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A generalized test variable approachfor grain yield comparisons of riceShin-Fu Tsaiaa Department of Statistics, Feng Chia University, Taichung 40724,TaiwanPublished online: 27 May 2014.

To cite this article: Shin-Fu Tsai (2014): A generalized test variable approach for grain yieldcomparisons of rice, Journal of Applied Statistics, DOI: 10.1080/02664763.2014.922169

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Journal of Applied Statistics, 2014http://dx.doi.org/10.1080/02664763.2014.922169

A generalized test variable approach forgrain yield comparisons of rice

Shin-Fu Tsai∗

Department of Statistics, Feng Chia University, Taichung 40724, Taiwan

(Received 19 August 2013; accepted 5 May 2014)

Traditionally, an assessment for grain yield of rice is to split it into the yield components, includingthe number of panicles per plant, the number of spikelets per panicle, the 1000-grain weight and thefilled-spikelet percentage, such that the yield performance can be individually evaluated through eachcomponent, and the products of yield components are employed for grain yield comparisons. However,when using the standard statistical methods, such as the two-sample t-test and analysis of variance, theassumptions of normality and variance homogeneity cannot be fully justified for comparing the grainyields, leading to that the empirical sizes cannot be adequately controlled. In this study, based on theconcepts of generalized test variables and generalized p-values, a novel statistical testing procedure isdeveloped for grain yield comparisons of rice. The proposed method is assessed by a series of numericalsimulations. According to the simulation results, the proposed method performs reasonably well in TypeI error control and empirical power. In addition, a real-life field experiment is analyzed by the proposedmethod, some productive rice varieties are screened out and suggested for a follow-up investigation.

Keywords: generalized p-value; generalized test variable; reproductive success; yield comparison;yield component

1. Introduction

Rice (Oryza sativa) is a major cereal grain consumed as a staple food in Asia, Africa and LatinAmerica. Traditionally, an assessment for yield performance is to split the grain yield into theyield components. For a comprehensive introduction to the yield components of rice, the readeris referred to Fageria [1]. More precisely, let Yi be the grain yield per plant of the ith variety,which can be decomposed into the yield components as follows:

Yi = Yi1 × Yi2 × Yi3

1000× Yi4

ni4, (1)

where Yi1 denotes the number of panicles per plant; Yi2 stands for the number of spikelets per pan-icle; Yi3 represents the 1000-grain weight; and Yi4 denotes the number of filled spikelets amongni4 spikelets. Note that Yi4 over ni4 in the expression of Equation (1) represents the percentage of

∗Email: sftsai@fcu.edu.tw

c© 2014 Taylor & Francis

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filled spikelets. According to the decomposition presented as Equation (1), grain yield per unitarea can be readily estimated by multiplying a constant, which equals the total of plants grown inthat area, to the grain yield per plant, when the spaces between plants are fixed. For example, ifthe plants are spaced at 30 cm × 30 cm, the grain yields per acre and hectare are equal to 43,560and 107,593 times of the grain yield per plant, respectively.

Typically, a routine approach for analyzing a product of a series of random variables describedas Equation (1) is to take logarithms to convert the product into sum. For example, Piepho[9] used an additive log-transformed model on the yield components for selecting parents inheterosis-breeding and stability analysis. Although logarithmic transformation significantly sim-plifies the problem, the interpretation for the obtained results on log-transformed data is usuallyless intuitive. On the other hand, the measurements of yield components are commonly assumedto be normally distributed, then the comparisons on yield components can be simply done byusing the standard two-sample t-test or z-test. Also, the two-sample t-test is frequently employedfor comparing the grain yields, which are estimated by the products of measurements of the yieldcomponents. Clearly, the distribution assumption is not fully justified for the two-sample t-testin comparing the grain yields, since the product of a series of normal random variables is notnormally distributed. In practical applications, however, the use of two-sample t-test is justifiedby its robustness against non-normality. The reader is referred to Posten [10] for a thorough dis-cussion regarding the robustness of the two-sample t-test. To our knowledge and experience, thetwo-sample t-test performs not really well in comparing the grain yields, especially in Type Ierror control. In this study, based on the concepts of generalized test variables and generalizedp-values, a novel statistical testing procedure is developed, so that the grain yields of differentvarieties can be compared efficiently.

