Analysis of Variance
Posted April 6, 2009
Analysis of variance (ANOVA) is a statistical technique that was invented by Fisher, and it is therefore sometimes called Fisher’s analysis of variance (ANOVA). In survey research, analysis of variance (ANOVA) is used to compare the means of more than two populations. Analysis of variance (ANOVA) technique can be used in the case of two sample means comparison.
Additionally, it can be used in cases of two samples analysis of variance (ANOVA) and results will be the same as the ttest. For example, if we want to compare income by gender group. In this case, ttest and analysis of variance (ANOVA) results will be the same. In the case of more than two groups, we can use ttest as well, but this procedure will be long. Thus, analysis of variance (ANOVA) technique is the best technique when the independent variable has more than two groups. Before performing the analysis of variance (ANOVA), we should consider some basics and some assumptions on which this test is performed:
Assumptions:
1. Independence of case: Independence of case assumption means that the case of the dependent variable should be independent or the sample should be selected randomly. There should not be any pattern in the selection of the sample.
2. Normality: Distribution of each group should be normal. The KolmogorovSmirnov or the ShapiroWilk test may be used to confirm normality of the group.
3. Homogeneity: Homogeneity means variance between the groups should be the same. Levene’s test is used to test the homogeneity between groups.
If particular data follows the above assumptions, then the analysis of variance (ANOVA) is the best technique to compare the means of two populations, or more than two populations. Analysis of variance (ANOVA) has three types.
One way analysis of variance (ANOVA): When we are comparing more than three groups based on one factor variable, then it said to be one way analysis of variance (ANOVA). For example, if we want to compare whether or not the mean output of three workers is the same based on the working hours of the three workers, then it said to be one way analysis of variance (ANOVA).
Two way analysis of variance (ANOVA): When factor variables are more than two, then it is said to be two way analysis of variance (ANOVA). For example, based on working condition and working hours, we can compare whether or not the mean output of three workers is the same. In this case, it is said to be two way analysis of variance (ANOVA).
K way analysis of variance (ANOVA): When factor variables are k, then it is said to be the k way of analysis of variance (ANOVA).
Key terms and concepts:
Sum of square between groups: For the sum of the square between groups, we calculate the individual means of the group, then we take the deviation from the individual mean for each group. And finally, we will take the sum of all groups after the square of the individual group.
Sum of squares within group: In order to get the sum of squares within a group, we calculate the grand mean for all groups and then take the deviation from the individual group. The sum of all groups will be done after the square of the deviation.
F –ratio: To calculate the Fratio, the sum of the squares between groups will be divided by the sum of the square within a group.
Degree of freedom: To calculate the degree of freedom between the sums of the squares group, we will subtract one from the number of groups. The sum of the square within the group’s degree of freedom will be calculated by subtracting the number of groups from the total observation.
BSS df = (g1) for BSS is between the sum of squares, where g is the group, and df is the degree of freedom.
WSS df = (Ng) for WSS within the sum of squares, where N is the total sample size.
Significance: At a predetermine level of significance (usually at 5%), we will compare and calculate the value with the critical table value. Today, however, computers can automatically calculate the probability value for Fratio. If pvalue is lesser than the predetermined significance level, then group means will be different. Or, if the pvalue is greater than the predetermined significance level, we can say that there is no difference between the groups’ mean.
Analysis of variance (ANOVA) in SPSS: In SPSS, analysis of variance (ANOVA) can be performed in many ways. We can perform this test in SPSS by clicking on the option “one way ANOVA,” available in the “compare means” option. When we are performing two ways or more than two ways analysis of variance (ANOVA), then we can use the “univariate” option available in the GLM menu. SPSS will give additional results as well, like the partial eta square, Power, regression model, post hoc, homogeneity test, etc. The post hoc test is performed when there is significant difference between groups and we want to know exactly which group has means that are significantly different from other groups.
Extension of analysis of variance (ANOVA):
MANOVA: Analysis of variance (ANOVA) is performed when we have one dependent metric variable and one nominal independent variable. However, when we have more than one dependent variable and one or more independent variable, then we will use multivariate analysis of variance (MANOVA).
ANCOVA: Analysis of covariance (ANCOVA) test is used to know whether or not certain factors have an effect on the outcome variable after removing the variance for quantitative predictors (covariates).
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