Analysis of Augmented Unreplicated Factorial
Designs Repeated in Time
Carla A. Vivacqua,
aAndré Luís Santos de Pinho
aand Linda Lee Ho
b*
†Unreplicated designs are fairly common in industrial applications; however, there is resistance to their use in agricultural science. In the agriculture community, there is still a belief that lack of replication may prevent the experimenter from getting useful conclusions. Nevertheless, sound statistical methods that permit valid comparisons in unreplicated studies are available for many types of designs. The objective of this paper is to present an analysis procedure for unreplicated designs combining typical characteristics found in industrial experimentation (factorial designs augmented with center points) and in agricultural applications (inclusion of control treatments and repeated measurements). We illustrate the method through a real experiment to evaluate the use of sugarcane by-products in chicken diet. Specifically, it is an unreplicated two-level factorial design with two additional runs (a center point and a control treatment), with experimental units measured in two periods of time. Replication was initially planned in the case study, but the actual treatment application led to an unreplicated design. The application of the proposed method allows interpretation of the data collected. We conclude that the appropriate use of unreplicated designs in agricultural and biological research may reduce overall costs and lessen the use ofin vivo testing. Copyright © 2015 John Wiley & Sons, Ltd.
Keywords: agricultural application; design of experiments; feed trial; industrial application; half-normal probability plot; repeated measurement
1. Introduction
R
eplication is one of the cornerstones of experimentation, and its importance is undeniable. However, unreplicated designs arise in practice for many reasons. In industrial settings, such experiments are frequently used as a way to save costs. Machado and Petrie1cite the following examples in agricultural science in which replication is impractical, expensive, or impossible: long-term experiments initiated before current understanding of statistics, ecological, and watershed studies, largefield-scale research trials, demonstration plots, geological research, and even unforeseen design mistakes.Unreplicated designs are very useful in industrial experimentation. Applications of these designs are found in the literature since the 1960s. For example, Michaels2 describes the experimental designs most used in Procter and Gamble Ltd., which includes unreplicated fractional factorial designs. Prvan and Street3present an annotated bibliography of application papers on fractional factorial designs with examples of unreplicated cases. Ilzarbe et al.4analyze 77 cases of practical design of experiments applications in the field of engineering published in important scientific journals between 2001 and 2005 and concluded that generally unreplicated designs are used.
Despite of the widespread use of unreplicated designs, they are uncommon in agricultural experiments (exceptions include Thies and Fery;5Astatkie, Joseph, and Martin;6Martin et al.7; Payne;8Perret and Higgins;9and Camara et al.10). Machado and Petrie1point out that many agricultural researchers consider unreplicated experiments to be unscientific and unacceptable for publication.
As a consequence of the lack of replication, a direct estimation of error variance is impossible from unreplicated experiments. This causes problems in the assessment of the significance of estimated effects. However, there are sound methods to overcome this difficulty (e.g., Miliken and Johnson11). Daniel12 proposes the use of normal probability plots for unreplicated two-level factorial experiments. Box and Meyer13and Lenth14provide alternative procedures to normal probability plots.
Experimental design techniques were originally developed for agricultural and biological research. According to Bisgaard,15 although there are early industrial applications of design of experiments, in a broad sense, engineers took a long time to envision that methods used for agricultural and biological studies had relevance to their work. Here, we call attention that methods which are currently widely used in industry can be applied in agricultural, biological, and environmental settings. We present a case study to evaluate the use of sugarcane by-products in chicken diet to show evidence of the utility of unreplicated designs. Specifically, the
a
Departamento de Estatística, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova 59078-970, Natal, RN, Brazil
bDepartamento de Engenharia de Produção, Escola Politécnica da Universidade de São Paulo, 05508090, São Paulo, SP, Brazil
*Correspondence to: Linda Lee Ho, Departamento de Engenharia de Produção, Escola Politécnica da Universidade de São Paulo, 05508090, São Paulo, SP, Brazil.
†E-mail: lindalee@usp.br
Research Article
(wileyonlinelibrary.com) DOI: 10.1002/qre.1796 Published online 26 March 2015 in Wiley Online Library
objective of this paper is to propose an analysis procedure for unreplicated designs combining typical design characteristics found in industrial experimentation (two-level factorial designs with center points) and in agricultural and animal research (inclusion of control treatments and repeated measurements).