The rest of this article is now organized as follows. Section 2 formulates the problem ofinterest. Section 3 gives the proposed testing procedure for comparing the grain yields of rice.Section 4 presents a series of numerical simulations for assessing the performance of the pro-posed method. In Section 5, a real-life field experiment is analyzed by the proposed method,some productive rice varieties are screened out and suggested for a follow-up investigation.Concluding remarks are given in the final section.

2. Problem formulation

Suppose that v varieties are now considered for pairwise yield comparisons. Let Yi1k , Yi2k andYi3k represent the kth measurement of the number of panicles per plant, the number of spikeletsper panicle and the 1000-grain weight of the ith variety, respectively. Furthermore, assume thatthe measurements of these traits are normally distributed. More precisely, let

Yijk ∼ N(μij, σ2ij )

for i = 1, . . . , v; j = 1, 2, 3; and k = 1, . . . , nij. Note that μij and σ 2ij separately represent the pop-

ulation mean and population variance of the jth yield component of the ith variety, and nij denotesthe number of observations. Alternatively, let Yi4 denote the number of filled spikelets among ni4

spikelets. It is natural to assume that Yi4 is a binomial random variable, that is,

Yi4 ∼ Binomial (ni4, pi4),

where pi4 represents the filled-spikelet percentage of the ith variety. Typically, the comparisonon a specific trait between different varieties is of particular interest. Therefore, the hypothesis

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testing problems, such as

H0 : μij = μi′j versus H1 : μij �= μi′j

for j = 1, 2 and 3; or

H0 : pi4 = pi′4 versus H1 : pi4 �= pi′4,

are practical issues in a grain yield comparison study. Under the assumption of normality,an investigator can determine whether the experimental data support the null hypothesis H0 :μij = μi′j by the standard two-sample t-test. On the other hand, the sampling distribution ofstandardized difference between two sample proportions can be approximated by the standardnormal distribution. Accordingly, the z-test can be employed for testing the null hypothesisH0 : pi4 = pi′4. In a yield comparison study, however, the most important task is to comparethe grain yields among different varieties. Let

μi = μi1 × μi2 × μi3

1000,

then the expected grain yield per plant of the ith variety can be expressed as μipi4, and thecomparison on the grain yields between the ith and i′th varieties can be described as

H0 : μipi4 = μi′pi′4 versus H1 : μipi4 �= μi′pi′4, (2)

or equivalently,

H0 : μi1μi2μi3pi4 = μi′1μi′2μi′3pi′4 versus H1 : μi1μi2μi3pi4 �= μi′1μi′2μi′3pi′4.

Clearly, the grain yield comparison can be explicitly formulated as the comparison between twoproducts of yield component means. Furthermore, define

θii′ = μipi4

μi′pi′4

= μi1μi2μi3pi4

μi′1μi′2μi′3pi′4,

the ratio of expected grain yields between the ith and i′th varieties. The grain yield compari-son between the ith and i′th varieties can be equivalently achieved through θii′ . For example,suppose that the 1000-grain weight of the ith variety is twice as that of the i′th variety, that is,μi3 = 2μi′3, and the performances of remaining traits are all identical. In this case, θii′ = 2, whichcan be interpreted as that the grain yield per plant of the ith variety is twice as that of the i′th vari-ety. Accordingly, the comparison of grain yields described as Equation (2) can be equivalentlyformulated as

H0 : θii′ = 1 versus H1 : θii′ �= 1. (3)

Under the normal and binomial assumptions for the measurements of yield components, the con-struction of a testing procedure with an explicit sampling distribution for the hypothesis testingproblem described as Equation (3) is quite a challenge. Throughout, our main objective is todevelop a novel statistical testing procedure for addressing this practical issue.

3. GTV-based approach

Before presenting the proposed testing procedure, some fundamentals regarding the generalizedtest variables and generalized p-values are first given.

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3.1 Generalized test variables and generalized p-values

Suppose that Y is a random sample from a distribution depending on an unknown parametervector ζ = (θ , ν), in which θ is the parameter of interest and ν is the vector of nuisance parame-ters. Let y be the realized value of Y . A data-based random quantity T = T(Y ; y, ζ ) is said to be ageneralized test variable, abbreviated as GTV hereinafter, for θ , if the following three conditionsare fulfilled simultaneously.

