The Independent Groups f-Test

The major steps in the calculation and interpretation of this test are summarized as follows  

Calculate the mean change score for each treatment group. 

Calculate the difference between the mean change score for the drug treatment group and the mean change score for the placebo group, i.e., the effect size. 

Divide the effect size by the error variance to give the test statistic /. 

Calculate the degrees of freedom associated with the /-value. 

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In Section 7.6, the steps involved in the calculation of the independent groups f-test were listed. The final three were ... 

In this case, some of the computational steps are different, since there is a relationship between pairs of data that does not exist in cases where the use of the independent groups f-test is appropriate. The computation of the error variance is performed in a different manner, and the value of the degrees of freedom associated with the test statistic is computed differently. Nevertheless, the basic objective, rejecting the null hypothesis or failing to reject the null hypothesis, is identical. 

As noted in Section 7.5, one-factor independent groups ANOVA can also be used in cases where the independent groups f-test is appropriate. The term independent groups is derived in exactly the same way as was independent groups f-test, in that independent groups of subjects are employed. The term one-factor relates to the fact that, in our ongoing example, there is only one factor that is of interest that factor is type of treatment administered. A factor is an influence that one wishes to study it is of interest to know whether the factor is a systematic source of influence on, and therefore a systematic source of variance in, the data collected in a study. An equivalent designation is not necessary in the case of the f-test, since it can only be used when there is just one factor of interest. 

The simple fact that there are more than two treatment groups means that the independent-group f-test introduced in Sect. 4.3.4 cannot be employed that test can only be employed when there are two treatment groups. In this case, an independent-group analysis of variance (ANOVA) is appropriate, since this analytical approach can encompass data from more than two groups. The test statistic in an ANOVA is... 

The simple fact that there are three treatment groups means that an independent groups f-test cannot be employed that test can only handle two treatment groups. In this case, a one-factor independent groups ANOVA is appropriate. From now on, the one-factor part of the name will be left off, since our examples focus on designs where only one source of influence is being investigated. 

There are two forms of the f-test, and each is applicable for sets of measurements that have been obtained in different ways. The method of data collection precisely and uniquely determines which of these two forms of statistical analysis is appropriate. Section 7.6 introduced the independent groups Mest, which is appropriate for the analysis of data collected during a study employing a parallel group study design. Another form of the Mest is called the dependent measures t-test. This test is sometimes called the related measures Mest, the repeated measures Mest, or the Mest for matched pairs. The name dependent measures Mest has been chosen here since the contrast with the word independent in the name independent groups Mest is clear. 

The dependent measures Mest uses the relationship between each pair of data in the steps necessary to calculate the test statistic f. Conceptually, the calculation of the test statistic is the same as for the independent groups Mest, as listed in Section 7.6 ... 

Two common statistical techniques that are typically used to analyze efficacy data in superiority trials are f-tests and ANOVA. In parallel group trials, the independent groups Mest and the independent groups ANOVA discussed in Chapter 7 would be used. Another important aspect of the statistical methodology employed in superiority trials, the use of CIs (confidence intervals) to estimate the clinical significance of a treatment effect, was discussed in Chapter 8. These discussions are not repeated here. Instead, some additional aspects of statistical methodology that are relevant to superiority trials are discussed. 

It should be noted that ANOVA can also be used when there are only two treatment groups. If one analyzed the same data set with an independent-group t-test and then with an independent-group ANOVA, the t statistic and the F statistic would be different numbers, but the p-vedue associated with each of them, i.e., the key value of interest in deteimining the attainment of statisticed significance, would be identical. 

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

ANOVAs have one more advantage over /-tests ANOVAs can compare mean scores of several groups of students based on differences in more than one independent variable. When two independent variables are studied, the test statistics are referred to as two-way ANOVAs (higher-order ANOVAs are possible, but are rarely used by chemical education researchers). The major advantage of performing two-way ANOVAs, instead of two separate one-way ANOVAs or /-tests, is that two-way ANOVAs can determine whether there is a difference due to each of the independent variables (called a main effect) and whether there is an interaction between the two independent variables. This occurs when the effects of one of the variables depends on the other variable (e.g., the effect of an instructional lesson may be different for males and females). The null hypodiesis is that there is no interaction between variables, and the research hypothesis is that there is some sort of interaction. For two-way ANOVAs, there are three F-values calculated The main effect for variable A (F ), the main effect for variable B (Fb), and the interaction between A and B (Faxb)- The dfwiaun value is the same for the three tests = N- kAX ks,... 

The role of Lewis acids in the formation of oxazoles from diazocarbonyl compounds and nitriles has primarily been studied independently by two groups. Doyle et al. first reported the use of aluminium(III) chloride as a catalyst for the decomposition of diazoketones.<78TL2247> In a more detailed study, a range of Lewis acids was screened for catalytic activity, using diazoacetophenone la and acetonitrile as the test reaction.<80JOC3657> Of the catalysts employed, boron trifluoride etherate was found to be the catalyst of choice, due to the low yield of the 1-halogenated side-product 17 (X = Cl or F) compared to 2-methyI-5-phenyloxazole 18. Unfortunately, it was found that in the case of boron trifluoride etherate, the nitrile had to be used in a ten-fold excess, however the use of antimony(V) fluoride allowed the use of the nitrile in only a three fold excess (Table 1). 

Figure 7.1 Analysis of serum IgE concentrations following exposure of mice to allergens or vehicle. Groups of mice (n=6) were exposed to 50 pi of 25 per cent TMA or 1 per cent DNCB bilaterally on the shaved flanks. Control mice received identical treatment with vehicle (AOO) alone. Seven days later, 25 pi of the same test chemical at half the application concentration used previously, or an equal volume of vehicle alone, were applied to the dorsum of both ears. Fourteen days following the initiation of exposure, mice were exsanguinated by cardiac puncture and serum prepared. Serum IgE was measured using a sandwich ELISA. Results are expressed as mean serum IgE concentration in pg. ml 1 SE (where these exceeded 0.075 pg. ml-1). A summary of six independent experiments a-f.

GROUPS 2,3, aid 4—The difference between two single and independent results obtained by different operators woridng in different laboratories on identical test material would in the normal and correct operation of this test method, exceed the values indicated in Fig. 6 (Manual, C) or Fig. 8 (Manual, F) or Table 9 (Automatic) in only one case in twenty. 

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