STATISTICAL ASPECTS OF SAFETY OF COSMETICS 149 SPECIAL PROBLEMS IN THE TESTING OF COSMETICS Up to this point, consideration has turned upon the question of the general notion of estimating the incidence of irritation due to a cosmetic or toiletry product. This concept has been presented as if it were a relatively simple task to translate into practice. It is not simple, how- ever, because a number of factors add complexity to the problem. If any investigation is to provide useful information, the object of such investigatien must be clearly formulated. That is to say a decision has to be made as to the precise question with which the investigator wishes to deal. The general question as to whether a substance produces irritation may be answered in several ways, since it is not unambiguously stated. The failure to state the question clearly accounts in part for the differences in the irritation rates reported by the dermatologists as opposed to those reported by the cosmetic manufacturers. Moreover, the clearly different motivation also affects the way in which the ques- tion is interpreted and, therefore, the manner in which the information is analyzed and inferences are made. The kind of confusion that may result from not clearly stating the question is evident in the followSng quotation: "In 1961, the cosmetic industry sold about two billion separate units in this country alone. A 5% irritation rate would therefore imply that up to 100,000,000 persons in this country developed positive eczematous reactions specifically attributable to cosmetics last year !" (1) Goldemberg is clearly correct when he denies the applicability of a clinical irritation rate based upon patch test reactions to cosmetic substances carried out on persons who already have a dermatological condition. Such a rate is incorrectly extrapolated to the population at large, since the latter does not meet the definition of the population from which the clinical irritation rate was obtained. On the other hand, Goldemberg makes the unwarranted as- sumption that each unit sold represents a unique consumer in applying the 5% clinical rate to come up with the figure of 100 million cases. He merely succeeds in knocking down a straw man. As opposed to the clinical dermatologist, the cosmetic producer is interested in the question of the incidence of irritation in the particular segment of the population which represents the potential consumers. More specifically, he is interested in determining whether the risk of irritation is greater than some threshold quantity which is related to the cost to him in terms of losses from indemnification of legitimate complaints, loss of business and reputation, and possible punitive meas- ures resulting from state intervention. This transforms the question

150 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS considerably. It forces the manufacturer to consider very concretely the threshold incidence which w-ill minimize his possible losses. That is, the investigator now asks not, "What is the incidence of irritation in the population of consumers of his product?" but, "Is the incidence greater than some specific quantity?" This latter quantity can be estimated by the producer from his experience and other information available to him. Once the question is formulated unequivocally, including the quanti- tative statement of the risk the cosmetic manufacturer can tolerate, determination of sample size is possible. However, since sample size may also be related to the design of the investigation, final decision is withheld until the design is decided upon. Decision as to the design and the sample size will be partly affected by the cost involved in carry- ing out the study. The author will not go into this aspect other than to mention that cost plays an important role that is frequently neglected by non-statisticians. Goldemberg states that if a cosmetic formula cannot be used safely by 99.8% of actual consumers, then the producer will be in trouble (1). This implies that the manufacturer is willing to risk the possibility of a maximum irritation rate of 2/1000. Although this implies something like 2000 cases of irritation if one million persons purchase his product, it is a convenient figure for the purpose of this discussion. If the true rate is 2/1000 and a sample size of 1000 persons is used, then one can expect a very large proportion of the time that the number of cases of irritation to be observed will be in the range from 0 to 4. If 2000 is the sample size, this range is diminished and 3 to 5 cases might be expected. The above comments hold only if there is an intermixture of a new ingredient with others whose irritation potential is known and negligible, that is, if a comparison is to be made between an existing preparation and a modification of that preparation. This seems to be the case, so that attention can be confined to this kind of situation. This leads directly to the problem of study design, on the supposition that one is concerned with comparing the irritation potential of a modi- fied product with that of the product as it was formerly constituted. One can make this comparison in one of two ways, if it can be assumed that for the purpose of the investigation the old and new cosmetics will be packaged identically so that a consumer is not aware of the dif- [erence. The first procedure would be to choose a random sample of 4000 in-