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Certain Statistical Considerations in Patch Testing1
CERTAIN STATISTICAL CONSIDERATIONS IN PATCH TESTING12 CHARLES R. HENDERSON AD EDWARD C. RILEY
Although patch testing procedures were not originally developed for bio-assay,
they have been adapted for the detection and evaluation of the skin-irritating and sensitizing properties of chemical agents and materials. Information thus derived is used to prevent the exposure of the general population to such substances. In using the patch test to obtain such information a certain group of individuals is tested under control conditions, and the number of positive reactions noted. If a high incidence of reactions occurs in a test group, it indicates that a given substance is prohibitively skin-sensitizing or irritating, but if no positive reactions occur in a sample, for example of 100, this does not necessarily guarantee the absence of dermatitis-producing factors. In order to interpret the results found on a test group of subjects in terms of the general population, certain statistical principles may be applied. The following discussion is confined solely to statistical considerations of sampling techniques as applied to patch tests, no attempt being made to discuss the quantitative relationship and variations between the reactions of persons subjected to a specific substance under test conditions and the reactions under normal exposures. From a statistical point of view two questions are of importance: (1) if no positive reactions or if a certain number of positive reactions to a patch test are observed in a given test group, what would be the maximum incidence likely to occur, under identical conditions, in a large number of individuals similar to the test group, and (2) how large a group must be tested to predict, with a given degree of cer-
tainty, that the rate of positive reactions in the general population lies within certain given limits. To consider a concrete example, suppose that in a certain test no reactions were observed in a group of 100 subjects and we wish to know what range of positive reaction rates in a large population could give rise to such a finding. Theoretically any value from 0 up to but not including 100% could do so. It is obvious, however, that the larger the positive reaction rate in the population, the less is the likelihood that a sample of 100 will be drawn which shows no positive reactions. It can therefore be inferred with a certain degree of confidence that the given sample of 100 came from a population with a small positive reaction rate. It can be shown that if no positive reactions occur in 100 subjects, the rate of positive reactions in the population is not likely to exceed approximately 2.9%, the likelihood being at the 95% level ;8 and that if one reaction is observed in 100 1From The Army Industrial Hygiene Laboratory Johns Hopkins School of Hygiene and Public Health, Baltimore, Md. 2
Received for publication December 5, 1944.
The likelihood level of 95% is a measure of the confidence with which it can be asserted
that the population rate does not exceed the stated value. The likelihood level may be chosen at any point, depending on how conservative one wishes to be in his estimates. If a higher level, say 99%, is used, then the maximum estimate of the population rate would be increaaed, and it could then be asserted with greater assurance that the population rate did not exceed that higher value. The most frequently used likelihood or confidence levels are 95% and 99%.
227
228
'ruE JOURNAL OF INVESTIGATIVE DERMATOLOGY
TABLE I Upper Limit of Po8itive Reaction Rate Estimated for the Population, with a 95% Likelihood,
from Results of 0, 1, or Positives in a S ample of ,Size n NO.
LILY MAXIMUM RA1Z 07 POSITIVE REACTIONS IN ALPRCE NOREEN 07 SUBJECTS IN $A]iPI2 ______________________________________________ ______________________ If no reaction in sample If one reaction in sample If two reactions in sample (f5)
30,000 20,000 10,000
% .01
.06
2,000 1,000
.15 .30 .37
800 600 500 400 300
.03 .06
.05 .09
.03
5,000 4,000
%
%
.15 .30
.23
.45 .56 .75
.50
.75
1.0 1.2 1.5
.75 .99 1.2 1.5
1.1 1.5
2.0 2.4
3.0 3.6
3.0 3.7 4.9 5.8
4.4 5.5 7.3 8.8
5.8 7.3 9.6
2.0 2.4 2.9
1.8 2.2
250 200 160 150 125 120 100 80 60 50 40 30 20
7.2 9.5
10.8 14.2
14.0
13.9
26.9
20.8 38.2
27.0
10
3.7 4.9
* 95% likelihood.
