Planning experiments statistical methods

Weights will be unconsciously applied if operating conditions are non-uniformly distributed in the experimental space. Estimated model parameters will then better reproduce the experimental data from that part of the space where the density of experimentation is greater. Therefore, statistical methods of planning of kinetic experiments, possibly modified by appropriate transformation of variables, are strongly recommended. 

In practice, most lifetime prediction is based on service experience. Depending on the industry concerned, this can take the form of planned examination of components at the end of their service life or be limited to the explanation of warranty returns. Experience with polymers is now sufficiently long for service experience to be a prime source of information for components with lifetimes of up to 35 years. The construction industry provides a good example of systematic listing of component lifetimes, related to minimum quality levels and modified according to the service conditions. The electrical industry applies statistical methods to life components and predict failures. This, however, strays into the general field of engineering component lifetimes. In this book we are concerned with materials rather than components. 

However, it should be emphasized that the statistical methods presented here are no cures for poor data. Irrelevant or erroneous measurement and poorly planned experiments will still be irrelevant, erroneous and poorly planned in spite of any statistical analysis. There are, however, many examples of excellent data that have been seriously mutilated by poor statistical analysis. The aim of this chapter is to present multivariate statistical methods for design and... 

Response Surface Methodology (RSM) is a statistical method which uses quantitative data from appropriately designed experiments to determine and simultaneously solve multi-variate equations (3). In this technique regression analysis is performed on the data to provide an equation or mathematical model. Mathematical models are empirically derived equations which best express the changes in measured response to the planned systematic... 

For a given rate expression this yields the optimal parameter values, but it has to be decided whether the rate expression is the most adequate one. This selection can be based on statistical analysis and on physical significance of the parameter values (for example they should be positive in many cases). Additional experimental effort can be used towards further model discrimination by carefully planning experiments - a kind of experimental design method. 

Many catalytic testing situations benefit from statistical methods of planning efficient sequences of experiments. 

Chemometrics has been defined as the chemical discipline that uses mathematical and statistical methods to design or select optimal measurement procedures and experiments and to provide maximum chemical information by analysing chemical data (Kowalski, 1978). It is a relatively new discipline that assists with (i) the planning of experiments, and (ii) the manipulation and interpretation of large data sets. Some aspects of chemometrics can be done using an appropriate speadsheet but the majority of applications require the use of dedicated software. The fundamental principles of most of the processes involved in chemometrics are those of statistics. You are therefore advised to become familiar with the material in Chapters 40 and 41 before proceeding. 

The observational areas and the oceanographic sampling plan used in this experiment are described here. The choice of the appropriate statistical methods to use in data analysis generally is dependent on the nature of the data and how the data were taken. 

Statistical procedures for planning an experiment, i.e. collecting appropriate data which, after analysis by statistical methods, result in valid conclusions. The design includes the selection of experimental units, the specification of the experimental conditions, i.e. the specification of factors whose effect will be studied on the outcome of the experiment, the specification of the level of the factors involved and the combination of such factors, the selection of response to be measured, and the choice of statistical model to fit the data [Box et ai, 1978 Carlson, 1992]. 

It is evident that experunents run at random also will give results at random. It is necessary to use planned experiments. It is, however, of tremendous importance how the experiments are planned and executed. There are no computational methods, no statistical tricks available which can extract the desired chemical information from experimental data, if the experiments have been run in such a way that there is no such information in the data. 

Planning of experiments and analysis by statistical methods will furnish precise and detailed answers only if precise and detailed questions are posed to the experimental system. For chemical problems, such questions are of chemical origin. Experimental design can therefore never substitute chemical reason or knowledge. With a good experimental design, statistics will provide the chemist with efficient tools ... 

If the method does not meet run time criteria or robustness requirements, it may require further optimization. Method development optimization is typically continued until the objectives outlined in the development plan are met. Method development experts often have the experience necessary to adequately optimize the methodology others may benefit from the documented and organized development approach of available software packages. Several commercial packages are available that combine classical chromatographic theory with statistical design to predict optimum separation conditions with a minimum number of experiments. 

A quasi steady-state solution for the tracer distribution in a soilpolutnn has been developed for the inlet boundary concentration being a constant plus a Sinusoidal component. Then an unsteady state solution for tracer distribution a soil column was developed for the same inlet boundary condition as above. The unsteady-state tracer concentration distribution applies to the section of a soil column that still remembers the initial condition. The two solutions may be applicable to those planning experiments to measure parameters such as the dispersion coefficient from tracer tests. A sinusoidal loading of tracer at the inlet boundary may enable one to obtain repeated data traces at the column outlet as part of an extended experiment. Continued collection of tracer concentration vs. time data at the column outlet over a number of periods would enable one to collect data from repeated experiments, for each period of the sine wave would represent another experiment. This should enable one to obtain more replicates of data to improve statistical estimates of the dispersion coefficient than could be obtained by experimental methods that use a slug loading or a step change of concentration at the column inleL"... 

Burner tests are complex, time-consuming, and expensive experiments. In the case of this category of experiments, one should always put a strong emphasis on the planning supported by statistical methods. Should the examined value be influenced by a low number of factors (< 3) and should the dependence between the values be known as linear, it is sufficient to use the so called factorial plan [29] for fhis fype of experiment, or a fractional factorial plan [29]. Nevertheless, most experiments are supposed to have a response variable without linear dependence on some of the factors, and it... 

This chapter gives an overview of relevant statistical methods for the analysis of bioimpedance measurements, with an aim to answer questions such as How do I begin with planning an experiment How many measurements do I need to take How do I deal with large amounts of frequency sweep data Which statistical test should I use How do I validate my results ... 

V. V. Nalimov, N. A. Tchernova/ Statistic Methods of Planning Extremal Experiments, Nauka, Moscow (1965). 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

Ghosh, S., Ed. (1990), Statistical Design and Analysis of Industrial Experiments, Dekker, New York, NY. Gibson, R.J. (1968), Experimental Design, or Happiness is Planning the Experiment, Bioscience, 18,223-225. Gitlow, H., Gitlow, S., Oppenheim, A., and Oppenheim, R. (1989), Tools and Methods for the Improvement of Quality, Irwin, Homewood, IL. 

One kind of innovation could be, to apply statistical mathematical methods or to develop design of experiments-DOB. DOE is a planned approach for determiniing cause and effect relationships. 

The specific problems discussed above emphasize that environmental chemistry poses considerably harder problems to the chemometrician than straight analytical chemistry [BRERETON, 1995]. The current state of environmental analysis often involves empirical planning of experiments and monitoring, as well as expensive and time-consuming analysis, with the result that only simple statistics are applied to the data obtained. In practice, simple comparison of data averaged in time or space with legally fixed thresholds or limits is often performed at the end of the environmental analysis process. Because environmental data contains so much information, chemometric methods should be used to extract the latent information from these data. 

The thorough treatment of the experimental data does allow one to obtain reliable values of the reactivity ratios. The results of such a treatment are presented in Table 6.3 for some concrete system let us form a notion about an accuracy of the reactivity ratios estimations. The detailed analysis of such a significant problem in the case of the well-studied copolymerization of styrene with methyl methacrylate is reported in Ref. [227]. Important results on the comparison of the precision of rj, r2 estimates by means of different methods are presented by O Driscoll et al. [228]. Such a comparison of six well-known linear least-squares procedures [215-218,222,223] with the statistically correct non-linear least-squares method leads to the conclusion that some of them [216, 217, 222] can provide rather precise rls r2 estimates when the experiment is properly planned. 

For obtaining interpolar formulas for Sn, w, Slw/ n we have used the method of statistic planning of experiment (11). 

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