Paired Sample Comparison
September 2006
In this issue:
Greetings,
Last month we looked at ways to compare the averages from two processes. In this case, the samples were independent. We pulled a number of samples from process 1 and a number from process 2. There are times when the observations will not be independent. For example, suppose you want to compare two analytical test methods to see if there are any differences in the two methods. One might take a series of samples, divide each sample in half, and run each sample half in one of the two test methods. In this case, the samples are not independent. The methods learned previously for comparing the difference between two averages cannot be used. Instead, the procedure below for these paired sample comparisons must be used.
William McNeese
Methodology for Paired Sample Comparison
Consider the following example. Suppose there are two methods for determining nickel content. One method is a wet chemical method; the other method is atomic absorption. The same 10 samples were analyzed by each method. We will designate the number of samples as n. The samples were mixed and split in half. One half of the sample was tested using the chemical method. The other half was tested using the atomic absorption method. Thus, the same sample is being tested by each method. We will use the results to determine if there is a difference in the two methods. The results are given below.
The first step in analyzing the results is to take the difference between the results for chemical (method 1) and atomic absorption (method 2) for each sample, retaining the sign:
D = X1 - X2
These differences have been calculated in the table above.
The next step is to find the average difference, Dbar, and the standard deviation, s, of the D values. The average and standard deviation of the D values are given in the table above.
The next step is to calculate a confidence interval for the paired sample comparison. This confidence interval will give us a range of possible D values based on the sample results. If this range contains zero, we will assume that there is no difference between the methods. If the range does not contain zero, we will assume that there is a difference between the methods.
The 100(1 - a)% confidence interval for the paired sample comparison is given by
Dbar +/- ts/SQRT(n)
where t is the value for the t distribution corresponding to n - 1 degrees of freedom. The value of t can be found in the t tables of many statistics books. It is also available in Microsoft Excel using the TINV function.
The number of degrees of freedom in this example is n -1 = 9. For a = 0.05, the value of t is 2.262.
The various values can now be substituted into the confidence interval equation:
Dbar +/- ts/SQRT(n)
0.01 +/- (2.262)(0.185)/SQRT(10)
0.01 +/- 0.132
-0.122 < Dbar < 0.142
Since the interval contains zero, we conclude that there is no evidence that the two methods are different.
This methodology provides a simple way of comparing two different methods when the same sample can be used in the two methods.
A Wealth of SPC Knowledge
It is hard to believe, but this is the 34th issue of our SPC newsletter. We started in January 2004. We hope you have found these issues informative. Please let us know if you have any suggestions for future newsletter topics (select Contact Us at the bottom of the newsletter). Our past issues are given below and are available on our website.
Variation (January 2004)
Leadership and Variation (February 2004)
Operational Definitions/Measurement Systems Analysis (March 2004)
Interpreting Control Charts (April 2004)
Problem Solving Model (May 2004)
Pareto Diagrams (June 2004)
c Control Charts (July 2004)
Control Strategies (August 2004)
Data Collection (September 2004)
Process Capability - Part 1 (October 2004)
Process Capability - Part 2 (November 2004)
Process Capability - Part 3 (December 2004)
Use of Control Charts (January 2005)
Scatter Diagrams (February 2005)
Xbar-R Charts - Part 1 (March 2005)
Xbar-R Charts - Part 2 (April 2005)
Rational Subgrouping and Xbar-R Charts (May 2005)
Rational Subgrouping and Xbar-R Charts - Part 2 (June 2005)
p Control Charts (July 2005)
Explaining Standard Deviation (August 2005)
Inspecting Supplier Material (September 2005)
Creating Cause and Effect Diagrams (October 2005)
Analyzing Cause and Effect Diagrams (November 2005)
Histograms - Part 1 (December 2005)
Histograms - Part 2 (January 2006)
The Impact of Statistical Control (February 2006)
Revisiting Variation (March 2006)
Selecting the Right Control Chart (April 2006)
Monitoring Test Methods Using SPC (May 2006)
Monitoring Customer Complaints Using SPC (June 2006)
Overcontrolling a Process: The Funnel Experiment (July 2006)
Comparing Two Processes: (August 2006)
To obtain the past newsletters, please click here.
Profound Knowledge
For the next four months, starting next month, you will be receiving two newsletters. One newsletter will contain the usual SPC technique. The other newsletter will be one of a four part series on Dr. W. Edwards Deming and his four bodies of profound knowledge. The picture shown here is of Dr. W. Edwards in 1957 (permission to use the Media Gallery photo of Dr. Deming granted by Diana Deming Cahill of the W. Edwards Deming Institute).
Our newsletters typically cover a topic on statistical process control (SPC). Implementation of SPC has been successful in many companies. Yet, there are probably many more unsuccessful implementation efforts. Why has this occurred? I believe the major reason is that the leadership of these companies did not understand that quality is more than SPC and attempting to put control charts on the floor. It represents a philosophy. And the philosophy that should have driven these leaders was that developed by Dr. W. Edwards Deming.
Dr. Deming spent years developing a theory for helping companies move forward into the twenty-first century. Remarkably, it all still applies today. Understanding Dr. Deming begins with understanding his "system of profound knowledge." This system is composed of four bodies of knowledge:
- Knowledge of a system
- Knowledge of variation
- Theory of knowledge
- Knowledge of psychology
We will be looking at one of these each month for the next four months.