TABLE II Number of Subjects Required, If There Are No Positives in the Sample to Predict with a 95% or 99% Likelihood That the Population Rate Does Not Exceed Certain Designated Values )SAXIMOM PERMISSIBLE PERCENTAGE OS REACTIONS IN THE POPULATION
NO. 07 SUBJECTS SON LTEELIN000 OF
95%
0.01
29,978
0.1
2,994
0.5 1
2 3
597 298 148
98
99%
46,083 4,603 918 459 228 151
229
STATISTICAL CONSIDERATIONS IN PATCH TESTING
subjects, the likelihood is 95% that the rate does not exceed approximately 4.5%.
Conversely it can be shown that in order to insure, with a likelihood of 95%, that the rate of positive reactions in the population does not exceed 2%, 147 500
400
NO REACTION IN THE SAMPLE
ONE REACTION IN THE SANPI.E — — — — —
TWO REACTIONS 14 THE SAMPLE -— -— -
(ii U)
z 300 0 Id
I
0 0
5
$0
UKELY' MXINUU PERCENTAGE
'5%
O
POSITIVE REACTIONS IN POPULATION
('LIKELIHOOD AT 95% LEVEL)
subjects must be tested and no reactions appear or 225 subjects must be tested and only one reaction appear. The equations used for deriving these results are part of the statistical theory
for the distribution of repeated results. If a certain number of positives are
230
THE JOITRNAL OF INVESTIGATIVE DERMATOLOGY
found in n individuals, and we want to estimate the reaction rate, p, so that the likelihood is 95% that the population value does not exceed p, the following equations4 may be used for the estimation: For 0 positive reactions in the sample of size n (1 — p)"
=
.05
For 1 positive reaction in the sample n(1 —
p)"p
= .05
For 2 positive reactions in the sample n(n — 1) (1 — p)n_2p2 = .05
Calculations resulting from substitution in these equations are shown in Table I and Graph I, which give for various sizes of samples the "maximum" reaction rate estimated if 0, 1 or 2 positive reactions appear in the sample. By "maximum" here, we mean that the likelihood is 95% that it would not be exceeded. Table II gives for the case of no positives in the sample, the size sample necessary to determine with an assurance of 95% or with an assurance of 99% that the population rate does not exceed certain specified values. The data my be used for two purposes. (1) If 0, 1 or 2 positive reactions are observed in a sample of a given size, an estimate can be made as to the likely (95% likelihood) maximum rate of positive reactions in the population from which the sample was drawn. (2) If it is desired to make reasonably certain (95% likelihood) that the rate of positive reactions in the population does not exceed a given figure, the size of sample necessary to accomplish this end may be determined (3 values will of course be obtained depending upon whether 0, 1, or 2 reactions in the sample is
to serve as the criterion). Acknowledgement is made to Dr. Margaret Merrell, Associate Professor of Biostatistics, Johns Hopkins University School of Hygiene and Public Health, for her assistance in the statistical considerations. 4 equation for extimating with a likelihood of P, that the population rate does not exceed p, if a positives are found in n individuals is:
ii!
(l._p)n—apadp=P
In the range of p's involved in the present problem, an approximate solution is given by the following formula, which was used in our estimates:
a! (n—a)!
(1—p)'p = 1—P
THE JOURNAL OF INVESTIGATIVE DERMATOLOGY
94
linolenic acid extract. Arch. This pdf is a scanned copy UV of irradiated a printed document.
