Statistical Methods in Medical Research - part 10 docx

generally provides a close approximation to maximum likelihood. In Does et al.
(1988) the jackknife (see §10.7) is applied to reduce the bias in the maximum
likelihood estimate: this issue is also addressed by other authors (see Mehrabi &
Matthews, 1995, and references therein).
How the dilutions used in the experiment are chosen is another topic that has
received substantial attention. Fisher (1966, §68, although the remark is in much
earlier editions) noted that the estimate of log u would be at its most precise when
all observations are made at a dilution d  1Á59=u. Of course, this has limited
value in itself because the purpose of the assay is to determine the value of u,
which a priori is unknown. Nevertheless, several methods have been suggested
that attempt to combine Fisher's observation with whatever prior knowledge the
experimenter has about u. Fisher himself discussed geometric designs, i.e. those
with d
j
 c
jÀ1
d
1
, particularly those with c equal to 10 or a power of 2. Geometric
designs have been the basis of many of the subsequent suggestions: see, for
example, Abdelbasit and Plackett (1983) or Strijbosch et al. (1987). Mehrabi
and Matthews (1998) use a Bayesian approach to the problem; they found
optimal designs that did not use a geometric design but they also noted that
there were some highly efficient geometric designs. Most of the literature on the
design of limiting dilution assays has focused on obtaining designs that provide
precise estimates of u or log u. Two aspects of this ought to be noted. First, the
designs assume that the single-hit Poisson model is correct and some of the
designs offer little opportunity to verify this from the collected data. Secondly,
the experimenters are often more interested in mechanism than precision, i.e.
they want to know that, for example, the single-hit Poisson model applies, with
precise knowledge about the value of associated parameter being a secondary

matter. Although the design literature contains some general contribution in this
direction, there appears to be little specific to limiting dilution assays.
Mutagenicity assays
There is widespread interest in the potential that various chemicals have to cause
harm to people and the environment. Indeed, the ability of a chemical or other
agent to cause genetic mutationsÐfor example, by damaging an organism's
DNAÐis often seen as important evidence of possible carcinogenicity. Conse-
quently, many chemicals are subjected to mutagenicity assays to assess their
propensity to cause this kind of damage. There are many assays in this field
and most of these require careful statistical analysis. Here we give a brief
description of the statistical issues surrounding the commonest assay, the Ames
Salmonella microsome assay; more detailed discussion can be found in Kirkland
(1989) and in Piegorsch (1998) and references therein.
The Ames Salmonella microsome assay exposes the bacterium Salmonella
typhimurium to varying doses of the chemical under test. This organism cannot
20.5 Some special assays 737
synthesize histidine, an amino acid that is needed for growth. However, muta-
tions at certain locations of the bacterial genome reverse this inability, so, if the
bacteria are grown on a Petri dish or plate containing only minimal histidine,
then colonies will only grow from mutated cells. If there are more colonies on
plates subjected to greater concentrations of the chemical, then this provides
evidence that the mutation rate is dose-dependent. It is usual to have plates with
zero dose of the test chemicalÐnegative controlsÐand five concentrations of the
chemical. There may also be a positive controlÐa substance known to result in a
high rate of mutationsÐalthough this is often ignored in the analysis. It is also
usual to have two, three or even more replicates at each dose.
The data that arise in these assays comprise the number of mutants in the jth
replicate at the ith dose, Y
ij
, j  1, , r

i
, i  1, , D. A natural assumption is
that these counts will follow a Poisson distribution. This is because: (i) there are
large numbers of microbes placed on each plate and only a very small proportion
will mutate; and (ii) the microbes mutate independently of one another. How-
ever, it is also necessary for the environment to be similar between plates that are
replicates of the same dose of test chemical and also between these plates the
number of microbes should not vary.
If the Poisson assumption is tenable, then the analysis usually proceeds by
applying a test for trend across the increasing doses. In order to perform a test of
the null hypothesis of no change of mutation rate with dose against an alter-
native of the rate increasing with dose, it is necessary to associate an increasing
score, x
i
, with each dose group (often x
i
will be the dose or log dose given to that
group). The test statistic is
Z
P


D
i1
x
i
r
i
Y
i

ÀY

p
YS
2
x

, 20:27
where
Y
i


j
Y
ij
=r
i
,Y 

i
r
i
Y
i
=

i
r
i

and S
2
x


i
r
i
x
i
À

x
2
, with

x


i
r
i

x
i
=

i
r
i

. Under the null hypothesis, (20.27) has approximately a stand-
ard normal distribution.
This straightforward analysis is usually satisfactory, provided the conditions
outlined above to justify the assumption of a Poisson distribution hold and that
the mutation rate does increase monotonically with dose. It is quite common for
one of these not to hold, and in such cases alternative or amended analyses must
be sought.
The mutation rate is often found to drop at the highest doses. This can be due
to various mechanismsÐe.g. at the highest doses toxic effects of the chemical
under test may kill some microbes before they can mutate. Consequently a test
based on Z
P
may be substantially less powerful than it would be in the presence
of monotone dose response. A sophisticated approach to this problem is to
738 Laboratory assays
attempt to model the processes that lead to this downturn; see, for example,
Margolin et al. (1981) and Breslow (1984a). A simpler, but perhaps less satisfying
approach is to identify the dose at which the downturn occurs and to test for a
monotonic dose response across all lower doses; see, for example, Simpson and
Margolin (1986).
It is also quite common for the conditions relating to the assumptions of a
Poisson distribution not to be met. This usually seems to arise because the
variation between plates within a replicate exceeds what you would expect if
the counts on the plates all came from the same Poisson distribution. A test of
the hypothesis that all counts in the ith dose group come from the same Poisson
distribution can be made by referring

r
i
j1

Y
ij
ÀY
i

2
Y
i
to a x
2
distribution with r
i
À 1 degrees of freedom. A test across all dose groups
can be made by adding these quantities from i  1toD and referring the sum to
a x
2
distribution with

r
i
À D degrees of freedom.
A plausible mechanism by which the assumptions for a Poisson distribution
are violated is for the number of microbes put on each plate within a replicate to
vary. Suppose that the mutation rate at the ith dose is l
i
and the number of
microbes placed on the jth plate at this dose is N
ij
. If experimental technique
is sufficiently rigorous, then it may be possible to claim that the count N