(A) The distribution of T is free of the nuisance parameters ν.(B) The observed value of T , say tobs = T(y; y, ζ ), does not depend on any unknown parameters.(C) For given y and ν, P(T < t0 | θ) is monotonic in θ for any fixed t0.

Note that T = T(Y ; y, ζ ) is said to be a data-based random quantity, since it depends on therandom sample Y , the observed value y, and the parameters ζ . A generalized extreme regionfor testing the null hypothesis H0 : θ ∈ �0 versus the alternative hypothesis H1 : θ ∈ �1 is thusdefined as C = {Y : T(Y ; y, ζ ) ≥ tobs}. Note that �0 is a subset of parameter space and �1 isits complement. In a statistical testing procedure, a small probability of C indicates that theempirical evidence against the null hypothesis. Accordingly, a generalized p-value is defined as

p = supθ∈�0

P(C | θ).

Suppose that the null hypothesis H0 : θ ≤ θ0 versus the alternative hypothesis H1 : θ > θ0 arenow considered. If T is stochastically increasing in θ , a generalized p-value is defined as

p = P(T ≥ tobs | θ0).

When T is stochastically decreasing in θ , then a generalized p-value is given by

p = P(T ≤ tobs | θ0).

For a given nominal significance level α, the null hypothesis is rejected if p < α. Typically, theexplicit sampling distribution of T is difficult to characterize, and the generalized p-value is usu-ally estimated by a Monte-Carlo algorithm. The generalized test variable is first proposed byTsui and Weerahandi [13], and it has been successfully used in many industrial and biomedicalhypothesis testing problems, especially for those with nuisance parameters, including Weera-handi and Johnson [15], Krishnamoorthy and Mathew [6], McNally et al. [8], Tian and Cappelleri[12], Krishnamoorthy and Guo [4] and Li et al. [7], among others. The reader is referred toWeerahandi [14] for a comprehensive introduction to the generalized test variables and relatedgeneralized statistical inferences.

3.2 The test based on generalized test variables

Let Yij and S2ij be the sample mean and sample variance for the jth yield component of the ith

variety, and yij and s2ij be the observed values of Yij and S2

ij, respectively. Under the normalityassumption, one has

Zij = Yij − μij

σij/√

nij∼ N(0, 1), (4)

and

Vij = (nij − 1)S2ij

σ 2ij

∼ χ2nij−1, (5)

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Journal of Applied Statistics 5

where N(0, 1) and χ2nij−1 represent the standard normal distribution and chi-square distribution

with nij − 1 degrees of freedom, respectively. Define

Tμi =⎧⎨⎩

3∏j=1

⎡⎣yij − Zij

√(nij − 1)s2

ij

nijVij

⎤⎦/ 1000

⎫⎬⎭− μi. (6)

First, it is straightforward to verify that Tμi in the expression of Equation (6) is free of the nui-sance parameters σ 2

i1, σ 2i2 and σ 2

i3. In other words, the condition (A) of a GTV is satisfied. Next,replacing Yij with yij in Equation (4) and S2

ij with s2ij in Equation (5), and plugging them into

Equation (6), it immediately follows that

tμi =⎧⎨⎩

3∏j=1

⎡⎣yij −

(yij − μij

σij/√

nij

)√√√√(nij − 1

)s2

ij

nij

σ 2ij(

nij − 1)

s2ij

⎤⎦/ 1000

⎫⎬⎭− μi

= μi1 × μi2 × μi3

1000− μi

= 0,

which does not depend on any unknown parameters. As a result, the condition (B) of a GTV isfulfilled. Furthermore, the distribution function of Tμi can be expressed as

Pr(Tμi ≤ t0) = Pr(Rμi ≤ t0 + μi),

where

Rμi =3∏

j=1

⎡⎣yij − Zij

√(nij − 1)s2

ij

nijVij

⎤⎦/ 1000.