24. Wynn, C. H. and Iqbal, M.: Isolation of rat
skin lysosomes and a comparison with liver Path., 80: 91, 1965. and spleen lysosomes. Biochem. J., 98: lOP, 37. Nicolaides, N.: Lipids, membranes, and the 1966.
human epidermis, p. 511, The Epidermis
Eds., Montagna, W. and Lobitz, W. C. Acascopic localization of acid phosphatase in demic Press, New York. human epidermis. J. Invest. Derm., 46: 431, 38. Wills, E. D. and Wilkinson, A. E.: Release of 1966. enzymes from lysosomes by irradiation and 26. Rowden, C.: Ultrastructural studies of kerathe relation of lipid peroxide formation to tinized epithelia of the mouse. I. Combined enzyme release. Biochem. J., 99: 657, 1966. electron microscope and cytochemical study 39. Lane, N. I. and Novikoff, A. B.: Effects of of lysosomes in mouse epidermis and esoarginine deprivation, ultraviolet radiation and X-radiation on cultured KB cells. J. phageal epithelium. J. Invest. Derm., 49: 181, 25. Olson, R. L. and Nordquist, R. E.: Ultramicro-
No warranty is given about the accuracy of the copy.
Users should refer to the original published dermal cells. Nature, 216: 1031, 1967. version of1965. the material. vest. Derm., 45: 448, 28. Hall, J. H., Smith, J. G., Jr. and Burnett, S. 41. Daniels, F., Jr. and Johnson, B. E.: In prepa1967.
Cell Biol., 27: 603, 1965.
27. Prose, P. H., Sedlis, E. and Bigelow, M.: The 40. Fukuyama, K., Epstein, W. L. and Epstein, demonstration of lysosomes in the diseased J. H.: Effect of ultraviolet light on RNA skin of infants with infantile eczema. J. Inand protein synthesis in differentiated epi-
C.: The lysosome in contact dermatitis: A ration. histochemical study. J. Invest. Derm., 49: 42. Ito, M.: Histochemical investigations of Unna's oxygen and reduction areas by means of 590, 1967. 29. Pearse, A. C. E.: p. 882, Histochemistry Theoultraviolet irradiation, Studies on Melanin, retical and Applied, 2nd ed., Churchill, London, 1960.
30. Pearse, A. C. E.: p. 910, Histacheini.stry Thearetscal and Applied, 2nd ed., Churchill, London, 1960.
31. Daniels, F., Jr., Brophy, D. and Lobitz, W. C.: Histochemical responses of human skin fol-
lowing ultraviolet irradiation. J. Invest. Derm.,37: 351, 1961.
32. Bitensky, L.: The demonstration of lysosomes by the controlled temperature freezing section method. Quart. J. Micr. Sci., 103: 205, 1952.
33. Diengdoh, J. V.: The demonstration of lysosomes in mouse skin. Quart. J. Micr. Sci., 105: 73, 1964.
34. Jarret, A., Spearman, R. I. C. and Hardy, J. A.:
Tohoku, J. Exp. Med., 65: Supplement V, 10, 1957.
43. Bitcnsky, L.: Lysosomes in normal and pathological cells, pp. 362—375, Lysasames Eds., de Reuck, A. V. S. and Cameron, M. Churchill, London, 1953.
44. Janoff, A. and Zweifach, B. W.: Production of inflammatory changes in the microcirculation by cationic proteins extracted from lysosomes. J. Exp. Med., 120: 747, 1964.
45. Herion, J. C., Spitznagel, J. K., Walker, R. I. and Zeya, H. I.: Pyrogenicity of granulocyte lysosomes. Amer. J. Physiol., 211: 693, 1966.
46. Baden, H. P. and Pearlman, C.: The effect of ultraviolet light on protein and nucleic acid synthesis in the epidermis. J. Invest. Derm.,
Histochemistry of keratinization. Brit. J. 43: 71, 1964. Derm., 71: 277, 1959. 35. De Duve, C. and Wattiaux, R.: Functions of 47. Bullough, W. S. and Laurence, E. B.: Mitotic control by internal secretion: the role of lysosomes. Ann. Rev. Physiol., 28: 435, 1966. the chalone-adrenalin complex. Exp. Cell. 36. Waravdekar, V. S., Saclaw, L. D., Jones, W. A. and Kuhns, J. C.: Skin changes induced by
Res., 33: 176, 1964.
