Ralph B. Dell, Steve Holleran, and Rajasekhar Ramakrishnan

1 Sample size DeterminationRalph B. Dell, Steve Holleran, and Rajasekhar RamakrishnanAbstractScientists who use animals in research must justify the num-ber of animals to be used, and committees that review pro-posals to use animals in research must review thisjustification to ensure the appropriateness of the numberof animals to be used. This article discusses when the num-ber of animals to be used can best be estimated from pre-vious experience and when a simple power and sample sizecalculation should be performed. Even complicated experi-mental designs requiring sophisticated statistical modelsfor analysis can usually be simplified to a single key orcritical question so that simple formulae can be used toestimate the required sample size . Approaches to samplesize estimation for various types of hypotheses are de-scribed, and equations are provided in the Appendix.

2 Sev-eral web sites are cited for more information and forperforming actual Words:number of animals; power calculation; samplesize; statistical analysisIn the United States and in most European countries, aninvestigator must provide the animal care committee withan explanation for the number of animals requested in aproposed project to ensure appropriateness of the numbersof animals to be used. This article is written for animal carecommittee members and veterinarians and for researcherswho are asked to provide statistical calculations for the pro-posed number of animals to be used in their project. Theproject s purpose may be to obtain enough tissue to dosubsequent analyses, to use a small number of animals for apilot experiment, or to test a hypothesis. In the text below,we discuss the statistical bases for estimating the number ofanimals (sample size ) needed for several classes of hypoth-eses.

3 The types of experiments that an investigator mightpropose and the methods of computing sample size arediscussed for situations where it is possible to do such of ExperimentsTypes of experiments include pilot and exploratory, thosebased on success or failure of a desired goal, and thoseintended to test a formal hypothesis. Each type is discussedbriefly and Exploratory ExperimentsIt is not possible to compute a sample size for certain typesof experiments because prior information is lacking or be-cause the success of the experiment is highly variable, suchas in producing a transgenic animal. For other types ofexperiments, complicated statistical designs can be simpli-fied to an important comparison wherein the sample sizeshould be large enough to have a good chance of findingstatistical significance (often called power; see Effect size ,Standard Deviation, Power, and Significance Level).

4 Pilotexperiments are designed to explore a new research area todetermine whether variables are measurable with sufficientprecision to be studied under different experimental condi-tions as well as to check the logistics of a proposed experi-ment. For example, suppose the investigator wishes todetermine whether a certain factor,x,is elevated in an ani-mal model of inflammation. The laboratory has developedan assay for factorxand now wishes to determine the varia-tion of factorxin a population of mice. In the protocol, theinvestigator proposes measuring the concentration of factorxin 10 animals before and after the induction of inflamma-tion. In a pilot experiment such as this, the number of ani-mals to be used is based on experience and guessworkbecause there are no prior data to use in estimating thenumber of animals needed for the study. The experiment isperformed to provide a rough idea of the standard deviationand the magnitude of the inflammatory statistical analysis of the results yields estimates ofthe mean and standard deviation of factorxconcentrationbefore and after the induction of inflammation as well asestimates of the mean difference and its standard estimates can then be used to compute the sample sizefor further experiments.

5 The investigator would be encour-aged if the standard deviation of factorxin the 10 animalsis relatively small compared with the concentration of thefactor. Suppose that the mean concentration of factorxin-creased twofold after inflammation was induced, a changethat should be easily detected if the variation of the changein the population is low. Then the pilot experiment will haveAll authors hold positions in the Department of Pediatrics, College ofPhysicians & Surgeons, Columbia University, New York, New B. Dell, , is Emeritus Professor of Pediatrics, Steve Holleran, is Staff Associate, and Rajasekhar Ramakrishnan , , is Director of theBiomathematics Division in 43, Number 42002207been encouraging in that the investigator may be able totrack the increase in the concentration of factorxover timeand determine changes in the concentration of the factorwith various forms of therapy.

6 The results of the pilot ex-periment can be used to estimate the number of animalsneeded to determine time trends and to study the effect ofvarious interventions on the concentration of factorxusingmethods described exploratory experiments are performed togenerate new hypotheses that can then be formally tested. Insuch experiments, the usual aim is to look for patterns ofresponse, often using many different dependent variables(characters). Formal hypothesis testing and the generationofpvalues are relatively unimportant with this sort of ex-periment because the aim will be to verify by additionalexperiments any results that appear to be of interest. Usuallythe number of animals used in such experiments is basedon a guess based on previous experience. Data collectedin exploratory experiments can then be used in samplesize calculations to compute the number of animals that willbe needed to test attractive hypotheses generated by Based on Success or Failure of aDesired GoalIn experiments based on the success or failure of a desiredgoal, the number of animals required is difficult to estimatebecause the chance of success of the experimental proce-dure has considerable variability.

7 Examples of this type ofexperiment are production of transgenic animals by geneinsertion into fertilized eggs or embryonic stem cells. Largenumbers of animals are typically required for several rea-sons. First, there is considerable variation in the proportionof successful gene or DNA incorporation into the cell sgenome. Then there is variability in the implantation of thetransferred cell. Finally, the DNA integrates randomly intothe genome and the expression varies widely as a functionof the integration site and transgene copy this variability, different strains of micereact differently to these manipulations, and different genesvary in their rates of incorporation into the genome. It isoften necessary to make several transgenic lines (see thediscussion of transgenic animals in the ARENA/OLAW In-stitutional Care and Use Committee Guidebook [ARENA/OLAW 2002]).

8 Using equation 1 below (Single-GroupExperiments) and assuming that the success rate for all ofthe steps just mentioned is 5%, then one would need to use50 animals, whereas a success rate of 1% would requireusing 300 animals. These numbers accord with the experi-ence of investigators in the field and are usually the range ofnumbers of mice required to produce a single the case of knockout or knockin mice produced byhomologous recombination, there is much less variability inthe results and fewer animals may have to be , it is difficult to predict the number required, espe-cially if investigating the effects of regulatory sequencesrather than of protein expression. The number of animalsrequired is usually estimated by experience instead of byany formal statistical calculation, although the procedureswill be terminated when enough transgenic mice have beenproduced.

9 Formal experiments will, of course, be requiredfor studying the characteristics of the transgenic animalsrequiring yet more to Test a Formal HypothesisMost animal experiments involve formal tests of hypoth-eses. In contrast to pilot experiments and the other types ofexperiments described above, it is possible to estimate thenumber of animals required for these experiments if a fewitems of information are available. Broadly, there are threetypes of variables that an investigator may measure: (1)dichotomous variable,often expressed as a rate or propor-tion of a yes/no outcome, such as occurrence of disease orsurvival at a given time; (2)continuous variable,such asthe concentration of a substance in a body fluid or a physi-ological function such as blood flow rate or urine output;and (3)time to occurrenceof an event, such as the appear-ance of disease or death.

10 Many statistical models have beendeveloped to test the significance of differences amongmeans of these types of data. Detailed discussions of themodels can be found in books on statistics (Cohen 1988;Fleiss 1981; Snedecor and Cochran 1989), in manuals forvarious computer programs used for statistical analyses(Kirkpatric and Feeney 2000; SAS 2000), and on websitesthat present elementary level courses on statistics ( ,< >). In this article,we describe methods for computing sample size for each ofthese types of the Hypothesis to Be TestedAlthough experimental designs can be complicated, the in-vestigator s hypotheses can usually be reduced to one or afew important questions. It is possible then to compute asample size that has a certain chance or probability of de-tecting (with statistical significance) an effect (or differ-ence) the investigator has postulated.