Statistical Methods in Chemistry Hubert
M. Hill and Robert H. Brown, Tennessee Eastman Company, Division o f Eastman Kodak Company, Kingsport, Tenn.
T
survey continues the attempt to provide a guide t o applications of statistics in chemistry and chemical engineering. It covers material published between October 1965 and October 1967. 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 2hemistry or describe techniques useful to the chemical industry. I n 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 t o statisticians rather than t o chemists and engineers have been omitted from this review. HIS
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 Jiathematical Statistics, Applied Statistics, Biometrics, Biometrika, Quality Progress, Journal of the American Statistical ilssociation, Journal of the Royal Statistical Society (Series A , Series B ) , Qualify Assurance, and Technometrics. Of these, Annals of Mathematical Statistics, Biometrika, and Journal of the American Statistical Association typically contain articles of interest primarily t o 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 Qualify Progress and Quality Assurance are devoted to articles on techniques involved in process control and product assurance, and Biometrics is addressed t o 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. Among recently published reviews of this type are two articles in Technometrics, one covering Evolutionary Operations (49) and one on Response Surface Methodology ( 4 1 ) ; a survey of articles on “Industry Use of Statistical Test De-
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sign” (91) and the application of interlaboratory techniques (90) printed in Znd. Quality Control; and a survey of techniques useful in “The Sampling of Bulk Materials” ( 9 ) . Abstracts of articles dealing with probability theory and statistical theory and methods are to be found in Mathematical Reviews (75-79), Statistical Theory and lilethods Abstracts (ll?),and Quality Control and Applied Statistics Abstracts (101). All of these include articles from foreign journals not included in this review. I n the chemical literature, annual reviews on mathematics and statistics (119, 120) appear in Industrial and Engineering Chemistry in addition to the present biennial series (40) in ANALYTICAL CHEMISTRY. Also a recent review article on mathematical methods ( 7 ) appeared in British Chemical Engineering as part of a series on Systems Engineering. Among the books published during this review period was a bibliography of statistical literature (56), indexed by author and covering the period 1940-9. This is the second volume of a series, the first of which covered the period 1950-8. (The 1950-8 bibliography was cross-indexed by subject by 1 4 . J. R. Healy in J. Roy. Statist. SOC.,1963, Series A, Vol. 126, 270-5.) Kew books covering statistical methodology include one written particularly for chemists ( I S ) , one for engineering students (44), one dealing with design and analysis of experiments (96), and one with regression analysis (24). Two books addressed t o students and workers in the biological sciences have applicability to other fields of science as well. One is an introductory text in statistics (94) and the other presents a very lucid explanation of the modern matrix techniques so widely used in computer-based applications of statistical analysis (114). The third edition of “Biometrika Tables” (95) appeared in 1966 with several new entries which should tend t o enlarge the already wide audience of users. STATISTICAL A N D QUALITY CONTROL METHODS
Statistical Approach to Problem Solving. Recent expository papers cover a wide variety of statistical techniques. Ku’s article (68) on “the law of propagation of error” presents a review of this topic and a discussion of
the uncertainties of final results when several sources of variability are present. Jaech (52) and Hahn and Shapiro (36) discuss the hazards facing the user of multiple regression. Jaech suggests that “expressing multiple regression estimators as linear combinations of observations leads t o a better understanding of what multiple regression is and does.” Hunter and Atkinson (47) discuss the relative value of data from planned and unplanned experiments, in the second part of a series on “Statistical Designs for Pilot-plant and Laboratory Experiments.” I n his final article of a series reported in the previous review (4O), Enrick (28) discusses “Overall Strategy in Experimentation.” An example of the use of a series of small factorial experiments in sequence t o increase research efficiency is given by Hunter and Hoff (48). Hinchen (43) emphasizes the value of designed experiments as a means for communicating existing knowledge a t the beginning of a n experimental program. As previously noted, Ostle (91) has prepared a survey of “Industry Use of Statistical Test Design” which contains a n extensive bibliography, concentrating particularly on articles from Technometrics and Industrial Quality Control. Hill and Hunter (41) concentrated their literature survey on Response Surface Methodology while Hunter and Kittrell (49) have a very complete review on Evolutionary Operations as well as a discussion of its successes. Statistical Analysis. I n the last four parts of his eight-part series on basic statistical methods, McCartney (81-84) covers the uses of t- and Fdistributions, the x*-distribution, nonparametric methods, and a general list of “do’s and don’ts” in experimental work. Wilkie (127) has investigated the effect of using the closest two of three observations in estimating the mean and standard deviation for Xormal populations. He concludes that this approach is not advisable unless other information exists which points to the presence of an outlier in the trio. On the other hand, Anscornbe and Barron ( 3 ) in the “Treatment of Outliers in Samples of Size Three” suggest that estimation by Least Squares should be tempered by application of a rejection rule and a modification rule. Ott (92) discusses graphical procedures for the analysis of either production or experimental data t o determine the relative importance of various