Although patch testing procedures were not originally developed for bio-assay,
they have been adapted for the detection and evaluation of the skin-irritating and sensitizing properties of chemical agents and materials. Information thus derived is used to prevent the exposure of the general population to such substances. In using the patch test to obtain such information a certain group of individuals is tested under control conditions, and the number of positive reactions noted. If a high incidence of reactions occurs in a test group, it indicates that a given substance is prohibitively skin-sensitizing or irritating, but if no positive reactions occur in a sample, for example of 100, this does not necessarily guarantee the absence of dermatitis-producing factors. In order to interpret the results found on a test group of subjects in terms of the general population, certain statistical principles may be applied. The following discussion is confined solely to statistical considerations of sampling techniques as applied to patch tests, no attempt being made to discuss the quantitative relationship and variations between the reactions of persons subjected to a specific substance under test conditions and the reactions under normal exposures. From a statistical point of view two questions are of importance: (1) if no positive reactions or if a certain number of positive reactions to a patch test are observed in a given test group, what would be the maximum incidence likely to occur, under identical conditions, in a large number of individuals similar to the test group, and (2) how large a group must be tested to predict, with a given degree of cer-
tainty, that the rate of positive reactions in the general population lies within certain given limits. To consider a concrete example, suppose that in a certain test no reactions were observed in a group of 100 subjects and we wish to know what range of positive reaction rates in a large population could give rise to such a finding. Theoretically any value from 0 up to but not including 100% could do so. It is obvious, however, that the larger the positive reaction rate in the population, the less is the likelihood that a sample of 100 will be drawn which shows no positive reactions. It can therefore be inferred with a certain degree of confidence that the given sample of 100 came from a population with a small positive reaction rate. It can be shown that if no positive reactions occur in 100 subjects, the rate of positive reactions in the population is not likely to exceed approximately 2.9%, the likelihood being at the 95% level ;8 and that if one reaction is observed in 100 1From The Army Industrial Hygiene Laboratory Johns Hopkins School of Hygiene and Public Health, Baltimore, Md. 2
Received for publication December 5, 1944.
The likelihood level of 95% is a measure of the confidence with which it can be asserted
that the population rate does not exceed the stated value. The likelihood level may be chosen at any point, depending on how conservative one wishes to be in his estimates. If a higher level, say 99%, is used, then the maximum estimate of the population rate would be increaaed, and it could then be asserted with greater assurance that the population rate did not exceed that higher value. The most frequently used likelihood or confidence levels are 95% and 99%.
227
228
'ruE JOURNAL OF INVESTIGATIVE DERMATOLOGY
TABLE I Upper Limit of Po8itive Reaction Rate Estimated for the Population, with a 95% Likelihood,
from Results of 0, 1, or Positives in a S ample of ,Size n NO.
LILY MAXIMUM RA1Z 07 POSITIVE REACTIONS IN ALPRCE NOREEN 07 SUBJECTS IN $A]iPI2 ______________________________________________ ______________________ If no reaction in sample If one reaction in sample If two reactions in sample (f5)
30,000 20,000 10,000
% .01
.06
2,000 1,000
.15 .30 .37
800 600 500 400 300
.03 .06
.05 .09
.03
5,000 4,000
%
%
.15 .30
.23
.45 .56 .75
.50
.75
1.0 1.2 1.5
.75 .99 1.2 1.5
1.1 1.5
2.0 2.4
3.0 3.6
3.0 3.7 4.9 5.8
4.4 5.5 7.3 8.8
5.8 7.3 9.6
2.0 2.4 2.9
1.8 2.2
250 200 160 150 125 120 100 80 60 50 40 30 20
7.2 9.5
10.8 14.2
14.0
13.9
26.9
20.8 38.2
27.0
10
3.7 4.9
* 95% likelihood.
