Which is the Faster Food?

Excellent! You have just performed the ANOVA procedure and have concluded that at least one of the means is not the same as the others. Now, which is different?? That is the real question we care about.

Post Hoc Testing

The famous Taco Bell chihuahua.

Let us pick up where we ended the last extensive example. Because the p-value of 0.0018 was less than our alpha value of 0.05, we rejected the null hypothesis and concluded that at least one (population) average was not the same as the others. To determine which, we perform post hoc tests.

The phrase “post hoc” is Latin for “after this.” You perform post hoc tests only after rejecting the ANOVA null hypothesis in order to see which population mean is different. The output will be substantive, because it will show you all possible differences and their p-values against the null hypotheses that there is no difference.

There are many different post hoc tests one can perform. The one we will use is Tukey’s HSD test. Here, HSD stands for “honestly significant difference.” Tukey was kind of an ***hole and he was making a dig at Pearson, who created the “least significant difference” post hoc test.

The Output

Unfortunately Excel does not offer functionality for post hoc tests. You can purchase add-ins, but there is nothing freely available (that I know of) that does Tukey. That is why I provide the output for this Connect assignment (only for STAT2023.503 course). Feel free to use the formulas in the book and Excel’s calculator-like functionality to get these results.

Here is the output from the statistical program StatCrunch for the fast food example:

What do we do with this table of results???

The famous Golden Arches.

First, look at the p-values. Find those which are less than α=0.05. There are two. That means Sonic and McDonalds are significantly different (p=0.0214), as are Sonic and Taco Bell (p=0.0014).

Second, we know that Sonic is significantly faster than McDonalds. We know this because “McDonalds subracted from” Sonic is -77.5; that is, our estimate of $\mu_S - \mu_M = -77.5$ seconds. This means $\mu_M \gt \mu_S$.

Third, we know that Sonic is significantly faster than Taco Bell. We know this because “Sonic subtracted from” Taco Bell is 121. Our estimate of $\mu_T - \mu_S = 121$ seconds. This means $\mu_T \gt \mu_S$, and Sonic is faster than Taco Bell, on average.

Finally, the students were unable to detect a difference between McDonalds and Taco Bell (p=0.2017). This could be due to two reasons: there is no difference or the sample size is too small to detect one. Most likely, the sample size is too small.

Discussion

And so, after all of that work, we have concluded that not only is there a difference, but that Sonic is significantly faster than the other two fast food restaurants. If speed is needed, go to Sonic. If taste is needed, that is a different question.

And that is it. This example showed how to test which population is different. Here, we were able to determine that Sonic was the fastest and that there was no detected difference between McDonalds and Taco Bell. Why did we detect no difference between McDonalds and Taco Bell? As usual, it comes down to two possibilities: there really is no difference in average service times, or our sample is too small to detect the difference that exists. Since we only took four measurements, I would tend toward the latter explanation.

While the topic of post hoc comparisons is beyond the scope of the course, it does offer a nice closure on the analysis of variance example. Thank you for reading this far. I appreciate it. I hope these things helped you better understand statistics.