Livestock is an important branch of economics in Sudan. Sudan has 30 million cows, which distributed among states and middle states have 30% of cows. Statistics play an important role in knowledge of the difference between the hybrid and local cows. Analysis of Variance and post hoc tests are the better statistical techniques to study the difference between the hybrid and local cows. Analysis of Variance (ANOVA) is common statistical analysis to know if there is difference between more than two groups. ANOVA tells that groups are different but does not tell which group is giving significant difference from the other groups; then more tests needed to know where the difference is from. For some research purposes, this might be sufficient. Post hoc tests have been designed for analysing and understanding the situation of null hypothesis rejection that all groups are same, they are usually done after the collection of data and ANOVA F test of hypothesis that all groups' means are equal is not significant thus, the null hypothesis is rejected. Rejection of null hypothesis required more tests to distinguish between groups in the study, which are the same, and which are different also, the differences between the groups are significant or not. Post hoc procedures give vulnerable to statistical decision when they are used in wrong way. If there are Multiple Comparisons, there is warring about type I error and how can control error rate while keeping the test has high power? There were many studies on hybrid cow but the data did not satisfy the assumptions of ANOVA table, researchers were located in using alpha in wrong way in multiple comparisons and the result statistically were wrong.

Problem

Since the cows are important in the economy of the Sudan and there were many studies on hybrid cow but the data did not satisfy the assumptions of post hoc procedures to measuring statistical significant differences between local and hybrid cows based on Calving Interval, Total Milk. Post Hoc Procedures may be the right way to examine this case. It’s appropriate to carry this study to correct the situation.

Research Questions

1. Whether post hoc procedure has controlled the error rate and high power test at the same time?
2. Whether post hoc procedure has good confidence interval?
3. Whether the best post hoc procedure is powerful to describe the difference between hybrid and local cows?
4. Whether all hybrid breeds are being same in Calving Interval and Total Milk?
5. Are hybrid cows best than local cows in Calving Interval and Total Milk?

To test whether post hoc procedures can be used in a right way to help non statistical researchers to choose the best post hoc procedure to support their ANOVA’s results throw actual data.

Specific objective

1. To give new approach to common post hoc procedures.
2. To control disadvantages of post hoc procedures.
3. To find the ways of differences between post hoc and the interpreted differences statistically.
4. To build scientific information about hybrid cows under condition of environment of Gezira state.
5. To explain the calving interval and total milk yield for hybrid and local cows and which of them is the best?

Justification

1. Post-hoc analyses are required when a significant relationship has been found between the dependent variable and independent variables on the study which have more than two levels. This allows researchers to ascertain to which levels, the significance can be ascribed too.
2. Most of the researchers are thinking that the p-value in ANOVA enough to give information that the groups are different, if they are testing multiple hypotheses of single sample of data, then they made inflate of type I error.
3. Researchers must have knowledge about how to use post hoc procedures correctly in simple way to support their result of ANOVA, and they must get help about how to get an optimum of post hoc procedure through controlling advantages and disadvantages of post hoc procedure.

Sudan is one of agricultural country that contributes to national income by 30.6% about 90% of the Livestock come from traditional sector where Livestock have great social economic importance. Sudan is considered one of the richest Arab and African countries with its animal wealth. Livestock (cows, goats….e.t.c?) are estimated with about 103 million heads and 30 millions cows. Animal wealth contribute to the Sudanese economy, providing food security and contributing to income. Sudanese cows belong to the Asian Zebu cattle and are mixed with some African breeds in addition to the imported pure breeds and its special facets, the Friesian with the Butana and the Kenana, explained [1-2]. In Sudan, the rural communities own 80% of the livestock and the nomadic tribes own 90% of the rural holdings with livestock playing a central role in their livelihoods. Attempts to infuse exotic improved blood into Sudanese dairy cattle population led in some cases to the extinction or near-extinction of the best local types of cattle (Butana and Kenana) in some areas of the country. Through experience, many herdsmen have come to understand that the best results are obtained by crossing the best local cattle (usually Butana and Kenana) with the exotic breeds (usually Friesian). This has led to widespread concern over the fate of Butana and Kenana types and to efforts for conservation of these strains for both present and future use. This concern is motivated by the fact that the genotypes of the improved indigenous breeds may be required to upgrade or replace low producing cattle in harsh nomadic environments where exotic cattle cannot survive. Another cause for concern is the fact that the directions of future demand cannot be predicted with any certainty.

