Statistical Methods in Chemistry Hubert
M . Hill and Robert H. Brown, Tennessee Eastman Co., Division of Eastman Kodak Co., Kingsport, Tenn.
T
continues the attempt to provide a guide to applications of statistics in chemistry and chemical engineering. It covers material published between October 1963 and October 1965. Emphasis has been placed on finding articles which provide examples of proper use of statistical procedures in scientific investigations reported in the chemical literature and articles from the statistical literature which contain examples applied to chemistry or describe techniques useful to the chemical industry. In addition to these are cited some recently published books which are being used in teaching statistics courses for scientists and engineers or are considered to be useful reference sources for those working in the chemical industry. As a result of applying these guidelines, many valuable books and articles addressed to statisticians rather than to chemists and engineers have been omitted from this review. HIS SURVEY
(95); and in “Quality Control and Applied Statistics Abstracts” (81). In the chemical literature, annual reviews on mathematics (96, 97) and process control (103) appear in Industrial and Engineering Chemistry in addition to the present review series (78)in ANALYTICAL CHEMISTRY. There are a number of recently published books in the field: two designed for use by investigators who may not have had formal training in statistics (14, 30); one dealing particularly with experimentation and data analysis (70); several introductory textbooks in statistical methods addressed to “scientists and engineers” [48, 61 (Vol. l ) , 65, l o g ] ; two books containing extensive material on experimental design [63, 61 (Vol. I I ) ] ; and two reference books in the sampling field (12, 69). New editions of tables for mathematicians and statisticians (1, 16, 43, 101) include an old favorite of researchers in the biological, medical, pharmaceutical, and related fields (43).
JOURNALS, BOOKS, REVIEWS, A N D ABSTRACTS
Statistical journals which publish articles of the type cited in this review are: The American Statistician, Annals of Mathematical Statistics, Applied Statistics, Biometrics, Biometrika, Industrial Quality Control, Journal of the American Statistical Association,Journal of the Royal Statistical Society (Series A , Series B ) , Quality Assurance, and Technometn’cs. Of these, Annals of Mathematical Statistics, Biometrika, and Journal of the American Statistical Association typically contain articles of interest primarily to mathematicians and statisticians, although articles of general interest can also be found in them. Applied Statistics and Technometrics are excellent sources of material on statistical methods in general, while Industrial Quality Control and Quality Assurance are devoted to articles on techniques involved in process control and product assurance, and Biometrics is addressed to workers in biological, medical, and related fields. A number of these journals are themselves sources of review material on books, meeting transactions, and technical articles on statistics, mathematics, computers, operations research, etc. More extensive abstracts are provided in Mathematical Reviews (72-74) , which has sections devoted to probability and statistics; in a publication of the International Statistics Institute 440 R
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STATISTICAL A N D QUALITY CONTROL METHODS
Statistical Approach to Problem Solving. A number of expository papers have dealt with the needs and uses for statistics in chemical research and development. Calder (18) makes a plea for chemists to become statisticians (within limits) , Davies has comments on project selection (26), Cram& (24) and Box and Hunter (12) discuss ideas related to choice of models, and Chanmugam and Jenkins (21) deal with means for efficient experimentation with operating chemical processes. Other papers treat specialized topics dealing with efficient experimentation (6, 33, 51, 52, 65, 75), model selection (58), and combination of multiple responses into a single value (50). Statistical Analysis. Box and Cox (10) have provided some useful ideas and techniques related to data transformations. This subject is also touched upon in three other papers, one dealing with textile experiments (6.29, one with radiochemical experiments (79), and one with material balance problems (86). A series by Enrick (SS-4l) provides brief descriptions and calculating procedures for a number of statistical techniques useful in experimental work and data analysis. The workhorse