process factors being studied. He claims that graphical methods are simpler to use and can be just as effective as Analysis of Variance. Applications of the Binomial Distribution are discussed by Larsen (70) in an article which also gives a full-page reproduction of a nomograph of the Cumulative Binomial Distribution. York (129) treats lines of best fit under various assumptions about error in X and in Y . Papovics (93) treats the special case of evaluating the fit of data to the line Y = X when Y , is a calculated value and X , is an experimental value. Hicks (39) discusses the assumptions involved in covariance analysis and presents a simple example to demonstrate the effect of removing a covariate in an Analysis of Variance problem. I n “The Orthogonalization of Undesigned Experiments,’’ Dykstra (25) gives methods for determining additional runs to make to increase the power of the experiment. Koenig (63) discusses numerical differentiation of experimental data and Smith (115) gives an iterative procedure for estimating the proportions of ingredients in a mixture. Church (16) advocates the use of Principal Component Analysis when the response is a curve. The same technique is used by Wernimont (126) for the purpose of evaluating the performance of spectrophotometers. Another example of the use of Principal Components is found in a paper by Jenkins (5.2‘). The multiresponse situation decribed in “A Report on a Simplex EVOP for Multiple Responses” (74) is an optimization problem where it is desired to reduce the responses t o a single “utility” value. Experimental Design. I n addition to the review articles and expository papers already quoted, several particular aspects of statistical design of experiments have been treated. These include, “Factorial, One R u n at a Time Experiments” (46) to minimize the number of preplanned runs, and fractional factorial designs useful when conducting experiments in which linear or quadratic time trends may be expected (19). Mixture designs have been treated by several authors. Kurotori (69) deals with mixtures of components having lower bounds. bIcLean and Anderson (85) present an algorithm for calcuIating design points for “Extreme Vertices Design of Mixture Experiments.” Gorman (36) discusses this paper in the same issue of Technometrics. Screening of process variables by the use of Plackett-Burman designs has been discussed by Stowe and Mayer (118). Several papers dealing with design considerations involved in developing and evaluating mechanistic models are reviewed in the following section. Building and Testing Models. The use of regression models to predict
grid points has been discussed by Unwin, et al. in “The Use of a Digital Computer for the Calculation of Successive Complex Formation Constants” (123) while Mcllllister, et al. (80) use stepwise multiple regression to test the plausibility and fit of alternative models for flooding of extraction columns. Atkinson (4) traces the steps of design, experimentation, and analysis leading to the estimation of parameters in kinetic models. Kittrell and Mezaki deal with questions related to the choice of a kinetic model in general (59) and in the particular case of “Reaction Rate Modeling in Heterogeneous Catalysis” (60). Two papers discuss the use of diagnostic parameters in model building (42, 6’7) and four papers deal with estimation procedures for nonlinear kinetic models (18, 58, 61, 86). Power function models are treated by Mezaki, Kittrell and Hill (87) and Koppel, Pate1 and Holmes study “Statistical Models for Surface Renewal in Heat and Mass Transfer” in two successive articles (66, 67). -4 stochastic model to obtain exact expressions for the probability of occurrence of all possible extents of reaction was used by Fredrickson (31) to study a cyclic system of first order reversible reactions. Modeling of the extrusion process is the subject of a paper by Klein and Marshall (62). Practical examples are given in almost all of these papers to demonstrate that these complex modeling problems can be dealt with effectively using the techniques described. Additional examples are found in references given in the Applications Section of this review. Statistical Quality Control. A complete description of statistical methods applicable in “The Sampling of Bulk Materials” has been prepared by Bicking (9). He includes an extensive bibliography dealing with this topic. Basic and simplified statistics for describing the quality of lot sampling units is the subject of a paper by Couden ( 1 7 ) . Weaver (124) reviews the use of the “Bruceton Staircase Method” for reducing the number of tests required in a destructive test situation. His example is the drop-testing of plastic bottles. SQC applications in the plastics industry are discussed by DeBell (22) while Kaldy covers application of basic statistical methods to quality control problems in food processing in a three part series (53-55). Roberts (103) has extended his previous work on the relative advantages of the newer forms of control charts and Hsi (45) describes an approach to the choice of a procedure to optimize the cost and reliability of numbers obtained from a control lab. Hurst (50) provides a discussion of a general approach to process control through the use of statistical decision theory.