Two-level factorial designs are very useful in practice. According to Box, Hunter, and Hunter,16 some of the reasons for the importance of these designs include the relative few runs required per factor studied, the interpretation of the data obtained by elementary arithmetic and graphics, and the fact that the design can be suitably augmented. When the factors investigated are quantitative, the factorial design can be augmented by the inclusion of center points. The levels of the center points are obtained by averaging the two levels of the corresponding factor. Center points allow the evaluation of the linearity of the response.
Experiments in agricultural, biological, and environmental research often involve a control treatment (e.g., Dunnett,17Ogbe et al.,18 and Vieira et al.19). When a factorial design is used, sometimes the control treatment is different from the factorial combinations. In
this case, the inclusion of the control treatment would lead to an augmented design. In this paper, we discuss a case study, which is an unreplicated two-level factorial design augmented with two treatments: a center point and a control. In a completely randomized factorial design, the estimates of main effects and interactions have the same variance. Therefore, the significance of the estimated effects in an unreplicated design can be evaluated through normal or Lenth’s plots. However, in augmented designs with a center point and a control treatment, estimates of some meaningful contrasts will have variance different from the estimates of factorial effects. The current available literature lacks of a method to analyze such a design.
Moreover, the unreplicated two-level factorial design augmented with a center point and a control considered here in the case study is repeated in two different times. Repeated measure data occur frequently in many differentfields (e.g., Kick et al.20). Everitt21
points out that many methods of analysis have been suggested including t-tests at each separate time point and multivariate analysis of variance. More recently, a mixed model approach has been used to analyze data with such structure (e.g., Ogliari and Andrade22).
Also, in some circumstances (e.g., when there are only two times), repeated measurements can be treated and analyzed as split-plot experiments in time (e.g., Hall23). It is common tofind applications of split-plot experiments in time, especially in the agriculture and
animal sciences (e.g., Andersen and Gorbet,24Hall and Weimer,25and Avila et al.26). However, if we extend the experimental design structure by considering altogether an unreplicated factorial plan augmented with one center point and a control treatment repeated in a two-time-period, an analysis procedure is unavailable in the literature. Hence, in this paper, an approach to evaluate the significance of meaningful contrasts is provided when the additional treatments evaluated over a two-time-period are also unreplicated. We present the method through a real experiment. The paper is organized as follows: the case study and the proposed analysis methodology are described in Section 2, the results of the analysis and a generalization of the proposed method to other experiments are presented in Section 3, and conclusions are pointed out in Section 4.
2.
Material and methods
Food contributes, in general, to more than 60% of the total cost of chicken production (e. g., Cordeiro et al.27and Tangendjaja28). The basis of the diet is corn, which becomes expensive in some seasons. For example, the drought of 2012 had a significant impact on US poultry industry. The corn production went down, and prices reached high levels (Price and Kim29). This leads to the search for new food options worldwide. An alternative is to use by-products of the sugarcane industry. This would be especially beneficial to the Brazilian poultry industry because the country is the largest producer of sugarcane in the world. Sugarcane can be grown in many regions of Brazil and can be produced in large or small scale. According to the Brazilian Census Bureau, it is expected to be produced over 743 millions of tons in 2014 in the country (IBGE30).
There are two potential beneficial aspects as outcomes from this study worth highlighting. The first one is in reducing the cost with chicken food by replacing the current diet based on corn with cheaper ingredients based on by-products of sugarcane industry. The second one is reducing the environmental pollution, because by-products of sugarcane have a high pollution potential (e.g., Cortez et al.31). Moreover, based on the Annual Report of the Brazilian Union of Aviculture (UBABEF32) in 2012, Brazil had a chicken meat production that exceeded 12 million tons, keeping its position as largest exporter and third largest producer, just after USA and China. From the total production of chicken, 69% was intended for domestic consumption, and 31% for exports. Therefore, the per capita consumption of chicken was 45 kg in 2012. Considering the importance of the poultry industry in Brazil, any effort to improve the conditions of chicken production may have a substantial impact in the Brazilian gross domestic product.
