Weibull accelerated life testing analysis with several variables using multiple linear regression

Abstract

In Weibull accelerated life test analysis (ALT) with two or more variables (<em>X<sub>2</sub>, X<sub>3</sub>, ... X<sub>k</sub></em>), we estimated, in joint form, the parameters of the life stress model r{X(t)} and one shape parameter β. These were then used to extrapolate the conclusions to the operational level. However, these conclusions are biased because in the experiment design (DOE) used, each combination of the variables presents its own Weibull family (β<sub>i</sub>, η<sub>i</sub>). Thus the estimated β is not representative. On the other hand, since (β<sub>i</sub>, η<sub>i</sub>) is determined by the variance of the logarithm of the lifetime data σ<sub>t</sub><sup>2</sup> , the response variance σ<sub>y</sub><sup>2</sup> and the correlation coefficient R<sup>2</sup>, which increases when variables are added to the analysis, β is always overestimated. In this paper, the problem is statistically addressed and based on the Weibull families (β<sub>i</sub>, η<sub>i</sub>) a vector Y<sub>η</sub> is estimated and used to determine the parameters of r{X(t)}. Finally, based on the variance σ<sub>y</sub><sup>2</sup> of each level, the variance of the operational level σ<sub>op</sub><sup>2</sup> is estimated and used to determine the operational shape parameter β<sub>op</sub>. The efficiency of the proposed method is shown by numerical applications and by comparing its results with those of the maximum likelihood method (ML).

Conditional Weibull Control Charts Using Multiple Linear Regression

“…In this section, two facts are addressed. The first one is that η is completely determined by the log-mean μ x and β parameters (see Eqn (16)). The second one is that β is completely determined by σ x (see Eqn (14)).…”

“…Step 5. Determine from the conditional lifetime data of step 4, or by using the addressed β and η value in Eqn (16), the log-mean μ x . Step 6.…”

“…If a plotted μ xi fall down of the limit, take action to correct the process. Here, remember that according to Eqn (16), the root cause could be either a down shift on μ x or an up shift on σ x , so the first action should be checking the behavior of σ x .…”

“…Note 2: Remember that while μ x mainly depends on the behavior of the significant variables of the used life/stress model (e.g. Arrhenius, Eyring and power law models; see Piña-Monarrez et al 16 Sections 2 and 5), σ x depends on both the behavior of the mentioned significant variables and on the strength of the used material or supplier's capability indices (cp and cpk). (see Piña et al 15 ).…”

“…Before present, the steps of the proposed β control chart, first, note from Eqn (16) and Eqn (2), that a down shift on β (or an increment on σ x ), diminishes both the η value and the R(t) index. Second, remember that because for the desired R(t) index, n is unique (see Piña et al 16 ), then, σ y in Eqn (14) is constant, and as a consequence, β only depends on σ x . Thus, in the proposed β control chart, σ x is set as the upper control limit.…”