ij
is
constant from plate to plate. If the environments of the plates are sufficien-
tly similar for the same mutation rate to apply to all plates in the ith dose
group, then it is likely that Y
ij
is Poisson, with mean l
i
N
ij
. However, it may be
more realistic to assume that the N
ij
vary about their target value, and vari-
ation in environments for the plates, perhaps small variations in incubation
temperatures, leads to mutation rates that also vary slightly about their
expected values. Conditional on these values, the counts from a plate will
still be Poisson, but unconditionally the counts will exhibit extra-Poisson varia-
tion.
In this area of application, extra-Poisson variation is often encountered. It is
then quite common to assume that the counts follow a negative binomial dis-
tribution. If the mean of this distribution is m, then the variance is m  am
2
, for
some non-negative constant a a  0 corresponds to Poisson variation). A crude
justification for this, albeit based in part on mathematical tractability, is that the
negative binomial distribution would be obtained if the l
i
N
ij

varied about their
expected values according to a gamma distribution.
If extra-Poisson variation is present, the denominator of the test statistic
(20.27) will tend to be too small and the test will be too sensitive. An amended
version is obtained by changing the denominator to 
p
Y1  ^aYS
2
x
, with ^a an
20.5 Some special assays 739
estimate of a obtained from the data using a method of moments or maximum
likelihood.
20.6 Tumour incidence studies
In §20.5, reference was made to the use of mutagenicity assays as possible
indicators of carcinogenicity. A more direct, although more time-consuming,
approach to the detection and measurement of carcinogenicity is provided by
tumour incidence experiments on animals. Here, doses of test substances are
applied to animals (usually mice or rats), and the subsequent development of
tumours is observed over an extended period, such as 2 years. In any one
experiment, several doses of each test substance may be used, and the design
may include an untreated control group and one or more groups treated with
known carcinogens. The aim may be merely to screen for evidence of carcino-
genicity (or, more properly, tumorigenicity), leading to further experimental
work with the suspect substances; or, for substances already shown to be carci-
nogenic, the aim may be to estimate `safe' doses at which the risk is negligible, by
extrapolation downwards from the doses actually used.
Ideally, the experimenter should record, for each animal, whether a tumour
occurs and, if so, the time of occurrence, measured from birth or some suitable
point, such as time of weaning. Usually, a particular site is in question, and only

the first tumour to occur is recorded. Different substances may be compared on
the `response' scale, by some type of measurement of the differences in the rate of
tumour production; or, as in a standard biological assay, on the `dose' scale, by
comparing dose levels of different substances that produce the same level of
tumour production.
The interpretation of tumour incidence experiments is complicated by a
number of practical considerations. The most straightforward situation arises
(i) when the tumour is detectable at a very early stage, either because it is easily
visible, as with a skin tumour, or because it is highly lethal and can be detected at
autopsy; and (ii) when the substance is not toxic for other reasons. In these
circumstances, the tumorigenic response for any substance may be measured
either by a simple `lifetime' count of the number of tumour-bearing animals
observed during the experiment, or by recording the time to tumour appearance
and performing a survival analysis. With the latter approach, deaths of animals
due to other causes can be regarded as censored observations, and the curve for
tumour-free survival estimated by life-table or parametric methods, as in Chap-
ter 17. Logrank methods may be used to test for differences in tumour-free
survival between different substances.
The simple lifetime count may be misleading if substances differ in their non-
tumour-related mortality. If, for instance, a carcinogenic substance is highly
lethal, animals may die at an early stage before the tumours have had a chance
740 Laboratory assays
to appear; the carcinogenicity will then be underestimated. One approach is to
remove from the denominator animals dying at an early stage (say, before the
first tumour has been detected in the whole experiment). Alternatively, in a life-
table analysis these deaths can be regarded as withdrawals and the animals
removed from the numbers at risk.
A more serious complication arises when the tumours are not highly lethal. If
they are completely non-lethal, and are not visible, they may be detected after the
non-specific deaths of some animals and subsequent autopsy, but a complete

count will require the sacrifice of animals either at the end of the experiment or
by serial sacrifice of random samples at intermediate times. The latter plan will
provide information on the time distribution of tumour incidence. The relevant
measures are now the prevalences of tumours at the various times of sacrifice,
and these can be compared by standard methods for categorical data
(Chapter 15).
In practice, tumours will usually have intermediate lethality, conforming
neither to the life-table model suitable for tumours with rapid lethality nor to
the prevalence model suitable for non-lethal tumours. If tumours are detected
only after an animal's death and subsequent autopsy, and a life-table analysis is
performed, bias may be caused by non-tumour-related mortality. Even if sub-
stances under test have the same tumorigenicity, a substance causing high non-
specific mortality will provide the opportunity for detection of tumours at
autopsy at early stages of the experiment, and will thus wrongly appear to
have a higher tumour incidence rate.
To overcome this problem, Peto (1974) and Peto et al. (1980) have suggested
that individual tumours could be classified as incidental (not affecting longevity
and observed as a result of death from unrelated causes) or fatal (affecting
mortality). Tumours discovered at intermediate sacrifice, for instance, are inci-
dental. Separate analyses would then be based on prevalence of incidental
tumours and incidence of fatal tumours, and the contrasts between treatment
groups assessed by combining data from both analyses. This approach may be
impracticable if pathologists are unable to make the dichotomous classification
of tumours with confidence.
Animal tumour incidence experiments have been a major instrument in the
assessment of carcinogenicity for more than half a century. Recent research on
methods of analysis has shown that care must be taken to use a method appro-
priate to the circumstances of the particular experiment. In some instances it will
not be possible to decide on the appropriate way of handling the difficulties
outlined above, and a flexible approach needs to be adopted, perhaps with

