sed

method w hich w ould approach the ideal requi rements mentioned above.

In order to rev i se the above method to conform to the aims l i sted, three disti nct

ty pes of problem w ere involved. The f i rst of these w as the problem of using

percentages and ari thmeti c values in a w ay exactl y equivalent to the use of

logari thms and probi ts. L ogari thm ic-probabi l i ty paper perm i ts plotti ng the

data in or iginal uni ts but leaves the problem of converti ng log-probi t equations

to thei r ari thmeti c equivalent.

The resul t of addi ti on and subtracti on of logar i thms can be obtained easi l y

by mul ti pl i cati on and div i sion of the numbers themselves. Sim i larl y , the resul t

of mul ti pl y ing or di v iding a logar i thm by an ari thmeti c value can be represented

by rai sing the number corresponding to the logar i thm to a pow er equal to the

ar i thmeti c value or by tak ing the root in the case of di v i sion . Such a calcula-

ti on cannot be made, how ever, w i thout recourse to logari thms or to the use of

log- log sl i de rule. Consequentl y , the second major problem ari ses, the need for

el im inating calculati ons w hich requi re logar i thms. In thi s parti cular case, a

nomograph w as constructed for obtaining f racti onal pow ers and roots of num -

bers com ing w i thin the scope of the method.

A f urther compl i cati on ari ses in the case of the product or quotient of tw o

l ogar i thms since thi s operation cannot be represented at al l as a purel y ari th-

meti c process. I n the tw o such cases w hich ar i se in the rev i sed method, nomo-

graphs w ere constructed to perm i t soluti on w i thout recourse to logari thms or

a log- log sl i de rule.

B y means of tw o of the above mentioned nomographs, a simple ar i thmeti c

soluti on of a dose-ef fect curve can be obtained w hich is equivalent to the soluti on

by the original method using logar i thms and probi ts. T he tw o soluti ons are

numeri cal l y equal but the rev i sed method is more rapid and perm i ts using the

data in i ts or iginal form .

The thi rd type of problem in the rev i sion consisted of f i nding the means for

adding to the method a simple test for heterogenei ty or goodness of f i t of the

l i ne, together w i th the correcti on of conf idence l im i ts for heterogenei ty ; a means

for ef f ecti v el y using 0 and 100 per cent ef fects in f i tti ng the l i ne to the data;


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E V A L U A T I N G D O S E -E F F E C T E X P E R I M E N T S

10 1

and lastly , the m eans for approx im ating the co nfidence lim its o f do ses o ther than

the M edian Effec tive D o se .

The com plete m etho d

including

the neces sary table s and nom o graphs is

presented belo w , tog ether

with

several exam ples

illustrating

its

application.

M E T H O D .

The fo llow ing sy m bols are used in this m ethod:

K the num ber o f doses plo tted

n

K 2

degrees o f freedom for (C hi)

t value of S tudents t for p .0 5

E D 5 0 M edian Effec tive D ose

S

Slope

function

fE D , 0

and fs fac tors for ED ,0 and 5 , respective ly

N to tal num ber of anim als used betw een 16 and 84 per cent expected effects

R the ratio o f largest to sm alles t dose plo tted

A a v alue derived from S and R

S .R . and P.R .

S lope functio n R atio and Potency R atio

f R

and f p . R .

fac tors fo r S .R . and P.R ., respectiv e ly

U nless o therw ise indicated

all

ratios are taken as: larger/sm aller value .

P R O C E D U R E .

A .

The data and graph.

1 . List the actual doses used, the number react-

ing /num ber tes ted, and the per cent e ffec ts. D o not list m ore than tw o consecutive 10 0 pe r

cent e ffec ts at the upper end or m ore than tw o consecutive 0 per cent e ffec ts at the lo w er

end of the curve .

2 . Plo t doses ag ainst per cent effec t on log arithm ic .probability paper (N o . 3128 , C odex

B oo k C o., Inc ., N o rw o od, M ass .) leav ing space for but om itting any 0 or 100 per cent e ffects .

W ith a transparent straight edge o r triang le fit a tem porary straight line through the

po ints, particularly those in the reg ion of 4 0 to 60 per cent e ffec t.

B .

Plotting 0 or 100 per cent effects.

1 . R ead and lis t the expected per cent e ffec t, as

indicated by the line draw n, for each dose tested. If the expected value for any dose is less

than .0 1 or greater than 99 .99 de le te such doses and effec ts from the list.

2 . U sing the ex pected effec t record and plo t from table 1 a co rrected value for each 0 or

10 0 per cent e ffec t w hich is lis ted. Inspect the fit o f the line to the co m plete ly plo tted data.

If it is obv ious ly unsatis fac to ry refit the line and repeat the preceding tw o steps to obtain

a new set o f ex pected and corrected values .

W hen the line appears to fit satisfac torily , as is alm ost alw ays the case w ith the first line,

proceed to the (C hi) tes t.

C.

The (C hi)

t es t 1 . Lis t the difference betw een each o bserv ed (or corrected) e ffec t

and the correspo nding expected effec t.

2 . U sing each difference and the corresponding ex pected effec t read and list the contribu-

tions to (C hi) from N om ograph N o.1 . (A straight edge connecting a value on the expected

per cent scale w ith a v alue on the difference scale, w ill indicate at the po int o f intersection

of the (C hi) scale, the contribution to (C hi)2 .)

3 . Total the contributions to (C hi) and m ultiply by the average num ber of anim als per

do se , i.e ., the to tal num ber of anim als/K , the num ber of doses. This is the (C hi)2 o f the

line . The degrees of freedom are tw o less than the num ber of do ses plo tted, i.e ., n K 2.

4 . If the (C hi) o f the line is less than the value o f (C hi)2 g iven in table 2 for n degrees

of freedom , the data are not significantly hetero geneous , i.e ., the line is a good fit. If the

(C hi) o f the curve exceeds the v alue of (C hi) g iven in table 2 , the data are s ignificantly

heterog eneous and the line is not a goo d fit. (If the (C hi) o f the line cannot be reduced

below the perm issible (C hi) by refitting the line , the value o f t in table 2 for n degrees of

freedom should be noted.)