technique, Analysis of Variance (ANOVA), is introduced in one of these papers (35) and is also employed by the same author in an earlier paper (32) dealing with variation introduced by each phase of a multistage process. Beazley (2) goes farther to describe methods which point out assignable causes for variation in each stage. Another aspect of Analysis of Variance is treated by Calvin in “Individual Comparisons in ASOVA” (19). Considerable attention is given to several different forms of regression analysis : simple linear regression (do), multiple linear regression (45, 60, 94, loo), stepwise multiple regression (99), and nonlinear regression (3, 6, 27, 87, 90). The paper by Shah and Khatri (90) is a particularly helpful introduction to the difficulties inherent in the use of nonlinear regression techniques. Graphical procedures in analysis (56, 67), correlation analysis (40, 4 1 ) , spectral analysis ( 5 ) , principal component analysis ( 7 l ) ,the use of the chisquared statistic ( g o ) , the logistic function (93), and nonparametric statistics (83, 84, 102) are among the statistical methods discussed in recent articles. Models and Experimental Design. The subject of model selection is treated in papers by Cram& (24), by Box and Hunter ( I @ , and by Hunter and Reiner (58). CramCr, summarizing work by many people, provides a survey of general models; Box and Hunter attempt to show how to develop a suitable mechanistic model by experimentation and model revision; Hunter and Reiner discuss designs for discriminating between two rival kinetic models. Factorial designs receive attention in their classical forms (2,8,36,37,62,89), as well as in sequentially applied form (51, 57, 81). Designs particularly suitable for screening (26, 59, 62), for studying mixture problems (77, 88), and for process optimization ( 4 , 11, 20, 1 1 , 39, 42) are to be found along with more general papers (9, 75, 98) on the usefulness of statistically designed experiments. Quality Control. Smith (98) has made an excellent addition to the literature on economic considerations in sampling inspection. The use of computers in process control is discussed by Eilon and Deziel (Sf),by Lee (64), and by Lieberman (66). Experimental designs useful in quality control investigations are described in other work
( 2 , 29, 38, 98). Diviney and David (28) are interested in the relationship between instrument measurement error and sampling plan selection. “Color and Shade Control of Ceramic Tile” is discussed by Dana and coworkers (25). Guthrie (47) discusses some fundamental methods for using statistics in meeting product specifications. EXAMPLES
OF STATISTICAL APPLICATIONS
h few excellent examples of the application of statistics in analytical chemistry are available. However, most authors still quote absolute error or relative error as a measure of the precision of an analytical procedure. In most of these experiments, the analytical method is being applied to samples of unknown composition; hence, the error values quoted are measures of accuracy rather than precision. The standard deviation of experimental procedures and the Confidence limits of effects are quoted infrequently. Tests of statistical significance to determine the probability that the observed bias is different from zero are rarely used. Precision. Evaluation of the precision of analytical methods has been discussed in a paper by Linnig and Mandel (68). This paper is concerned with linear calibration curves and discusses replication error, scatter about the calibration line, and uncertainty of the calibration line. Hana and McLaughlin (49) have discussed the development of precision statements for several hSTbI methods and presented statistical procedures with examples. Hinchen and Koernes (54) have discussed the normalization of data when a mixture of compounds is analyzed for the components present. They recommend improved techniques which depend upon knowledge of the reproducibility of the original measurements. The adjustment of process data to close material and energy balances was discussed by Ripp (86), who proposed a method of adjustment which considers both gross errors and small random errors. Interlaboratory Comparisons. An article by Lashoff (63) presents methods for ranking laboratories and for evaluating methods of measurements in roundrobin tests. He discusses the procedures to be used and shows their relationship to the linear model presented by hfandel and Wernimont (78). Another example of interlaboratory cornparision was presented by Cook, Crispi, and Menczewski (23). They compared trace element analyses for copper, manganese, chromium, and mercury in nine laboratories using emission spectrography, absorption spectrography, polarography, and neutron activation. After normalization, the mean and 95y0confidence limits for