alternative analyses making different assumptions about the unknown factors.
For more detailed discussion, see Peto et al. (1980) and Dinse (1998).
20.6 Tumour incidence studies 741
Appendix tables
743
z
P
2
1
Table A1 Areas in tail of the normal distribution
Single-tail areas in terms of standardized deviates. The function tabulated is
1
2
P, the probability of obtaining a
standardized normal deviate greater than z, in one direction. The two-tail probability, P, is twice the tabulated value.
z 0Á00 0Á01 0Á02 0Á03 0Á04 0Á05 0Á06 0Á07 0Á08 0Á09
0Á00Á5000 0Á4960 0Á4920 0Á4880 0Á4840 0Á4801 0Á4761 0Á4721 0Á4681 0Á4641
0Á10Á4602 0Á4562 0Á4522 0Á4483 0Á4443 0Á4404 0Á4364 0Á4325 0Á4286 0Á4247
0Á20Á4207 0Á4168 0Á4129 0Á4090 0Á4052 0Á4013 0Á3974 0Á3936 0Á3897 0Á3859
0Á30Á3821 0Á3783 0Á3745 0Á3707 0Á3669 0Á3632 0Á3594 0Á3557 0Á3520 0Á3483
0Á40Á3446 0Á3409 0Á3372 0Á3336 0Á3300 0Á3264 0Á3228 0Á3192 0Á3156 0Á3121
0Á50Á3085 0Á3050 0Á3015 0Á2981 0Á2946 0Á2912 0Á2877 0Á2843 0Á2810 0Á2776
0Á60Á2743 0Á2709 0Á2676 0Á2643 0Á2611 0Á2578 0Á2546 0Á2514 0Á2483 0Á2451
0Á70Á2420 0Á2389 0Á2358 0Á2327 0Á2296 0Á2266 0Á2236 0Á2206 0Á2177 0Á2148
0Á80Á2119 0Á2090 0Á2061 0Á2033 0Á2005 0Á1977 0Á1949 0Á1922 0Á1894 0Á1867
0Á90Á1841 0Á1814 0Á1788 0Á1762 0Á1736 0Á1711 0Á1685 0Á1660 0Á1635 0Á1611
1Á00Á1587 0Á1562 0Á1539 0Á1515 0Á1492 0Á1469 0Á1446 0Á1423 0Á1401 0Á1379
1Á10Á1357 0Á1335 0Á1314 0Á1292 0Á1271 0Á1251 0Á1230 0Á1210 0Á1190 0Á1170
1Á20Á1151 0Á1131 0Á1112 0Á1093 0Á1075 0Á1056 0Á1038 0Á1020 0Á1003 0Á0985

1Á30Á0968 0Á0951 0Á0934 0Á0918 0Á0901 0Á0885 0Á0869 0Á0853 0Á0838 0Á0823
1Á40Á0808 0Á0793 0Á0778 0Á0764 0Á0749 0Á0735 0Á0721 0Á0708 0Á0694 0Á0681
1Á50Á0668 0Á0655 0Á0643 0Á0630 0Á0618 0Á0606 0Á0594 0Á0582 0Á0571 0Á0559
1Á60Á0548 0Á0537 0Á0526 0Á0516 0Á0505 0Á0495 0Á0485 0Á0475 0Á0465 0Á0455
1Á70Á0446 0Á0436 0Á0427 0Á0418 0Á0409 0Á0401 0Á0392 0Á0384 0Á0375 0Á0367
1Á80Á0359 0Á0351 0Á0344 0Á0336 0Á0329 0Á0322 0Á0314 0Á0307 0Á0301 0Á0294
1Á90Á0287 0Á0281 0Á0274 0Á0268 0Á0262 0Á0256 0Á0250 0Á0244 0Á0239 0Á0233
744 Appendix tables
2Á00Á02275 0Á02222 0Á02169 0Á02118 0Á02068 0Á02018 0Á01970 0Á01923 0Á01876 0Á01831
2Á10Á01786 0Á01743 0Á01700 0Á01659 0Á01618 0Á01578 0Á01539 0Á01500 0Á01463 0Á01426
2Á20Á01390 0Á01355 0Á01321 0Á01287 0Á01255 0Á01222 0Á01191 0Á01160 0Á01130 0Á01101
2Á30Á01072 0Á01044 0Á01017 0Á00990 0Á00964 0Á00939 0Á00914 0Á00889 0Á00866 0Á00842
2Á40Á00820 0Á00798 0Á00776 0Á00755 0Á00734 0Á00714 0Á00695 0Á00676 0Á00657 0Á00639
2Á50Á00621 0Á00604 0Á00587 0Á00570 0Á00554 0Á00539 0Á00523 0Á00508 0Á00494 0Á00480
2Á60Á00466 0Á00453 0Á00440 0Á00427 0Á00415 0Á00402 0Á00391 0Á00379 0Á00368 0Á00357
2Á70Á00347 0Á00336 0Á00326 0Á00317 0Á00307 0Á00298 0Á00289 0Á00280 0Á00272 0Á00264
2Á80Á00256 0Á00248 0Á00240 0Á00233 0Á00226 0Á00219 0Á00212 0Á00205 0Á00199 0Á00193
2Á90Á00187 0Á00181 0Á00175 0Á00169 0Á00164 0Á00159 0Á00154 0Á00149 0Á00144 0Á00139
3Á00Á00135
3Á10Á00097
3Á20Á00069 Standardized deviates in terms of two-tail areas
3Á30Á00048 P 1Á00Á90Á80Á70Á60Á50Á4
3Á40Á00034 z 00Á126 0Á253 0Á385 0Á524 0Á674 0Á842
3Á50Á00023 P 0Á30Á20Á10Á05 0Á02 0Á01 0Á001
3Á60Á00016 z 1Á036 1Á282 1Á645 1Á960 2Á326 2Á576 3Á291
3Á70Á00011
3Á80Á00007
3Á90Á00005
4Á00Á00003
Reproduced in part from Table 3 of Murdoch and Barnes (1968) by permission of the authors and publishers.

Appendix tables 745
χ
2
ν,
P
P
V
Table A2 Percentage points of the x
2
distribution
The function tabulated is x
2
v
,P
, the value exceeded with probability P in a x
2
distribution with n degrees of
freedom (the 100P percentage point).
Probability of greater value, P
Degrees of
freedom, n
0Á975 0Á900 0Á750 0Á500 0Á250 0Á100 0Á050 0Á025 0Á010 0Á001
1Ð0Á02 0Á10 0Á45 1Á322Á71 3Á84 5Á02 6Á63 10Á83
20Á05 0Á21 0Á58 1Á392Á77 4Á61 5Á99 7Á389Á21 13Á82
30Á22 0Á58 1Á21 2Á374Á11 6Á25 7Á81 9Á3511Á3416Á27
40Á48 1Á06 1Á92 3Á365Á397Á78 9Á49 11Á14 13Á28 18Á47
50Á83 1Á61 2Á67 4Á356Á63 9Á24 11Á07 12Á83 15Á09 20Á52
61Á24 2Á20 3Á45 5Á357Á84 10Á64 12Á59 14Á45 16Á81 22Á46
71Á69 2Á83 4Á25 6Á359Á04 12Á02 14Á07 16Á01 18Á48 24Á32
82Á18 3Á49 5Á07 7Á3410Á22 13Á3615Á51 17Á53 20Á09 26Á12