D .

The ED ,0 and

I E D O 1 . R ead from the line on the graph the dose for 16 , 5 0 , and 84

pe r cent ef fects (ED 15 , ED ,0 and ED 84 ).

2 . C alculate the s lope function, 5 , as:


4/15

10 2


E D 54/E D ,o + E D ,0 /E D ,.

2

3 . O bta in f rom the da ta tabu la tion , N , the to ta l num ber of an im als tes ted a t th ose do ses

w h o s e expected effects w ere between 16

an d

84 p er cen t.

4 . C a lcu la te the exp onen t in the expression :

fE D, 0 = 52 . 77 / N = 5exponent

T o carry ou t th is s tep , ob ta in firs t the s ,/ f rom a square roo t tab le , o r w ith a slide ru le , o r

by m eans o f N om o graph N o. 2 . T hen so lv e 2 .7 7 /V p expo nen t. N ex t, using th is ex -

pon en t an d the va lue of S , read the

f E D , 0

on the cen te r scale o f N om og raph N o. 2 by lay in g a

stra igh t edg e

across the correc t sca le va lues .

5 . C a lcu la te the con fid ence lim its o f the E D ,0 as:

E D ,0 X fE n , 0 = u pper) lim it fo r 1 9 /20 pro bab ility .

E D ,o / fE D ,0 low erf

E.

Sandfs

1 . C a lcu la te th e dosag e range as a ratio , as fo llo w s:

R

la rges t/sm alles t do se p lo tted

2 . U s in g th is va lue of R and tha t o f S from step D 2) , read the va lue des ig na ted as A from

N om og raph N o . 3 by lay ing a stra igh t edg e across the correc t sca le v a lu es .

3 . So lve fo r the expo nen t in the fo llo w ing expression , us ing K from step C 3 an d s , /F . r

fro m step D 4.

fs =

A b 0 ( 1) S A e x P o n e t

T hen w ith th is expon en t and th e va lue of A , read fa fro m N o m ograph N o.2 .

4 . C a lcu la te the confidence lim its o f S as:

S X fs

=

upper)

?

lim it fo r 19 /20 p robab ility .

S/fs

=

low er)

F.

The factor s for 8igni fi cantl y heter ogeneous data.

W h en th e C h i) tes t in d ica tes

s ign ifican t h e te ro gene ity the va lue of t from tab le 2 is no ted and th e

fo rmulas

be low are

used fo r the fac to rs ins tead o f th ose in steps D 4 and E 3 . T he p rocedure fo r so lu tion con-

s ists o f so lv ing firs t fo r the va lues o f the exp onen t an d then w ith the va lue o f S from step D 2

and tha t o f A from step E 2 , the fac to rs a re read from N om og raph N o . 2 .

1. f E D , 0 51 . 4 t1( 2/i =

2 . fs AE5.1t i)V 2h1N]11 =

A 50 n 5 1 t

G. The test for par al lel i sm of two l i nes and the estimate of r el ati ve potency.

T he fo llow in g

va lues w hich represen t the param eters and fac to rs o f a dose-per cen t e ffec t lin e a re to b e

com pared to a sim ila r se t o f va lues fo r a second line :

ED,0 an d f E D , 0 )

i fo r each lin e.

Sandfs

1 . T he test fo r para lle lism : the slope func tion ra tio , S .R .

a) C a lcu la te : S .R .

=

S1 /S2 w here S i is the la rger va lu e .

b ) U sing fs1 an d f8 , read from the cen te r scale o f N om ograp h N o. 4 by lay ing a

s tra ig h t edge acro ss the correc t scale va lues.

c ) If the v alue of S .R . exceed s th e va lue of f s . R . the curv es dev ia te sign ifican tly 1 9 /20

probab ility ) f rom para lle lism . If S .R . is less than f O R , th e curves m ay be co nsidered p ara l-

le l w ith in experim en ta l e rro r and the po tency ratio m ay then be co m puted as fo llow s:


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E V A L U A T IN G D O S E -E F F E C T E X P E R IM E N T S

1 0 3

2 . T h e Po ten cy R atio , P.R .

(a) C alcu late: P.R . E D ,0 1 /E D ,,, w h ere E D ,0 , is th e larg er v alu e .

(b ) U s in g f E D, 0 , an d f E D, 0 , read

fp . R ,

f ro m th e cen te r scale o f N o m o g rap h N o .4 .

(c) T h e v alu e o f P.R . m u s t ex ceed th e v alu e o f

f p , R .

f o r th e tw o su b s tan ces b e in g co m -

p ared to d if f e r sig n if ican tly in p o ten cy .

3 . T h e co n f id en ce lim its o f th e s lo p e an d th e p o ten cy ratio are g iv en b y :

S .R . X f s . R . = upper)

(a) S .R ./f 5.R . = lo w erf lim it f o r 1 9 /2 0 p ro b ab ility .

P.R . X fp . i = upper)

(b ) P R /fp R = io w erf lim it f o r 1 9 /2 0 p ro b ab ility .

T h e f o llo w in g ex am p le illu s trate s th e u se o f th e m eth o d . O n th e w o rk sh ee t b e lo w th e

v ario u s s tep s are in d icated b y A 1 -E 4 . T h e g rap h (f ig . 1 ) co rre sp o n d s to step A 2 .

Solution of the D ose-Effect C urve of Tagathen (Chlorothen Citrate) against H istamine

A l A l

.

A l B I

C i

O B S E R V E D

C 2 ( No xo ol Ap E

N O . 1 )

D O S E A L IV E /T E S T ED O B S E R V E D

A L IV E EX PE C T E D

A L I V E

M I N U S

E X P E C T E D

C O N T R I B U T I O N

TO

2

mgtn /kgris

1 .0 8 /8

1 00 (99 .5 )

B 2 (tab le 1 )

9 8 .6 0 .9 0 .0 0 6

0. 5

0 .25

0 . 1 2 5

0 . 0 6 2 5

7 /8 8 8 90 .5 2.5

1 7 .0

1 6 . 0

2 . 5

0 .0 0 7

0 .1 1 0

0 . 1 0 5

0 .0 0 7

4/ 8

4 / 8

50

5 0

67 I

1 D3

3 4 ]

1 / 8 1 2 9 . 5

T otal an im als = 4 0 T o tal 0 .235

C3 Nu mb e r o f Do s e s , K

=

5

(Chi) 0 .235

X 8

1 .88

An i ma l s / Do s e = 4 0 /5 = 8 De g r e e s o f Fr e e d o m, n K

2 3

fro m t a b l e 2 f o r n o f 3

=

7 . 8 2 .

