Analysis of Variance (ANOVA)

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Analysis of variance (ANOVA) is a statistical technique and analysis tool used to analyze the differences among means as well as analyze variation in a response variable measured under conditions defined by discrete factors. There are two main types of ANOVA: One-Way ANOVA and Two-Way ANOVA, however, this blog post will only cover the One-Way ANOVA.

One-Way ANOVA

The single factor or one-way analysis of variance (ANOVA) is used to compare and determine the means between two or more independent or unrelated groups of values whether there is statistical evidence that the associated population means are significantly different. . This one-way ANOVA is used to test the null hypothesis that the means of several populations are all equal.

Load the Data Analysis ToolPak in Excel

To access the analysis of variance (ANOVA) functions in Excel, you need a free Microsoft add-in called the Analysis ToolPak.

ASSUMPTIONS FOR PERFORMING ONE-WAY ANOVA

1. Assumption of Normality

ANOVA requires that the dependent variable is normally distributed in each group.

a. Histograms

To test the assumption of normality, inspect the data visually using a histogram. If your data looks like a bell curve: then it’s probably normal.

Histogram of normal data, a normally distibuted sample with 30 observations

b. QQ-plot

The quantile-quantile plot or q-q plot is a graphical tool to determine if a set of data came from some theoretical distribution such as a Normal or exponential.

Normal QQ plot of normal data, a normally distibuted sample with 30 observations

2. Assumption of Homogeneity of Variance

ANOVA assumes that the variances of the distributions in the populations are equal. 

This can be done using Levene’s test, it is the most common assessment for homogeneity of variance.  If the variances of groups are equal, the p-value should be greater than 0.05.

Example of Levene’s test  from the Real Statistics Using Excel

3. Assumption of Independence

ANOVA assumes that the observations are random and that the samples taken from the populations are independent of each other. 

IMPLEMENTING ONE-WAY ANOVA

Example:

Four brands of flashlight batteries are to be compared by testing each brand in five flashlights. Twenty flashlights are randomly selected and divided randomly into four groups of five flashlights each. Then each group of flashlights uses a different brand of battery. The lifetimes of the batteries, to the nearest hour, are as follows:

Step 1: Write your null hypothesis statement. null hypothesis: H0 : µ1 = µ2 = µ3 = µ4 (The mean lifetimes are equal)

Step 2: Write your alternate hypothesis. alternative hypothesis: Ha : Not all of the means are equal.

Step 3: Use MS Excel to perform one-way ANOVA: On the Data tab,  in the Data Analysis dialog box, select “Anova: Single Factor” and click the OK button to display the Anova: Single Factor dialog box.

Step 4: In the dialog window, specify the Input Range and Output Range, Select “Columns” or “Rows”, depending on whether you entered data in columns or rows. Indicate whether you included labels in the first column or row of the input range. And finally, enter a value for Alpha, we’ll use a significance level of 0.05, as shown below, then click OK.

ANOVA: Single Factor Results

The Results shows a summary of the descriptive statistics and the One-way ANOVA test results.

Descriptive Statistics Results

1. Count – is simply the number or count of how many items or observations you have. 

2. Sum – is the sum (or total) of the data values.

3. Average – is the average or mean is a central tendency of the data, it is the sum of the observations divided by the total number of observations.

4. Variance – is the sample variance is a popular measure of dispersion. It is the simple mean of the squared distance from the mean.

One-way ANOVA Test Results

1. SS – is the sum of the squares of the deviations from the means. 

2. df – stands for degrees of freedom.

3. MS –  or the Mean Square, is a kind of average variation and is found by dividing the variation by the degrees of freedom.

4. F – stands for an F variable. F was the ratio of two independent chi-squared variables divided by their respective degrees of freedom.

5. P values – is the probability of observing a result (F critical) as big as the one which is obtained in the experiment, assuming that the null hypothesis is true. 

TEST RESULTS INTERPRETATION

When analyzing the results of the One-Way ANOVA test, we can use two measures to make our conclusions:

1. Look at the p-value: If the p-value is lower than the significance level, reject the null hypothesis; otherwise, the null hypothesis is not rejected.

Let’s start with the p-value. The model shows a p-value of 0.47 which is higher than the significance level of 0.05. Since the p-value is higher than the significance level, we accept the null hypothesis. This value also suggests a high level of confidence that the null hypothesis gives an adequate model for the data.

2. Compare the F-test to the F-critical value: If the F value is higher than the critical F value, the null hypothesis is rejected; otherwise, the null hypothesis is not rejected.

Now, let’s take a look at the F-test result and the F-critical value. The F value is 0.88, which is way below the critical F value of 3.24. These values also suggest that we have no reason to reject the null hypothesis.

STATE CONCLUSIONS

At the α = 0.05 level of significance, there is not enough evidence to conclude that the mean lifetimes of the brands of batteries differ.

Related Links:

Excel – One-Way ANOVA Analysis Toolpack

One-way ANOVA in Excel

One-Way ANOVA (ANOVA: Single Factor) using Excel 2016 Data Analysis Tools

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