92Á70 4Á17 5Á90 8Á3411Á3914Á68 16Á92 19Á02 21Á67 27Á88
10 3Á25 4Á87 6Á74 9Á3412Á55 15Á99 18Á3120Á48 23Á21 29Á59
11 3Á82 5Á58 7Á58 10Á3413Á70 17Á28 19Á68 21Á92 24Á72 31Á26
12 4Á40 6Á308Á44 11Á3414Á85 18Á55 21Á03 23Á3426Á22 32Á91
13 5Á01 7Á04 9Á3012Á3415Á98 19Á81 22Á3624Á74 27Á69 34Á53
14 5Á63 7Á79 10Á17 13Á3417Á12 21Á06 23Á68 26Á12 29Á14 36Á12
15 6Á27 8Á55 11Á04 14Á3418Á25 22Á3125Á00 27Á49 30Á58 37Á70
746 Appendix tables
16 6Á91 9Á3111Á91 15Á3419Á3723Á54 26Á3028Á85 32Á00 39Á25
17 7Á56 10Á09 12Á79 16Á3420Á49 24Á77 27Á59 30Á19 33Á41 40Á79
18 8Á23 10Á86 13Á68 17Á3421Á60 25Á99 28Á87 31Á53 34Á81 42Á31
19 8Á91 11Á65 14Á56 18Á3422Á72 27Á20 30Á14 32Á85 36Á19 43Á82
20 9Á59 12Á44 15Á45 19Á3423Á83 28Á41 31Á41 34Á17 37Á57 45Á32
21 10Á28 13Á24 16Á3420Á3424Á93 29Á62 32Á67 35Á48 38Á93 46Á80
22 10Á98 14Á04 17Á24 21Á3426Á04 30Á81 33Á92 36Á78 40Á29 48Á27
23 11Á69 14Á85 18Á14 22Á3427Á14 32Á01 35Á17 38Á08 41Á64 49Á73
24 12Á40 15Á66 19Á04 23Á3428Á24 33Á20 36Á42 39Á3642Á98 51Á18
25 13Á12 16Á47 19Á94 24Á3429Á3434Á3837Á65 40Á65 44Á3152Á62
26 13Á84 17Á29 20Á84 25Á3430Á43 35Á56 38Á89 41Á92 45Á64 54Á05
27 14Á57 18Á11 21Á75 26Á3431Á53 36Á74 40Á11 43Á19 46Á96 55Á48
28 15Á3118Á94 22Á66 27Á3432Á62 37Á92 41Á3444Á46 48Á28 56Á89
29 16Á05 19Á77 23Á57 28Á3433Á71 39Á09 42Á56 45Á72 49Á59 58Á30
3016Á79 20Á60 24Á48 29Á3434Á80 40Á26 43Á77 46Á98 50Á89 59Á70
40 24Á43 29Á05 33Á66 39Á3445Á62 51Á80 55Á76 59Á3463Á69 73Á40
50 32Á3637Á69 42Á94 49Á33 56Á33 63Á17 67Á50 71Á42 76Á15 86Á66
60 40Á48 46Á46 52Á29 59Á33 66Á98 74Á40 79Á08 83Á3088Á3899Á61
70 48Á76 55Á33 61Á70 69Á33 77Á58 85Á53 90Á53 95Á02 100Á42 112Á32
80 57Á15 64Á28 71Á14 79Á33 88Á13 96Á58 101Á88 106Á63 112Á33 124Á84
90 65Á65 73Á29 80Á62 89Á33 98Á64 107Á56 113Á14 118Á14 124Á12 137Á21
100 74Á22 82Á3690Á13 99Á33 109Á14 118Á50 124Á34 129Á56 135Á81 149Á45

Condensed from Table 8 of Pearson and Hartley (1966) by permission of the authors and publishers.
Appendix tables 747
t
υ,

P
−t
υ,

P
Total for
both tails, P
Table A3 Percentage points of the t distribution
The function tabulated is t
n,P
, the value exceeded in both directions with probability P in a t distribution with n degrees of
freedom (the 100P percentage point).
Degrees of
freedom, n
Probability of greater value, P
0Á90Á80Á70Á60Á50Á40Á30Á20Á10Á05 0Á02 0Á01 0Á001
10Á158 0Á325 0Á510 0Á727 1Á000 1Á376 1Á963 3Á078 6Á314 12Á706 31Á821 63Á657 636Á619
20Á142 0Á289 0Á445 0Á617 0Á816 1Á061 1Á386 1Á886 2Á920 4Á303 6Á965 9Á925 31Á598
30Á137 0Á277 0Á424 0Á584 0Á765 0Á978 1Á250 1Á638 2Á353 3Á182 4Á541 5Á841 12Á924
40Á134 0Á271 0Á414 0Á569 0Á741 0Á941 1Á190 1Á533 2Á132 2Á776 3Á747 4Á604 8Á610
50Á132 0Á267 0Á408 0Á559 0Á727 0Á920 1Á156 1Á476 2Á015 2Á571 3Á365 4Á032 6Á869
60Á131 0Á265 0Á404 0Á553 0Á718 0Á906 1Á134 1Á440 1Á943 2Á447 3Á143 3Á707 5Á959
70Á130 0Á263 0Á402 0Á549 0Á711 0Á896 1Á119 1Á415 1Á895 2Á365 2Á998 3Á499 5Á408
80Á130 0Á262 0Á399 0Á546 0Á706 0Á889 1Á108 1Á397 1Á860 2Á306 2Á896 3Á355 5Á041
90Á129 0Á261 0Á398 0Á543 0Á703 0Á883 1Á100 1Á383 1Á833 2Á262 2Á821 3Á250 4Á781