1 .88

is le ss th an 7 .8 2 , th e re f o re, th e d ata are n o t

ig n if ican tly h e te ro g en eo u s .

E D , 4 m g m . / k g m . = 0 .3 9 0

D l E D ,o m g m ./k g m . = 0 .1 7 5

E D,5 m gm ./k gm . = 0 .0 8 0

5 E D s,/E D ,o + E D ,o /E D ,5

2

D 2 0 .3 9 0 /0 .1 7 5 + 0 .1 7 5 /0 .0 8 0 2 2

2

D 3 (N o te B o ld f ace lim its ab o v e ) N = 16

D 4 fE D , O = ( S ) 2 7 7 N 2 .2 2 .h h / T h (2 .2 ) { 1 7 6 } 1 .7 2 (f ro m N o m o g rap h N o . 2 )

lE D , 0

X

fE D . O = 0 .1 7 5 X 1 .7 2 = 0 .3 0 m gm ./k gm .

D 5 E D , 0

/ f E D , ,

0 .1 7 5

/

1 .7 2

0 .1 0 m gm ./k gm .

I. E D ,0 an d 1 9 /2 0 co n f id en ce lim its: 0 .1 8 (0 .1 0 to 0 .3 0 ) m g m ./k g m .

E l R

=

larg es t/sm alle st d o se

1 .0 /0 .0 6 2 5

16

E 2 A 1 .2 7 (f ro m N o m o g rap h N o . 3 , u sin g S 2. 2 and R 16 )

E 3 f (A ) E

( l , 27) lOX4/5Vi

(1 .2 7 )2 .0

1 .6 0 (f ro m N o m o g rap h N o . 2 )

IS X fs 2 .2 X 1 .6 = 3. 5

E 4 S / f -2 .2 /1 .6 = 1 .4

S a n d 1 9 /2 0 co n f id en ce lim its: 2 .2 (1 .4 to 3 .5 )


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1 0 4

T . L I T C H F I E L D , J R .

A N D F. W . W IL COX ON

I n practi ce the w ork sheet can be greatl y condensed in compari son to the example by

om ission of the vari ous step symbols, f ormulae and by recordi ng on the graph the figures

F I G U R E 1

D O S E M G./K G .

.J

.

used in the cal culations. I n the same experiment i n w hi ch the antih istam ine acti v i ty of

T agathen w as studied, tr i pel ennam ine Pyri benzam ine w as tested sim i l arl y i n order to


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E V A L U A T I N G

D OS E -E FFE CT E X PE RIM E N T S

1 0 5

determ ine the po tency ratio of the tw o drugs . The so lution of the dose effec t curv e

of

Pyribenzam ine illustrates the com bined w ork sheet and graph.

The param eters and facto rs of the tw o dose effec t curv es are sum m arized be lo w .

g then

Pj i5enzamirte

ED ,0 0 .1 8 m gm ./kgm . 0 .60 m gm ./kgm .

fED, 0

1 .72 1 .6 0

S 2 .20 2. 34

f5 1 .60 1 .57

The curves are tested fo r paralle lism and the potency ratio o btained as fo llo w s:

01 The s lope ratio , S .R . = S , /S , = 2 .34 /2 .20

=

1 .06

fS . R .

=

1 .92 (fro m N om ograph N o. 4 , us ing the tw o fs v alues)

S .R . of

1 .06

is less than fs.R. o f 1 .9 2 , there fore, the dev iation from paralle lism is not

significant.

02 The po tency ratio , P.R . = E D ,, , /E D ,0 , =

0 .60 /0 .18

= 3. 3

fp . R .

=

2 .0 5 (from N om og raph N o .4 , us ing the tw o fED ,0 value s)

P.R . o f 3 .3 ex ceeds

f p . R . of 2 .05 , the refo re, the tw o drugs dif fer sig nif icantly in po tency .

03 C o nfidence lim its o f

S .R . and P.R .

S .R . X fs.i = 1 .06 X 1 .92 = 2 .03

S.R./fs.R. = 1 .06 /1 .92 = 0 .55

The s lope ratio , S .R ., and 19 /20 co nfidence lim its: 1 .06 (0 .55 to 2 .03).

P.R . X

=

3. 3 x 2 .05

=

6 .75

P.R./fP.R.

=

3 .3 /2 .05

=

1 .60

The potency ratio , P.R ., and 1 9 /20 confidence lim its : 3 .3 (1 .6 to 6 .8 ). Tagathen w as

s ig nificantly m ore active than Py ribenzam ine and fo r confidence lim its o f 1 9 /20 , its re lative

activ ity lie s betw een 1 .6 and 6 .8 tim es that o f Pyribenzam ine .

Occasionally the ex perim enter is interested in a dose other than the ED ,o; for exam ple ,

he m ay w ish to kno w the ED ,0 and its confidence lim its. The dose for any des ired per cent

ef fect, Y , can be read from the g raph. The 19 /20 confidence lim its o f this dose , ED y , can

be appro x im ated by increasing the value of fED ,0 by an amo unt determ ined by the value of

fs and X , (the

deviation

in standard dev iation units , o f Y from 50 per cent). V alues of X

for co m m on v alues of Y are g iv en in table 3 .

The pro cedure for obtaining the 19 /2 0 confidence lim its o f ED is as fo llow s:

1 . O btain

( f5 )X

using N om og raph N o. 2 to raise the base , fs (fro m step E3) to the ex po-

n e n t X

(from table 3) .

2 . O btain f E D from center scale o f N o m ograph N o. 4 , us ing the v alue obtained fo r ( f5 )X

and the value o f fmi,0 (from step D 4 ). If the scale lim its o f the nom ograph are exceeded,

the co nfidence lim its are like ly to be so w ide that the ED y v alue is rather m eaning less. The

confidence lim its are obtained in the usual w ay using the f E D Y .

