One Way ANOVA (Analysis of Variance)

When to Use One Way ANOVA (Analysis of Variance)

In a manufacturing environment, you might wonder if changing a formula or a raw material might deliver a better product. How can you compare the old formula with a new one and be certain that you have an opportunity to improve? You can use one-way ANOVA (also known as single factor ANOVA) to determine if there's a statistically significant difference between three or more alternatives.

One-Way ANOVA Example

This example comes from Montgomery's Intro to SPC book.

Imagine that you manufacture paper bags and that you want to improve the tensile strength of the bag. You suspect that changing the concentration of hardwood in the bag will change the tensile strength. You measure the tensile strength in pounds per square inch (PSI). So, you decide to test this at 5%, 10%, 15% and 20% hardwood concentration levels. These "levels" are also called "treatments."

Since we are only evaluating a single factor (hardwood concentration) this is called one-way ANOVA.

The null hypothesis is that the means are equal:

• H₀: Mean1 = Mean2 = Mean3 = Mean4

The alternate hypothesis is that at least one of the means are different:

• H_a: At least one of the means is different

To conduct the one-way ANOVA test, you need to randomize the 24 trials (assumption #1). Imagine that we've conducted these 24 trials at each of the four levels of hardwood concentration.

You'll find the results of these trials in the ANOVA test data provided with the QI Macros at c:\qimacros\testdata\anova.xls.

Select the data with your mouse and click on the QI Macros Menu to choose:
Anova and Analysis Tools - Anova: Single factor.

The QI Macros will prompt you for the significance level you desire.
While the default is 0.05, in this example we want to be even more certain, so we use 0.01.

The QI Macros will use Excel's Data Analysis tools to analyze the results:

The "null" hypothesis assumes that there is no difference between the hardwood concentrations. So, first we look at the P-value:

Interpreting the Anova One Way test results

The P-value of 3.59E-06 (Excel often uses scientific notation when numbers are very small; 3.59E-06 equals 0.00000359) which is less than the significance level (0.01), so we can reject the null hypothesis and safely assume that hardwood concentration affects tensile strength.

F (19.60521) is greater than F crit, so again, we can reject the null hypothesis that the variances are all equal.

Now we can look at the average tensile strength and variances:

The average tensile strength increases, but we cannot say for certain which means differ. The variance at the 15% level looks substantially lower than the other levels. We might need to do additional analysis.

If we reran the one way Anova test with just 10% and 15%, we'd discover there is no statistically significant difference between the two means.

The P value (0.349373) is greater than the signficance level (0.01), so we cannot reject the null hypothesis that the means are equivalent. And F (0.963855) is less than F crit (10.04423) so we cannot reject the null hypothesis that the variances are the same.

Based on this analysis, if we were aiming for a tensile strength of 15 PSI or greater, the 10% level might be more cost effective.

Another way to look at this data might be to use a box and whisker diagram which shows the distribution of each level:

The 10% and 15% levels are close.

Excel and the QI Macros can also perform:

• Two-Way ANOVA analysis without replication
• Two-Way ANOVA analysis with replication

One Way Anova and other ANOVA tests are just some of the statistical tools included in the QI Macros Statistical Process Control software. Other statistical tests include:

• t test assuming equal variances
• t test assuming unequal variances
• f test
• Chi square
• Overvew of all hypothesis tests in Excel

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