10 0Á129 0Á260 0Á397 0Á542 0Á700 0Á879 1Á093 1Á372 1Á812 2Á228 2Á764 3Á169 4Á587
11 0Á129 0Á260 0Á396 0Á540 0Á697 0Á876 1Á088 1Á363 1Á796 2Á201 2Á718 3Á106 4Á437
12 0Á128 0Á259 0Á395 0Á539 0Á695 0Á873 1Á083 1Á356 1Á782 2Á179 2Á681 3Á055 4Á318
13 0Á128 0Á259 0Á394 0Á538 0Á694 0Á870 1Á079 1Á350 1Á771 2Á160 2Á650 3Á012 4Á221
14 0Á128 0Á258 0Á393 0Á537 0Á692 0Á868 1Á076 1Á345 1Á761 2Á145 2Á624 2Á977 4Á140
15 0Á128 0Á258 0Á393 0Á536 0Á691 0Á866 1Á074 1Á341 1Á753 2Á131 2Á602 2Á947 4Á073
748 Appendix tables
16 0Á128 0Á258 0Á392 0Á535 0Á690 0Á865 1Á071 1Á337 1Á746 2Á120 2Á583 2Á921 4Á015
17 0Á128 0Á257 0Á392 0Á534 0Á689 0Á863 1Á069 1Á333 1Á740 2Á110 2Á567 2Á898 3Á965
18 0Á127 0Á257 0Á392 0Á534 0Á688 0Á862 1Á067 1Á330 1Á734 2Á101 2Á552 2Á878 3Á922
19 0Á127 0Á257 0Á391 0Á533 0Á688 0Á861 1Á066 1Á328 1Á729 2Á093 2Á539 2Á861 3Á883
20 0Á127 0Á257 0Á391 0Á533 0Á687 0Á860 1Á064 1Á325 1Á725 2Á086 2Á528 2Á845 3Á850
21 0Á127 0Á257 0Á391 0Á532 0Á686 0Á859 1Á063 1Á323 1Á721 2Á080 2Á518 2Á831 3Á819
22 0Á127 0Á256 0Á390 0Á532 0Á686 0Á858 1Á061 1Á321 1Á717 2Á074 2Á508 2Á819 3Á792
23 0Á127 0Á256 0Á390 0Á532 0Á685 0Á858 1Á060 1Á319 1Á714 2Á069 2Á500 2Á807 3Á767
24 0Á127 0Á256 0Á390 0Á531 0Á685 0Á857 1Á059 1Á318 1Á711 2Á064 2Á492 2Á797 3Á745
25 0Á127 0Á256 0Á390 0Á531 0Á684 0Á856 1Á058 1Á316 1Á708 2Á060 2Á485 2Á787 3Á725
26 0Á127 0Á256 0Á390 0Á531 0Á684 0Á856 1Á058 1Á315 1Á706 2Á056 2Á479 2Á779 3Á707
27 0Á127 0Á256 0Á389 0Á531 0Á684 0Á855 1Á057 1Á314 1Á703 2Á052 2Á473 2Á771 3Á690
28 0Á127 0Á256 0Á389 0Á530 0Á683 0Á855 1Á056 1Á313 1Á701 2Á048 2Á467 2Á763 3Á674
29 0Á127 0Á256 0Á389 0Á530 0Á683 0Á854 1Á055 1Á311 1Á699 2Á045 2Á462 2Á756 3Á659
300Á127 0Á256 0Á389 0Á530 0Á683 0Á854 1Á055 1Á310 1Á697 2Á042 2Á457 2Á750 3Á646
40 0Á126 0Á255 0Á388 0Á529 0Á681 0Á851 1Á050 1Á303 1Á684 2Á021 2Á423 2Á704 3Á551
60 0Á126 0Á254 0Á387 0Á527 0Á679 0Á848 1Á046 1Á296 1Á671 2Á000 2Á390 2Á660 3Á460
120 0Á126 0Á254 0Á386 0Á526 0Á677 0Á845 1Á041 1Á289 1Á658 1Á980 2Á358 2Á617 3Á373
I 0Á126 0Á253 0Á385 0Á524 0Á674 0Á842 1Á036 1Á282 1Á645 1Á960 2Á326 2Á576 3Á291
Reproduced from Table III of Fisher and Yates (1963) by permission of the authors and publishers.
Appendix tables 749
P
F

P,v
1
,v
2
Table A4 Percentage points of the F distribution
The function tabulated is F
P
, n
1
, n
2
, the value exceeded with probability P in the F distribution with n
1
degrees of freedom
for the numerator and n
2
degrees of freedom for the denominator (the 100P percentage point). The values for P  0Á05 and
0Á01 are shown in bold type.
DF for
denominator, n
2
DF for numerator, n
1
P 123456781224I
10Á05 161Á4 199Á5 215Á7 224Á6 230Á2 234Á0 236Á8 238Á9 243Á9 249Á1 254Á3
0Á025 647Á8 799Á5 864Á2 899Á6 921Á8937Á1 948Á2 956Á7 976Á7 997Á2 1018
0Á01 4057 5000 5403 5625 5764 5859 5928 5981 6106 6235 6366
0Á005 16211 20000 21615 22500 23056 23437 23715 23925 24426 24940 25465
20Á05 18Á51 19Á00 19Á16 19Á25 19Á30 19Á33 19Á35 19Á37 19Á41 19Á45 19Á50
0Á025 38Á51 39Á00 39Á17 39Á25 39Á3039Á33 39Á3639Á3739Á41 39Á46 39Á50

0Á01 98Á50 99Á00 99Á17 99Á25 99Á30 99Á33 99Á36 99Á37 99Á42 99Á46 99Á50
0Á005 198Á5 199Á0 199Á2 199Á2 199Á3 199Á3 199Á4 199Á4 199Á4 199Á5 199Á5
30Á05 10Á13 9Á55 9Á28 9Á12 9Á01 8Á94 8Á89 8Á85 8Á74 8Á64 8Á53
0Á025 17Á44 16Á04 15Á44 15Á10 14Á88 14Á73 14Á62 14Á54 14Á3414Á12 13Á90
0Á01 34Á12 30Á82 29Á46 28Á71 28Á24 27Á91 27Á67 27Á49 27Á05 26Á60 26Á13
0Á005 55Á55 49Á80 47Á47 46Á19 45Á3944Á84 44Á43 44Á13 43Á3942Á62 41Á83
40Á05 7Á71 6Á94 6Á59 6Á39 6Á26 6Á16 6Á09 6Á04 5Á91 5Á77 5Á63
0Á025 12Á22 10Á65 9Á98 9Á60 9Á369Á20 9Á07 8Á98 8Á75 8Á51 8Á26
0Á01 21Á20 18Á00 16Á69 15Á98 15Á52 15Á21 14Á98 14Á80 14Á37 13Á93 13Á46
0Á005 31Á33 26Á28 24Á26 23Á15 22Á46 21Á97 21Á62 21Á3520Á70 20Á03 19Á32
750 Appendix tables
50Á05 6Á61 5Á79 5Á41 5Á19 5Á05 4Á95 4Á88 4Á82 4Á68 4Á53 4Á36
0Á025 10Á01 8Á43 7Á76 7Á397Á15 6Á98 6Á85 6Á76 6Á52 6Á28 6Á02
0Á01 16Á26 13Á27 12Á06 11Á39 10Á97 10Á67 10Á46 10Á29 9Á89 9Á47 9Á02
0Á005 22Á78 18Á3116Á53 15Á56 14Á94 14Á51 14Á20 13Á96 13Á3812Á78 12Á14
60Á05 5Á99 5Á14 4Á76 4Á53 4Á39 4Á28 4Á21 4Á15 4Á00 3Á84 3Á67
0Á025 8Á81 7Á26 6Á60 6Á23 5Á99 5Á82 5Á70 5Á60 5Á375Á12 4Á85
0Á01 13Á75 10Á92 9Á78 9Á15 8Á75 8Á47 8Á26 8Á10 7Á72 7Á31 6Á88
0Á005 18Á63 14Á54 12Á92 12Á03 11Á46 11Á07 10Á79 10Á57 10Á03 9Á47 8Á88
70Á05 5Á59 4Á74 4Á35 4Á12 3Á97 3Á87 3Á79 3Á73 3Á57 3Á41 3Á23
0Á025 8Á07 6Á54 5Á89 5Á52 5Á29 5Á12 4Á99 4Á90 4Á67 4Á42 4Á14
0Á01 12Á25 9Á55 8Á45 7Á85 7Á46 7Á19 6Á99 6Á84 6Á47 6Á07 5Á65
0Á005 16Á24 12Á40 10Á88 10Á05 9Á52 9Á16 8Á89 8Á68 8Á18 7Á65 7Á08
80Á05 5Á32 4Á46 4Á07 3Á84 3Á69 3Á58 3Á50 3Á44 3Á28 3Á12 2Á93
0Á025 7Á57 6Á06 5Á42 5Á05 4Á82 4Á65 4Á53 4Á43 4Á20 3Á95 3Á67
0Á01 11Á26 8Á65 7Á59 7Á01 6Á63 6Á37 6Á18 6Á03 5Á67 5Á28 4Á86
0Á005 14Á69 11Á04 9Á60 8Á81 8Á307Á95 7Á69 7Á50 7Á01 6Á50 5Á95
90Á05 5Á12 4Á26 3Á86 3Á63 3Á48 3Á37 3Á29 3Á23 3Á07 2Á90 2Á71
0Á025 7Á21 5Á71 5Á08 4Á72 4Á48 4Á324Á20 4Á10 3Á87 3Á61 3Á33
0Á01 10Á56 8Á02 6Á99 6Á42 6Á06 5Á80 5Á61 5Á47 5Á11 4Á73 4Á31

