ANOVA and the T-Test - Statistics Project Example

Introduction A simple description of analysis of variance (ANOVA) would be that it is a set of statistical models, together with their related procedures, whereby the variable to be looked at is divided into components capable of being attributed to diverse sources of variation (Urdan 98). This makes it considerably complex and subtle since each of the different variations applies in a given experimental context. The application of ANOVA is therefore a test to find out whether the means of a number of groups presented are equal or not (Urdan 103). This means that ANOVA tries to generalize t-test into two or more groups. What makes ANOVAs so useful is the fact that they help eradicate chances of committing type 1 errors. This advantage is what sets them apart from a two-sample t-test since one is able to compare more than two means easily. The invention of ANOVA is attributed to Sir Ronald Fisher in 1920. He was both a statistician and geneticist. Although ANOVA and the t-test share some similarities, ANOVA manages to compare several groups; something the t-test is not capable of doing. Applications of ANOVAs, thus, have to be consistent since the likelihood of using the wrong type in the interpretation of given data will result into erroneous conclusions (Acock 183). The different types on analysis of variance are one-way ANOVA, two-way ANOVA, Nested Anova and Friedman. All these types have unique applications and this minimizes the occurrence of errors during data analysis (Russo and Roberts 59). 1. One-way Analysis of variance One-way ANOVA is applied when one wants to determine if any considerable differences exist among three or more autonomous groups that are also unrelated (Urdan 106). It therefore compares means between the groups of interest and tries to determine if these means are in any significant way different. The null hypothesis is what is specifically tested and incase one-way ANOVA gives a considerable result, then the alternative hypothesis denoted as HA is accepted (Girden 84). This would mean that, at the least, the means of two groups are considerably different. The formula for one-way ANOVA is F = Where F = coefficient for Anova, MST = mean sum of squares as a result of treatment and MSE = mean sum of square as result of errors. Example The following data is given: Plant Name  Number of plants Average Flowers Standard Deviation Mary 5 12 2 Jane  5 16 1 Rose 5 20 4 Calculate the Anova coefficient. Solution:   First we construct a table like the one below in order to find S2: Plant name   n      x      S     S2  Mary 5 12 2 4 Jane  5 16 1 1 Rose 5 20 4 16 p = 3 n = 5 N = 15 x¯ = 16 SST = ∑n(x−x¯)2 SST = 5(12−16)2+5(16−16)2+11(20−16)2 = 160 MST = SSTp−1 MST = 1603−1 = 80 SSE = ∑(n−1)S2 SSE = 4*4 + 4*1 + 4*16 = 84 MSE = SSEN−p MSE = 8415−3 MSE = 7 F = MSTMSE F = 807 = 11.429 Thus the coefficient of Anova is 11.429 According to Girden (86) when applying the one-way Anova test, it is important to note that the F value can only be proven reliable under certain assumptions. These assumptions are: values in each group assume a normal curve possibilities of different population means standard deviations of populations are equal 2. Two-way Analysis of variance The two-way Anova test is more or less an extension of one-way Anova (Kothari 256). It is employed to examine the kind of influence various independent variables have on just a single dependent variable. As one-way Anova determines the considerable effect one independent variable has, two-way Anova measures the same effects in cases involving several independent variables each with its own multiple observations (Kothari 257). In addition to determining the major effects each independent variable contributes, the two-way Anova test identifies any considerable interaction effect existing among independent variable. A researcher might for instance want to examine the effect that sex and race have on wages. The researcher may further want to elaborate if disparities in wages are due to differences in sex alone or differences in race alone or if differences are attributable to certain combinations of both race and sex – do interaction effects exist? Example: A research was carried out to determine the impact of high protein meal on physical performance in adolescents during a fitness test. Half of the students were given high protein breakfast while the other have had low protein breakfast. All subjects underwent a fitness test afterwards and the results recorded. Better performance was represented by high scores. Question: Are there significant effects or interaction effect? Answer: We first create an Anova table; From the values of F, it is clear that there are considerable effects for protein levels (F=8.89 (1,16), p Read More