Clearly, the distribution function of Tμi is stochastically increasing in μi for a fixed t0. Therefore,the condition (C) of a GTV is also satisfied. Accordingly, Tμi presented as Equation (6) is thusverified to be a GTV for μi. On the other hand, let

pi4 = Yi4

ni4, (7)

which stands for the sample proportion of filled spikelets of the ith variety. Based on the gener-alized test variable Tμi described as Equation (6), the proposed test statistic for pairwise grainyield comparison is now given as

Tθii′ = Tμi pi4

Tμi′ pi′4. (8)

The random quantity presented as Equation (8) is not a GTV for θii′ , since the two randomquantities pi4 and pi′4 are involved. However, the performance of Equation (8) is close to that ofa GTV for θii′ . The use of Equation (8) is now justified as follows. By the strong law of largenumbers, it follows that pi4 converges to pi4 with probability one, when ni4 is large enough. Inpractical applications, the number of spikelets ni4 for estimating the filled-spikelet percentage isfrequently large. Typically, the number of spikelets ni4 is often greater than 1000. Therefore, it isreasonable to replace pi4 and pi′4 by pi4 and pi′4, respectively. In addition, an appealing propertyof GTVs is that a ratio between two GTVs is also a GTV. More precisely, if Tμi and Tμi′ areseparately GTVs for μi and μi′ , then ciTμi/ci′Tμi′ is a GTV for ciμi/ci′μi′ , where ci and ci′ are

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two constants. According to these facts, the behavior of power function of Equation (8) might besimilar to that of a GTV for θii′ , when ni4 and ni′4 are sufficiently large.

For comparing the grain yields between the ith and i′th varieties, that is, the null hypothesisH0 : θii′ = 1 versus the alternative hypothesis H1 : θii′ �= 1 are now considered, the followingMonte-Carlo algorithm is provided for estimating the required generalized p-value.

Step 1 Choose a sufficiently large size of Monte-Carlo samples, say M = 10,000. For 1 ≤ m ≤M , generate mutually independent standard normal random variables Zij and Zi′j, and chi-squarerandom variables Vij and Vi′j separately with nij − 1 and ni′j − 1 degrees of freedom for j = 1, 2and 3.

Step 2 Compute Tθii′ ,m by Equations (8) through (6) and (7) under H0 : θii′ = 1.Step 3 The required generalized p-value can then be estimated by pθii′ = 2qθii′ /M , where qθii′ =

min[∑M

m=1 I(Tθii′ ,m > 1),∑M

m=1 I(Tθii′ ,m < 1)], and I(·) stands for an indicator function.For a nominal significance level α, the null hypothesis H0 : θii′ = 1 is rejected, if the resulting

generalized p-value pθii′ is less than α. When the number of varieties v is greater than 2, thenumber of pairwise comparisons is equal to ( v

2 ). To effectively control the familywise TypeI error rate, a multiple testing adjustment is required to determine the nominal level for eachindividual comparison. In this study, the Bonferroni correction is recommended, primarily dueto its easy implementation. For a comprehensive introduction to the Bonferroni correction andother multiple testing adjustments, the reader is referred to Hochberg and Tamhane [3].

4. Simulation studies

To assess the performance of the proposed GTV-based method, a series of numerical simulationsis implemented. Note that the settings of simulation studies are mainly based on a real-life grainyield comparison study presented in the next section. The baseline values of yield componentsare set as μi′1 = 10, μi′2 = 100, μi′3 = 20 and pi′4 = 0.75, respectively. Recall that θii′ is definedas

θii′ = μi1μi2μi3pi4

μi′1μi′2μi′3pi′4,

and θii′ = 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0 are considered in the simulation studies, respectively. Fora given θii′ , the value of μij is set as μij = θii′μi′j separately for j = 1, 2 and 3. Specifically,θii′ = 1.0 and 1.2 are considered for pi4 = θii′pi′4 = 0.75θii′ , since pi4 must less than 1. The aver-age performance over these cases is reported for a fixed θii′ , since their simulation results exhibita similar pattern. Moreover, the numbers of measurements for yield components are set to beequal, that is, nij = ni′j = n. To observe the impact of sample size, the values of n are sepa-rately selected as n = 20, 30 and 50. In addition, ni4 and ni′4 are fixed at 10,000. For assessingthe impact of variances of yield components on the behavior of power function, the scenarios

Table 1. The standard deviations of yield components for simulation studies.