0Á005 13Á61 10Á11 8Á72 7Á96 7Á47 7Á13 6Á88 6Á69 6Á23 5Á73 5Á19
10 0Á05 4Á96 4Á10 3Á71 3Á48 3Á33 3Á22 3Á14 3Á07 2Á91 2Á74 2Á54
0Á025 6Á94 5Á46 4Á83 4Á47 4Á24 4Á07 3Á95 3Á85 3Á62 3Á373Á08
0Á01 10Á04 7Á56 6Á55 5Á99 5Á64 5Á39 5Á20 5Á06 4Á71 4Á33 3Á91
0Á005 12Á83 9Á43 8Á08 7Á346Á87 6Á54 6Á306Á12 5Á66 5Á17 4Á64
Continued on p. 752
Appendix tables 751
Table A4 (continued)
DF for
denominator, n
2
DF for numerator, n
1
P 123456781224I
12 0Á05 4Á75 3Á89 3Á49 3Á26 3Á11 3Á00 2Á91 2Á85 2Á69 2Á51 2Á30
0Á025 6Á55 5Á10 4Á47 4Á12 3Á89 3Á73 3Á61 3Á51 3Á28 3Á02 2Á72
0Á01 9Á33 6Á93 5Á95 5Á41 5Á06 4Á82 4Á64 4Á50 4Á16 3Á78 3Á36
0Á005 11Á75 8Á51 7Á23 6Á52 6Á07 5Á76 5Á52 5Á354Á91 4Á43 3Á90
14 0Á05 4Á60 3Á74 3Á34 3Á11 2Á96 2Á85 2Á76 2Á70 2Á53 2Á35 2Á13
0Á025 6Á304Á86 4Á24 3Á89 3Á66 3Á50 3Á383Á29 3Á05 2Á79 2Á49
0Á01 8Á86 6Á51 5Á56 5Á04 4Á69 4Á46 4Á28 4Á14 3Á80 3Á43 3Á00
0Á005 11Á06 7Á92 6Á68 6Á00 5Á56 5Á26 5Á03 4Á86 4Á43 3Á96 3Á44
16 0Á05 4Á49 3Á63 3Á24 3Á01 2Á85 2Á74 2Á66 2Á59 2Á42 2Á24 2Á01
0Á025 6Á12 4Á69 4Á08 3Á73 3Á50 3Á343Á22 3Á12 2Á89 2Á63 2Á32
0Á01 8Á53 6Á23 5Á29 4Á77 4Á44 4Á20 4Á03 3Á89 3Á55 3Á18 2Á75
0Á005 10Á58 7Á51 6Á305Á64 5Á21 4Á91 4Á69 4Á52 4Á10 3Á64 3Á11
18 0Á05 4Á41 3Á55 3Á16 2Á93 2Á77 2Á66 2Á58 2Á51 2Á34 2Á15 1Á92
0Á025 5Á98 4Á56 3Á95 3Á61 3Á383Á22 3Á10 3Á01 2Á77 2Á50 2Á19
0Á01 8Á29 6Á01 5Á09 4Á58 4Á25 4Á01 3Á84 3Á71 3Á37 3Á00 2Á57
0Á005 10Á22 7Á21 6Á03 5Á374Á96 4Á66 4Á44 4Á28 3Á86 3Á40 2Á87

