Journal of Agricultural Science and Technology A 7 (2017) 474-478 doi: 10.17265/2161-6256/2017.07.003 D DAVID PUBLISHING

Using Contrasts in One-Way Analysis of Variance with Control Groups and an Application

Demet Çanga1 and Ceren Efe2 1. Department of Food Processing, University of Osmaniye Korkut Ata, Osmaniye 80050, Turkey 2. Department of Biostatistics, Faculty of Medicine, Çukurova University, Adana 01330, Turkey

Abstract: In this study, it was shown that, same comparisons can be made by using contrast coefficients instead of Dunnett’s test in the experiments with control groups. It was also shown that, in situations with an ordinal scale and equal spacing quantitative grouping, a trend investigation could be done by contrast coefficients. For this purpose, a small part of the data from a TUBITAK project in the Soil Science Department, Agriculture Faculty, Kahramanmaras Sutcu Imam University, was used with permission. The soils were absorbed to natural zeolite in concentration of 0, 2.5, 5 and 10 mg/kg, and after two years, the available Zinc (Zn) amounts in the soil were analyzed. As a result, in both Dunnett’s test and contrast methods, the Zn amounts in control and 2.5 mg/kg concentration groups were found similar (P > 0.01); but were different (P < 0.01) between control and 5 mg/kg concentration groups, and control and 10 mg/kg concentration groups. Furthermore, when orthogonal polynomial contrast coefficients were investigated, linear effects were found significant (P < 0.01) and cubic effects were obtained significant (P < 0.05), but quadratic effect was obtained insignificant (P > 0.05).

Key words: Analysis of variance (ANOVA), comparisons, Dunnett’s test, contrast, trend analysis.

1. Introduction tests, such as Duncan’s and Tukey’s. In experiments with a control group, if the researcher desires only to In experimental research with more than two groups, compare the control group with the other groups, one-way analysis of variance (ANOVA) is a common Dunnett’s test is preferred [2, 5, 6]. These three tests method to compare means [1]. In analysis of variance, are called post-hoc (or unplanned) tests. the null hypothesis indicating that the group means are On the other hand, if the researchers would like to equal is tested using F test. If the null hypothesis is investigate the pairwise relations between particular rejected, further investigations to find out the pair groups instead of all the groups, they should use causing the difference are conducted by using multiple planned comparisons, for which the contrast comparison tests [2]. These tests are divided into two coefficients of the hypotheses are needed to be defined main categories called unplanned (post-hoc or [7, 8]. posteriori) and planned (priori) tests [3]. Dunnett’s In this study, the usage of contrast in one-way test, which is one of the unplanned comparisons, is ANOVA with control groups is thoroughly examined. used to compare the control group with all other Usage of contrast enables the researchers to consider groups [4]. the possible relationships between the groups more If one finds a difference between treatment means detailed [9, 10]. It is an alternative way of finding the by using F test, a detailed investigation can be carried difference between specific pairs of groups faster and out to determine the particular pair of groups that therefore a faster decision-making [11-13]. There are causes the difference, by using multiple comparison not many/enough publications about the usage of contrast methods. The aim of this study is to present Corresponding author: Demet Çanga, Ph.D., research that it is possible to make the same comparisons by fields: biometry and genetics, statistic.