Scenario {σi1, σi2, σi3} {σi′1, σi′2, σi′3}

1 {1.25, 12.5, 2.5} {1.25, 12.5, 2.5}2 {2.5, 25, 5} {2.5, 25, 5}3 {5, 50, 10} {5, 50, 10}4 {2.5, 25, 5} {1.25, 12.5, 2.5}5 {5, 50, 10} {1.25, 12.5, 2.5}6 {5, 50, 10} {2.5, 25, 5}

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Journal of Applied Statistics 7

presented in Table 1 are considered in the simulation studies, respectively. The scenario 1–3 con-sider the cases of homogeneous variances, and the scenario 4–6 take the cases of heterogeneousvariances into account. For each combination of scenario and θii′ , n observations are repeatedlygenerated 2500 times, and within each of the 2500 random samples, 10,000 Monte-Carlo runsare generated for computing the generalized p-value using the algorithm provided in Section 3.2.The empirical power is defined as the proportion that θii′ is found to be not equal to one amongthe 2500 simulated data, and the empirical powers over the homogeneous and heterogeneous

Table 2. The empirical powers of the GTV-based method and two-sample t-test.

θii′

Scenario n Method 1.0 1.2 1.4 1.6 1.8 2.0

1

20 GTV 0.0469 0.7318 0.9991 0.9999 0.9999 0.9999t 0.0398 0.7233 0.9980 0.9999 0.9999 0.9999

30 GTV 0.0478 0.8979 0.9999 0.9999 0.9999 0.9999t 0.0456 0.8948 0.9998 0.9999 0.9999 0.9999

50 GTV 0.0502 0.9872 0.9999 0.9999 0.9999 0.9999t 0.0483 0.9873 0.9999 0.9999 0.9999 0.9999

2

20 GTV 0.0440 0.2481 0.6787 0.9323 0.9926 0.9994t 0.0378 0.2222 0.6488 0.9128 0.9880 0.9988

30 GTV 0.0471 0.3521 0.8584 0.9924 0.9999 0.9999t 0.0445 0.3337 0.8464 0.9880 0.9996 0.9999

50 GTV 0.0488 0.5567 0.9796 0.9998 0.9999 0.9999t 0.0469 0.5339 0.9727 0.9996 0.9999 0.9999

3

20 GTV 0.0408 0.0862 0.2132 0.3973 0.5671 0.7287t 0.0361 0.0740 0.1804 0.3221 0.4824 0.6140

30 GTV 0.0439 0.1121 0.3177 0.5719 0.7707 0.8973t 0.0399 0.0959 0.2683 0.4737 0.6716 0.8231

50 GTV 0.0469 0.1808 0.5043 0.8047 0.9465 0.9870t 0.0442 0.1517 0.4239 0.7128 0.8915 0.9675

4

20 GTV 0.0427 0.3432 0.8543 0.9904 0.9994 0.9999t 0.0564 0.3164 0.8383 0.9884 0.9993 0.9999

30 GTV 0.0474 0.4980 0.9677 0.9998 0.9999 0.9999t 0.0550 0.4799 0.9588 0.9992 0.9999 0.9999

50 GTV 0.0492 0.7462 0.9980 0.9999 0.9999 0.9999t 0.0543 0.7166 0.9988 0.9999 0.9999 0.9999

5

20 GTV 0.0453 0.1057 0.3129 0.5803 0.7739 0.8923t 0.0735 0.0755 0.2347 0.4552 0.6572 0.8009

30 GTV 0.0474 0.1543 0.4949 0.7844 0.9329 0.9805t 0.0674 0.1066 0.3753 0.6769 0.8684 0.9511

50 GTV 0.0491 0.2700 0.7465 0.9591 0.9961 0.9997t 0.0590 0.1855 0.6372 0.9112 0.9835 0.9980

6

20 GTV 0.0420 0.0957 0.2779 0.5131 0.7243 0.8604t 0.0640 0.0753 0.2181 0.4281 0.6231 0.7767

30 GTV 0.0464 0.1462 0.4321 0.7260 0.9013 0.9691t 0.0612 0.1106 0.3508 0.6307 0.8301 0.9337

50 GTV 0.0496 0.2383 0.6836 0.9316 0.9925 0.9985t 0.0566 0.1766 0.5671 0.8745 0.9724 0.9955

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variances cases are separately reported in Table 2. Specifically, the empirical power is termed asthe empirical size, when the null hypothesis is true, that is, θii′ = 1.0. In addition, the standardtwo-sample t-test is employed to test the hypotheses described as (2) for each simulated data set.For heterogeneous variance cases, the degrees of freedom of the t-statistic is approximated bySatterthwaite’s [11] method. Note that the programs for the simulation studies are written by thefree statistical software R, and are implemented on a personal computer with Intel Core i7-3770CPU 3.40 GHz.

