1. Introduction

Analysis of variance is one of the most popular models in statistics in which interest may be mainly in testing the homogeneity of the different group means using the classical Analysis of variance (ANOVA). In many practical situations the standard assumption of homogeneity of error variances are often violated. In general, ANOVA assumes that the sample data sets have been drawn from populations that have normal distributions.

One – way or more complex analysis of variance usually evaluates the effect of a factor (s) on a single response variable. For example, a pharmaceutical industry may be interested in comparing a newly introduced product into the market for treating a particular ailment e.g Drugs use for controlling Blood pressure in patients with high blood pressure.

To satisfy the assumptions, the sales of drugs are randomly monitored in some wholesale pharmaceutical shops, the value of the sales for the shops for each sampled within a given period. The observations can be assumed to have the normal distribution having variances that are group dependent. see Ott, 1984 and Abidoye et al (2014).

The analysis of variance supports unequal sample sizes. This is important because designers of experiments seldom have complete control over the ultimate sample sizes in their studies. Each of the factors (products) must have at least two or more readings or sample observations. However, it is recommended that the error degrees of freedom should not be below ten (10).

The conventional analysis of variance (ANOVA) is usually based on the assumption of normality, independence of errors and equality of the error variances. Past studies have shown that the F – test is not robust under the violation of equal error variances, especially if the sample sizes are not equal and some authors have developed an exact Analysis of variance for testing the means of independent normal populations by using one or two stage procedures. See Jonckhere (1954), Abidoye et. al (2013a), (2013b), (2013c), Barlow et al (1971), Dunnett (1980), Dunnett and Tamhane (1997), Gupta et al (2006) Behren – Fisher’s problem is one of the notable complications that arise in this regard.

The hypothesis of homogeneity of means is to be tested. In some medicine researches where the interest is to investigate the effectiveness of certain brands of a drug meant for a particular ailment , there might be a pre- conceived belief that certain drug (s) are more effective than others. Thus, if where, i = 1, 2, …, h represents the mean measurement of i^th brand of drug, the test hypothesis here would be of the ordered type given above. See Barlow et al (1971), Dunnett (1980), Dunnett and Tamhane (1997), Gupta et al (2006) and Abidoye et. al (2013a), (2013b), (2013c).

In some instances, however, it may be of interest to investigate the claim that some drugs perform better than others. Hence, in an attempt to compare these differing performances, we resort to the use of hypothesis against non - directional alternative. See Abidoye 2012, Yahya and Jolayemi (2003), Ott (1984), Montgomery (1981), Jonckheere (1954) and Bartholomew (1959).

2. Development of the Test Procedure

The One – way analysis of variance (ANOVA) model equation is:

where

is the j^th observation for the i^th drug

is a constant value

is the effect of i^th drug.

j = 1, 2, …,n_i .

We are interested in developing a suitable test procedure to test the hypothesis:

against the alternative

(2.1)

as demonstrated in Abidoye et. al (2014)

(2.2)

(2.3)

(2.4)

(2.5)

where is the sample variance of the i^th group. has been shown to have the distribution with r degrees of freedom (not necessary an integer) and is given by r = 22.096 + 0.266(n-h) - 0.000029 See Abidoye (2012) and Abidoye et. al 2013a. is likened to the common MSE as defined in one- way Analysis of variance (ANOVA). It is a stabilized “pooled” variance, unaffected seriously by some outliers in group variances. The one – way ANOVA is presented in Table 2.1 below. In Table 1 the sum of squares between groups and adjusted sum of squares do not necessarily sum to total sum of squares.

Analysis of Variance Table

Table 2.1. One – way analysis of variance with unequal group variances

Simulation study show that without any loss of generality, nearest integer values can be used to approximate the r degrees of freedom.

3. Application of the Test Procedure

The data used in this study considered three drugs in use for treatment of Blood pressure (hypertension). The data used is a secondary data, collected from pharmaceutical premises in Ilorin, Kwara State, covering the period of six months ( April to September, 2014).

Table 3.1. Wholesales of three drugs in five pharmaceutical companies in Ilorin, Kwara State, covering the same period

We need to verify the equality of the variances between these three drugs. This is shown in Table 3.2 using Levene’s test. Thus the group variances cannot be assumed equal (P = 0.039). Thus the regular ANOVA procedure can not be implemented. testing the hypothesis

Table 3.2. Levene test for equality of variances

Computation on Sales of Data on Blood Pressure

From the data set on sales of drugs for treatment of hypertension, the following summary statistics were obtained:

Lisinopil:

Losartan:

Cardesarten:

In the above, data set, n_i = 5, h = 3,

The main hypothesis is

Note that F- statistic using equal group variances produced the p – value of 0.323 which is higher than the one in the modified F – statistic obtained from harmonic mean of sample variances, showing that regular ANOVA test is conservative even though the same conclusion is obtained here.

Analysis of variance shows that the three drugs used for high blood pressure collected from some pharmaceutical premises in Ilorin, Kwara State are not significantly different.

Table 3.3. Analysis of Variance Table (ANOVA)

4. Conclusions

In this work, we examined a test statistic for testing equality of means under unequal population variances in a comparative drug experiment. We adopted the proposed modified F – test statistic, where the harmonic mean of variances replaced the sample pooled variance in order to avoid the Beheren – Fisher’s problem. Because the sample harmonic mean of variances has the chi – square distribution, the modified F –test is found to be appropriate for the data set used on the sale of hypertensive drugs. We can therefore conclude that the new drug (cardesartan) compete favourably well with the existing drugs for the treatment of Blood pressure (BP) as recommended by the caregivers.