Investigation of full observation and missing observation in randomized block design

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Investigation of full observation and missing observation in randomized

block design,

G.T. Kayaalp, M. Çelik Güney

School of Animal Science, University of Çukurova, Adana, Turkey

Introduction

There are many design used in agricultural studies. Of these design, completely randomized design (CRD) is used mostly. In CRD, experimental material must be generally homogeneous. If the experimental material is not homogeneous, randomized block design (RBD) is used. There is no missing observation in experiment which is expressed as full observation. Missing observation consists of unexpected events occurring during the experiment. Sometimes, data may be lost in part of the study, some observations may be damaged in some blocks or error can be made when saving datas. In such cases, it is not right to discontinue the investigation or continue working without considering that unit. The only thing to do is to estimate the missing observation and then continue to study (Düzgüneş et al., 1987; Kayaalp and Polat, 2001; Çelik, 2012). The aim of this study was to compare the F test results and the relative efficiency both full observation and estimated missing observations in RBD.

Materials and methods

The milk yield of Alpin and Saanen dairy goat breeds were used as a material in Çukurova University, Faculty of Agriculture, Department of Animal Science, Research and Application Farm. Eight animals from two goat breeds were milked twice a day for 5 weeks in the morning and evening and milk yields were calculated separately. The average milk yields weekly are given in Table 1.

Table 1. The average milk yields weekly

Breed/Week 1 2 3 4 5

Alpin 13.042 15.285 14.171 13.842 14.350 70.69

Saanen 15.592 15.642 16.428 16.657 16.042 80.361

28.634 30.920 30.599 30.499 30.392

The mathematical model of RBD is as follows.

Yij= µ +αi + βj + eij i= 1 , 2, . . . , t ; j=1, 2, . . . , b (1)

Where:

µ: The mean effect,

αi: The ith treatment effect, βj: The jth block effect, and

eij: The term of error (Montgomery, 2001).

Allan and Wishart (1930) developed method for the estimation of a single missing observation in RBD. Yates (1933) extended these method for several missing observations. The missing observation is found as follows.

Missing Observation = (2) Where:

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m:The number of treatment,

B: The sum of items in same block as missing observation,

M:The sum of items in same treatment as missing observation, and G: The sum of all observations in that experimental (Bek and Efe, 1988).

The estimated relative efficiency (RE) of RBD vs. CRD is found as follows.

RE= (3) Where:

b: The number of block, m: The number of treatment,

: The mean square of block, and

: The mean square of error (Düzgüneş et al., 1987).

Comparison between design is also provided by using follows the formula.

= or n=b (4)

Where:

: The relative efficiency of the RBD, : The RBD error variance,

: The CRD error variance (Düzgüneş et al., 1987; Çelik, 2012).

Results Variance analysis was performed full observation in RBD and the results are

given in Table 2.

Table 2. ANOVA of data set with full observation

Source of variation D.F. S.S M.S F Blocks 4 1.633 0.408 0.858 Treatments 1 9.353 9.353 19.69* Error 4 1.902 0.475 Total 9 12.888 *_{: p<0.05}

The blocks (weeks) were determined statistically not significant (p>0.05) and the treatments (goat breeds) statistically significant (p<0.05).

Relative efficiency of the RBD vs. CRD and n have been identified using Eq. (3) and Eq. (4).

RE = = 0.937.

n= 5(0.937) = 4.68.

Then, two observation were removed randomly. The average milk yields weekly with missing observation are given in Table 3.

Table 3. The average milk yields weekly

Breed/Week 1 2 3 4 5

Alpin 13.042 15.285 14.171 X 14.350 56.848

Saanen 15.592 Y 16.428 16.657 16.042 64.719

28.634 15.285 30.599 16.657 30.392 121.567=

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G =121.567+15.73= 137.297 = = 14.92 G=121.567+14.92=136.487 = = 17.35 G = 121.567+17.35=138.917 = = 14.52 G = 121.567+14.52=136.087 = = 17.45 G = 121.567+17.45=139.017 = = 14.50 G = 121.567+14.50=136.067 = = 17.45

So, Y is fixed. Accordingly, these observation values were estimated as =14.50 and = 17.45. Variance analysis was performed with estimated observation values in RBD and the results are given in Table 4.

Table 4. ANOVA of data set with estimated observation values

*_{: p<0.05}

Since there are 2 missing observations, the degree of freedom of error is reduced by 2. The blocks (weeks) were determined statistically not significant (p>0.05) and the treatments (goat breeds) statistically significant (p<0.05).

Conclusion

The blocks (weeks) were determined statistically not significant (p>0.05) both full observation and estimated missing observations design. The mean square of block (0.408) was lower than the mean square of the error (0.475) for full observations. In addition, the relative efficiency of RBD vs. CRD was found to be less than 1 (0.937). n was found 4.68. In other words, if the experiment is carried out in CRD to achieve the same efficiency as RBD, there should be approximately 5 replicates. This means that the same amount of labor and expense will be made. For these reasons, it was decided that it would be more appropriate to carry out the experiment in CRD.

References

Allan, F.E. and Wishart, J. (1930). A Method of estimating the yield of a missing plot in field experimental work. The Journal of Agricultural Science, 20(3):399-406. Bek, Y. and Efe, E. (1988). Araştırma ve deneme metodları I. Çukurova Üniversitesi,

Ziraat Fakültesi Ders Kitabı.

Source of variation D.F. S.S M.S F Blocks 4 4.350 1.087 10.97 Treatments 1 11.657 11.657 117.74* Error 2 0.198 0.099 Total 7 16.205

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Çelik, Ş. (2012). Tesadüf blokları deneylerde tam gözlemle kayıp gözlemi tahmin ederek nispi etkinliğin karşılaştırılması: tarım verilerinde uygulaması. Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2(4): 49-54.

Düzgüneş, O., Kesici, T., Kavuncu O. and Gürbüz, F. (1987). Araştırma ve deneme metodları (İstatistik Metodları II). Ankara Üniversitesi, Ziraat Fakültesi Yayınları, 1021. Ders Kitabı: 295. Ankara, s39.

Kayaalp, G.T. and Polat, S. (2001). Tüm gözlemli ve eksik gözlemli regresyon modelinde klorofil-a miktarının tahmini. Ege Üniversitesi Su Ürünleri Dergisi, 18(3-4):529-535.

Montgomery, D.C. (2001). Design and analysis and experiments. John Wiley-Sons, New York, pp. 141-142.

Yates, F. (1933). The Analysis of replicated experiments when the field results are incomplete. Empire Journal of Experimental Agriculture, 1:129–142.