For a nominal significance level 0.05, a simulation study with 2500 random samples indicatesthat if a testing procedure can adequately control the size, the margin error is equal to 0.0085.In other words, under the 95% confidence level, the empirical sizes should be within 0.0415 and0.0585. According to Table 2, under the six scenarios considered in the simulation studies, 9out of 18 empirical sizes are adequately controlled by the two-sample t-test. Furthermore, it canbe found that the two-sample t-test is relatively conservative for some homogeneous variancecases, and it appears to be relatively liberal for some heterogeneous variance cases. This mightbe due to the fact that the normality assumption does not exactly hold for the current hypothesistesting problem. On the other hand, the empirical sizes are effectively controlled by the proposedGTV-based method over most of the scenarios, only 1 out of 18 empirical sizes, which equals0.0408, is less than the lower limit 0.0415. Accordingly, the GTV-based method appears to havea better capacity in Type I error control.

Alternatively, according to Table 2, as expected, the empirical power increases as the samplesize n increases, and it decreases as the variances increase. Moreover, under whether the homo-geneous or heterogeneous variance cases, it can be found that the GTV-based approach is moresensitive than the two-sample t-test. For example, under the scenario 5, the empirical powerof the GTV-based method is 13% higher than that of the two-sample t-test, when n = 20 andθii′ = 1.6. Overall, the proposed testing procedure not only adequately controls the size at thenominal level, but also provides more sufficient power in comparing the grain yields. Although ageneral conclusion cannot be drawn from these limited evaluations, the proposed method appearsto be quite competitive for practical applications.

5. Numerical example

Belize is located on the Caribbean coast of northern Central America, which has a tropical cli-mate with pronounced dry and rainy seasons. Belizean farmers, especially those small-scalefarmers, traditionally grow rice under rainfed condition, primarily due to the scattered fields andconsiderable costs for constructing the irrigation systems. The rainfed system depends on the nat-ural cycle of dry and rainy seasons for planting and harvesting, and produces only one crop peryear. To increase the productivity of rainfed rice fields, a field experiment was implemented atPoppy Show Farm by Taiwan Technical Mission in Belize and Ministry of Natural Resourcesand Agriculture of Belize. The main objective of this experiment was to screen out severalproductive varieties, which give high yields under rainfed condition, for a series of follow-upinvestigations, such as drought tolerance, disease and pest resistance. Note that the raw data ofrainfed experiment are available upon request from the author.

In this experiment, 11 major rice varieties in Central America were considered, includingCARDI-70, Taichung Sen-10, Cypress, Cocodrie, Pricilla, Cica-4, Virginia, Bonita, Delrose,Quirinqua and Chinandega. To achieve the objective, the observations of individual yield com-ponents were separately evaluated for each variety. For an overview of the yield performance, thesample means of yield components are summarized in Table 3. According to Table 3, the grainyield per plant can be readily estimated by multiplying the sample means of yield components

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Table 3. Summary statistics of the yield components in rainfed experiment.

Rice variety Numberpanicles(panicle)

Numberspikelets(spikelet)

1000-Grainweight

(g)

Filledspikelets

(%)

Grainyield(g)