Kenana Breed

A Kenana cattle is considered as one of the most important dairy cattle breeds in the Sudan. They represent about 83% of the total cattle in Sudan spread in western bank of Blue Nile on the region from Sennar in the north to A’lli Nile States in South they have dual purposes milk and beef on the first milk production about 6.6 to 7.9 kg per day and lactation length 244-273 days, age at first calving 38.4-43.4 month on high 51 months, calving interval is 368-432 days and pregnancy length is 283 days.

Butana Breed

Butana cattle are found in the Butana plain in the central Sudan, between the Blue Nile and Atbara Rivers. The breed is known as good milk producer in a rather harsh climate. They represent about 8.7% of the total of cattle in Sudan in tringle bounded by in East Atbara River, in western Blue Nile, Nile River and south of the line 14^? north. The location of this breed called Sahel el Butana. Milk yield 5.5 to 7.4 kg per day and lactation length 267 days, age at first calving 50.3 months to high 51 months. Analysis of Variance: The Analysis of Variance (ANOVA) is general statistical technique. It is appropriate procedure for testing the equality of several means. The ANOVA approach is partitioning and analyzing the variation in continuous response variable (study variable) into unexplained (experimental error) and explained (treatments) by one or more categorical predicators called factors. ANOVA is probably the most useful technique in the field of statistical inference. However; the analysis of variance has a much wider application. Explained [3-4]. The categories of each factor are groups or experimental treatments or levels and the focus is often comparing response variable means between groups. Generally, the main aims of Anova:

1. To examine the relative contribution of different sources of variation to the total amount of variability in the response variable.
2. To test the null hypothesis H₀ that population groups or treatments means are equal.

Assumption of ANOVA

The assumptions of ANOVA for estimating parameters and testing hypotheses depended on

• The sample must be chosen randomly.
• The data has normality distribution this made reliability of estimators and decision based on tests hypotheses.

There are two tests to diagnosis normality

• Kolomogorov-Simanov test for small samples.
• Shapiro and Wilk that is comparatively quite sensitive to wide range departures from normality such as skewed kurtosis of the distribution.
• Barrlett’s test.

3-Ahomogeneity-of-variance test that is less dependent on the assumption of normality than most tests. The variance must be homogeneity or equal for all groups and tested with many procedures to test homogeneity like:

1. Hartley’s involving the ration of largest to the smallest of the variance within groups.
2. Cochran’s test involving the ration of largest to the sum of the variances.
3. Leveno test: checks for homogeneityof variances and the null hypothesis is that all variances are equal.

To solve the heterogeneity is to use the transformation of data and find suitable form. The number of the factors that contain in the experiment determine the classification of ANOVA model, if the experiment contain one factor it called one –way and if it have two factors it called two-way ANOVA and so on.

Multiple Comparisons: Multiple comparison procedures go against this basic philosophy in that they appear to formulate and test hypotheses in the same study simultaneously. In fact, the multiple comparison controversy is resolved if the procedures are thought of as hypothesis generators rather than as methods for simultaneous generation and test in [5]. Statistical testing of comparisons among a number of groups remains a common occurrence in behavioral science. However researchers do not provide calculated test values where appropriate, but simply give P-value [6]. Thus for pairwise comparison and to test

           H₀: µ_i = µ_j      i, j=1… a

It become

Where:

t_diff: test statistic for the difference of two groups.

 : The mean of the i th group.

 : The mean of the j th group.

MSE: the estimation of sample variance.

Ci: the confection of the i th group's mean in the comparison.

n_i: the sample size of the i th group.

For equal sample size

If there is more than one comparison t must be referred to appropriate critical value for the Multiple Comparisons Procedures. The selection of Multiple Comparisons Procedures will depend upon the choice of error rate.

Bonferroni procedure

The most widely applicable familywise control procedure for small families is the Bonferroni correction. One of the fundamental and historically utilized adjustments for type I error is the Bonferroni correction. The Bonferroni correction adjusts the alpha level at which a statistical test considered to be significant based on the total number of analyses being conducted. It is most common procedure to adjusting significant level to control Type I error rate in multiple testing situations. Specifically, the utilized alpha level is quantified as being the original alpha level divided by the number of comparisons being made. Implicitly, the Bonferroni adjustment assumes that these test statistics are independent. So each comparison is tested at α/c where:

α: significant level.

c: is number of comparison in family.