EXAMPLES OF STATISTICAL APPLICATIONS
A larger number of examples of the application of statistics are appearing in the chemical literature. However, reports on experimental work would be improved by a discussion of the precision of the data and by the application of statistical techniques in data analysis. There are numerous examples of large factorial experiments which were reported in the literature but without showing the analysis of the data t o support the conclusions. Also, many experimenters show straight lines through experimental data with obvious curvature. I n the area of kinetics, many experimenters apparently are not yet familiar with the excellent work which is appearing on the design and analysis of experiments for nonlinear models. Some chemistry departments are beginning to teach statistical techniques t o their students. Chapin and Burns (15) discussed statistical analysis of laboratory data by students. Simple statistics were calculated by the students using data from their own experiments in general chemistry. Analytical Methods. Design and statistical procedures for the evaluation of an automatic Gamma-ray point source calibrator were presented by Garfinkle, hlann, and Youden (33). This is an excellent example of the design of experiments and data analysis for an apparatus which can hold multiple samples. X study of the detection limits in Emission Spectroscopy was reported by Boumans and JIaessen (12). They discuss practical us. theoretical detection limits. Practical limits were set using statistical techniques. Sanborn (107) studied the precision and accuracy of the determination of lattice parameters by step-scaning X-ray diffractometry. Stat is tical met hods were used. A good discussion of random and systematic error in mass spectrometry was presented by Avery, Cuthbert, Prosser, and Silk ( 5 ) . Eckhoff (26)discussed the use of histogram plots in determining the solid volume percentage below detectable size in Coulter-Counter Analyses. Blackburn (10) discusses errors in spectral analysis and the application of data transformation before applying least squares analysis. The classification and analysis of NMR spectra were discussed by Dischler (23). He discussed various methods of fitting observed spectra to reference spectra. The use of multiple regression techniques for relating characteristic X-ray emission line intensities and ingredient percentages in mixtures was discussed by Alley and Myers ( 2 ) . A statistical criterion for testing the quality of chromatographic data was presented by Ledin, Gustavson, and Furst (72). The use of statistical techniques in activation analysis was VOL. 40, NO. 5, APRIL 1968
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the topic of a paper by Smith (115). He discussed the model and method of iterative calculation of the best estimate of the proportion of ingredients in a mixture. A discussion of the error in parameter estimation by least squares methods on gas electron diffraction data was presented by Murata and hforino (88). A study of several methods of fitting curves t o infrared band envelopes was reported by Pitha and Jones (99). Nonlinear least squares approximations were used on Gauss and Cauchy functions. The methods were compared with respect to degree of convergence and computational time. New statistical tests for distinguishing between centro- and noncentro-symmetric structures were described by Srinivasan and Srikrishnan (116). The tests are based on the possibility of generation of a random set of intensities from data available for a given crystal t o give a pair of independent variables. The probability distribution and statistical parameters of such a pair are illustrated with a practical case. Emerson and Cohen (27) used weighted least-squares to adjust ABC type NMR Spectral line position data which contained experimental uncertainties. The effect of molecular weight upon the heterogeneous nucleation of crystallization in polypropylene was studied by Beck ( 8 ) . The precision of the test used was determined statistically, and least squares methods were used to obtain a calibration line. An excellent article on the application of Factor Analysis to the prediction of activity