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Modeling Interactions Between Variables in
Accelerated Life Testing or Life-Stress AnalysisWhen analyzing accelerated test life data or life-stress data, interactions between factors are often wrongly ignored or poorly understood and yet have significant impact. This article presents a simple methodology for analyzing the significance of interactions among different factors and incorporating their effects into the reliability analysis.
A paper manufacturer is investigating the effect of the percentage of hardwood concentration in raw pulp, the vat pressure and the cooking time of the pulp on the strength of paper. A life-stress experiment was conducted to assess the strength of the paper and the data set presented in Figure 1 was collected. The manufacturer wants to find the optimum settings for the production factors that result in a desirable paper strength. Note: In addition to being an accelerated life data analysis software, ALTA Pro can be used to analyze life-stress relationships similar to Design of Experiments (DOE) tests analyzed with various DOE software packages. The advantage of using ALTA Pro for this type of analysis is the ability to involve life distributions and to use physical properties knowledge to relate the stress variables to the life (or other characteristics) of the product. In this example, the paper (produced under different settings) is put under test (normal conditions) and the strength of the paper is recorded.
Because this is a multivariate test example, the General Log-Linear model can be used to analyze the data. (Note that the General Log-Linear model is available only in ALTA Pro, not in ALTA Standard.) From the equation of the General Log-Linear model, we obtain the following. where: *α*are model parameters._{0}, α_{1}, α_{2 }, α_{3 }, α_{4}, α_{5}, α_{6 }, α_{7 }*X*(lb/in_{1}^{2}) represents the Pressure (P) variable.-
*X*(%) represents the Hardwood Concentration (H) variable._{2} -
*X*_{3}(h) represents the Cooking Time (C) variable. *X*_{4}=*X*represents the Pressure/Hardwood Concentration interaction term._{1}X_{2 }*X*_{5}=*X*represents the Pressure/Cooking Time interaction term._{1}X_{3 }*X*_{6}=*X*represents the Hardwood Concentration/Cooking Time interaction term._{2}X_{3 }*X*=_{7}*X*represents the Pressure/Hardwood Concentration/Cooking Time interaction term._{1}X_{2}X_{3 }*X*is a vector of n (n=7) stresses.*L(X)*is the expected strength.
In ALTA, additional columns representing
the interaction term are required. In this example, the interaction columns
are obtained through the multiplication of the stress values of the
interaction term's variables. For example, (PH)
interaction term, the interaction column is to be filled by multiplying the
values of the P and C columns. With the interaction columns, the data sheet
looks like the one shown next._{2 }First, choose Select Using Weibull as the life distribution, the parameter values with 80% two-sided confidence are:
This table shows that the 80%
bounds on
αinclude
0,
therefore the _{7 }
X_{1}X_{2 }(PH)_{
}
,
X
(PHC) interaction terms can be ignored. The
analysis is repeated without these terms, as shown next._{1}X_{2}X_{3 }The new model is:
where: *X*(N) represents the Pressure (P) variable._{1}-
*X*(%) represents the Hardwood Concentration (H) variable._{2} -
*X*_{3}(h) represents the Cooking Time (C) variable. *X*=_{4}*X*represents the Pressure/Cooking Time interaction term._{1}X_{3 }*X*=_{5}*X*represents the Hardwood Concentration/Cooking Time interaction term._{2}X_{3 }
The new 80% bounds on the parameter values are:
This table shows that the 80%
bounds on
α
include
0,
therefore the _{4 }
X (P)
and the
_{1 }
X
(PC)
interaction term can be ignored. The analysis is repeated without these
terms._{1}X_{3 }The new model is:
where: -
*X*(%) represents the Hardwood Concentration (H) variable._{1} -
*X*_{2}(h) represents the Cooking Time (C) variable. -
*X*=_{3}*X*represents the Hardwood Concentration/Cooking Time interaction term._{2}X_{3 }
The new 80% bounds on the parameter values are:
None of the above bounds include 0, therefore all the included terms are considered to be significant. The following figure displays the Standardized vs. Fitted Value Residuals plot. It shows no obvious abnormality, which confirms that the new model is valid.
The purpose of this experiment is to
determine the settings that produce a paper strength greater than 200. The
estimated percentage of population with a strength greater than 200 is
calculated at different settings of Hardwood Concentrations and Cooking
Times. This is performed using the Quick Calculation Pad (QCP),
The following table summarizes the results. Therefore, the optimum settings for the production factors are found to be at Cooking Time = 4 hr and Hardwood Concentration = 2%. Under these conditions, 99.98% of the paper produced will have a strength greater than 200. |
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