A s an exam ple of this procedure applied to the Tagathen line , the confidence lim its o f

E D ,0

=

0 .48 m gm ./kgm . are obtained as fo llow s:

1.

( f8 )X

= 1 .60) = 1 .85 (from N om ograph

N o . 2

and table 3).

2, fmi,0

=

2 .25 (from N om o graph N o . 4 , us ing ( f8 )X

=

1 .85 and fE D , Q

=

1 .72) .

ED ,0 and 1 9 /20 confidence lim its = 0 .48 (0 .21 to 1 .08 ) m gm ./kgm .

D Iscuss Io N . The m ethod presented fulfills , for the m os t part, the aim s of a

satisfacto ry approx im ate m ethod. A t the risk o f m aking the m ethod appear

rather co m plex the instruc tions have been m ade as com plete as po ssible . W e

hav e fo und that these instruc tions in the hands o f an inexperienced person

perm it a com plete so lution o f data, such as that in fig . 1 , to be o btained in 2 0 to

3 0 m inutes . A n experienced perso n o n the other hand requires le ss than half


8/15

m

.175 ..35

.7 i.4 2 .5

1 0 6

J

T . L I T C U F I E L D , J R . A N D F . W . W I L C O X O N

this tim e. A lthoug h the accuracy of this m ethod has not been exam ined,

it

canno t be le ss than that o f the Litchfie ld-Fertig m etho d w hich has been show n

-

..

: ?J

. .

: I

I , . I

DOSE EFFECTL E I N i OF P Y RI B E NZ A MI N E

P

i I STAMIN1 i ; v

,

L

. . , .

L

i:

D O 5 F

.

O - O

-

f

I

,

-

-

I

)

/e 00(98.9) 96.S .4.

4

l

7 1 /8 L 1 2 .

.

n

L:;

- 2:3 } js t, ro: ( % 3 . s 6

2

{ 182} 8

r .

2 8 ,, 16

S 4 .68 /2 2 ;34 L,

R :. .175

. , . # {1 49 } # {1 49 } . .7- -

:-

-

A :

f

. : , 2 .77 /4 .9 , . :5 - - ;- . ,

;H S

. :

t I I j J

,

h

I

a

L X M G . / K G .

FIGURE

2

to be satisfac tory for all ordinary purposes. The new m ethod in som e respec ts

is undo ubtedly m ore accurate since not only can a po orly fitted line be de tec ted

and im pro ved but also sig nificant he tero gene ity , if present, w ill be found.


9/15

E V A L U A T I N G D O S E -E F F E C T E XPE RIME NT S

1 0 7

T A B L E

1

Cor r ected Values* of 0 or 100

per cent

Effect Body of Table)

Cor r esponding to Expected Values M ar gins)

EXPECTED 0

1 2

3 4 5 6 7

8 9

0

0 .3 0 .7 1 .0 1 .3

1 .6 2 .0 2 .3 2 .6

2. 9

10 3 .2 3 .5 3 .8 4 .1 4 .4 4 .7 4 .9 5 .2

5 .5 5 .7

20

6 .0 6 .2 6 .5 6 .7 7 .0 7 .2 7 .4 7 .6 7 .8 8 .1

30 8 .3 8 .4 8 .6 8 .8 9 .0 9 .2 9 .3 9 .4 9 .6 9 .8

40

9 .9 10 .0 10 .1

10 .2 1 0 .3 1 0 .3

10 .4 10 .4 10 .4 1 0 .5

50 8 9 .5 89 .6 89 .6 89 .6 89 .7 8 9 .7

8 9 .8 89 .9 90 .0

60

90 .1 9 0 .2

90 .4 90 .5 90 .7 90 .8

91 .0 9 1 .2 9 1 .4 91 .6

7 0 9 1 . 7 9 1 . 9 9 2 . 2 9 2 . 4 9 2 . 6

9 2 . 8 9 3 . 0 9 3 . 3 9 3 . 5

9 3 . 8

80 94 .0 9 4 .3 94 .5 94 .8 95 .1 95 .3 95 .6 9 5 .9 9 6 .2 96 .5

90 96 .8

97 .1 97 .4 97 .7 98 .0

98 .4 98 .7 9 9 .0 9 9 .3 99 .7

*

These values are derived from the m axim al and m inim al corrected probits o f B liss (1 ) .

TA B LE 2

Values* of t and Chi) for p

.06

DEGREES 0 7 p i z E Do M

t

c R 1 ,

1 12 .7

3 .84

2 4 .3 5 .99

3 3 .18 7 .82

4 2 .78 9 .49

5 2 .57 1 1 .1

6 2 .45

12 .6

7 2 .36 1 4 .1

8 2 .31 15 .5

9

2 .26 1 6 . 9

10 2 .23 18 .3

*

Va l u e s o f s t u d e n t s

t and (C hi) for p

.05 are the sam e as m ay be found in m ore

extens ive tables such as those in (1 7).

TA B LE 3 *

%EmcT,Y

X

16or84

1 .00

lOo r9O 1.30

5or95

1 .65

2or98

2 .05

1 o r 9 9

2 . 3 5

* Other values of X m ay be o btained from any table re lating dev iations and areas o f the

n o r ma l c u r v e , s u c h a s 1 7 .

A ll approx im ate m ethods w ithout exception have one o r m ore w eak po ints .

