Analysis of Variance (ANOVA) is a parametric statistical technique used to compare data sets. This technique was invented by R.A. Fisher; hence, it is also referred to as Fisher’s ANOVA. It is similar to techniques such as t-test and z-test to compare means and also the relative variance between them.
Similarly, a t-test can be used to compare two sample means. What if we want to compare more than two means? Analysis of variance (ANOVA) is best applied where more than 2 populations or samples are meant to be compared.
It is used to test the statistical significance of the relationship between a dependent variable (“Y”) and single or multiple independent variables (“X’s”).
Types of ANOVA
- One-way
- Measures single factor from multiple sources
- Uses only one technician / one measurer
- Two-way (without replicates)
- Measures 2 factors
- Uses only one technician (unless technicians are one of the factors)
- Two-way (with replications)
- Measures 2 factors but has multiple repetitions of each combination.
- Uses only one technician (unless the technicians are one of the factors)
ANOVA Sum of Squares Correction Factor
- Grand total of all runs (G) = ΣX
- N= Total number of runs
- Correction factor (CF)= (ΣX)2 /N = (G)2 /N
Terms used in ANOVA
- Degrees of Freedom (df): The number of independent conclusions that can be drawn from the data.
- SSFactor: It measures the variation of each group’s mean to the overall mean across all groups.
- SSError: It measures the variation of each observation within each factor level to the mean of the level.
- Main effect: A main effect is an effect where the performance of one variable is considered in isolation by neglecting other variables in the study.
- Interaction: An interaction effect occurs where the effect of one variable is different across levels of one or more other variables.
- Mean Square Error (MSE): The mean square of the error (MSE) is divided by the sum of squares of the residual error by the degrees of freedom.
- F-test Statistic: The null hypothesis that the category means are equal in the population is tested by F Statistic based on the ratio of mean square related to X and mean square related to the error.
- P-value: It is the smallest level of significance that would lead to rejection of the null hypothesis (Ho). If α = 0.05 and the p-value ≤ 0.05, then reject the null hypothesis. Similarly if the p-value > 0.05, then fail to reject the null hypothesis.
Analysis of Variance (ANOVA) has three types:
- One-way analysis
- Two-way analysis
- K-way analysis: K-Way ANOVA can be two-way ANOVA or three-way ANOVA, or multiple ANOVA
One way ANOVA
One-way ANOVA (one-way analysis of variance) is a statistical method to compare the means of two or more populations.
Assumptions of
One-way ANOVA
- The sample data drawn from k populations are unbiased and representative.
- The data of k populations are continuous.
- The data of k populations are normally distributed.
- The variation within each factor or factor treatment combination is the same; hence, it is also called homogeneity of variance.
- Finally, the variances of k populations are equal.
Steps for Computing one-way ANOVA:
- Establish the hypotheses. H0: µ1= µ2= µ3 and H1: At least one group means differs from the others.
- In ANOVA, the total variance is subdivided into two independent variances: the variance due to the treatment and the variance due to random error.
- SST =
SSb + SSw - Calculate the ANOVA table with degrees of freedom (df), and calculate the group’s error and total sum of squares.
- SSb= sum of squares between treatments
- SSw= sum of squares due to error
- MSb= mean square for treatments
- MSW= mean square for error
- SST= total sum of squares
- T= number of treatment levels
- n= number of runs at a particular level
- N= total number of runs
- F= the calculation of the F Statistic with k-1, and N-k is the degrees of freedom.
- Determine the critical value. F critical value from the F distribution table.
- Finally, Draw the statistical conclusion. If Fcalc< Fcritical, fail to reject the null hypothesis, and if Fcalc > Fcritical, reject the null hypothesis.
Example of One-Way ANOVA
A car manufacturer planned to conduct tests to determine the performance of 3 different brands of 12V batteries, so he selected five batteries from each brand and discharged them under controlled conditions. Assuming the lifetime of batteries is normally distributed at a 95% confidence level. The hypothesis is that the three brands have no difference in lifetime.
H0: µ1= µ2= µ3
H1: At least one of the brands mean life is different from the others.
k = 3, n = 5, N = 15, T1 = 71, T2 = 85, and T3 = 116, and G = 272 ΣX2= 1021 + 1489 + 2730 = 5240
Correction factor (CF) = (ΣX)2 /N = (G)2 /N = (272)2/15
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One-Way ANOVA Excel Download
A: analysis of variance (ANOVA). The other options don’t make any sense. In other words, An ANOVA is designed to compare three or more factors against each other- This is what happens in a designed experiment.
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