It provides great control over Type I error but conservative when there are lots of comparisons. The advantage is that can be applied to many situations where there is a family of tests [7].

Dunn-Sidak procedure

It modification of the Bonferroni procedure which improved power for each comparison which test by α= (1-(1- α) ^1/c). The Šidàk-Bonferonni approach becomes very conservative when the number of comparisons becomes large and the tests are not independent.

Sequential Bonferroni Holm

An equivalently more powerful statistical procedure is the Holm–Bonferroni technique. The Holm-Bonferroni technique is a successively rejective adaptation of the simpler Bonferroni adjustment for multiple analyses and strongly controls the alpha level. The Holm-Bonferroni adjustment ranks all of the observed p-values in order. It is major improvement on Bonferroni procedure where the c number test statistics (F, t) or p-values are ranked from largest to smallest (smallest to largest) p value is tested at α/c. if the first p-value is greater than or equal to α/c procedure is halted and the null hypothesis is accepted. If the first p-value is found to be significant the second p-value is compared to the α/c-1. Testing stops when nonsignificant result occurs. This procedure provides more power for individual test and it is recommended for any situation in which Bonferroni adjustment is applicable. Hochberg described similar procedure that worked inverse. The largest p is tested at α rejecting all other tests if this one is significant. If it is not significant the next largest at α/2 and soon. Shaffer stated that Hochberg’s procedure is slightly more powerful than Holm’s. The sequentially rejection Bonferroni test is an easily applied, versatile statistical tool that enables researchers to make simultaneous inferences from their data without risking an unacceptably high overall type I error rate [8].

Table 1: The normality measurements due to calving interval and total milk.

Tukey (Honestly Significant Difference) HSD Procedure

Tukey`s Honestly Significant Difference(HSD) is most popular simple, easy and reliable comparison test which compares each group mean with every group mean in pair wise manner and control Type I error rate to no more than nominal level. The logic of Tukey’s HSD:

1. Using t test and calculate the stander error for the difference between two means.

   Using harmonic mean of the sample sizes when the groups have unequal simple size this is some time called Tukey –Kramer modification and reduced to 1/n in equal sample size.

2.  

   Determine appropriate value of q distribution at the chosen significant level using df_Error and number of means being compared.
3. 
4. Compare HSD with t_diff:

   If t_diff> HSD then rejected H₀ that in the respective population means are not equal. Otherwise, accept H₀. Repeat this for all pairs of means.

   The power of the test may be low for less than all pairwise comparisons it prefer for all pairwise comparisons [9].

The Scheffe multiple: comparison has constant critical value for all comparisons on all group mean and it is appropriate for all possible comparison not just for pairwise comparison. Scheffe procedure controlling of p (at least one Type I error) at α using ERFW will do pairwise, nonpairwise comparison, orthogonal polynomial and so on. Scheffe procedure is very conservative test based on F-ratio statistic. The comparison suggested by the data.

Reject H₀ if:

Otherwise, accept H₀. The Scheffe procedure conserves the power of test unless there are large number of comparisons then it a good for large number.

Fisher’s protected t test LSD

This test is often called the Least Significant Difference (LSD), and it is popular multiple comparisons and usually attributed to Fisher. It is protected t test because it is compute usual t after only if the overall F ANOVA test is significant, Least Significant Difference is due to that α level t critical value that t statistic must exceed in order to be significant when only a single comparison is considered. Unfortunate because it controls p (at least Type I error) at α using ERFW for full null hypothesis. To make decision rule by two steps first: Test overall H₀ with ANOVA F if F test is significant then second step else if F test is not significant then fail to reject H₀ for all comparisons. Second is to reject H₀ for a comparison if

Otherwise, fail to reject H₀. LSD does not require equal sample sizes.

The power of test calculated as:

Where

ES: effect size.

α: significant level.

n: sample size.

σ: the standard division of population

The study carried in the Gezira State from two farms El Nisheishiba Dairy Farm of Gezira University and Bashaier Dairy Farm in ElShuckaba in South of Gezira State also, the tow farms have the same environment and managements

Data Source

Cow’s records from two farms are 162 cows. All cows have more than or equal three calving. The records for Calving interval are 954 and for Total Milk, Yield are 699.

Data Variables

The independent variables are type of cow (Kenana, Butane, hybrid cows50%, hybrid cows 62.5%, hybrid cows 75%and hybrid cows 87.5%), age at first calving, parity number and type of farm. Dependent variables are calving interval and total milk.