20 0Á05 4Á35 3Á49 3Á10 2Á87 2Á71 2Á60 2Á51 2Á45 2Á28 2Á08 1Á84
0Á025 5Á87 4Á46 3Á86 3Á51 3Á29 3Á13 3Á01 2Á91 2Á68 2Á41 2Á09
0Á01 8Á10 5Á85 4Á94 4Á43 4Á10 3Á87 3Á70 3Á56 3Á23 2Á86 2Á42
0Á005 9Á94 6Á99 5Á82 5Á17 4Á76 4Á47 4Á26 4Á09 3Á68 3Á22 2Á69
752 Appendix tables
300Á05 4Á17 3Á32 2Á92 2Á69 2Á53 2Á42 2Á33 2Á27 2Á09 1Á89 1Á62
0Á025 5Á57 4Á18 3Á59 3Á25 3Á03 2Á87 2Á75 2Á65 2Á41 2Á14 1Á79
0Á01 7Á56 5Á39 4Á51 4Á02 3Á70 3Á47 3Á30 3Á17 2Á84 2Á47 2Á01
0Á005 9Á18 6Á355Á24 4Á62 4Á23 3Á95 3Á74 3Á58 3Á18 2Á73 2Á18
40 0Á05 4Á08 3Á23 2Á84 2Á61 2Á45 2Á34 2Á25 2Á18 2Á00 1Á79 1Á51
0Á025 5Á42 4Á05 3Á46 3Á13 2Á90 2Á74 2Á62 2Á53 2Á29 2Á01 1Á64
0Á01 7Á31 5Á18 4Á31 3Á83 3Á51 3Á29 3Á12 2Á99 2Á66 2Á29 1Á80
0Á005 8Á83 6Á07 4Á98 4Á373Á99 3Á71 3Á51 3Á352Á95 2Á50 1Á93
60 0Á05 4Á00 3Á15 2Á76 2Á53 2Á37 2Á25 2Á17 2Á10 1Á92 1Á70 1Á39
0Á025 5Á29 3Á93 3Á343Á01 2Á79 2Á63 2Á51 2Á41 2Á17 1Á88 1Á48
0Á01 7Á08 4Á98 4Á13 3Á65 3Á34 3Á12 2Á95 2Á82 2Á50 2Á12 1Á60
0Á005 8Á49 5Á79 4Á73 4Á14 3Á76 3Á49 3Á29 3Á13 2Á74 2Á29 1Á69
120 0Á05 3Á92 3Á07 2Á68 2Á45 2Á29 2Á17 2Á09 2Á02 1Á83 1Á61 1Á25
0Á025 5Á15 3Á80 3Á23 2Á89 2Á67 2Á52 2Á392Á302Á05 1Á76 1Á31
0Á01 6Á85 4Á79 3Á95 3Á48 3Á17 2Á96 2Á79 2Á66 2Á34 1Á95 1Á38
0Á005 8Á18 5Á54 4Á50 3Á92 3Á55 3Á28 3Á09 2Á93 2Á54 2Á09 1Á43
I 0Á05 3Á84 3Á00 2Á60 2Á37 2Á21 2Á10 2Á01 1Á94 1Á75 1Á52 1Á00
0Á025 5Á02 3Á69 3Á12 2Á79 2Á57 2Á41 2Á29 2Á19 1Á94 1Á64 1Á00
0Á01 6Á63 4Á61 3Á78 3Á32 3Á02 2Á80 2Á64 2Á51 2Á18 1Á79 1Á00
0Á005 7Á88 5Á304Á28 3Á72 3Á353Á09 2Á90 2Á74 2Á361Á90 1Á00
Condensed from Table 18 of Pearson and Hartley (1966) by permission of the authors and publishers.
For values of n
1
and n
2

not given, interpolation is approximately linear in the reciprocals of n
1
and n
2
.
Appendix tables 753
α
Q
p,

α
Table A5 Percentage points of the distribution of studentized range
The function tabulated is Q
p, a
, the value exceeded with probability a in the distribution of studentized range, for
p groups and f
2
DF within groups (the 100a percentage point). The values for a  0Á05 are shown in bold type.
Number of groups, p
f
2
 2345678910
5 0.05 3Á64 4Á60 5Á22 5Á67 6Á03 6Á33 6Á58 6Á80 6Á99
0.01 5Á70 6Á98 7Á80 8Á42 8Á91 9Á329Á67 9Á97 10Á24
6 0.05 3Á46 4Á34 4Á90 5Á30 5Á63 5Á90 6Á12 6Á32 6Á49
0.01 5Á24 6Á33 7Á03 7Á56 7Á97 8Á328Á61 8Á87 9Á10
7 0.05 3Á34 4Á16 4Á68 5Á06 5Á36 5Á61 5Á82 6Á00 6Á16
0.01 4Á95 5Á92 6Á54 7Á01 7Á377Á68 7Á94 8Á17 8Á37
8 0.05 3Á26 4Á04 4Á53 4Á89 5Á17 5Á40 5Á60 5Á77 5Á92
0.01 4Á75 5Á64 6Á20 6Á62 6Á96 7Á24 7Á47 7Á68 7Á86

9 0.05 3Á20 3Á95 4Á41 4Á76 5Á02 5Á24 5Á43 5Á59 5Á74
0.01 4Á60 5Á43 5Á96 6Á356Á66 6Á91 7Á13 7Á33 7Á49
10 0.05 3Á15 3Á88 4Á33 4Á65 4Á91 5Á12 5Á30 5Á46 5Á60
0.01 4Á48 5Á27 5Á77 6Á14 6Á43 6Á67 6Á87 7Á05 7Á21
12 0.05 3Á08 3Á77 4Á20 4Á51 4Á75 4Á95 5Á12 5Á27 5Á39
0.01 4Á325Á05 5Á50 5Á84 6Á10 6Á326Á51 6Á67 6Á81
754 Appendix tables
14 0.05 3Á03 3Á70 4Á11 4Á41 4Á64 4Á83 4Á99 5Á13 5Á25
0.01 4Á21 4Á89 5Á325Á63 5Á88 6Á08 6Á26 6Á41 6Á54
16 0.05 3Á00 3Á65 4Á05 4Á33 4Á56 4Á74 4Á90 5Á03 5Á15
0.01 4Á13 4Á79 5Á19 5Á49 5Á72 5Á92 6Á08 6Á22 6Á35
18 0.05 2Á97 3Á61 4Á00 4Á28 4Á49 4Á67 4Á82 4Á96 5Á07
0.01 4Á07 4Á70 5Á09 5Á385Á60 5Á79 5Á94 6Á08 6Á20
20 0.05 2Á95 3Á58 3Á96 4Á23 4Á45 4Á62 4Á77 4Á90 5Á01
0.01 4Á02 4Á64 5Á02 5Á29 5Á51 5Á69 5Á84 5Á97 6Á09
30 0.05 2Á89 3Á49 3Á85 4Á10 4Á30 4Á46 4Á60 4Á72 4Á82
0.01 3Á89 4Á45 4Á80 5Á05 5Á24 5Á40 5Á54 5Á65 5Á76
40 0.05 2Á86 3Á44 3Á79 4Á04 4Á23 4Á39 4Á52 4Á63 4Á73
0.01 3Á82 4Á374Á70 4Á93 5Á11 5Á26 5Á395Á50 5Á60
60 0.05 2Á83 3Á40 3Á74 3Á98 4Á16 4Á31 4Á44 4Á55 4Á65
0.01 3Á76 4Á28 4Á59 4Á82 4Á99 5Á13 5Á25 5Á365Á45
120 0.05 2Á80 3Á36 3Á68 3Á92 4Á10 4Á24 4Á36 4Á47 4Á56
0.01 3Á70 4Á20 4Á50 4Á71 4Á87 5Á01 5Á12 5Á21 5Á30
I 0.05 2Á77 3Á31 3Á63 3Á86 4Á03 4Á17 4Á29 4Á39 4Á47
0.01 3Á64 4Á12 4Á40 4Á60 4Á76 4Á88 4Á99 5Á08 5Á16
Condensed from Table 29 of Pearson and Hartley (1966) by permission of the authors and publishers.
Appendix tables 755
Table A6 Percentage points for the Wilcoxon signed rank sum test
The function tabulated is the critical value for the smaller of the signed rank sums, T


and T
À
.An
observed value equal to or less than the tabulated value is significant at the two-sided significance
level shown (the actual tail-area probability being less than or equal to the nominal value shown). If
ties are present, the result is somewhat more significant than is indicated here.
Sample size, n
H
(excluding zero
differences)
Two-sided significance level
0Á05 0Á01
60Ð
72Ð
830
951
10 8 3
11 10 5
12 13 7
13 17 9
14 21 12
15 25 15
16 29 19
17 34 23
18 40 27
19 46 32
20 52 37
21 58 42
22 66 48
23 73 54