TABLE II Number of Subjects Required, If There Are No Positives in the Sample to Predict with a 95% or 99% Likelihood That the Population Rate Does Not Exceed Certain Designated Values )SAXIMOM PERMISSIBLE PERCENTAGE OS REACTIONS IN THE POPULATION
NO. 07 SUBJECTS SON LTEELIN000 OF
95%
0.01
29,978
0.1
2,994
0.5 1
2 3
597 298 148
98
99%
46,083 4,603 918 459 228 151
229
STATISTICAL CONSIDERATIONS IN PATCH TESTING
subjects, the likelihood is 95% that the rate does not exceed approximately 4.5%.
Conversely it can be shown that in order to insure, with a likelihood of 95%, that the rate of positive reactions in the population does not exceed 2%, 147 500
400
NO REACTION IN THE SAMPLE
ONE REACTION IN THE SANPI.E — — — — —
TWO REACTIONS 14 THE SAMPLE -— -— -
(ii U)
z 300 0 Id
I
0 0
5
$0
UKELY' MXINUU PERCENTAGE
'5%
O
POSITIVE REACTIONS IN POPULATION
('LIKELIHOOD AT 95% LEVEL)
subjects must be tested and no reactions appear or 225 subjects must be tested and only one reaction appear. The equations used for deriving these results are part of the statistical theory
for the distribution of repeated results. If a certain number of positives are
230
THE JOITRNAL OF INVESTIGATIVE DERMATOLOGY
found in n individuals, and we want to estimate the reaction rate, p, so that the likelihood is 95% that the population value does not exceed p, the following equations4 may be used for the estimation: For 0 positive reactions in the sample of size n (1 — p)"
=
.05
For 1 positive reaction in the sample n(1 —
p)"p
= .05
For 2 positive reactions in the sample n(n — 1) (1 — p)n_2p2 = .05
Calculations resulting from substitution in these equations are shown in Table I and Graph I, which give for various sizes of samples the "maximum" reaction rate estimated if 0, 1 or 2 positive reactions appear in the sample. By "maximum" here, we mean that the likelihood is 95% that it would not be exceeded. Table II gives for the case of no positives in the sample, the size sample necessary to determine with an assurance of 95% or with an assurance of 99% that the population rate does not exceed certain specified values. The data my be used for two purposes. (1) If 0, 1 or 2 positive reactions are observed in a sample of a given size, an estimate can be made as to the likely (95% likelihood) maximum rate of positive reactions in the population from which the sample was drawn. (2) If it is desired to make reasonably certain (95% likelihood) that the rate of positive reactions in the population does not exceed a given figure, the size of sample necessary to accomplish this end may be determined (3 values will of course be obtained depending upon whether 0, 1, or 2 reactions in the sample is
to serve as the criterion). Acknowledgement is made to Dr. Margaret Merrell, Associate Professor of Biostatistics, Johns Hopkins University School of Hygiene and Public Health, for her assistance in the statistical considerations. 4 equation for extimating with a likelihood of P, that the population rate does not exceed p, if a positives are found in n individuals is:
ii!
(l._p)n—apadp=P
In the range of p's involved in the present problem, an approximate solution is given by the following formula, which was used in our estimates:
a! (n—a)!
(1—p)'p = 1—P
THE JOURNAL OF INVESTIGATIVE DERMATOLOGY
94
linolenic acid extract. Arch. This pdf is a scanned copy UV of irradiated a printed document.