each laboratory, each procedure, and each element was calculated and plotted. Analytical Methods. I n their paper on the use of infrared spectrometry in the analysis of ethylene-propylene copolymers, Brown, Tryon, and Mandel ( I S ) present an excellent example of the use of statistical methods in the evaluation of an analytical method. Statistical procedures were also used by Burns and Muraca (15) in their study of the determination of water in fuming nitric acid. Butlu and coworkers (17) compared three methods for the determination of magnesium in blood serum. Theirs is an excellent example of good experimental design, data analysis, and reporting of conclusions. Other examples of interest are Blackburn’s (5) use of the least squares method in the resolving of components of a complex spectrum, Boonstra’s ( 7 ) use of statistics in the design of balanced x-ray filters, and Mulford and Burrows’ (76) use of statistics in plasma probe measurements. One paper “Statistical Methods and Beer’s Law” (44) makes rather extensive use of statistical procedures and can be misleading to the statistical neophyte. The authors conclude that a correlation coefficient ( T ) greater than or equal to 0.995 is evidence of linearity in Beer’s Law, while a value lower than this is evidence of nonlinearity. The correlation coefficient is not the correct statistic to use for testing lack of fit of a model. The test of the adequacy of a mathematical model is properly approached through regression analysis. For problems of this type, experimenters should consult the paper by Linnig and Mandel (68). Ghosh, Schaad, and Bush (45) discussed the application of the Pascal distribution to single, simultaneous, and diamond-pattern withdrawal in countercurrent distribution. They show how binomial distribution tables can be used to calculate the theoretical distribution of solute from either phase in a withdrawal series. Soni and Little (93) described conditions for use of the logistic function with an example in fatigue data. Chemical Processes. A designed experiment (4.2) in five variables was used to develop a response surface to describe the efficiency of perforated tray columns in a methanol-air-water system. Hunter and Mezaki (59) discussed catalyst selection by screening techniques. Fourteen catalysts were grouped into sets, and a mixture of the catalysts in each set was screened for activity. Regression analysis was used to study production data on nuclear fuel by Glass (46). Limitations of the method are discussed. Shah and Khatri (90) also gave two simple examples of the use of regression analysis for one particular regression model. Reis and Smith (86) used Chi-square to test the
independence of several variables in a study of preference for a consumer product. Hetzel (52) presents 10 steps to be followed in determining process capability, which he defines as the natural tolerance or forecasted variability of a process operating under normal, in-control conditions. Optimization. Optimization by statistical techniques was the purpose of a starch vinylation study ( 4 ) using response surface techniques. Designed experiments using both factorial designs and paired comparisons for preference rating were used by Schneider and Stockett (89)to study gas scrubbing to remove an objectionable odor from a household product. Lee (64) discussed control of a catalytic process and process optimization to maximize profits by using an on-line computer. Kinetics. One of the newest areas of the application of statistical techniques is in the development of kinetic models of chemical reactions. Box and Hunter (12) discussed the development of mechanistic models by experimentation and model revision. Hunter and Reiner (58) discussed kinetic model selection by sequential experimentation and presented experimental designs for discriminating between rival models. Blakemore and Hoerl (6) discussed the fitting of data to kinetic equations by nonlinear least squares methods and the problems and values of this method. Behnken (3) developed special designs for the determination of reactivity ratios in copolymerization, and used nonlinear least squares methods for calculating them. Textile. Several