The inadequacy of these m ethods becom es ev ident w hen they are applied to an


10/15

1 0 8

T . L IT CH FIE L D,

JR . A N D P. W . W U C O X O N

un balan ced or trun cated se t of d ata. T hu s in th e m ore ex ac t m e tho d o f B liss

(1 ) , the con f id en ce lim its are co rrec ted f or th e dev iation (caused b y u nb al an ce ),

o f the m ean p ro b it f ro m 5 .0 . D ue to the nature o f the w e igh ting coe f f ic ien ts

th is co rrec tio n is o f little sign if icance un les s the degree of tru ncatio n is rath er

large . T h us, in th e case o f any approx im ate m e tho d , the con f idence lim its w ill

ten d to b e und erestim ated w hen the m ax im um ob serv ed ef f ec t is 70 p er cen t or

le ss . T h is w eak ness com m on to all su ch m eth ods is u sually o v erlo ok ed an d the

ex p erim en ter sho u ld av o id app ly ing an ap prox im ate m eth od to v ery p oorly

b alanced ex perim en ts. In such cases repe tition o f the ex p erim en t or the us e

o f the m o re ex ac t proced ure o f B liss (1 ) is in d icated . It cann o t be su f f icien tly

em phasiz ed , h ow ev er, that a statistical m e tho d is in n o w ay a substitu te for a

g ood ex perim en t.

T h e nam e slo pe fu nc tion has been app lied to S , th e an tilo garith m of th e

q uan tity design ated as s in B liss no tatio n , o r X in G addu m s notation (18), w h ere

s or X is th e standard d ev iatio n o f the logarithm s o f th e ind iv idual ef f ec tiv e do ses.

O u r

cho ice w as

based

o n the use o f S f or purposes cus tom arily serv ed b y the

slo pe co nstan t b , an d w e are n o t aw are o f an y ex istin g design atio n fo r the

an tilog arithm of the standard d ev iation .

S in ce the d ose -p er cen t e f f ec t curv e is en coun tered so f req uen tly in biological

and occas ionally ev en in n on-b io lo g ical f ield s, th is rap id

ap p ro x im ate m e th o d

sho u ld be o f he lp to the m any ind iv iduals w ho h av e n o t the tim e, d esire no r

f ac ilitie s f or com p lex m ath em atical treatm en t o f th is k ind o f d ata.

S U M M A R Y

1 . A rap id graph ic m e thod f or ap pro x im ating th e

M edian

E f f ec tiv e D o se and

the S lo pe o f dose-per cen t ef f ec t cu rv es is presented. Confidence l imi t s of bo th

o f these p aram eters f or

1 9 /2 0

probab ility are g iv en by the m eth od . In add itio n ,

con f idence lim its f or any o th er p ro bab ility or f or a d ose o ther th an th e M edian

E f f ec tiv e D ose are read ily estim ated .

2.

T he data are us ed thro ugh ou t the m ethod in th eir orig inal f orm w itho u t

tran sfo rm atio n to log arithm s and prob its .

3 . A n ef f ec tiv e m ean s f or p lo tting and u sing 0 and 10 0 per cen t e f f ec ts is

prov ided .

4 . T he calcu lations hav e been sim p lif ied b y m ean s o f n om ograp hs to the

ex ten t that a slide ru le is a co nv en ien ce b u t n o t a necessity .

5 . A sim p le

m e a n s

is p rov ided f or de tec ting a p oorly

f itted lin e

or

significantly

he tero geneou s data, In the fo rm er case , the lin e

m ay

b e re f itted ; in the latte r,

the con f id en ce lim its are correc ted fo r the d eg ree o f he terog ene ity .

6 . T he m etho d prov id es m ean s fo r the rap id test o f paralle lism of tw o curv es

an d easy co m pu tatio n o f re lativ e p o tency w ith its con f id en ce lim its .

7 . A lth oug h the m etho d is rap id (10-15 m in u tes), its accu racy is co m m ensu rate

w ith th e n atu re o f dose -per cen t ef f ec t d ata.


11/15

EVALUATI NG

DOS E -E FFE CT EXPERIMENTS 1 0 9

A P P E ND I X

A. S ou rce o r d eriv a tion o f fo rm u lae u sed in the m e thod . The rev ised m ethod uses

in m oth-

fled form : (1 ) The appro x im ations deve loped by Litchfie ld and Fertig (13) for obtaining

co nfidence lim its o f the param eters o f a dose-per cent e ffec t curv e, and (2) the m etho d for

(C hi) proposed by W ilcoxon and M cC allan (1 4). The corrections used i n the event of

hetero gene ity , the m ethod for using 0 and 10 0 per cent ef fects , and the tes t for significant

differences be tw een values are derived from conventional procedures (1 ). The fo rm ula for

o btaining approx im ate confidence lim its o f doses g iv ing per cent e ffec ts o ther than 50 per

cent is derived from that for the variance o f the log ED ,, as g iv en by B liss (1 ).

The fo llow ing table show s som e of the re latio nships betw een fo rm ulae used in the rev ised

m ethod and the ir equivalents after transform ation to the log arithm -probit sys tem .

A ri thm e tic m etho d L og -p ro b it m etho d

E D, , l o g E D, ,

fED,0 1 .96SE1o5ED,0

S logS orsorl /b

f s 1 . 9 6 S . E . . o r 1 . 9 6 s S . E . b

A/ B l o g A- l o g B

f A / B 1 .96V(S.E .i0

A 2 + S . E . i 0 B

fE D y %

1 .96 S.E .2 iogED,o + [S .E. y 5 .0)1

C orrected effec t for 0 or 10 0 per cent M axim al or m inim al corrected pro bit

B .

T he p aram eters an d co n fid ence lim its o f a do se p er cen t effec t line o n loga rithm ic -p rob -

a b ility pa per.

1.

T he m ed ian e ffec tive dose : E D 0 .

This is

the do se indicated by the line

to cause 50 per cent of the anim als or item s to react o r no t, to live or die , to be pos itiv e or

neg ative , to fit into a catego ry o r not, e tc . D ose is used in the abstract sense and m ay be

dose , tim e, s ize , dis tance, e tc .

2. T he s lope func tion o f the line : S . This is the fo ld change in do se required to produce a

unit s tandard dev iatio n chang e in response along the

line . Thus:

X, - X

S = antilo g o f: 1/b, 5 , o r

Yl Yl

w here b and s are , respective ly , the slope constant and standard dev iation o f a line re lating

lo g dose X , and pro bit per cent e ffec t Y . S ince s is actually the difference betw een tw o par-

ticular log doses , its antilo g , the slope function 5 , is the

ratio o f the arithme tic value o f

those doses .