24. Wynn, C. H. and Iqbal, M.: Isolation of rat
skin lysosomes and a comparison with liver Path., 80: 91, 1965. and spleen lysosomes. Biochem. J., 98: lOP, 37. Nicolaides, N.: Lipids, membranes, and the 1966.
human epidermis, p. 511, The Epidermis
Eds., Montagna, W. and Lobitz, W. C. Acascopic localization of acid phosphatase in demic Press, New York. human epidermis. J. Invest. Derm., 46: 431, 38. Wills, E. D. and Wilkinson, A. E.: Release of 1966. enzymes from lysosomes by irradiation and 26. Rowden, C.: Ultrastructural studies of kerathe relation of lipid peroxide formation to tinized epithelia of the mouse. I. Combined enzyme release. Biochem. J., 99: 657, 1966. electron microscope and cytochemical study 39. Lane, N. I. and Novikoff, A. B.: Effects of of lysosomes in mouse epidermis and esoarginine deprivation, ultraviolet radiation and X-radiation on cultured KB cells. J. phageal epithelium. J. Invest. Derm., 49: 181, 25. Olson, R. L. and Nordquist, R. E.: Ultramicro-
No warranty is given about the accuracy of the copy.
Users should refer to the original published dermal cells. Nature, 216: 1031, 1967. version of1965. the material. vest. Derm., 45: 448, 28. Hall, J. H., Smith, J. G., Jr. and Burnett, S. 41. Daniels, F., Jr. and Johnson, B. E.: In prepa1967.
Cell Biol., 27: 603, 1965.
27. Prose, P. H., Sedlis, E. and Bigelow, M.: The 40. Fukuyama, K., Epstein, W. L. and Epstein, demonstration of lysosomes in the diseased J. H.: Effect of ultraviolet light on RNA skin of infants with infantile eczema. J. Inand protein synthesis in differentiated epi-
C.: The lysosome in contact dermatitis: A ration. histochemical study. J. Invest. Derm., 49: 42. Ito, M.: Histochemical investigations of Unna's oxygen and reduction areas by means of 590, 1967. 29. Pearse, A. C. E.: p. 882, Histochemistry Theoultraviolet irradiation, Studies on Melanin, retical and Applied, 2nd ed., Churchill, London, 1960.
30. Pearse, A. C. E.: p. 910, Histacheini.stry Thearetscal and Applied, 2nd ed., Churchill, London, 1960.
31. Daniels, F., Jr., Brophy, D. and Lobitz, W. C.: Histochemical responses of human skin fol-
lowing ultraviolet irradiation. J. Invest. Derm.,37: 351, 1961.
32. Bitensky, L.: The demonstration of lysosomes by the controlled temperature freezing section method. Quart. J. Micr. Sci., 103: 205, 1952.
33. Diengdoh, J. V.: The demonstration of lysosomes in mouse skin. Quart. J. Micr. Sci., 105: 73, 1964.
34. Jarret, A., Spearman, R. I. C. and Hardy, J. A.:
Tohoku, J. Exp. Med., 65: Supplement V, 10, 1957.
43. Bitcnsky, L.: Lysosomes in normal and pathological cells, pp. 362—375, Lysasames Eds., de Reuck, A. V. S. and Cameron, M. Churchill, London, 1953.
44. Janoff, A. and Zweifach, B. W.: Production of inflammatory changes in the microcirculation by cationic proteins extracted from lysosomes. J. Exp. Med., 120: 747, 1964.
45. Herion, J. C., Spitznagel, J. K., Walker, R. I. and Zeya, H. I.: Pyrogenicity of granulocyte lysosomes. Amer. J. Physiol., 211: 693, 1966.
46. Baden, H. P. and Pearlman, C.: The effect of ultraviolet light on protein and nucleic acid synthesis in the epidermis. J. Invest. Derm.,
Histochemistry of keratinization. Brit. J. 43: 71, 1964. Derm., 71: 277, 1959. 35. De Duve, C. and Wattiaux, R.: Functions of 47. Bullough, W. S. and Laurence, E. B.: Mitotic control by internal secretion: the role of lysosomes. Ann. Rev. Physiol., 28: 435, 1966. the chalone-adrenalin complex. Exp. Cell. 36. Waravdekar, V. S., Saclaw, L. D., Jones, W. A. and Kuhns, J. C.: Skin changes induced by
Res., 33: 176, 1964.