of freedom in this case is (n� 1) where n is thenumber of pairs of data.

Comparisons between means (and standarddeviations) can be extended to the study of three ormore sets of data. Such comparisons require the veryimportant statistical method called analysis of vari-ance (ANOVA).

Application of Statistical Techniques

Many of the significance tests and other proceduressummarized in this article (and many others) are veryreadily performed with the aid of Microsoft Excels,Minitabs, and other widely available programs.Such software also gives instant access to the mostimportant descriptive statistics (mean, median,standard deviation, s.e.m., confidence limits, etc.).In practice, the major problems are therefore (1) ac-curate entry of the experimental data into theprogram (this problem may not arise if an analyti-cal instrument is directly interfaced to a PC); and(2) choice of the appropriate test once data entry hasbeen successfully completed. Guidance on the latter

crucial issue is provided by the resources listed in theFurther Reading section.

See also: Chemometrics and Statistics: ExperimentalDesign; Optimization Strategies; Multivariate Classifica-tion Techniques; Multivariate Calibration Techniques;Expert Systems; Multicriteria Decision Making; SignalProcessing; Spectral Deconvolution and Filtering.

Further Reading

De Levie R (2001) How to Use Excels in AnalyticalChemistry. Cambridge: Cambridge University Press.

ISO (1993) Guide to the Expression of Uncertainty inMeasurement. Geneva: International Organisation forStandardisation.

Meier PC and Zund RE (2000) Statistical Methods in An-alytical Chemistry, 2nd edn. New York: Wiley.

Miller JN and Miller JC (2000) Statistics and Chemomet-rics for Analytical Chemistry, 4th edn. Harlow: PearsonEducation.

Neave HR (1981) Elementary Statistics Tables. Routledge:London.

Experimental Design

G Hanrahan, J Zhu, S Gibani, and D G Patil,California State University, Los Angeles, CA, USA

& 2005, Elsevier Ltd. All Rights Reserved.

This article is a revision of the previous-edition article by J Goupy,

pp. 659–666, & 1995, Elsevier Ltd.

Introduction

Experimental design methods allow the experimenterto understand better and evaluate the factors thatinfluence a particular system by means of statisticalapproaches. Such approaches combine theoreticalknowledge of experimental designs and a workingknowledge of the particular factors to be studied.Although the choice of an experimental design ulti-mately depends on the objectives of the experimentand the number of factors to be investigated, initialexperimental planning (as shown in Figure 1) isessential.

The relationship between the various factorsand response within a given system can be shown

mathematically as follows:

y ¼ f ðx1; x2; x3;y; xkÞ ½1�

where y is the response of interest in the system and xare the factors that affect the response when theirvalues change. In general, the following types of fac-tors can be distinguished: (1) continuous, e.g., tem-perature; and (2) discrete, e.g., experimenters.Factors are considered to be independent if there isno relationship between them and dependent if a re-lationship exists. The values or settings attributed foreach factor are called levels. Each experimental runin an experimental design study requires that one ormore treatments (stimulus applied to one or morefactors) be applied to the system and the responsemeasured. The experimenter then employs statisticaldesign methods to determine if the treatment of in-terest or combination of treatments was significant ininfluencing the response of the system under study.Calculation of the treatment effects can then be usedto identify which variables lead to an optimalresponse.

8 CHEMOMETRICS AND STATISTICS /Experimental Design

The two main applications of experimental designare screening, in which the factors that influence theexperiment are identified, and optimization, in whichthe optimal settings or conditions for an experimentare found. A screening experiment is a systematicapproach to identifying the key input parameters of aprocess or product that affect the output perform-ance. The usual approach is to start with a screeningdesign including all controllable factors that maypossibly influence the experiment, identify the mostimportant ones, and then proceed with an experi-mental optimization design.

