Development of New LSD Formula when Numbers of Observations Are Unequal

In the case of comparison between samples with unequal replications [n1 # n2 # n3 # ∙∙∙ # nk], the statistician should use LSD (Least Significant Difference) many times to achieve these comparisons, this method consumes more time and effort. The purpose of this research is to find out a new method to do easy and reliable comparison between population means when the experiment or data involving different numbers of replications. In order to do this, a new formula was designed; the aim of this new formula is to develop the multiple comparisons into one step instead of doing it with many steps that consume more effort and time. In this work, the researcher used the mean of the replication number in the LDS formula instead of doing the calculation many times for each two means. The results of this study found that the new formula give similar results for multiple comparisons with minimal effort. It was concluded that the new formula (named Al-fahham’s Formula) achieves the same results with less time and efforts. It was recommended that Al-fahham’s Formula can be approved in the statistical textbooks and procedures.


Introduction
The analysis of variance ANOVA is the parametric method for identify whether differences occur significantly in an experiment including more than conditions.
It is an example of simultaneous inference in which multiple comparisons are achieved and investigate only two-tailed hypotheses. The null hypothesis indicates that differences are not found between the populations represented by these conditions [1] [2].
In general, for any ANOVA with k levels, the null hypothesis is: H 0 : = µ 1  When F test is significant, it reveals only that somewhere among the means of the samples at least two of them differ significantly. However, F test does not specify which specific means shows the significant difference. Thus, if F test for the multiple treatment study is significant, we will indicate that there are significant differences among the means of samples or populations, but we can't identify where they are or how much the power of the difference [2].
Therefore, when F test is significant we make a second procedure, called post hoc comparisons. Post hoc comparisons are similar to t-tests in which we do comparisons between all possible pairs of means for the studied factor, one pair comparison at a time, to indicate which means differ significantly. For analyzing the differences between means, the statistician often do specific comparisons, the most common of which is comparing two means, i.e. what we called "pairwise comparisons" [3]. Seven post hoc comparisons methods are used: 1) least significance difference (LSD) 2) preliminary one-way ANOVA F-test is done as pairwise comparisons by the (LSD), 3) Bonferroni inequality method, 4) Tukey Studentized range, 5) Duncan multiple range, 6) Newman-Keuls multiple range, and 7) Scheffè F-projection method [4].
The first pairwise comparison method was invented by Ronald Aylmer Fisher (1890-1962), who tried to indicate which treatments had a significant different effect in the ANOVA test, he establish his work on the least squares of the standard deviations of the treatment means with equal replications [5].
LSD is well-known methods of simultaneous inference. The main idea of this method is to be used in the comparisons between the populations as pairs. It is then used to continue in a two-way analysis of variance, indicating that the null hypothesis has already been rejected [6].
The main advantage of the LSD is to calculate the smallest significant difference between two means; it helps to determine the populations that are statistically different in means. If the overall F test is significant, LSD, a procedure analogous to ordinary Student's t test, is used to test any pair of means. If the difference between two population means is more than the LSD, then those means differ significantly at the a specific level of confidence, only one LSD value is required to test all the possible comparisons between the population means when replications are equal [4].
In the case of different numbers of observations for treatment or samples

The Aim of the Study
This study aims to reduce the time and effort for the statisticians and researchers when doing multiple comparisons for treatment with different replications or different number of observations. This is performed by using one LSD value for all comparisons, instead of calculating LSD many times.

The Suggested Development
If the difference between numbers of observations are not very large, statistician can use the mean number of observations (i.e. the mean of replications of treatment) to calculate LSD value, so that we can use the formula one time only, as follows: where: k = number of groups, n = objects in group, i = row, j = column, x = group mean, x = grand mean So that: − ≥ so that reject null hypothesis and accept alternative, the conclusion is that the means of the two populations are different.

Conclusion
This research confirmed that the new LSD formula named Al-fahham's Formula (LSDd) achieves the same results in the pairwise comparisons as do the other methods with less time and efforts.

Recommendations
Al-fahham's Formula (LSD d ) can be approved in the textbooks of statistics and used in the statistical procedures.