In the search for reducing costs in chicken production, an experiment was carried out to evaluate the use of three by-products of sugarcane (Factors A, B, and C) in substitution to corn in the feed of chickens. The objective was tofind a cheaper diet composition, which would provide satisfying results. The three factors are left unspecified for proprietary reasons. It was decided to run a two-level factorial design with these factors augmented with a center point. It was also included the usual chicken diet as a control treatment. This control treatment has 0% of the three by-products of sugarcane, 65% of corn, and 35% of soy and other substances. All treatments were measured in two periods of time, 21 and 42 days after the beginning of the study.
One-day-old chicks were allocated to cages and fed for 42 days with one of the treatment diets. Each cage accommodated 10 chicks. The chicks were subjected to two growing stages: initial (from 1 to 21 days) andfinal (22–42 days). Water and food were permanently available to the chicks during the experiment. Each chick was individually weighted three times: at 1, 21, and 42 days. Some responses of interest in evaluating the performance of the diet are body weight, food consumption, food conversion, and weight gain.
The experiment was brought to statisticians after execution, and according to the experimenter, it was a replicated design. However, after the description of how the experiment was performed, it became clear that it was an unreplicated design. The reason for the mismatch involving the planning and the execution was due to the confusion between genuine replication and repetition, and the incorrect identification of experimental and observational units by the experimenter. According to Hinkelmann and Kempthorne,33 genuine replication requires that the same treatment is applied to different experimental units, whereas in
repetition, the same treatment is measured on the same experimental unit. Moreover, the experimental unit is the piece of material to which a treatment is assigned and applied, and observational unit is the piece of material on which observations are made or measured. In many situations, experimental and observational units are identical, but we need to be careful with the situations in which they are different. Thus, the actual treatment application led to an unreplicated two-level three-factor (Factors A, B, and C) factorial experiment augmented with a center point and a control treatment measured at two different periods of time. Therefore, it involves a total of 10 experimental units (cages), each associated with one of the treatments. Table I shows the 10 treatments used. Note that treatments 9 and 10 are, respectively, the center point and the control treatment. Each cage is measured at two different periods of time.
The design can be represented as a (23+ 2) × 2. LetY(10 × 2) × 1be the response variable vector of the chicken food experiment with
Yijthe response variable of ith experimental unit (cage) measured at jth time with i = 1,…,10; j = 1, 2. In this case, the model for the
response variable may be expressed as
Yij¼ μijþ εiþ δij;
and the ith unit has two sources of error,εi, the experimental error, andδij, the time measurement error. Letε and δ be the vectors
representing the experimental error vector and the time measurement error vector, respectively. It is assumed E(ε) = 0 with a covariance matrixσ2
WI10, whereI10is a 10 × 10 identity matrix; E(δ) = 0 with the covariance matrix σ2SI2, whereI2is a 2 × 2 identity matrix
and Cov(εi,δij) = 0 for i = 1,…, 10; j = 1, 2.
Explicitly, the error vectorε can be written as
ε ¼ ε1þ δ11 ε1þ δ12 ε2þ δ21 ε2þ δ22 ⋮ ε10þ δ101 ε10þ δ102 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 102 ð Þ1
and its covariance matrix is a block diagonal matrix given by
Covð Þε 2020¼ V ¼ Σ1 0 ⋯ 0 0 Σ2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ Σ10 2 6 6 6 4 3 7 7 7 5¼ ⊕10i¼1Σi¼ diag Σð 1; Σ2…; Σ10Þ; with
Table I. Treatments for the chicken production experiment in Brazil
Treatments A (%) B (%) C (%) 1 25 3 2 2 50 3 2 3 25 6 2 4 50 6 2 5 25 3 4 6 50 3 4 7 25 6 4 8 50 6 4 9 37.50 4.5 3 10 0 0 0
857
Σi¼ σ 2 Wþ σ2S σ2W σ2 W σ2Wþ σ2S " # 22 ¼ σ2 SI2þ σ2W1212
and 12represents a 2 × 1 vector, with all elements equal to one. Because, in this particular case,Σihave the same expression for all i,
we can simplify the notation by dropping out the subscript i. Therefore, with all the error structure assumptions employed earlier Cov(Y)20 × 20= Cov(ε)20 × 20= diag(Σ, Σ, …, Σ)20 × 20. This model is the same used for a standard split-plot design (e.g., Jones and
Nachtsheim34). And, because the measurements are made in only two periods of time, here, we consider the design as a split-plot
for analysis purposes (Hall25).
