In a manufacturing or service environment, you might wonder if changing a
formula, process or material might deliver a better product at a lower cost. Saving a penny a pound on five million pounds a month can really add up. Saving ten minutes of wait time in hospital might add $100,000 to the bottom line and deliver better patient outcomes. Comparing two or more drug formulations might pinpoint the best drug for a desired result.
How can
you compare the old formula with a new one and be certain that you
have an opportunity to improve? Use one-way ANOVA (also
known as single factor ANOVA) to determine if there's a statistically
significant difference between two or more alternatives.
One-Way ANOVA Example
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:
Ha: At least one of the means is different
To conduct the one-way ANOVA test, you need to randomize the
trials (assumption #1). Imagine that we've conducted these 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 (95% confident), in this example we want to be even more
certain, so we use 0.01 (99% confident).
The QI Macros will perform the calculations for you and will
analyze the results. It will also tell you whether you should accept
or reject the null hypothesis.
What's cool about QI Macros ANOVA? When you run ANOVA, you don't have to think. Unlike other statistical software, the QI Macros is the only SPC software that compares the p-value (0.000) to the signficance (0.01) and tells you to "Reject the Null Hypothesis because p<0.01" and that the "Means are Different".
Interpreting the Anova One Way test results
The QI Macros automatically compares the p value to a, but you might want to know how to do this manually. The "null" hypothesis assumes that there is no difference
between the hardwood concentrations.
If
Then
test statistic > critical value
(i.e. F> Fcrit)
Reject the null hypothesis
test statistic < critical value
(i.e. F< Fcrit)
Accept the null hypothesis
p value < a
Reject the null hypothesis
p value > a
Accept the null hypothesis
The P-value of 0.000 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 (4.938193), so
again, we can reject the null hypothesis.
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 (0.349) 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 (10.04429) so we cannot reject
the null hypothesis.
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, but 10% shows more variability with some of the bags below the 15 PSI target.
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