coefficients in nonelectrolytes was presented by Funke, et al. (32). They suggested application to gasliquid chromatography and prediction of possible systems for separation. Bloor, Gilson, and Daykin (11) discussed a restricted Hartree-Fock Perturbation Method for calculating spin densities. Regression analysis was used for model fitting and testing. Two papers were found which compared sensory test scores with quantitative analytical methods (SO, 65). Both papers present the use of statistical techniques for comparing the different test methods used. The application of principal components in choosing which properties to measure on lubricating oil basestocks was discussed by Jenkins (52). His analysis showed that many tests may be superfluous. Different applications of principal components were presented by Church (16) and Wernimont (126). Interlaboratory Studies. The application of statistical procedures in interlaboratory comparisons was discussed by Nelson (90). This paper should be read by anyone expecting to carry out an interlaboratory study. An extensive interlab experiment on international comparison of working 378 R
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standards in Colorimetry was reported by Robertson and Wright (104). This experiment involved thirty laboratories and four glossy ceramic tile standards. The three responses measured were spectral reflectance, luminescent reflectance, and chromaticity reflectance. Analysis of the data and conclusion were presented. Another interlaboratory study was concerned with the laboratory variations in analysis of fertilizers (100). This paper presents an extensive description, discussion, and tabulation of results from a large study. Another paper (34) reported a study of detergency with artificially soiled cloth. The laboratories and detergent types were ranked using confidence bounds and the least-significant difference. A discussion of several round robin experiments on experimental stress rupture of Type I11 polyethylene was presented by Larsen (71). A paper by Yokoyama (128) reported the results of a collaborative study on the determination of 1malic acid in lemon juice. The results of a round robin test on the microbiological assay for chlortetracycline in feeds was reported by Abbey (1). Chemical and Physical Processes. Several excellent papers describing the application of factorial experiments have appeared. A paper by Hunter and Hoff (48) gives an excellent elementary discussion with examples of the use of small factorial experiments in sequence. Stowe and Mayer (118) presented a very readable paper on the use of Plackett-Burnam designs for screening a large number of variables in a chemical system. il. paper by Takaya, Koga, and Hara (121) on the oxidation of p-xylene gives two excellent examples of factorial experiments and shows the analysis of variance for each. An eightfactor experiment studying the deposition of vinyl polymer on wool was described by D'Arcy, Hall, and Watt (20). This paper discusses model selection, choice of levels, and analysis of the results. 4 yield prediction equation is presented. A paper by Weinand (185) discussed two factorial experiments which were run in sequence. The results were used to pick the best conditions for several responses. rl split-plot experiment to evaluate the effects of various test chamber conditions on fading and break strength of fabrics was described by Peters and Saville (97). Komor and Beiswanger (64) studied the relationship between nonionic surfactant structure and solubility and textile wetting. Wetting speed was related to molecular diffusion rate, critical micelle concentration, area per molecule, extent of absorption, and temperature. Regression techniques were used to fit all relationships. A factorial experiment studying chemical machining was described by Schneider and Lucas (113). Thompson (122) discussed the