Experimental Designs

Full Factorial Designs (Two Levels per Factor)

The most general two-level designs is a full factorialdesign and described as 2k-designs where the base 2stands for the number of factor levels and k thenumber of factors each with a high and low value. Ina full factorial design, the levels of the factors arechosen in such a way that they span the completefactor space. Often, only a lower and upper level ischosen. With two factors, this defines a square in thefactor space, and with three factors, this defines acube. The lower level is usually indicated with a ‘� ’sign; the higher level with a ‘þ ’ sign.

The method can be graphically illustrated in asimplified example: the effects of reaction tempera-ture and pH in determining the spectrophotometricresponse (absorbance) of a standard analyte solution.Figure 2 shows a graphical definition of the exper-imental domain, with the reaction temperaturevarying from 401C (low level) to 601C (high level)

and the reaction pH varying from 1 (low level) to 3(high level). The best experimental points in the do-main are located in the corners A, B, C, and D asfollows: A (401C, pH 1); B (601C, pH 1); C (401C,pH 3); D (601C, pH 3).

The four trials of experimental matrix used in thisexperiment are shown in Table 1, with the results ofeach experiment indicated in the response columnand the factor levels in the rows below the experi-mental matrix. Note that � 1 is used for the lowlevel of each factor and þ 1 for the high level.

If we introduce another variable (e.g., reagentconcentration) in the experiment, it is then possibleto represent the factors as faces on one or more cubeswith the responses at the points. The distribution ofexperimental points within this type of experimentaldomain (23 design) is shown schematically in Figure 3.

Fractional Factorial Design

Fractional factorial designs are arguably the mostwidely used designs in experimental investigations,

Define objectives/variables

Planning/designingprocess

Interpretation ofresults

Conclusions

System under study

Experimentationprocedure

Analysis/modeling

Figure 1 Essential criteria during early experimental planning.

Fac

tor

2 ( P

H)

Factor 2 (temperature °C)

40 60

1

3

A B

DC

Figure 2 Graphical definition of the effects of reaction temper-

ature and pH in determining the spectrophotometric response of

a standard analyte solution.

Table 1 Experimental matrix: spectrophotometric response of a

standard analyte solution

Experiment number Temperature pH Response

A � 1 � 1 y1B þ 1 � 1 y2C � 1 þ 1 y3D þ 1 þ 1 y4Factor levels

(� ) 401C pH 1

(þ ) 601C pH 3

CHEMOMETRICS AND STATISTICS /Experimental Design 9

and mainly used for the screening portion of exper-iments. Such designs are good alternatives to a fullfactorial design, especially in the initial stage of aproject, and considered a carefully prescribed andrepresentative subset of a full factorial design. Infractional factorial designs, the number of experi-ments is reduced by a number p according to a 2k� p

design. In the most commonly employed fractionaldesign, the half-fraction design (p¼ 1), exactlyone-half of the experiments of a full design areperformed.

Suppose a situation occurs in which three factors,each at two levels are of interest, but the experi-menter does not want to run all eight treatmentcombinations (23¼ 8). A design with four treatmentcombinations can then be performed when consider-ing the one-half fraction of the 23 design (23� 1¼ 4).

The fractional factorial design is based on analgebraic method of calculating the contributions offactors to the total variance with less than a full fac-torial number of experiments. Such designs are usefulwhen the numbers of potential factors are relativelylarge because they reduce the total number of runsrequired for the overall experiment. However, byreducing the number of runs, a fractional factorialdesign will not be able to evaluate the impact of someof the factors independently.

Latin Squares

A Latin square is a block design with the arrangem-ent of v Latin letters into a v� v array (a table with vrows and v columns). Latin square designs are oftenused in experiments where subjects are allocatedtreatments over a given time period where time is

thought to have a major effect on the experimentalresponse. Suppose the treatments are labeled A, B,and C. In this particular situation, the design would be

Day 1 A B C

Day 2 C A B

Day 3 B C A

This type of design allows the separation of an ad-ditional factor from an equal number of blocks andtreatments. If there are more than three blocks andtreatments, then a number of Latin square designs arepossible. It can be noted that Latin square designsare equivalent to specific fractional factorial designs(e.g., the 4� 4 Latin square design is equivalent toa 43� 1 fractional factorial design).

