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17

Paired t-Test for Assessing the Average

of Differences

KEY WORDS confidence intervals, paired t-test, interlaboratory tests, null hypothesis, t-test, dissolved
oxygen, pooled variance.

A common question is: “Do two different methods of doing A give different results?” For example, two
methods for making a chemical analysis are compared to see if the new one is equivalent to the older
standard method; algae are grown under different conditions to study a factor that is thought to stimulate
growth; or two waste treatment processes are tested at different levels of stress caused by a toxic input.
In the strict sense, we do not believe that the two analytical methods or the two treatment processes are
identical. There will always be some difference. What we are really asking is: “Can we be highly confident
that the difference is positive or negative?” or “How large might the difference be?”
A key idea is that the design of the experiment determines the way we compare the two treatments.
One experimental design is to make a series of tests using treatment A and then to independently make
a series of tests using method B. Because the data on methods A and B are independent of each other,
they are compared by computing the average for each treatment and using an independent t-test to assess
the difference of the two averages.
A second way of designing the experiment is to pair the samples according to time, technician, batch
of material, or other factors that might contribute to a difference between the two measurements. Now
the test results on methods A and B are produced in pairs that are not independent of each other, so the
analysis is done by averaging the differences for each pair of test results. Then a paired t-test is used
to assess whether the average of these difference is different from zero. The paired t-test is explained
here; the independent t-test is explained in Chapter 18.
Two samples are said to be paired when each data point in the first sample is matched and related to
a unique data point in the second sample. Paired experiments are used when it is difficult to control all
factors that might influence the outcome. If these factors cannot be controlled, the experiment is arranged
so they are equally likely to influence both of the paired observations.
Paired experiments could be used, for example, to compare two analytical methods for measuring
influent quality at a wastewater treatment plant. The influent quality will change from moment to moment.
To eliminate variation in influent quality as a factor in the comparative experiment, paired measurements
could be made using both analytical methods on the same specimen of wastewater. The alternative approach
of using method A on wastewater collected on day one and then using method B on wastewater collected
at some later time would be inferior because the difference due to analytical method would be over-
whelmed by day-to-day differences in wastewater quality. This difference between paired same-day tests
is not influenced by day-to-day variation. Paired data are evaluated using the paired t-test, which assesses
the average of the differences of the pairs.
To summarize, the test statistic that is used to compare two treatments is as follows: when assessing
the difference of two averages, we use the independent t-test; when assessing the average of paired
differences, we use the paired t-test. Which method is used depends on the design of the experiment.
We know which method will be used before the data are collected.
Once the appropriate difference has been computed, it is examined to decide whether we can confidently
declare the difference to be positive, or negative, or whether the difference is so small that we are uncertain

© 2002 By CRC Press LLC