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93 V. STATISTICAL MODELING OF HISTOPATHOLOGICAL PROBABILITIES Charles K. Cayne* In 1968, the Smoking and Health Program was initiated by the National Cancer Institute to identify cigarette characteristics that have a major influence on the carcinogenic risk of smoking. Since 1968, four series of experiments have been conducted to determine the specific carcinogenic activity of smoke condensates on mouse skin. Smoke condensate for each cigarette in the four series was generated,and painted on the backs of mice. From these biological experiments, the number of histopathological verified tumors were used to estimate the probabilities of a mouse being tumor free. These histopathological probabilities were used as a measure of the bio- logical activity of the cigarette smoke condesnates. A corresponding sample of each of the smoke condensates was also analyzed for the amount of chemical constituents in the smoke condensates. Relating the chemical and the biological data acts to identify constituents with im- portant biological impact and can be used to formulate the design of less hazardous cigarette models. This relationship is formulated as a mathemat- ical model that predicts histopathological probabilities as a function of the smoke condensate concentrations applied to the mice and the amount of the chemical constituents in the smoke condensates. The prediction model for the histopathological probabilities (Pf ) is based on biological and chemical data from cigarettes with at least 70°% tobacco content tested in the Series I-IV experiments. Each of the 196 data points used in this study consiste~ of a vector of Pf values, applied condensate concentration to*the mice, and ten chemical constituents of the smoke condensate measured on all four series. *Mathematics and Statistics Research Department Computer Sciences Division

94 Examination of Pearson's correlation coefficients in Table 1-7 for this data reveals that there is no variable that has a correlation greater than r= 0.4 with the Pf values. Since correlation coefficients are an indication of linear relationships, a prediction model for histopathological probabilities will require more complex terms than just linear terms of the independent variables to account for the majority of the variation in the observed Pf values. Correlation coefficients among the independent variables indicate that these variables are not independent with the largest correlation of r = 0.9 between the phenol measurements and colorimetric phenol measurements. Therefore, the true mathematical function which relates the Pf values and the ten independent variables (after dropping the colorimetric phenol variable) was assumed to be approximated by a second order Taylor series. This approxi- mation of an intercept, linear terms, squared terms, and all interactions between two variables consisted of 66 terms. This model was considered too unwieldy and of limited practical use and smaller models were investigated which consisted of a subset of the 66 terms which were also`considered ade- quate to describe the Pf values. The problem of determining the "best" subset of variables has received considerable attention from applied statisticians but the problem is still unresolved. The "best" model found.to predict Pf values consisted of the 17 terms listed in Table 1-8. This model was chosen from many competitors after considering the statistical criteria of the standard deviation of the residuals, the percentage of variation accounted for by the prediction model, Mallows' Cp statistic, and an F-statistic to compare the predicition model with the full second order model. The 17 term model found had the best value for each of the first three criteria and the F-statistic indi- cated that the model was not significantly different at 5~ level from the full second order model.

95 In Figure I-1 the observed Pf values from Series I-IV are plotted against the predicted values. If there was perfect agreement between the observed Pf values and the prediction equation, the observed values would fall on the prediction line drawn from the lower lefthand corner to the upper righthand corner. One standard deviation(s) from the prediction line is s = 0.08 and the percentage of data points 0-15, 15-25, and 25-35 from the prediction line are 70.4%, 27.0%, and 2.6%", respectively. References National Cancer Institute, Smoking and Health Program, Toward Less Hazardous Cigarettes, Report No. 1: The First Set of Experimental Cigarettes, DHEW Publication No. (NIH) 76-905, U.S. Government Printing Office, Washington, DC, 1976. National Cancer Institute, Smoking and Health Program, Toward Less Hazardous Cigarettes, Report No. 2: The Second Set of Experimental Cigarettes, DHEW Publication No. (NIH) 76-1111, U.S. Government Printing Office, Washington, DC, 1976. National Cancer Institute, Smoking and Health Program, Toward Less Hazardous Cigarettes, Report No. 3: The Third Set of Experimental Cigarettes, DHEW Publication No. (NIH) 77-1280, U.S. Government Printing Office, Washington, DC, 1977. Hoeking, R. R. (1976), "The Analysis and Selection of Variables in Linear Regression," Biometrics 32, 1-49.

TABLE 1-7 Pearson's Correlation Coefficients PF C N BAP BAA P 0-C MP-C CP _2H WA VWA Histopathological Probabilities (PF) 1.00 Concentration (C) -0.39 1.00 Nicotine (N) -0.32 -0.25 1.00 Benzo (a) Pyrene (BAP) -0.03 -0.27 0.11 1.00 Benza (a) AnthracePe (BAA) -0.02 -0.25 0.19 0.78 1.00 Phenol (P) -0.11 0.08 0.22 0.04 -0.01 1.00 0-Cresol (0-C) 0.05 0.11 0.12 0.13 0.03 0.65 1.00 ~ M & P Cresol (MP-C) -0.12 0.06 0.16 0.05 0.01 0.77 0.79 1.00 ~ Calorimetric Phenols,(CP) -0.07 0.05 0.17 0.01 -0.07 0.90 0.72 0.81 1.00 pH -0.32 -0.24 0.48 -0.01 -0.01 -0.03 -0.25 0.14 -0.07 1.00 Weak Acid (WA) 0.32 -0.01 -0.04 0.04 -0.01 0.05 0.35 -0.01 0.10 -0.63 1.00 Very Weak Acid (VWA) 0.29 0.12 -0.20 0.01 -0.01 0.28 0.59 0.30 0.36 -0.49 0.65 1.00 s~q4C1.,66

97 TABLE 1-8 The 17 Terms Used to Predicted Histopathological Probabilities I Intercept 2 Concentration 3 Ni coti ne 4 pH 5 Weak Acid 6 Benzo (a) Pyrene 7 (concentration)2 8 (Nicotine)2 9 (pH)2 10 (Benzo (a) Pyrene)2 11 Nicotine X Phenol 12 Benzo (a) Pyrene X Phenol 13 0-Cresol X Phenol 14 0-Cresol X MP-Cresol 15 0-Cresol X Benzo (a) Pyrene 16 0-Cresol X Benza (a) Anthracene 17 MP-Cresol X Benza (a) Anthracene

98 FIGURE I-1 HISTOPATHOLOOICAL PR®BABILITIES T H Q m 0 n 0 a -F' a CiI ~ 0= ~ 0 0 1~ O 0.0 0.1 4 4. A 14 + f 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PREDICTED PR®BABILITIEB • ZH f 0.9 1.0 OBSERVED NISTCJPATN(lLQGICAL PROBABILITIES VERSUS PREDICTED HISTC9PATF4QLOGICAL PROBABILITIES