STATISTICAL ASPECTS OF SAFETY OF COSMETICS 147 Now if on the average a user of this product purchases three units per year, then there are, on the average, one million individuals at risk of having an irritation due to the use of the product, and the incidence of irritation is 120/1,000,000 or 12/100,000. Note the difference in the two rates--one is three times the other--when the unit consumer is used rather than the unit product. The effect of using the product number is to dilute the actual risk. The tendency to use the units sold as the base for an irritation rate prevails because it is easily determined. The consumers are not so easily counted. This difficulty arises not only from the fact that a consumer may use more than one unit but also because consumers may shift from the product of one maker to another for a variety of reasons. However, if one is able to determine the average units used per individual per year then one may estimate the risk on a population base. One of the problems associated with an estimate of the kind proposed above is that the numerator comprises only those cases which have been brought to the attention of the producer. Such complaints may repre- sent instances in which the irritation was severe. They may also in- clude individuals who tend to exaggerate their complaints or persons who have a litigious tendency. There are also individuals, probably in the majority, who never register a complaint. If they have an adverse reaction to the product, they change brands. There is no knowledge of the number of these cases. It follows that information provided by the incidence rate mentioned above represents only a minimum estimate of the true risk. VALUE OF THE RANDOM SAMPLE In order to get complete information about the effects of a product, one would have to study the entire population of users, and this is, of course, not possible. But one can, by appropriate methods, study a small part of this population and arrive at remarkably good estimates. The use of information obtained from a sample is common practice. The statistician attempts to refine this procedure by first clearly defining the population about which information is needed. The sample itself must be obtained by a random process, i.e., a mechanical procedure for selecting the members of a sample, based on each individual having a known probability of being included in the sample. The process needs to be mechanical in order to avoid bias in the selection process. An example of one kind of bias is that which the cosmetic producer corn-

148 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS plains occurs if irritation rates are determined by a dermatologist. The people who come to the office of the dermatologist are people who for, one reason or other, may be transiently, if not permanently, sensitive to many substances. Thus, incidence rates obtained from such experience are apt to be those for a special segment of the total population of in- terest to the cosmetics producers. The purpose of the random sample is to provide for the representation of all the disparate elements in a population in their approximately true proportions. This can be done by means of a number of different strategies. The most common procedure when one has no information regarding the variation inherent in the population is to take a simple random sample in which every element or member of the population has the same chance of being drawn into the sample. However, the population of consumers for any given cosmetic prepara- tion is not only large but unidentified. How can one make certain that every individual in the population at risk will have the same chance of being included in the sample? For, in order to determine that every individual has an equal chance, one must know how many people the population comprises. As an example, let us define the population of users of lipstick as all females who are 12 or more years of age living in the United States. Since the last census was in 1960, there is no current and accurate count of these females in the year 1964. It must be admitted that the size of the population of lipstick con- sumers is not known. But this does not concern the statistician. He will select a small city for which he has information sufficiently accurate to believe it to be representative of the population of interest with re- spect to factors such as age distribution, and other factors he has reason to believe to be relevant. For such a small community, he can deter- mine the numbers of individuals who meet the definition of the popula- tion at risk. It is then possible to obtain a sample of the size desired by use of a table of random numbers which meets the condition that every individual has the same chance of being included. The calculated incidence obtained from this sample is then an estimate of the incidence for the city, and also one for the identical population of the United States, subject to the qualification that it is a valid estimate if and only if the community from which the sample is selected is representative of the entire population. The "test market" procedure used often by cosmetic manufacturers prior to introducing a new product is based on this approach, although the author cannot guarantee that the sample used is always a random one.