24 81 61
25 89 68
Condensed from the Geigy Scientific Tables (1982), by
permission of the authors and publishers.
756 Appendix tables
Table A7 Percentage points for the Wilcoxon two-sample rank sum test
Define n
1
as the smaller of the two sample sizes (n
1
n
2
). Calculate T
1
as the sum of the ranks in sample 1, and E(T
1
) 
1
2
n
1
n
1
 n
2
 1. Calculate T
H
as T
1
if

T
1
ET
1
 and as n
1
n
1
 n
2
 1ÀT
1
if T
1
> ET
1
. The result is significant at the two-sided 5% (or 1%) level if T
H
is less than or equal to the upper (or lower)
tabulated value (the actual tail-area probability being less than or equal to the nominal value). If ties are present, the result is somewhat more significant than is
indicated here.
Smaller sample size, n
1
n
2
P 456789101112131415
40Á05 10
0Á01 Ð
50Á05 11 17
0Á01 Ð 15

60Á05 12 18 26
0Á01 10 16 23
70Á05 13 20 27 36
0Á01 10 16 24 32
80Á05 14 21 29 38 49
0Á01 11 17 25 34 43
90Á05 14 22 31 40 51 62
0Á01 11 18 26 35 45 56
10 0Á05 15 23 32 42 53 65 78
0Á01 12 19 27 37 47 58 71
11 0Á05 16 24 34 44 55 68 81 96
0Á01 12 20 28 38 49 61 73 87
12 0Á05 17 26 35 46 58 71 84 99 115
0Á01 13 21 30 40 51 63 76 90 105
13 0Á05 18 27 37 48 60 73 88 103 119 136
0Á01 13 22 31 41 53 65 79 93 109 125
14 0Á05 19 28 38 50 62 76 91 106 123 141 160
0Á01 14 22 32 43 54 67 81 96 112 129 147
15 0Á05 20 29 40 52 65 79 94 110 127 145 164 184
0Á01 15 23 33 44 56 69 84 99 115 133 151 171
Condensed from the Geigy Scientific Tables (1982) by permission of the authors and publishers.
Appendix tables 757
Table A8 Sample size for comparing two proportions
This table is used to determine the sample size necessary to find a significant difference (5% two-sided
significance level) between two proportions estimated from independent samples where the true
proportions are p
1
and p
2
, and d  p

1
À p
2
is the specified difference p
1
> p
2
. Sample sizes are
given for 90% power (upper value of pair) and 80% power (lower value). The sample size given in the
table refers to each of the two independent samples. The table is derived using (4.42) with a continuity
correction.
Note: If p
2
> 0Á5, work with p
H
1
 1 À p
2
and p
H
2
 1 Àp
1
.
Smaller
probability,
d  p
1
À p
2

p
2
0Á05 0Á10Á15 0Á20Á25 0Á30Á350Á40Á45 0Á5
0Á05 621 207 113 75 54 42 33 27 23 19
475 160 88 59 43 33 27 22 19 16
0Á1 958 286 146 92 65 48 38 30 25 21
726 219 113 72 51 38 30 24 20 17
0Á15 1252 354 174 106 73 54 41 33 27 22
946 270 134 82 57 42 33 26 21 18
0Á2 1504 412 198 118 80 58 44 34 28 23
1134 313 151 91 62 45 35 27 22 18
0Á25 1714 459 216 127 85 61 46 35 28 23
1291 349 165 98 66 47 36 28 23 18
0Á3 1883 496 230 134 88 62 46 36 28 23
1417 376 176 103 68 49 36 28 23 18
0Á35 2009 522 240 138 90 63 46 35 28 22
1511 396 183 106 69 49 36 28 22 18
0Á4 2093 538 244 139 90 62 46 34 27 21
1574 407 186 107 69 49 36 27 21 17
0Á45 2135 543 244 138 88 61 44 33 25 19
1605 411 186 106 68 47 35 26 20 16
0Á5 2135 538 240 134 85 58 41 30 23 18
1605 407 183 103 66 45 33 24 19 15
758 Appendix tables
Table A9 Sample size for detecting relative risk in case±control study
This table is used to determine the sample size necessary to find the odds ratio statistically significant
(5% two-sided test) in a case±control study with an equal number of cases and controls. The specified
odds ratio is denoted by OR, and p is the proportion of controls that are expected to be exposed. For
each pair of values the upper figure is for a power of 90% and the lower for a power of 80%. The
tabulated sample size refers to the number of cases required. The table is derived using (4.22) and

(4.42) with a continuity correction.
Note: If p > 0Á5, work with p
H
 1 Àp and OR
H
 1=OR.
Proportion of
controls exposed, p
OR (odds ratio)
0Á51Á52Á02Á53Á04Á05Á010Á0
0Á05 1369 2347 734 393 259 150 105 43
1044 1775 560 301 200 117 82 34
0Á1 701 1266 402 219 146 87 62 27
534 958 307 168 113 68 48 22
0Á15 479 913 295 163 110 67 48 23
366 691 225 125 85 52 38 19
0Á2 370 743 244 136 93 58 43 21
282 562 187 105 72 45 34 17
0Á25 306 647 216 122 85 53 40 21
233 490 165 94 66 42 32 17
0Á3 264 590 200 115 80 51 39 22
202 447 153 88 62 40 31 18
0Á35 236 556 192 111 79 51 39 23
180 421 147 86 61 40 31 18
0Á4 216 538 188 111 79 52 41 24
165 407 144 85 61 41 32 20
0Á45 203 533 189 112 81 54 43 26
155 403 145 87 63 43 34 21
0Á5 194 538 194 116 85 58 46 29
148 407 148 90 66 45 36 23

Appendix tables 759
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