3. T he fac tor o f fE D f or ob ta in ing 19 /0 con fidence lim its o f th e E D . This factor,

using the notatio n of Litchfie ld and Fertig , is de riv ed as fo llow s:

S.E.IO1ED,0 = i , /N /2 1 )

w here s is the difference betw een tw o log do ses w hose expec ted

ef fects,

a s ind ica ted by th e line

differ by 1 .0 probit and N is the to tal num ber of anim als or item s tes ted betw een the log

dose lim its corresponding to expec ted probits 4 .0 and 6 .0 .

M ultiply ing (1 ) by 1 .96 , sim plify ing and taking the antilo g g ives:

fE D, 0 = 5 2 . 7 7 h , / N (2 )

w here S = antilog

s a n d N i s n o w

the to tal num ber o f anim als or item s tes ted betw een

arithm etic dose lim its corresponding to expec ted

16 and 84 per ce nt effect.

The slope

function S

c a n be obtained from the line on log arithm ic-pro bability paper by any

o f t h e f o l l o wi n g e x p r e s s i o n s b u t 3 ) i s p r e f e r a b l e :

S

=

ED ,4 /ED ,, o r E D, , / E D, , o r E D, , / ED, ,

ED S e /ED S O + ED ,,/ED ,, (3 )

2


12/15

1 1 0


Th e f a c t o r f o r t h e ED, 0 c a n b e r ed u c e d t o t h e e x p r es s i o n :

fED,0

=

wh e r e t he e x p o n e n t i s

2.77/vi?.

T he value of fED ,, can then be read from the fractional

p o we r No mo g r ap h No . 2 .

4 . The fac tor fs fo r ob ta ining lim its for 19 0 probab ility o f the s lope func tion S. This

f a c t o r i s

derived

f r o m

the approximation of L itchfield and Fertig to the standard error of

the slope constant, b.

Since:

s

=

1 / b a n d

2

=

1/b

By d i f f e r e n t i a t i o n:

ds

=

- b d b

-

- 1/b ) db

T hen by substi tuti on:

ds

-

d b o r

SE.1 - sS.E.b 4)

T h e mi n u s s i g n c a n be dropped because it merely signifies the reciprocal relation between

s a n d b .

T he approximation to the standard error of b for limits for 19/20 probability is given by:

7.85

1 . 9 6 S . E . b

=

77F L K/ K - 1

wh e r e L a n d K r e f er , respectively, to the logarithmic dosage range of the experiment and

t he n u mb e r o f d o s e s t es t ed . N i s t h e s a me a s d e f i n e d a b o v e . Th i s c a n

be simplif ied to

give:

1 1 . 1 K - 1 )

1 . 9 6 S . E . b

=

LK W

By substitution of 5 ) i n e q u a t i o n 4 ) :

and

r e wr i t i n g:

1 1 . 1 s K

-

1 ) 1 . 1 a

1 0 K

-

1 )

1 . 9 6 S. E. 1

=

LK

=

L X

1.1 8

L et: A

=

a n t i l o g

R

-

antilog L

and, since S

=

antilog s,

1 . 1 l o gS)

t he n A

-

a n t i l o g 6 )

l o g R

a n d f s

=

a n t i l o g 1.96S.E.,

=

A b 0 ( 1 ) / K % l

-

AS E P . 7 )

T h e v a l ue o f S i s k n o wn f r o m

equation 3) and the value of R is given by the ratio: R

-

largest/smallest dose plotted. I n order to eliminate the use of logarithms, N omograph

N o.3, having scale values in units of S and R ,

was

constructed to solve equatiou 6) to

give

t h e v a l ue o f A. K i s t h e n u mb e r o f

doses plotted and N

has a l r e a d y

been

d e f i ne d . T h e

f ac t o r f o r S c a n t he n b e r ea d f r o m No mo g r ap h No . 2 u s i n g t he v a l u e o f A a n d i t s ex p o n e n t .

T h e c o n f i de n c e l i mi t s f o r t he p a r ame t e r s ED, , a n d S a r e o b t a i n e d b y u s i n g

the f actors

f E D , , and fs as follow s:

Pa r a me t e r X f

=

upper

? limit for

19/20

probability.

Parameter/f

= lower

)


13/15

E V A L U A T IN G D O S E -E F F E C T E X P E R I M E N T S

11 1

C. A dd itions to the bas ic m e th od . 1 . U se o f 0 and 100 per cen t e ffec ts . Fisher (cited by

B liss , 1 ) has sho w n that the m o st like ly value fo r 0 or

10 0

per cent e ffects is the m inim al o r

m axim al correc ted probit, the exac t value of w hich is de term ined by the expec ted pro bit

obtained from the line on the lo g do 8e probit graph. The equivalent procedure for 0 or 1 00

pe r c e n t e f f ec t i n t he c a s e of p l o t t i ng o n l o ga r i t hmi c p r ob a b i l i t y p a p e r i s m ade po ssible by

me a n s of a t a b l e r e l a t i n g t he e x p e c t e d

per cent e ffect, indicated by the line , to the m inim al

or ma x i ma l c or r e c t e d p e r c e nt e f f ec t . T he s e c o r r e c t e d v a l u e s ha v e b e e n i n t e r pol a t e d a n d

c o nv e r t e d to percentages fro m the o rig inal table o f correc ted probits (1 ).

Th e pr oc e d ur e f or us i ng 0 or

10 0

p e r c e nt e f f ec t s c on s i s t s o f : a ) p l o t t i n g t he d a t a on

l o ga r i t h mi c probability paper, om itting 0 or 10 0 per cent e ffects , and fitting a tem porary

line w ith transparent straight edge o r triang le ; (b) reading the ex pec ted per cent effec t

indicated by the s traight edg e at do ses w here 0 o r

1 00 per cent e ffec t w as observed; (c ) con-

verting the ex pec ted per cent e ffec t to a correc ted v alue by m eans of table 1 and plo tting this

co rrected v alue; (d) draw ing a line throug h the com ple tely plo tted data.

2 .

R eco gn itio n o f he te rogeneou s da ta : T es t o f the lin e for G ood ness o f F it.

