Statistical Methods in Medical Research - part 10 docx

1994 Modelling Survival Data in Medical Research.. 1984 Regression models and non-proportional hazards in the analysis ofbreast cancer surviv-al.. 1980 The Statistical Analysis of Failur

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 ˆ 1
59=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 djˆ cj 1d1, 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, 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.

muta-The data that arise in these assays comprise the number of mutants in the jth replicate at the ith dose, Yij, j ˆ 1, , ri, i ˆ 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, xi, with each dose group (often xi will be the dose or log dose given to that group) The test statistic is

ZP ˆ

PDiˆ1xiri…Yi Y†

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 ZP 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

Prijˆ1…Yij Yi†2

Yi

to a x2distribution with ri 1 degrees of freedom A test across all dose groups can be made by adding these quantities from i ˆ 1 to D and referring the sum to

a x2 distribution with P ri 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 li and the number of microbes placed on the jth plate at this dose is Nij If experimental technique

is sufficiently rigorous, then it may be possible to claim that the count Nij 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 Yij is Poisson, with mean liNij However, it may be more realistic to assume that the Nij 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 ‡ am2, for some non-negative constant a …a ˆ 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 liNij 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†S2

xŠ, 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 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

non-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

ν,P

P V

Table A2 Percentage points of the x2distributionThe function tabulated is x2

,P, the value exceeded with probability P in a x2 distribution with n degrees offreedom (the 100P percentage point)

Probability of greater value, PDegrees of

Condensed from Table 18 of Pearson and Hartley (1966) by permission of the authors and publishers.

For values of n1and n2not given, interpolation is approximately linear in the reciprocals of n1and n2

Table A6 Percentage points for the Wilcoxon signed rank sum test

observed value equal to or less than the tabulated value is significant at the two-sided significancelevel shown (the actual tail-area probability being less than or equal to the nominal value shown) Ifties are present, the result is somewhat more significant than is indicated here

Sample size, n0(excluding zerodifferences)

Two-sided significance level

Table A7 Percentage points for the Wilcoxon two-sample rank sum test

Define n1as the smaller of the two sample sizes (n1 n2) Calculate T1as the sum of the ranks in sample 1, and E(T1) ˆ1n1…n1‡ n2‡ 1† Calculate T0as T1if

T1 E…T1† and as n1…n1‡ n2‡ 1† T1if T1> E…T1† The result is significant at the two-sided 5% (or 1%) level if T0is 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 isindicated here

Smaller sample size, n1

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-sidedsignificance level) between two proportions estimated from independent samples where the true

given for 90% power (upper value of pair) and 80% power (lower value) The sample size given in thetable refers to each of the two independent samples The table is derived using (4.42) with a continuitycorrection

Note: If p2> 0
5, work with p0

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 specifiedodds ratio is denoted by OR, and p is the proportion of controls that are expected to be exposed Foreach pair of values the upper figure is for a power of 90% and the lower for a power of 80% Thetabulated sample size refers to the number of cases required The table is derived using (4.22) and(4.42) with a continuity correction

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