The Methodology

Completely Randomize Design (CRD) model constructed to show the relationship between dependent variable and independent variables.

The data had been checked to satisfying all the assumption of ANOVA.

ANOVA F test was used to enable the rejection of null hypothesis.

Then post hoc tests (Benoforrni, Dunn-Sidak, LSD, Tukey, Scheffe ,Gabrial, Hochberg) were carried . The data was analyzed by SPSS 20.0 software.

The Model

Y_ijklm: the study trait calving interval or total milk yield.

µ: the general mean.

τ_i: type of cow (1: Kenana, 2: Butana,3: 50% ,4: 62.5% ,5: 75% ,6: 87.5%)

β_j: type of dairy farm(1=nishishiba,2=bashaier).

γ_k: age group (1=1-2,2=2-3,3=3-4,4=4-5,5=5-6) year.

δ_l: parity) 1=1,2=2ed,3=3,4=4^th,5=5^th ,6= 6^th ,7=7^th,8=8^th,9=9^th).

(τβ)_ij: interaction between ith type of cow and jth type of dairy farm.

(τγ)_ik: interaction between ith type of cow and kth age group

(τδ)_il: interaction between ith type of cow and lth parity number.

(γδ)_kl: interaction between kth age group and lth parity number.

ε_ijklm: the error term which satisfy constrain that ε_ijklm~NIID (0, σ²)

The Data preparation

The important assumptions of ANOVA must be check before using the multiple comparison as following:

1-The normalityof the data by using skewness and kurtosis tests.

2- Homogeneity of variance by using Levene test. Levenetested that the group’s variances are equal.

(Table 2) shewed the results of Levene test due to total milk and calving interval.

The result of two tests that the data not satisfy the assumptions. Then log10 was taken to transform the data.

Table 2: The results of Levene test due to total milk and calving interval.

ANOVA Tables

(Table 3) shows analysis of variance table the dependent variable is Calving Interval. The less than 0.05. The factors explained 51.7 % of the variation on Calving Interval. The factors farm, type, parity, type*farm and age group*parity have significant effect p-value is less than 0.05. The factors age group, type*age group and age group*parity have non-significant effect p value is greater than 0.05. R Squared = .517 (Adjusted R Squared = .465) Source: SPSS of study data. (Table 4) shows the analysis of variance table the dependent variable is Total Milk yield (kg). The model that used to interpret the Total Milk data is corrected because p value (sig) is less than 0.05. The factors explained 52.8 % of the variation on Total Milk. The factors type, farm, and interaction type*age group have significant effect on Total Milk p-value is less than 0.05. The factors, age group, parity, the interaction type*farm, age group*parity and type*parity have non-significant effect p value is greater than 0.05. Results (Table 5) shows the means and standard deviation (std) for two traits Calving Interval (month) and Total Milk (kg) overall the study due to the type of cow. (Table 6) was showed the mean and standard deviation (std) for two traits Calving Interval (month) and Total Milk (kg) overall the study due to parity. After the ANOVA tells that there are differences then post hoc tests was done. Post hoc tests tested the hypothesis:

H₀: µ_a= µ_b

H_a: µ_a≠ µ_b

As pairwise comparisons. The post hoc tests that can be made confidence interval are be used in this study. (Table 7) shows the post hoc tests compared value and the power of test.

Table 3: The ANOVA due to calving interval transformed data.

Table 4: ANOVA table due to total milk yield (kg) the transformed data.

Table 5: Calving interval and total milk means due to type of cow breeding. [i]

Table 6: Calving interval and total milk means due to parity.

Table 7: post hoc tests ' comparedvalue and power.