factors influencing the field stability of soil-aggregate mixtures. Statistical methods were used to determine the mix properties which influence field stability as measured by the Burgraff shear test. The structure of random beds packed with spheres or irregular-shaped particles was characterized both theoretically and experimentally by statistical methods in an article by Debbas and Rumpf (21). They were able to demonstrate that the packing can be considered as a two-component mixture of solids and voids. Several good papers on the application of statistics in optimization were presented. Fein, Paustian, and Sarakwash (29) studied the optimization of a process for preparing l,&bis(acetoxymethy1)carborane. This paper presents two factorial experiments and the application of steepest ascent methods to locate a more desirable set of operating conditions. The maximization of yield for the production of the dyestuff, Indanthrone, was discussed by Balasubramanian, et al. (6). Schindler and Aris (109,110,111)presented three papers on the optimization of questing control of stirred reactors. The Li-Box-Chanmugan technique of adaptive optimization was used in simulation studies. I n addition to the numerous papers on model selection and fitting (see Building and Testing Models), several examples of the use of statistics in kinetics experiments are available. Peterson and Lapidus (98) discussed the application of nonlinear estimation to the kinetics of catalytic ethanol dehydrogenation. The method mas applied directly to the integral conversion data. Linear regression techniques were applied in kinetic studies by Nebeker and Pings (89),Hausmann and King ( S T ) , and Schaleger, Richards, and Watamori (108). Klein and 11Iarshall (62) developed a mathematical model of extrusion. Statistically designed experiments and nonlinear estimation were used in developing estimates of parameters in the model. Stepwise multiple regression was used with empirical data from a number of sources to test the plausibility and fit of alternate models of flooding in pulsed, perforated-plate extraction columns (80). Rose and English (105) investigated the strengths of sacks containing granular materials. Their experimental data were fitted to a theoretical distribution of drop failures. Statistical methods were used to test the goodness of fit of the model. The use of statistical techniques in analyzing spinning and breakage data was described by Little, Fiori, and Louis (73). Schmitz and Metcalfe (112) discussed the effect of test conditions on breakage of glass fibers. Their analysis showed that data from different labo-
ratorirs cannot be compared unless the test conditiolis are known. Canipbell and I3auer (14) suggested, in a study of demixing of solid materials, that the sample-to-sample standard deviation for various particle-size groups be used as a criterion for demixing. The correlation of fatigue limit and true stresa-true behavior was studied by Reemsnyder (102). Heap (38)suggested the mode as a good estimator for central tendency in distributions of tensile strength data in rubber. The relationships between tenacity ( T ) and elongation ( E ) of teytile fibers was studied by Roseiithal (106). He suggested using T d / E as a toughness index which has better statktical properties than the usual T E index. LITERATURE CITED
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(114) Searle, S. R., “Mat;$ Algebra for the Biological Sciences, John Wiley, New York, 1967. (115) Smith, L. H., Anal. Chim. Acta 36, 149 (1966). VOL. 40, NO. 5, APRIL 1968
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(116) Srinivasan, R., Srikrishnan, T., Acta Cryst. 21, 648 (1966). (117) “Statistical Theory and Methods
Abstracts,” International Statistical Institute, Oliver and Boyd, Edinburgh, Scotland. (118) Stowe, R. A,, Mayer, R. P., Znd. Eng. Chem. 58 (2), 36 (1966). (119) Sweeney, R. F., Davis, R. S., Hendrix, C. D., Zbid., 57 (12), 72 (1965).
(120) Zbid., 59 (a), 71 (1967). (121) Takaya, T., Koga, T., Hara, T., Bull. Chem. SOC.Japan 39, 654 (1966). (122) Thompson, iM. R., Mater. Res. Std. 7, 6 (1967). (123) Unwin, E. A., Beimer, R. G., Fernando, Q., Anal. Chim. Acta 39, 95 (1967). (124) Weaver, 0. R., Mater. Res. Std. 6,285 (1966).
(125) Weinand, L. H., Rubber World 154 (a), 73 (1966). (126) Wernimont, G., ANAL.CHEM.39, 554 (1967). (127) Wilkie, T. A,, J. Res. Nat. Bur. Std. 70B, 149 (1966). (128) Yokoyama, H., J. Assoc. Ojic. Anal. Chemists 49,621 (1966). (129) York, D., Can. J. Phys. 44, 1079 (1966).