CARDI-70Mean 16.80 125.33 27.37 78.56 45.27S.D. 3.10 31.22 2.77

Taichung Sen-10Mean 17.87 103.93 25.37 72.81 34.31S.D. 3.02 35.11 2.21

CypressMean 18.20 136.87 23.54 71.88 42.15S.D. 4.02 30.12 1.02

CocodrieMean 16.20 132.60 21.76 68.88 32.20S.D. 4.77 39.84 1.96

PriscillaMean 9.67 182.73 22.49 73.42 29.18S.D. 2.82 58.73 2.96

Cica-4Mean 15.00 130.40 21.49 75.32 31.66S.D. 5.18 31.75 1.31

VirginiaMean 26.47 143.40 23.75 66.42 59.88S.D. 8.36 26.55 1.74

BonitaMean 16.00 109.27 26.35 70.41 32.44S.D. 6.09 39.46 4.33

DelroseMean 17.27 126.20 23.02 87.08 43.69S.D. 7.02 26.18 1.97

QuiringuaMean 17.73 148.40 22.50 74.64 44.19S.D. 6.30 44.04 0.99

ChinandegaMean 16.80 121.87 27.65 76.54 43.33S.D. 6.04 40.97 1.05

through Equation (1). For example, the grain yield per plant of CARDI-70 can be estimated by

45.27 = 16.8 × 125.33 × 27.37

1000× 78.56%.

Similarly, all the remaining grain yields can be computed in a similar manner and are presentedin Table 3. Moreover, the sample standard deviations of the measurements of yield componentsare also provided as a reference.

Based on the proposed GTV-based method and Monte-Carlo algorithm provided inSection 3.2, the generalized p-values are separately computed for pairwise grain yield compar-isons. The resulting generalized p-values are displayed in the upper triangle of Table 4. Note thatthe generalized p-values, which are less than the nominal significance level 0.05, are marked with∗, respectively. On the other hand, the rainfed data are analyzed by the two-sample t-test underthe heterogeneous variance assumption, and the degrees of freedom of t-statistic is approximatedby Satterthwaite’s [11] method. The resulting p-values of two-sample t-tests are presented in thelower triangle of Table 4. According to Table 4, under the significance level .05 , there are 13and 17 significant pairs declared by the GTV-based approach and two-sample t-test, respectively.Clearly, the two-sample t-test identifies more significant pairs. This coincides with the conclusiondrawn in the previous section that the two-sample t-test appears to be relatively liberal under theassumption of variance heterogeneity. Furthermore, to control the familywise Type I error rateunder a nominal level 0.05, the Bonferroni correction is employed, and the nominal level is setat 0.05/( 11

2 ) = 0.0009 for each individual pairwise comparison. By the Bonferroni adjustment,the pairs declared to be significant are marked with † in Table 4, respectively.

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Table 4. The p-values and generalized p-values for the rainfed experiment.

Variety CARDI-70 Taichung Sen-10 Cypress Cocodrie Pricilla Cica-4 Virginia Bonita Delrose Quiringua Chinandega

CARDI-70 – 0.0551 0.5783 0.0272∗ 0.0070∗ 0.0195∗ 0.0543 0.0645 0.8142 0.8705 0.7835Taichung Sen-10 0.0774 – 0.1371 0.6903 0.3365 0.6147 0.0009† 0.7542 0.1671 0.1435 0.1942Cypress 0.5422 0.1302 – 0.0685 0.0183∗ 0.0516 0.0158∗ 0.1317 0.8245 0.7755 0.8779Cocodrie 0.0249∗ 0.5536 0.0324∗ – 0.5725 0.9163 0.0004† 0.9813 0.0948 0.0822 0.1154Priscilla 0.0078∗ 0.2670 0.0060∗ 0.6159 – 0.6364 0.0000† 0.6219 0.0345∗ 0.0292∗ 0.0441∗Cica-4 0.0476∗ 0.7442 0.0753 0.8119 0.4736 – 0.0003† 0.9183 0.0785 0.0635 0.0953Virginia 0.0774 0.0016∗ 0.0165∗ 0.0005† 0.0002† 0.0010∗ – 0.0014∗ 0.0571 0.0672 0.0583Bonita 0.0279∗ 0.5368 0.0404∗ 0.9473 0.6948 0.7751 0.0006† – 0.1449 0.1271 0.1646Delrose 0.8386 0.1300 0.7213 0.0469∗ 0.0165∗ 0.0825 0.0566 0.0505 – 0.9509 0.9648Quiringua 0.9971 0.1690 0.6421 0.0816 0.0412∗ 0.1185 0.1348 0.0820 0.8658 – 0.9187Chinadega 0.8371 0.1693 0.7592 0.0708 0.0298∗ 0.1127 0.0681 0.0731 0.9876 0.8616 –

Note: Significance codes: ∗0.05; †0.05 with Bonferroni correction.