Using Contrasts in One-Way Analysis of Variance with Control Groups 475 and an Application using planned contrasts in place of Dunnett’s test, relationship between response and treatment for the which is a post-hoc test used for studies with control one-way ANOVA (Table 2) is given by the following groups. Eq. (1): (1) 2. Materials and Methods where, Yij: the j-th observation (j = 1, 2, ..., ni) on the 2.1 Materials i-th treatment (i = 1, 2, ..., k levels); : the common effect for the whole experiment; : the i-th treatment Dataset is obtained from a project carried out in the effect; eij: the random error present in the j-th Department of Soil Science, Faculty of Agriculture, observation on the i-th treatment [2, 6, 14]. Kahramanmaras Sutcu Imam University. It consists of Instead of comparing all the groups pairwise, each available zinc (Zn) amounts of the soils. In the project, group’s mean is compared only to the control group’s Zn sorption of natural zeolite and intake of wheat mean using Dunnett’s test. Dunnett’s test makes this plants are investigated. Project is conducted as a comparison using a single critical value, like least completely randomized one-way experimental design significant difference (LSD), Tukey’s test [5]. There with three replications. If a significant difference is are tD tables for one-way and two-way comparisons. If detected as a result of ANOVA, instead of a multiple the number of repetition of control and other groups pairwise comparison of all the groups in the are equal, Dunnett’s test statistic is as Eq. (2): experiment, Dunnett’s test, that is a test for comparing 2S 2 each group individually with the control group, is used. d=tD  (2) And Dunnett’s test results are evaluated along with n where, tD: t-Dunnett table value with v degrees of the contrast method results. freedom of error; S2: mean square of error; n: number For this purpose, the soil obtained from of repetition. Kahramanmaras province was applied of natural Two-sided tD value is set as tD = tp, v, 1 - α = t3, 8, 1 - 0.05 = zeolite containing 0 (D-0), 2.5 (D-2.5), 5 (D-5) and 10 t3, 8, 0.995 = 2.88 for the test statistic used in the mg/kg (D-10) Zn, respectively. Obtained data of the comparisons for the difference of control group and the four groups are provided in Table 1. other means. (p: number of groups, except control; : significance level and v: degrees of freedom of error). 2.2 Methods This value is substituted in Eq. (2) and the obtained The mathematical model that describes the result is compared to the means of other groups [2, 5].

Table 1 Experiment/Trial results of natural zeolite affiliated with Zn (mg/kg). Replications D-0 (control) D-2.5 D-5 D-10 1 0.97 1.48 2.06 2.79 2 1.13 1.14 2.31 2.69 3 1.2 1.24 2.28 2.36 Mean 1.10 1.29 2.22 2.61

Table 2 ANOVA results of natural zeolite affiliated with Zn. Source of variation (SV) Degrees of freedom (DF) Sum of squares (SS) Mean squares (MS) F Between groups 3 4.7657 1.5886 55.8862** Within groups (error) 8 0.2274 0.0284 Total 11 4.9931

** Significant at the 0.01 probability level F0.05 (3, 8) = 4.07, F0.05 (1, 8) = 5.32, F0.01 (1, 8) = 11.26.

476 Using Contrasts in One-Way Analysis of Variance with Control Groups and an Application

If the same comparisons are desired to be made coefficients; n: number of conditions (or groups) using contrast coefficients, the researcher asks the repetition [15-18]. three questions below: 3. Results and Discussion (1) Is there any difference between control group and D-2.5 mg/kg group means? ANOVA results of the data in Table 1 are given in (2) Is there any difference between control group Table 2. From the ANOVA results in Table 2, it is and D-5 mg/kg group means? concluded that there is a statistically significant (3) Is there any difference between control group difference between the group means (P < 0.05). And and D-10 mg/kg group means? Dunnett’s test results showed that the means of group These hypotheses can be expressed also as following: D-5 and D-10 are different from the control group’s

(1) H0, 1: µcontrol – µD-2.5 = 0 mean (P < 0.05), whereas the mean of the group D-2.5

(2) H0, 2: µcontrol – µD-5 = 0 was found to be equal to the control group’s mean (P >

(3) H0, 3: µcontrol – µD-10 = 0 0.05). Contrast coefficients for these hypotheses and Results of contrast analysis are summarized in group means are summarized in Table 3. According to Table 4. When Table 4 is examined, it is seen that Table 3, sum of squares (SS) for the i-th contrast similar results are obtained with the Dunnett’s definition is calculated as Eq. (3): test. The difference between the means of the groups D-5 and control, D-10 and control is found significant nMC ()2 SS SS ii (3) (P < 0.01); however it is not significant between contrast estimation Ci C 2  i D-2.5 and control group (P > 0.05). This means where, SSCi: the sum of square of contrasts; Mi: that the Zn application must be at least 5 means of conditions (or groups); : contrast mg/kg.