Greco-Latin Squares

The Greco-Latin square design involves two Latinsquares that are superimposed on each other. It con-tains two treatment factors instead of one and con-tains four factors overall instead of three. Anexample design would look as follows:

A1 A2 A3 A4

B1 C1D3 C2D4 C3D1 C4D2

B2 C4D2 C1D1 C2D3 C3D4

B3 C3D1 C4D2 C1D3 C2D4

B4 C2D4 C1D3 C3D2 C4D1

The analysis for the Greco-Latin square design issimilar to that of a Latin square design. However,one noticeable difference is that two treatment sumof squares have to be computed (factors C and D)by listing two sets of means outside the designtable. As an additional note, Greco-Latin squaresare most effective if replicated and are subject tothe same randomization rules as for the Latinsquares.

Response Surface Designs (More than TwoLevels for One or More Factors)

Response surface methodology is designed to allowexperimenters to estimate interactions, thereforegiving them an idea of the shape of the response sur-face they are investigating. This approach is oftenused when simple linear and interaction models arenot adequate, e.g., experimentation far from theregion of optimum conditions. Here, the experiment-er can expect curvature to be more prevalent and willneed a mathematical model, which can represent thecurvature. The simplest such model has the quadratic

A B

G H

−1−1

−1

+1

+1

+1

C D

E F

X2

X1

X3

Figure 3 Full factorial design at two levels, 23 design.

10 CHEMOMETRICS AND STATISTICS /Experimental Design

form:

Y ¼ b0 þ b1X1 þ b2X2 þ b12X1X2 þ b11X21 þ b22X

22 ½2�

which contains linear terms for all factors, squaredterms for all factors, and products of all pairs offactors. The two most common designs generallyused in response surface modeling are central com-posite designs and Box–Behnken designs. In thesedesigns the inputs take on three or five distinct levels,but not all combinations of these values appear in thedesign.

Central composite designs contain imbedded fac-torial or fractional factorial design with center pointsthat is augmented with a group of axial (star) pointsthat allow estimation of curvature (Figure 4). Onecentral composite design consists of cube points atthe corners of a unit cube that is the product of theintervals [� 1, 1], star points along the axes at oroutside the cube, and center points at the origin.Points A, B, C, and D are the points of the initialfactorial design with points E, F, G, and H being thestar points at the central 0.

A central composite design always contains twiceas many star points as there are factors in the design.The star points represent new extreme values (lowand high) for each factor in the design. There arethree types of central composite designs, ultimatelydepending on where the star points are placed. Cir-cumcised central composite designs are the originalforms of the central composite design. The starpoints are at some distance a from the center basedon the properties desired for the design and thenumber of factors in the design. The a values dependon the number of factors in the factorial part of thedesign.

Inscribed central composite designs are a scaleddown version of the circumcised version in that theyuse the factor settings as the star points and createfactorial or fractional factorial design within thoselimits. In other words, inscribed designs are thosewhere each factor of the circumcised version isdivided by a for their generation. Face centered cen-tral composite designs occur when the star points areat the center of each face of the factorial space, wherea¼71.

Box–Behnken Designs

The Box–Behnken is considered an efficient option inresponse surface methodology and an ideal alter-native to central composite designs. It has three levelsper factor, but avoids the corners of the space, andfills in the combinations of center and extreme levels(Figure 5). It combines a fractional factorial with in-complete block designs in such a way as to avoid the

extreme vertices and to present an approximatelyrotatable design with only three levels per factor.A design is rotatable if the variance of the predictedresponse at any point x depends only on the distanceof x from the design center point. It must be noted,however, that Box–Behnken designs should be con-fined to uses where the experimenter is not interestedin predicting response at extremes (corners of thecube).

