Consider the following example of a proportion equality test.
(example)
A metal processing company processes the same metal parts on two production lines, A and B, and would like to show that the proportion of nonconforming products is the same.
The results of randomly sampling metal parts produced at the same time and in the same environment from Line A and Line B and investigating the number of nonconforming products (defective products) are as follows.
Line A: 33 out of 290 items are nonconforming items
Line B: 28 out of 250 items are non-conforming.
If the margin of equivalence for the difference in nonconformity rate (defect rate) is 0.03 (3%), can it be said that the nonconformity rate of production line A and the nonconformity rate of production line B are equivalent?
Equivalence test for two sample proportions
ahead of time Part 1 of this blog series In this article, we discussed the equivalence test for means, but in Part 2 of this article, we will discuss Equivalence of proportions I will take up. As related to quality control in the manufacturing industry, as in this example, Nonconformity rate (defect rate) may be mentioned.
In this example, we are comparing the non-conformity rates on two production lines, as shown below. ``The difference in nonconformance rates is 0'' indicates that the so-called nonconformity rates are the same.
(Production line A nonconformity rate) - (Production line B nonconformity rate) = 0
However, if it is not a 100% inspection, it is necessary to calculate the nonconformity rate from samples taken from each line and determine whether they are equivalent. that time, Equivalence tolerance range (margin) Decide in advance.
When comparing the proportions of two samples, specify the acceptable range (margin) for the difference between the two proportions. In this example, it is 0.03 (3%).
Working with JMP and interpreting reports
Let's run an equivalence test in JMP.
First, as data, you need to enter the frequency of each production line and compliance/nonconformity in four lines as shown in the figure below. Leave the columns ``Production Line'' and ``Conformance/Nonconformance'' on a nominal scale, and ``Frequency'' on a continuous scale.
Analyze > Bivariate Relationships Then, specify Y, X, and degrees as shown below.
A report of the contingency table analysis is displayed.
The report shows that line A has a nonconformance rate of 11.38% and line B has a nonconformance rate of 11.20%. The difference is 0.18%. Since the margin is 3%, I think that if the difference is this much, it can be said that they are equivalent.
To perform an equivalence test for the difference in proportions (risk difference), click the red triangle button at the top left of the report. [Equivalence test] > [Risk difference] Choose.
In the dialog that appears after that, specify the target category for Y as "Nonconforming" and the margin (difference) as 0.03 (= 3%).
The "Risk Difference Equivalence Test" report will be displayed. This report is basically the same as the mean equality test. On the right side of the "TOST test" report, there is a statement that says "Substantial equivalence cannot be determined." This can be seen from the fact that the 90% confidence interval for the difference in proportions is not entirely included in the ±0.03 margin range (blue interval) in the Forest Plot.
The difference between the two proportions is 0.018, which is close to 0, so it seems like we can say that they are equivalent, but the 90% confidence interval is -0.044 to 0.047, so if there is a margin of 0.03, we cannot say that they are equivalent. .
Case with increased sample size
In the above example, we were not able to say equivalence, but what if we had the following case?
Line A: 330 out of 2,900 items are nonconforming items
Line B: 280 out of 2,500 items are nonconforming items
This is simply the number in the first example multiplied by 10. Therefore, the nonconformity rates for line A and line B will be the same as in the first example.
Let's take a look at the report for the "Risk Difference Equivalence Test" for this case. Looking at the forest plot, we can see that the 90% confidence interval for the difference is completely included in the interval considered to be equivalent, so we can say that they are equivalent.
This is due to the fact that the error in the "proportion difference estimate" is smaller due to the larger sample size.
When testing equivalence, it is difficult to determine how to determine the margin and how many samples to take for that margin. There may be times when you absolutely need past exam results or specific knowledge.
Over the course of this two-part series, we have shown examples of "equivalence tests" that are useful in the manufacturing industry, but if you have the opportunity to use them in practice, please try using them.
by Naohiro Masukawa (JMP Japan)
Reference: On-demand seminar
“Overview and usage examples of statistical testing methods using JMP for the manufacturing industry” on demand (Japanese)