The no m o-

graphic calculation of (Chi) , prev io usly repo rted by W ilcox on and M cCallan (14) , w as

m o dif ied slig htly and inco rporated into the rev ised m ethod. In this portio n o f the pro-

cedure , the expec ted per cent effec ts from the line are listed opposite the observed

per cent

e ffect and a list o f differences be tw een observed and ex pec ted per cent e ffec ts m ade . Fo r

each se t o f a difference and the corresponding expec ted per cent e ffec t, a (C hi) value based

on one anim al or item is read from N om ograph N o . 1. The to tal o f these (Chi) values

m ultiplied by the average num ber of anim als

o r item s per dose is

the (Chi) o f the dose .

e f f ec t l i ne . T he de gr ee s o f f r e e d om, n , a r e

tw o less than the num ber of po ints plo tted, i.e .,

n K 2 . B y co m parison of this to the value o f (C hi) for probability o f .05 and n degrees

of freedom , signif icant heterogene ity can be recognized. In the event of s ignificant hetero-

gene ity a better f itting line can o ften be draw n, and if not, the equations for the facto rs of

the param eters are m odified to inc lude this additio nal variation. For this m odificatio n

the value of students t for a probability o f .05 and n degrees o f freedom m ust be used.

For convenience in using the m ethod, the im portant values of students t, and (Chi)

fo r p .05 and various degrees of freedom are g iven in table 2 . Other values m ay be found

in m ore extens iv e tables such as are g iven by S nedecor (1 5).

W hen significant heterogene ity is found the facto rs of the param eters are obtained by the

fo llo w ing equations , w hose nom og raphic so lutio n is the sam e, how ever, as described above .

Fo r heterog eneous data

fE D , 0 S1 . 4 t / ( (1 ) t /N = 5 0 x P .

8 )

fs

A 1 1 t

-1)v(Cbr,N]/K

= EP

(9 )

A ll

sy m bo ls hav e the sam e sig nif icance as no ted abov e.

T he c ha ng e whi c h has been m ade i n t he e x p on e n t s o f bo t h o f the factors in order to co r-

rect for heterog ene ity is the co nventional m ultiplier (16) w hile the v alue of t replaces

the 1 .9 6 w hich w as prev io usly introduced into the expo nent and m ust now be div ided o ut

ag ain. Thus, for lim its fo r 1 9 /20 pro bability ,

ts,f(Chi)/n

X 1

Ex p. , , , , . , 0 5 01 1 .

N o m ograph N o . 1 co m putes (C hi) for a sing le item as:

( Ch #{ 1 49 } ) ( Ob se rv ed ex pected per cent e ffec t) (10)

- (Expected effec t) (100-expected effec t)

w hich is deriv ed from the expression used by W ilcox on and M cC allan (14) for nom o graphic

so lution of (C hi) for 10 0 item s.


14/15

1 1 2

J . T . L ITC HP IE LD ,

JR .

A N D F.

W . W IL C O X O N

I

3 .

Compar ison of two dose-effect curves: The test for parallelism and the ratio of p0-

tencies.

I n t h e me t h o d o f Bl i s s 1 ) a n d t h e a p p r o x i ma t e me t h o d o f L i t c h f i e l d a n d F e r t i g

1 3 ) , t h e s i g n i f i c a n c e o f d i f f e r e n c e s b e t we e n t h e p a r a me t e r s o f t wo curves wa s

tested

b y

c o mp u t i ng t h e s t a n d a r d e r r o r o f t h e d i f f e r e n c e a s :

S .E .D I f t . -

I n t h e r e v i s e d me t h o d , t h e e q u i v a l e n t a r i t h me t i c p r oc e d u r e i s u s e d ; t h a t i s , i n p l a c e o f a

difference betw een tw o lo garithm ic quantitie s , the ratio of the quantitie s them se lves is

used. Furtherm ore, in place o f the s tandard error of a lo g quantity , the factor of the

quantity itse lf is used s ince , as already noted, the fac tor is the antilog of the S tandard Error.

Thus, for lim its for 19 /20 p r o b a b i l i t y , i n p l a c e o f I .96S .E .D lf f .

=

s , / l o g

fi)

+ (lo g f ,) , the

f ol l o wi n g e q u a t i o n i s u s e d :

f 1 t i o - antilog /( log f,) + l o g f , )

To elim inate lo garithm s, N om ograph N o.

4 wa s c o n s t r u c t e d h a v i n g s c a l e v a l u e s i n u n i t s of

f, and f, w hich, if co nnected by a straight edge , perm it reading f u 0 o n t h e i n t e r s e c t e d c e n -

t e r s c a l e . T h e p r oc e d u r e i s t h e sam e for both the fac tor o f the ratio o f the slope func tions

1 or the ratio of potenc ies ED ,,1 1ED ,o ,.

Th e fac tor o f the ratio m ay be used as already described t o o b t a i n t h e l i mi t s o f t h e ratio

f o r

19 /20 probability . If this is done, it is ev ident that if the low er lim it is g reater

than 1 . 0

the ratio is significant. Ho w ever, the low er lim it c a n e x c e e d 1 . 0 o n l y i f t h e v a l u e o f t h e

ratio exceeds that o f the factor.

Therefore , tw o curv es m ay be cons idered parallel if S .R ., the slope functio n ratio ,

does

no t exceed

i ts f ac to r, f s . R . ; a n d t wo potencies ma y b e c o n s i de r e d s i gn i f i c a n t l y d i f f e r e n t i f

P . R. , t h e p o t e n c y

ratio,

exceeds

its

factor f p . .

4 .

Confidence limits for 19/20 probability of doses other than the median effective dose,

ED, , .