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Journal of Applied Statistics 11

According to Table 3, Virginia is found to be the variety of best grain yield, whose estimatedgrain yield per plant is equal to 59.88. On the other hand, CARDI-70, Quiringua, Delrose, Chi-nandega, Cypress and Bonita are also known as high-yielding varieties with estimated grainyields separately given by 45.27, 44.19, 43.69, 43.33, 42.15 and 32.44. Based on the pairwisecomparisons provided by the proposed GTV-based method with Bonferroni correction, thereseems no significant difference in grain yields between Virginia and the six varieties mentionedabove. In other words, the empirical evidence is not strong enough to conclude that Virginia is ofbest grain yield. Based on the analysis result of the GTV-based method and some domain knowl-edge regarding these rice varieties of interest, the six varieties, including Virginia, CARDI-70,Quiringua, Delrose, Chinandega and Cypress, are suggested as the candidates in a follow-upexperiment for a more detailed investigation.

6. Concluding remarks

Based on the concepts of generalized test variables and generalized p-values, a novel statisticaltesting procedure is developed for pairwise grain yield comparisons, and a series of numericalsimulations is implemented for assessing the performance of the proposed method in Type I errorcontrol and empirical power. In practical applications, the yield components can be defined inmany different ways. For example, Yoshida [16] described an alternative relationship betweengrain yield and yield components as follows:

Grain yield (t/ha) = spikelet number (m2) × 1000-grain weight (g)

× filled-spikelet percentage (%) × 10−5.

Note that only three yield components are defined here to characterize the grain yield per unitarea. The proposed GTV-based method can be easily modified, such that the pairwise yield com-parisons can be taken into consideration. However, the proposed method can be used for pairwisecomparisons only, and the Bonferroni correction is suggested to control the familywise Type Ierror rate. In practical applications, an effective and efficient method to control the overall errorrate is using an ANOVA-type testing procedure. To the best of our knowledge, however, thereseems no systematic approach for simultaneously comparing the overall difference of a seriesof products with normal and binomial random components. Hannig [2] introduced the general-ized fiducial distributions for a normal variance components model, a multinomial distribution,and a normal mixture distribution. Based on these generalized fiducial distributions, a simulta-neous testing procedure for addressing a series of products with normal and binomial randomcomponents could be developed. On the other hand, the sample proportions of filled-spikeletpercentages pi4 and pi′4 are simply plugged into Equation (8) for the consequent statistical infer-ence. The proposed method could be refined by using the generalized pivots for pi4 and pi′4instead of using the point estimates. Recently, several articles addressed the generalized fiducialand approximate fiducial pivots for a population proportion. For example, Hannig [2] presented ageneralized fiducial quantity for a binomial success probabilities. Alternatively, Krishnamoorthyand Lee [5] explored the confidence intervals for several functions of binomial success probabili-ties. However, there is no guarantee that these fiducial pivots work well for any given real-valuedfunction of parameters. Therefore, a comprehensive study is required to explore whether thesefiducial pivots are suitable for comparing the grain yields of rice. These important issues are nowunder investigation and will be published in a future communication.

In practice, it could be more reasonable to characterize the yield components by a jointdistribution function, so that the correlations, if exist, between different traits can be charac-terized. However, this requires six more parameters for describing the correlations between theyield components. Typically, the number of observations for each trait, except the filled-spikelet

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percentage, is small in a yield comparison study. Under the constraint of limited numbers ofmeasurements, it is quite a challenge to efficiently estimate all the parameters of interest. Inother words, some tradeoff between the probability model complexity and estimation efficiencycannot be avoided. As a result, the correlations between different traits are not considered by ajoint distribution function in this study, and the yield components are modeled individually by aseries of univariate distribution functions. It could be an interesting issue to take the correlationsinto account, however, this requires a further investigation.

Acknowledgements

The author is grateful to the three referees for their constructive comments and suggestions that resulted in a muchimproved article. The author thanks Malcolm D. Castillo, a grain officer of Ministry of Natural Resources and Agricultureof Belize, and Yaw-Chuan Lin, a specialist of Taiwan Technical Mission in Belize, for their fruitful discussions on theanalysis of rainfed data. The rainfed experiment was implemented, when the author worked with Taiwan TechnicalMission in Belize as an assistant specialist.

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