Mixture Designs

In a mixture design experiment, the independent fac-tors are proportions of different components of a

A B

H

G

+1

+1

−1

−1

C D

E 0 F X1

X2

Figure 4 Central composite design consisting of a full factorial

two-level and star design.

+1

0

−1

−1 +1 −1

0

+1

X2

0

X1

X3

Figure 5 Box–Behnken design with three levels per factor.

CHEMOMETRICS AND STATISTICS /Experimental Design 11

blend and often measured by their portions, whichsum to 100% or normalized to 1, i.e.,

XN

i¼1

Xi ¼ 1 for xiX0 ½3�

As shown, mixture components are subject to theconstraint that they must equal to the sum of one. Inthis case, standard mixture designs for fitting stand-ard models such as simplex-lattice and simplex-cent-roid designs are employed. When mixtures aresubject to additional constraints, constrained mix-ture designs (extreme-vertices) are then appropriate.Like the factorial experiments discussed above, mix-ture experimental errors are independent and iden-tically distributed with zero mean and commonvariance. In addition, the true response surface isconsidered continuous over the region being studied.Overall, the measured response is assumed to dependonly on the relative proportions of the components inthe mixture and not on the amount.

See also: Chemometrics and Statistics: OptimizationStrategies; Multivariate Calibration Techniques.

Further Reading

Box GEP, Hunter WG, and Hunter JS (1987) Statistics forExperimenters: An Introduction to Design, Data Anal-ysis and Model Building. New York: Wiley.

Cornell JA (1990) Experiments with Mixtures. New York:Wiley.

Dean A and Voss D (1999) Design and Analysis of Exper-iments. New York: Springer.

Goupy J (1993) Methods for Experimental Design. Prin-ciples and Applications for Physicists and Chemists. Am-sterdam: Elsevier.

Kempthorne O (1976) Design and Analysis of Experi-ments. New York: Wiley.

Martens H and Martens M (2001) Multivariate Analysis ofQuality. Chichester: Wiley.

Massart DL, Vandeginste BGM, Buydens LMC, et al.(1997) Handbook of Chemometrics and Qualimetrics,Part A. Amsterdam: Elsevier.

Miller JN and Miller JC (2000) Statistics and Chemomet-rics for Analytical Chemistry, 4th edn. New York:Prentice-Hall.

Morgan E (1991) Chemometrics: Experimental Design.Chichester: Wiley.

Myers RH and Montgomery DC (2002) Response SurfaceMethodology: Process and Product Optimization UsingDesigned Experiments. New York: Wiley.

Otto M (1999) Chemometrics: Statistics and ComputerApplication in Analytical Chemistry. Chichester: Wiley-VCH.

Sharaf MA, Illman DL, Kowalski BR (1986) Chemomet-rics. Chemical Analysis Series, vol. 82. New York: Wiley.

Walters FH, Parker LR, Morgan SL, and Deming SN(1991) Sequential Simplex Optimization. Boca Raton,FL: CRC Press.

Glossary

Blocking procedure by which experimental unitsare grouped into homogeneous clustersin an attempt to improve the compari-son of treatments by randomly alloca-ting the treatments within each clusteror ‘block’.

Control A control is a treatment, which is includ-ed to provide a reference set of datawhich can be compared with data ob-tained from the experimental treatments.

Experiment An investigation in which the investiga-tor applies some treatment(s) to exper-imental units to be observed andevaluated by measuring one or more re-sponse variables.

Experimentalunit

A physical entity or subject subjected tothe treatment independently of otherunits.

Factor A categorical explanatory variable stud-ied in an experiment, e.g., pH, flow rate.

Factorial designs A factorial design is used to evaluatetwo or more factors simultaneously. Thetreatments are combinations of levels ofthe factors. The advantages of factorialdesigns over one-factor-at-a-time exper-iments are that they are more ef-ficient and allow interactions to be de-tected.

Levels The different values assigned to a factor.