The main challenge of the analysis is that due to the lack of replication, traditional methods of hypothesis testing and analysis of variance are inappropriate. We propose a method that adapts strategies commonly used in industrial experimentation. Many methods are available for the analysis of unreplicated factorial designs, but normal probability plots, proposed by Daniel12, have been
the standard procedure (Hamada and Balakrishnan35). The approach enables the detection of active effects using contrasts, even though there are none degrees of freedom to estimate the error variance. The goal of the proposed methodology is to analyze a set of estimated contrasts through half-normal probability plots.
Thefirst step is to identify a set of orthogonal contrasts of interest. Because there are 10 cages measured twice, we can estimate 19 orthogonal contrasts. There are the usual seven contrasts associated with the 23factorial design (to measure the main effects and interactions among the three sugarcane by-products), one contrast associated with the control treatment (to compare the regular chicken diet with all the other alternatives), one contrast associated with the center point (to evaluate the linearity of the model), one contrast associated with the time effect, and nine contrasts representing the interaction of thefirst nine contrasts with time. Table II shows the coefficients of some of these contrasts.
The evaluation of the significance of the contrasts can be done using full or half-normal probability plots. However, only contrasts with the same variance can be plotted on a same graph. All estimated effects and main effects of a completely randomized two-level factorial design are contrasts, which have the same variance, so all contrasts of interest are plotted on one normal graph. For a split-plot design, there is a need for two separate normal plots.12Therefore, the next step is to calculate the variance of the contrasts of interest. Because the design of the chicken experiment can be viewed in a split-plot structure, the estimated effects associated with the factorial treatments have a variance different from the estimated effects associated with time and the interaction among time and the other estimated effects. We have to understand how the variance of the contrasts associated with the center point and the control treatment compared with the variance of the estimated factorial effects.
Let us consider the contrast associated to the main effect of Factor A (a factorial factor), which coefficients can be represented by the vector
cA’ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0) (See Table II).
Table II. Error term and coefficients of some contrasts for each observation of the experiment
Cage Treatment Time Error A D A × D Control Center Control × D Central × D
1 1 1 ε1+δ11 1 1 1 1 1 1 1 1 2 ε1+δ12 1 1 1 1 1 1 1 2 2 1 ε2+δ21 1 1 1 1 1 1 1 2 2 ε2+δ22 1 1 1 1 1 1 1 3 3 1 ε3+δ31 1 1 1 1 1 1 1 3 2 ε3+δ32 1 1 1 1 1 1 1 4 4 1 ε4+δ41 1 1 1 1 1 1 1 4 2 ε4+δ42 1 1 1 1 1 1 1 5 5 1 ε5+δ51 1 1 1 1 1 1 1 5 2 ε5+δ52 1 1 1 1 1 1 1 6 6 1 ε6+δ61 1 1 1 1 1 1 1 6 2 ε6+δ62 1 1 1 1 1 1 1 7 7 1 ε7+δ71 1 1 1 1 1 1 1 7 2 ε7+δ72 1 1 1 1 1 1 1 8 8 1 ε8+δ81 1 1 1 1 1 1 1 8 2 ε8+δ82 1 1 1 1 1 1 1 9 9 1 ε9+δ91 0 1 0 1 8 1 8 9 2 ε9+δ92 0 1 0 1 8 1 8 10 10 1 ε10+δ101 0 1 0 9 0 9 0 10 2 ε10+δ102 0 1 0 9 0 9 0
Obs.: D represents contrasts related to time.