Thermal Analysis C . 6. Murphy, Xerox Corp., Websfer, New York
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covers the major trends in thermal analysis from the period covered by the previous review (302) to October 1967. As is evident from this and previous reviews on the subject, variability in equipment and techniques is enormous. At the First International Conference on Thermal Analysis (ICTA), a Committee on Standardization was established. An initial report has been made (272) by this committee, and a full report will be presented at the Second ICTA later this year. As might be expected, the committee held no brief for specific instrumentation, but did recommend its description, as well as proper identification of materials, listing of experimental variables, and reproduction of original records in accord with a standard format. If these recommendations are followed, resolution of many thermal analysis problems would result. Keattch ($OW), a member of this committee, has recommended 31 compounds as thermal standards for TGA. The Second Toronto Symposium on Thermal Analysis was held in 1967 (276), with presentations covering advances in thermoanalytical techniques (306); kinetic and thermodynamic data from effluent gas analysis (190); applications to high polymers (8), textile materials (3’75), and metallurgical processes (421); torsional braid analysis (153); thermal analysis of natural fuels (389); and mass spectrometric thermal analysis (224). Two new books have appeared: “Gas Effluent hnalysis” (237), and “Thermal Analysis,” Volume 1, in a series entitled “Techniques and l l e t h ods of Polymer Evaluation” (587). An extensive bibliography on pyrolysis of polymers, listing 321 publications from 1862 to 1963, together with subject and author indexes, has been prepared by Kohl (213). Computer treatment of thermal analysis data is acquiring more attention. The theory of DTA transformations has been examined (281) and formulas were developed and solved by computer HIS REVIEW
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ANALYTICAL CHEMISTRY
which allow heats of transition to be calculated from DTA thermograms. Beraiiek (36) has described TGA equipment with graphical and electronic recording of weight changes, instantaneous values being read on an analog-to-digital converter. The latter can be encoded for computer processing. A digital computer program has been written for determination of activation energy and pre-exponential factor of the Arrhenius equation from raw data obtained from TGX curves (372). The program data for CaC204decomposition provided log IC values in excellent agreement with those calculated manually, and activation energies were 69.0 and 70.7 kcal/ mole from manual and computer calculation, respectively. A system has been described (365) for recording data from DTA and other techniques on magnetic tape in digital form, in anticipation that the system will permit complete identification or quantitative calculations to be made rapidly. Gornik (158) has applied computer analysis to DTA cooling curves of polymers to estimate rates of crystallization. THERMOGRAVIMETRIC ANALYSIS ( T G A )
A general review of this subject has been presented (304),while others have been written covering application to polymeric materials (9, 85). Doyle (114) has reviewed quantitative measurements as applied to polymers, with particular emphasis on kinetics. TGA equipment has been described (22, 29, 66, 7 5 ) . Special equipments have been described for use with radioactive samples (425), including microbalance equipment (257), and for high temperature corrosion studies in SO2 and H2S atmospheres (447). Microbalance equipment has been designed for ultrahigh vacuum use (70) and for high capacity (100 grams), high precision of load) operation (76). Fischer and King (134) have deposited polymeric films on piezoelectric quartz and have followed weight losses by measurement of the frequency change.
This equipment, stated to be rugged and to have a sensitivity in excess of 10-9 gram, was applied to the oxidation of rubber at 150’ C. Equipment for simultaneous infrared absorption spectroscopy and TGA (174) was described and applied to MnCO8 decomposition. Application of the individual techniques to definition of the 160’ C transformation in hydroxyethylethylenediaminetriacetic acid as lactonization (280) indicates the value of such equipment with organic and polymeric materials. Apparatus for simultaneous DTA and TGA has been described (161, 448), incorporating semimicrosample holders. Erdey et al. (326) described a polyplate sample holder that permits more rapid attainment of equilibrium by spreading out the sample. A means of accurate sample temperature measurement has been presented by Chatfield (82). Mettler Instrument (282) has described its commercial derivative TGA equipment with several applications. Rupert (363) has developed equipment for application in a dynamic gas system to provide measurement of weight, weight loss, and rate of weight loss very sensitively (-1.5 x lop4 lb/second) from large samples (250 lb). Experimental conditions and their implications on TGA results have been discussed with respect to sample holders (115, 169, 326), sample size (115, 169), sample comminution (215, 162, 175, 188), heating rate (115, 169), sample atmosphere (115, 169), gas pressure (dol), buoyancy (523), and convection (77, 323, 398). The use of derivative TGA to distinguish between evaporation and decomposition has been demonstrated by Adonyi (3). n’ewkirk and Simons (310) have pointed out that the designation of peak temperatures of derivative TGA curves is incorrect. Use of various atmospheres for differentiation of thermal activity is well known and has been exemplified by polymer degradation studies in O2 and K2 (400) and, in the inorganic field, by a study