Randomization A random assignment of experimentalmaterial to treatments prior to the startof the experiment. Randomization isvital in the experimental design processand provides: (1) the basis for a validinterpretation of the experimental out-comes in terms of a test of statisticalsignificance, and (2) the basis for co-mputing a valid estimate of experimen-tal error by justifying the assumption ofindependence of responses over experi-mental units.

Replication When a given combination of factors ispresent in a system, replication can beused to: (1) demonstrate that results arereproducible, (2) provide a degree of as-surance against erroneous results due tounforeseen reasons, (3) provide themeans to estimate experimental error,and (4) provide the capacity to increase

12 CHEMOMETRICS AND STATISTICS /Experimental Design

the precision for proper estimates oftreatment means.

Responsevariable

A characteristic of an experimental unitmeasured after treatment and analyzed

to address the objectives of a particularexperiment.

Treatments The set of circumstances created for anexperiment.

Optimization Strategies

R G Brereton, University of Bristol, Bristol, UK

& 2005, Elsevier Ltd. All Rights Reserved.

Introduction

Optimization has a significant role in analytical sci-ence. There are many reasons for finding an opti-mum. For example, it may be important to maximizethe extraction efficiency of a compound from a ma-trix; there may be a large number of factors involvedin the extraction procedure. Other examples involveimproving chromatographic separations and opt-imizing the factors that influence signal intensity inatomic spectroscopy.

Traditional methods involve studying the influenceof one factor at a time. For example, we might wantto look at chromatographic resolution of two com-pounds using isocratic conditions, as a function ofproportion of water (using a mobile phase of waterand acetonitrile) and acidity (as controlled by thenature and amount of buffer). We do not know howthe resolution varies as a function of proportion ofwater and pH prior to the experiment. Figure 1Aillustrates a possible underlying response surface. Wemight try to set pH at a constant level, then vary thepercentage water, until we reach an optimum, thenset this percentage constant, varying the pH until afresh optimum is chosen, which we use as the bestconditions. The problem with this strategy is illus-trated in Figure 1B. The change in resolution as afunction of proportion of water differs according topH, so a different optimum proportion of water isfound according to the pH that is used. Hence, if weperformed an optimization in which the initial ex-periments were performed at pH 4, we would obtaina different result for the best conditions to an opti-mization at pH 8 and so, in both cases a false op-timum unless we happen by accident to have hit onthe correct pH in the original experiments. The rea-son for this is that the influence of pH and mobilephase composition are said to interact and so cannotbe considered independent factors. If there areseveral different factors involved in an optimization

it can be extremely hard to find true optima usingtraditional methods.

In order to overcome these difficulties, it is necessaryto employ systematic approaches for determination

2

1

1.8

1.6

1.4

1.2

0.8

0.6

0.410 15 20 25 30 35 40 45 50 55 60

pH 8

pH 4

pH 6

9

8

7

6

5

4

3

210 15 20 25 30 35 40 45 50 55 60

1.11.1

1.1

1.1

1.1

1.10.80.6 1

1

11

1

0.4

1.21.2

1.2

1.2

1.2

1.2

1.31.3

1.3

1.3

1.3

1.4

1.41.4

1.4

1.41.3

1.5

1.5

0.80.6

1.5

1.5

1.5

1.6

1.6

1.61.6

1.61.7

1.71.68

1.68

1.68

1.64

1.64

1.64

1.64

Res

olut

ion

Proportion water

1.81.61.41.2

0.80.60.40.2

1

9 8 7 6 5 4 3 2 1020

3040

5060

pHPercentage water

Res

olut

ion

(A)

(B)

(C)

Figure 1 (A) Response surface for a typical chromatographic

resolution as pH and proportion of water (in a water–acetonitrile

mixture) is altered. (B) Resolution as a function of proportion of

water at three different values of pH. (C) Representation as a

contour plot.

CHEMOMETRICS AND STATISTICS /Optimization Strategies 13