Calculating its variance results in
VarðcA’YÞ ¼ 32σ2wþ 16σ2s The main effect of factor A is estimated as
Eff Að Þ ¼cA’Y 8 and consequently, Var cA’Y 8 ¼σ2w 2 þ σ2 s 4
All the other six usual factorial estimated main effects and interactions have this same variance. Another contrast of interest is related to the control treatment. The coefficients of this contrast (Table II) may be expressed by the vector
cCONTROL¼ 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 9; 9ð Þ The variance of this contrast is equal to
VarðcCONTROL’YÞ ¼ 360σ2wþ 180σ 2 s
Note that the variance of this contrast is quite different (but with a common structure) from the variance of the other estimated main effects and interactions. So, plotting this contrast with the other effects in a single half-normal probability plot may lead to incorrect conclusions. Therefore, an adjustment factor is proposed. Using the contrast representing the estimated main effect of A as a baseline, an adjustment factor is found in order to keep both variances equal. Therefore, a constant for the contrast of control treatment may be found in such a way that
360σ2 Wþ 180σ2s K2 CONTROL ¼σ2w 2 þ σ2 s 4: For the contrastcCONTROL’ Y, the adjustment constant is KCONTROL¼
ffiffiffiffiffiffiffiffi 720 p
. Similarly, an adjustment factor can be found for the center point contrast and for the contrasts associated with the estimated time effect.
3. Results and discussions
Table III presents expressions for the variances and adjustment factors for each contrast of Table II.
After the adjustment, the evaluation of the estimated effects can be made by two half-normal plots. Each plot is suitable for its correspondent group of effects, all having the same variance expression. Figure 1 presents the results related to the weight gain response. Figure 1a (at left hand) shows the half-normal plot of the estimated effects of the factorial factors and their interactions (A, B, C, AB, AC, BC, ABC) plus the estimated effects of the augmented points: the center point and control treatment. Figure 1b (at right hand) presents the half-normal plot of the estimated effects of the time factor (D), its interaction with the factorial factors (AD, BD, CD, ABD, ACD, BCD, ABCD) plus its interaction with the augmented points (Center × D and Control × D). By analyzing the half-normal plot, the estimated effects of the control treatment are active as also the time factor plus the estimated interaction of the time versus control treatment. A more detailed analysis of this estimated interaction led to the conclusion that for thefirst 21 days, the weight gain of the chickens under the control treatment is similar to the ones receiving the alternative diets. On the other hand, for the last 21 days, the weight gain is significantly higher for the control diet. Therefore, considering weight gain, the use of by-products of sugarcane is only viable during thefirst 21 days.
Table III. Variances and adjustment factors for the contrasts in Table II
Type Baseline Contrast Variance
Adjustment factor
Treatment estimated effects Main effect A A VarðcA’YÞ ¼ 32σ2wþ 16σ2s 8
Center Varc’CENTERY¼ 288σ2
wþ 144σ2s
ffiffiffiffiffiffiffiffi 576 p Control Varc’CONTROLY¼ 360σ2
wþ 180σ2s
ffiffiffiffiffiffiffiffi 720 p Treatment and time estimated
effects AD interaction effect A × D VarðcAD’YÞ ¼ 16σ2s 8 D VarðcD’YÞ ¼ 20σ2s ffiffiffiffiffi 80 p Center × D VarðcCENTERD’YÞ ¼ 144σ2
s
ffiffiffiffiffiffiffiffi 576 p Control × D VarðcCONTROLD’YÞ ¼ 180σ2
s
ffiffiffiffiffiffiffiffi 720 p Obs.: D represents contrasts related to time.
In the next paragraphs, the method illustrated for the chicken experiment in Section 2 is presented for a general case. Consider an experiment with nwexperimental units and ns= 2 time measurements on each experimental unit. Let YðnWnSÞ1 be the response
variable vector with Yijthe response variable of ith experimental unit measured at jth time. As earlier introduced in Section 2, the
response variable Yijcan be expressed as
Yij ¼ μijþ εiþ δij
There are two sources of error for the ith unit: the experimental errorεiand the time measurement errorδij. As in Section 2, it is
assumed E(ε) = 0 with a covariance matrix σ2WInW, whereInWis a nW× nWidentity matrix; E(δ) = 0 with the covariance matrix σ
2
SInS, where
InS, a nS× nSidentity matrix, and Cov(εi,δij) = 0 for i = 1,…, nw; j = 1,…, ns.