Table 3 Group means and contrast coefficients of the hypotheses. Coefficients Control D-2.5 D-5 D-10 Mean 1.10 1.29 2.22 2.61 ∑Ci ∑Ci2

C1 1 -1 0 0 0 2

C2 1 0 -1 0 0 2

C3 1 0 0 -1 0 2 2 ∑Mi Ci (∑Mi Ci)

Mi C1 1.10 -1.29 0.00 0.00 -0.18 0.0324

Mi C2 1.10 0.00 -2.22 0.00 -1.12 1.2544

Mi C3 1.10 0.00 0.00 -2.61 -1.51 2.2801

Table 4 Contrast analysis results. SV DF SS MS F Between groups 3 4.7657 1.5886 55.8862** Contrast 1_estimation 1 0.0523 0.0523 1.8388 Contrast 2_estimation 1 1.8704 1.8704 65.8018** Contrast 3_estimation 1 3,4353 3.4353 12.8537** Within groups (error) 8 0.2274 0.0284 Total 11 4.9931 * ** Significant at the 0.05 probability level, significant at the 0.01 probability level F0.05 (3, 8) = 4.07, F0.05 (1, 8) = 5.32, F0.01 (1, 8) = 11.26.

Using Contrasts in One-Way Analysis of Variance with Control Groups 477 and an Application

Table 5 Orthogonal polynomial coefficients and means. Means 1.10 1.29 2.22 2.61 Linear effects -3 -1 1 3 Quadratic effects 1 -1 -1 1 Cubic effects -1 3 -3 1

Table 6 ANOVA using trend contrasts. SV DF SS MS F Between groups 3 4.7657 1.5886 55.8862** Contrast _linear 1 4.4882 4.4881 157.8939** Contrast _quadratic 1 0.0331 0.0331 1.1636 Contrast _cubic 1 0.2445 0.2445 8.6009* Within groups (error) 8 0.2274 0.0284 Total 11 4.9931

* ** Significant at the 0.05 probability level, significant at the 0.01 probability level F0.05 (3, 8) = 4.07, F0.05 (1, 8) = 5.32, F0.01 (1, 8) = 11.26.

It is possible to observe trend of grouping variable effect is found statistically insignificant (P > 0.05). It using trend contrasts. For this, spacing should be is seen that linear effect explains 94% of the total equal and at least at an ordinal scale. If k groups are variance, whereas cubic effect explains only 5% of it compared in one trial, sum of squares between groups [24]. can be divided into (k – 1) parts, each with 1 degree of These results show that the wheat plant takes the Zn freedom [19]. Mean square for each part is compared absorbed by the soil in the interval of 0-10 mg/kg in a with mean square error individually by F test [20, 21]. linear manner. Within this interval, amount of the Zn In this study, using orthogonal polynomial in the plant does not decrease with the increase of coefficients for four groups, sum of squares between dosage. In other words, the quadratic effect does not groups are partitioned into linear, quadratic and cubic exist. effects, each with 1 degree of freedom. The reason 4. Conclusions that the orthogonal contrast is preferred is that they are used to test unrelated hypotheses. However, if the In this study, Zn sorption of natural zeolite and researcher determines some of the pairs among all intake of wheat plants is examined. In studies with possible pairwise comparisons intuitively, and wants control groups, while Dunnett’s test makes all to compare only those, non-orthogonal contrast choice possible pairwise comparisons, using contrasts to would be advantageous. Assuming that grouping investigate only the pairs that are of interests would criteria is equally spaced and quantitative in the save time. On the other hand, orthogonal comparisons dataset used in this study, and since it is necessary to are simpler to interpret due to trend investigation and use independent contrasts, trend (polynomial) unrelated contrasts. Therefore, using planned contrasts orthogonal contrast coefficients are used [22, 23]. when comparing group means is recommended for Coefficients and means are given in Table 5. Using researchers looking for more accurate results in a these coefficients, effect of sum of squares is found by shorter period. Eq. (2). ANOVA results with trend effects of sum of Acknowledgments squares are given in Table 6. According to Table 6, linear and cubic effects are The authors would like to express their special found significant (P < 0.01). However, quadratic thanks to Prof. Dr. Ercan Efe, supervisor of the first

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