A satisfac tory approx im ation to the lim its o f errors of ED (w here Y is a response

o t h e r t h a n 5 0 per cent) c a n be deriv ed from the expression for the (standard error) o f a log

d o s e wh o s e p r o b i t r e s po n s e y dev iate s fro m the m ean probit, 5 o f t h e experiment. This

ex press ion w hich B liss (1 ) g ives can be w ritten as:

S.E.D 7

=

(S.E.bs2)2(y 5k)

+

V 1 s

It has been show n that:

S . E . b s =

F u r t h e r mo r e , t h e f i r s t o f t h e a p p r o x i ma t i o n s o f t h e L i t c h f i e l d a n d F e r t i g me t h o d wa s

derived as:

V, s 2 S.E.?o,ED,,

B y substitution of 4 ) , 1 A) , and the app ro xi mati on :

(y r (y 5 .0 ) = X

e q u a t i on 1 2 ) b e c o me s :

S E ? o g E D y = ES .E..) (X )] + S.E ogxn,,

wh i c h is the sam e as:

l o g f E D ,) = l o g [ ( f5 ) X J ) 2 + ( log f E D, , )

t h e a n t i l o g a r i t h m o f t h e r oo t o f this expression is in a form suitable for use in the rev ised

m etho d dealing w ith arithm etic doses and per c e n t r e s po n s e s , n a me l y :

f E D ,

=

antilog V (lo g [fJ) + (log fE D, 0 ) 2


15/15

E V A L U A T I N G D O S E -E F F E C T E X P E R I M E N T S

1 1 3

s i n c e No mo g r a p h No . 4 so lv es ex p ress io n s o f th is ty p e . T h e v alu e o f X can b e read f ro m

an y tab le relatin g d ev iatio n s an d areas o f th e n o rm al cu rv e (1 7 ). T h u s, 1 6 an d 8 4 p er cen t.

ef f e ct b o th d ep art f ro m 5 0 b y 3 4 . T h e area, 3 4 p er cen t, co rre sp o n d s to a d ev iatio n X , o f

1 .0 0

in the tab le c ited abo v e . Fo r con v en ience certain com m on ly u sed v alu es o f X

a r e g iven

in tab le 3 .

In certain cases o n e m ay d es ire to ad o p t co n f id en ce lim its f o r p ro b ab ilitie s o th er th an

1 9 /2 0 . T h is m ay b e d o n e eas ily b y u sin g th e p ro p er m u ltip lie r f o r th e ex p o n en ts o f S an d A

b ef ore re ad in g

f E D ,0 an d f 5 f ro m N o m o g rap h N o . 2 . Fo r p ro b ab ilitie s co m m o n ly u sed th ese

m u ltip lie rs are lis ted b e lo w :

Fo r

p of:

M ul tiply exponent by:

0 . 3 2 2 / 3 o d d s 0 . 5 1

0 . 1 0 9 / 1 0 o d d s 0 .8 4

0 . 0 5 1 9 / 2 0 o d d s 1 .0 0

0 . 0 2 (4 9 /5 0 o d ds 1 . 1 9

0 . 0 1 (9 9 /1 0 0 o d d s) 1 . 3 1

0 .0 0 1 (9 9 9 /1 0 0 0 o d d s) 1 .6 8

R E F E R E N C E S

(1 ) B L I S S ,

C . I.: Q uart.

J. Ph arm . an d Ph arm aco l., 1 1 : 1 9 2 , 1 9 3 8 .

(2 ) B E R K 5 O N , J.: J. A m . S tat. A sso c., 3 9: 3 57 , 1 9 4 4 ; 4 1 : 4 0 , 1 9 4 6 .

(3 )

K N U D S E N ,

L. F . , A N D Cu RTI s , J . M : J . Am. S t a t . A sso c ., 4 2 :

282,

1 9 4 7 .

4 DRAGSTEDP, C. A. ,

A N D L A N G ,

V. F . : T a is J O U R N A L , 3 2: 2 1 5, 1 9 2 8 .

(5 )

BEHEENS,

B .: A rch . ex per. Path . u . Ph arm ak o l.,

1 4 0 :

237,

1 9 2 9 .

6

R E E D ,

L. J . : Bi o l o g i c a l

E f f ec ts o f R ad iation , 227 -5 1 , 193 6 , M cG raw -H ill B ook C o., Inc .

(7 ) R E E D , L. J . ,

AN D

MUENCH, H. : Am. J. H y g ien e , 2 7 : 4 9 3 , 1 9 3 8 .

8 K A R B E R , G. : Ar c h . e x p e r .

Path . u . Ph arm ak o l., 162 : 4 8 0 , 1931 .

(9 ) I R W I N ,

J . 0 . ,

A N D C HE ES E MA N, E . A .: J. H y g iene , C o m b. 39 :

5 74 , 1 93 9 .

1 0

T H O M P S O N ,

W R. : Ba c t . Re v i e ws , 1 1 : 1 1 5 , 1 9 4 7 .

1 1

M I L L E R ,

L. C. ,

A ND T AI N T ER ,

M L. : P r o c . S o c . e x p e r . Bi o l . a n d Me d . , 5 7 : 2 6 1 ,

1 9 4 4 .

1 2

D E B E E R , E . J.: T H IS J O U R N A L ,

8 5 : 1 , 1 9 4 5 .

(1 3) L IT CH FI E LD ,

J . T. , J R. , AND FERTI G, J. W .: B u ll. Jo h n s H o p k in s H o sp ., 6 9 : 2 7 6 , 1 9 4 1 .

(1 4) W iL cox oN , F., AN D

Mc CALLAN, S . E . A .: C o n trib . o f th e B o y ce T h o m p so n In s t., 1 0 :

3 2 9 , 1 9 3 9 .

(1 5 )

S N E D E C O R ,

G . W .: S tatis tical M e th o d s, Io w a S tate C o lleg e Press , A m es, Io w a, p ag es

5 8 a n d

163,

1 9 4 0 .

1 6 F IN N E Y , D. J . : Pr o b i t A n aly sis , C am b rid g e U n iv ersity Press, p ag e 6 0 , 1 9 4 7 .

(17) M ath em atical T ab le s f ro m H an d b o o k o f C h em is try an d Ph y s ics , C h em ical R u b b er

Pu b l i s h i ng Co mp a n y , Cl e v e l a n d , Oh i o , 7 t h Ed . , p . 2 0 0 , 1 9 4 1 . I n this t a b l e X i s

lis ted as t.

1 8 GADDUM J . H. : Me d .

R es . C ou nc il S p ec . R eport,

1 8 3 , 1 9 3 3 .

0 C T hD B

N C T IC E L: 7

u s

C O D

cci

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j Pharmacol Exp Ther 1949 Litchfield 99 113