So the error vectorε is
ε ¼ ε1þ δ11 ⋮ ε1þ δ1nS ε2þ δ21 ⋮ ε2þ δ2nS ⋮ εnWþ δnW1 ⋮ εnWþ δnWnS 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 nWnS ð Þ1
and its covariance matrix is a block diagonal matrix given by
Covð Þε n WnS ð Þ nðWnSÞ¼ V ¼ Σ1 0 ⋯ 0 0 Σ2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ ΣnW 2 6 6 6 4 3 7 7 7 5¼ ⊕ni¼1WΣi¼ diag Σ1;Σ2;…; ΣnW with Σi¼ σ2 Wþ σ2S σ2W ⋯ σ2W σ2 W σ2Wþ σ2S ⋯ σ2W ⋮ ⋮ ⋱ ⋮ σ2 W σ2W ⋯ σ2Wþ σ2S 2 6 6 6 6 4 3 7 7 7 7 5 nSnS ¼ σ2 SInSþ σ 2 W1nS1 ’ nS
1nSrepresents an nS× 1 vector, and consequently, Cov(Y) = Cov(ε).
A contrast of interest can be written as a linear combination of the vectorY, expressed as c ’ Y, with vector c ¼ cð 11c12…c1nsc21c22…a2ns…cnw1cnw2…cnwnsÞ
representing the coefficients of the contrast of interest. Therefore,
Figure 1. Half-normal plot of (a) treatment estimated effects, and (b) treatment and time estimated effects
Cov c’Y ¼ c’Covð Þc ¼ cY ’Covð Þcε ¼ ∑nw i¼1 ∑nj¼1s cij 2 σ2 Wþ ∑ nw i¼1∑nj¼1s cij2σS2¼ cwσ2Wþ csσ2s where Cw¼ ∑ nw i¼1 ∑ ns j¼1cij !2 and cs¼ ∑ nw i¼1∑ ns j¼1c 2
ij. Thus, a matrixC can be built in such a way that the columns of C are formed by each contrast of interest. For inference purposes, the elements of diag[Cov(C ’ Y)] of the matrix Cov(C ’ Y) play an important role in any experiment analysis. An example ofC matrix is comprised of columns 5 to 11 of Table II. Note that the contrasts associated with all estimated factorial effects (main effects and their interactions) have the same variance; however, the contrasts associated with the estimated effects of center point and control treatment have a different variance. So, misleading decisions can be taken if the same normal probability plot or the Lenth’s plot is used to assess the significance of the contrasts. Suitable adjustment factors should be identified so that contrasts can be evaluated in the same plot.
4. Conclusions
In this paper, an approach to evaluate the significance of the contrasts from unreplicated augmented designs with a center point and a control treatment measured in a two-time-period is presented. The adjustment method proposed consists of dividing each contrast of interest by a suitable constant to equalize their variances according to the correspondent baseline estimated effect. This approach enables the analysis of the estimated effects altogether in the statistical procedure of one’s choice to decide which effects are active. This approach can also be applied to experiments with restrictions on the randomization, such as split-plot designs.
Price and Kim28comment that, due to high cost, fewer feed trials are being conducted at universities that are under tight budget constraints. The methodology presented here contributes to reducing costs of experimentation by requiring less experimental units.
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Authors' biographies
Dr Carla Almeida Vivacqua is a professor in the Department of Statistics of the Universidade Federal do Rio Grande do Norte in Brazil. Her main work deals with the design and analysis of cost-efficient experiments for many areas of application. Her research interests include the use of statistical thinking and statistical tools for quality improvement.
Dr André Luís Santos de Pinho is a professor in the Department of Statistics of the Universidade Federal do Rio Grande do Norte in Brazil. His main work deals with the design and analysis of cost-efficient experiments for many areas of application. His research interests include Bayesian approach and dispersion effects to analyze unreplicated two-level experiments.
Dr Linda Lee Ho is a full professor in the Department of Production Engineering of the Universidade de São Paulo in Brazil. Her main work deals with the design of experiment and statistical process control.
