Tukey's Test for Unequall Sample Sizes

When the samples are not equal or not having same size, Then the Tuky's Test is not applicable. Spotvoll and Stoline have extended Tukey Procedure to the case where the sample sizes are different.Their Procedure , which is applicable for experiments involving three or more treatments and significance levels of 0.05 or less, consists of replacing n in the following equation;

the smallest of the two sample sizes associated with the two sample means that are to be compared. If we designate the new quantity by HSD, we have as the new test criterion

ANOVA EXAMPLE

Miller and Vanhoutte conducted experiments in which adult ovariectomized female mongrel dogs were treated with estrogen, progesteron, or estrogen plus progesteron. Five untreated animals served as controls. A Variable of interest was concentration of progesterone in the serum of the animals 14 to 21 days after treatment. We wish to know if the treatments have different effects on the mean serum concentration of progesterone.
Solution:-

One-Way ANOVA

The simplest type of the analysis of variance is that known as one-way ANOVA, in which only one source of variation, or factor, is investigated. It is an extension to three or more samples of the t-test procedure for use with two independent samples. We can say that the t test for use with two independent samples is a special case of one-way ANOVA.
In a typical situation we want to use one-way ANOVA to test the null hypothesis that three or more treatments are equally effective. The necessary experiment is designed in such a way that the treatments of intrest are assigned completely at random to the subjects or objects on which the measurements to determine treatment effectiveness are to be made. For this reason design is called completely randomized experimental design.

Assumptions of ANOVA

The valid use of analysis of variance as a tool of statistical inference are a set of fundamental assumptions. We refer to the paper of Eisenhart. Although an experimenter must not expert to find all the assumptions met to perfection, it is important that the user of analysis variance techniques be aware of the underlying assumptions and be able to recognize when they are substanially unsatisfied.The consequences of the failure to meet the assumptions are discussed by Cochran in a companion paper to that of Eisenhart. Because experiments in which all the assumptions are perfectly met are rare, Cochran suggests that ANOVA results be considered as approximate rather than exact. These assumptions are pointed out at appropriate points in the following.
We discuss ANOVA as it is used to analyze the results of two different experimental designs, the completely randomized and the randomized complete  block design. In addition to these , the concept of a factorial experiment is given through its use in a completely randomized design. These do not exhaust the possibilities.