interesting papers on the application of statistical techniques in the textile industry have appeared. Koehler (6.2) discussed the application of designed experiments to screen process variables and the transformation of responses to aid in the analysis of data. Slinger (91) applied extreme-value theory and the Weibull distribution to study of yarn breakage values. In a paper by Verbeck (loo), statistical methods were used to characterize the spinning quality of yarn. He discussed the fitting of data to an equation which relates count strength to twist multiplier and count, with parameters for obliquity, draft, intrinsic strength, ineffective twist, and cohesion. Hoffman (56), in a paper on measuring the aesthetic appeal of textile fabrics, discusses subjective characteristics and representation of the relationships between them by use of a proximity map, a device borrowed from psychology. LITERATURE CITED
(I), Abramowitz, M., Stegun, I. A., Handbook of Mathematical Functions,” U. S. Govt. Printing Office, Washington, 1964. VOL. 38, NO, 5, APRIL 1966
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(2) Beazley, C. C., Chem. Eng. Progr., Symp. Ser. 59, 28 (1963). (3) Behnken, D. W., J. Polymer Sci., 2, 645 (1964). (4) Berry, J. W., Tucker, H., Deutschman. A. J.. Ind. Ena. Chem.. Prod. Res. Devdop. 2,‘318 (1963). (5) Blackburn, J. A., ANAL. CHEM.37, 1000 (1965). (6) Blakemore, J. W., Hoerl, A. E., Chem. Eng. Progr., Symp. Ser. 59, 14 (1963). (7) Boonstra, E. G., J . Sci. Znstr. 42, 563 (1965). (8jBotieril1, J. S. &I., Brit. Chem. Eng. 9, 155 (1964). (9) Box, G. E. P., Technometrics 5, 247 (1963). (10) Box, G. E. P., Cox, D. R., J . Roy. Statist. SOC.(Ser. B ) 26, 211 (1964). (11) Box, G. E. P., Draper, N. R., Biometrika 50,335 (1963). (12) Box, G. E. P., Hunter, W. G., Technometrics 7, 23 (1965). (13) Brown, J. E., Tryon, PYI., Mandel, J., ANAL.CHEM.35,,?172 (1963). (14) Brownlee, K. A., Statistical Theory and Methodology in Science and Engineering,” 2nd ed., John Wiley, New York, 1965. (15) Burns, E., A., Muraca, R. F., ANAL. CHEM.35, 1967 (1963). (16) Burrington, R. S., ed., “Handbook of Mathematical Tables and Formulas,” 4th ed., McGraw-Hill, New York, 1965. (17) Butlu, E. J., Farbes, D. H. S., Munro, C. S., Russell, J. C., Anal. Chim. Acta 30, 524 (1964). (18) Calder, W. G., ANAL. CHEM. 39 (No. 9), 25A (1964). (19) Calvin, T. W., Trans. Ann. Tech. Conj., Am. SOC.Quality Control, 559 (1965). (20) Carpenter, B. H., Sweeney, H. C., Chem. Eng. 72, 117 (1965). (21) Chanmugam, J., Jenkins, G. M., Chem. Eng. Progr., Symp. Ser. 59, 108 (1963). (22) Cochran, W. G., “Sampling Techniques,” 2nd ed., John Wiley, New York, 1963. (23) Cook, G. B., Crespi, M., Menczewski, J., Talanta 10, 917 (1963). (24) Cram&, H., Technometrics 6, 133 (1964). (25) Dana, R., Bayer, H. S., McElrath, G. W., Znd. Quality Control 21 (No. 12), 608 (1965). (26) Davies, 0. L., Technometrics 5, 481 (1968,. (27) Dickinson, A. W., Chem. Eng. Progr., Symp. ser. 59, 84 (1963). (28) Diviney, T. E., David, N. A., Qualztu Assurance 2 (No. 12). 29 (1963). (2$) Duhcan, A. J., Ind. Quaiity Control 22, (No. 2), 55 (1965). (30) Dunn, 0. J., “Basic Statistics, A Primer for the Biomedical Sciences,” John Wiley, New York, 1964. (31) Eilon, S., Deziel, D. P., Operational Res. Quart. 16, 341 (1965). (32) Enrick, N. L., Quantity Assurance 2 (No. ll), 20 (1963). (33) Zbid., 4 (No. 3), p. 25 (1965). (34) Zbid., (No. 4), p. 23.
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(56) Hoffman, R. M., Teztile Res. J . 35,428 (May 1965). (57) Hunter. J. S., Technometrics 6, 41 ’ (1964). ’ f.58) Hunter. W. G.. Reiner. A. M.. Ibid., 7, 307 (1965): (59) Hunter, W. G., Mezaki, R., Znd. Eng. Chem. 56, 29 (1964). (60) Jaech, J. L., Trans. Ann. Tech. Conf.. Am. SOC.Quality - - Control, 539 (1965). (61) Johnson, N. L., Leone, F. C., “Statistics and Experimental Design in Engineering and the Physical Sciences,” Vol. I and 11, John Wiley, New York, 1964. (62) Koehler, T. L., Am. Dyestuf Reptr. 54 (No. lo), 50 (1965). (63) Lashoff, T. W., Mater. Res. Std. 4, 397 (1964). (64) Lee, W. T., Appl. Statist. 13 (No. 3), 195 (1964). (65) Li, C. C., “Introduction to Experimental Statistics,” McGraw-Hill, New York, 1964. (66) Lieberman, G. J., Technometrics 7, 283 (August 1965). (67) Lind, E. E., Young, W. R., Trans. Ann. Tech. Conj., Am. SOC.Quality Control, 545 (1965). (68) Linnig, F. J., Mandel, J., ANAL. CHEM.36 (No. 12), 25A (1964). (69) Mace, A. E., “Sample Size Determinations,” Reinhold, New York, 1964. \ - - I
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