The study’s means due to calving interval for Kenana and Butana greater than 15.20±0.04 and 12.60±0.06 respectively. Butana total milk mean also less than and Kenana total milk means was greater than but study’s mean of hybrid 50% is less than 14.80±0.12 [10-12]. Found that the blood group had reversed effects on calving interval. study found that there is no significant differencebetween: 1-Kenana and Butana breed(local cows)on two traits (Calving Interval and Total Milk). (Table 8) shows that there is no difference between Kenana breed and Butana breed cows. Also, Bonferroni, Sidak and Hochberg had the same length confidence interval except LSD has narrow one and Scheffe had wider one on two traits (Calving Interval and Total Milk). Hybrid cow 50% and (62.5, 75 ,87.5) % in Calving Interval but they have significant differences in Total Milk. (Table 9) shows that there is no difference between Hybrid cow 50% and Hybrid cow 62.5% when HSD, Scheffe, LSD, Bonferroni Sidak, Gabriel and Hochberg are used on Calving Interval. Also (Tables 10-11) shows that there is no difference between Hybrid cow 50% and Hybrid cow 62.5% when HSD, Scheffe, Bonferroni Sidak, Gabriel and Hochberg are used but LSD shows there is difference between Hybrid cow 50% and Hybrid cow 62.5% on Total Milk and LSD had a narrow Confidence Interval in two traits (Calving Interval and Total Milk). In the same way the differences between 50 and 75, 50 and 87.5. 3-Hybrid cow cows 62.5 %and (75, 87.5) %. 4-Hybrid 75% and 87.5% in Calving Interval but they have significant differences on Total Milk. However, there are significant differences between: 1-Hybrid cow (50, 62.5,75 , 87.5) % and local cow Kenana. (Table-10) showed that there was different between Kenana breed and Hybrid cow 50%. Also, all pairwise comparisons approximately have the same confidence interval except LSD has narrow one on two traits (Calving Interval and Total Milk0. 2-Local cow Butana and Hybrid cow (50, 62.5 , 75 , 87.5) %. (Table 11) shows that there is difference between Butana and Hybrid cow 50% when HSD, Scheffe, LSD, Bonferroni, Sidak, Gabriel and Hochberg ware used but Scheffe had wider interval and LSD has a narrow length of Confidence Interval. The Hybrid 50% has better reproduction traits than the other Hybrid cow which agree with. Also, the study shows that there are no differences on two traits (Calving Interval and Total Milk) between: parity one and parity four, parity one and parity six, parity two and parity three, parity two and parity five, parity three and parity four, parity four and parity five, parity five and parity eight, parity six and parity seven, parity six and parity eight, parity six and parity nine, parity seven and parity eight, parity seven and parity nine. Parity eight and parity nine. Otherwise, there are significant differences between the other parities on two traits (Calving Interval and Total Milk).

Also, the tables from 7 to 11, showed that:

1-α decreased as the number of comparisons increased.

• Bonferroni is good for 6 comparisons or less.
• All post hoc procedures have approximately the same tabulated value if the data are balanced.
• Scheffe has the longest confidence interval.
• LSD is the best for balanced and unbalanced data and has the shortest confidence interval.

In practice, an applied statistician cannot afford to recommend an inconsistent procedure, so the unrestricted LSD procedure is the only procedure that can be safely recommended to researchers. In addition to its consistency, the unrestricted LSD procedure has many practical advantages over the alternative procedures because it is simple, provides a natural extension to the two-population case, flexible enough to cater easily for unequal replication or heterogeneity of variance and also because LSD is consistency, simplicity, flexibility in calculations and Type I error rate is constant, maximum Power. This explanation is compatible with the results obtained by this study.

Table 8: The comparisons between Kenana and Butana on Calving Interval and Total Milk.

Table 9: Hybrid 50% and Hybrid 62.5% cow comparisons on calving interval and total milk.

Table 10: Kenana and Hybrid cow 50% comparisons on calving interval and total milk.

Table 11: Butana and Hybrid cow 50% comparisons on calving interval and total milk.

After ensuring that the data satisfied the conditions of the analysis of variance then at significant level 0.05: LSD has a narrow interval. Scheffe is good for pair wise and nonpairwise but has wide interval. Bonferroni is good for small comparisons. Kenana and Butana have the same reproductive traits. Hybrid 50% cow has good reproductive traits than other Hybrid cows. The cow can be stay at the herd until parity six. There are differences between tow farms on tow traits.

From this study, it could finally recommended that: LSD is the best post hoc procedure to study the difference between groups. It has been established that dairy cattle managers at ElNishishiba and El Bashaier dairy farms, may be advised to keep crossbred dairy cows of the intermediate exotic blood (50%Friesian inheritance) for higher production and reproduction traits. The cow may be culled after parity six. More work in evaluation crosses with the intermediate blood level with large number of records and for longer period was so needed at the two farms. The tow farms must be established scientific data sets for each cow. For future research development of software of statistical analysis that first checks the data that satisfy the assumption of any statistical techniques before it is used is may be place.

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