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8.4.2.1. Graphical estimation Note that the lines are somewhat straight (a check on the lognormal model) and the slopes are approximately parallel (a check on the acceleration assumption). The cell ln T50 and sigma estimates are obtained from the FIT function as follows: FIT Y1 X1 FIT Y2 X2 FIT Y3 X3 Each FIT will yield a cell Ao, the ln T50 estimate, and A1, the cell sigma estimate. These are summarized in the table below. Summary of Least Squares Estimation of Cell Lognormal Parameters Cell Number 1 (T = 85) 2 (T = 105) 3 (T = 125) ln T50 8.168 6.415 5.319 Sigma .908 .663 .805 The three cells have 11605/(T + 273.16) values of 32.40, 30.69 and 29.15 respectively, in cell number order. The Dataplot commands to generate the Arrhenius plot are: LET YARRH = DATA 8.168 6.415 5.319 LET XARRH = DATA 32.4 30.69 29.15 TITLE = ARRHENIUS PLOT OF CELL T50'S http://www.itl.nist.gov/div898/handbook/apr/section4/apr421.htm (4 of 6) [5/1/2006 10:42:28 AM] 8.4.2.1. Graphical estimation With only three cells, it is unlikely a straight line through the points will present obvious visual lack of fit. However, in this case, the points appear to line up very well. Finally, the model coefficients are computed from LET SS = DATA 5 35 24 WEIGHT = SS FIT YARRH XARRH This will yield a ln A estimate of -18.312 (A = e-18.312 = .1115x10-7) and a H estimate of .808. With this value of H, the acceleration between the lowest stress cell of 85°C and the highest of 125°C is which is almost 14× acceleration. Acceleration from 125 to the use condition of 25°C is 3708× . The use T50 is e-18.312 x e.808x11605x1/298.16= e13.137 = 507383. A single sigma estimate for all stress conditions can be calculated as a weighted average of the 3 sigma estimates obtained from the experimental cells. The weighted average is (5/64) × .908 + (35/64) × .663 + (24/64) × .805 = .74. Fitting More Complicated models http://www.itl.nist.gov/div898/handbook/apr/section4/apr421.htm (5 of 6) [5/1/2006 10:42:28 AM] 8.4.2.1. Graphical estimation Models involving several stresses can be fit using multiple regression Two stress models, such as the temperature /voltage model given by need at least 4 or five carefully chosen stress cells to estimate all the parameters. The Backwards L design previously described is an example of a design for this model. The bottom row of the "backward L" could be used for a plot testing the Arrhenius temperature dependence, similar to the above Arrhenius example. The right hand column could be plotted using y = ln T50 and x = ln V, to check the voltage term in the model. The overall model estimates should be obtained from fitting the multiple regression model The Dataplot command for fitting this model, after setting up the Y, X1 = X1, X2 = X2 data vectors, is simply FIT Y X1 X2 and the output gives the estimates for b0, b1 and b2. Three stress models, and even Eyring models with interaction terms, can be fit by a direct extension of these methods. Graphical plots to test the model, however, are less likely to be meaningful as the model becomes more complex. http://www.itl.nist.gov/div898/handbook/apr/section4/apr421.htm (6 of 6) [5/1/2006 10:42:28 AM] 8.4.2.2. Maximum likelihood 8. Assessing Product Reliability 8.4. Reliability Data Analysis 8.4.2. How do you fit an acceleration model? 8.4.2.2. Maximum likelihood The maximum likelihood method can be used to estimate distribution and acceleration model parameters at the same time The Likelihood equation for a multi-cell acceleration model starts by computing the Likelihood functions for each cell, as was described earlier. Each cell will have unknown life distribution parameters that, in general, are different. For example, if a lognormal model is used, each cell might have its own T50 and . Under an acceleration assumption, however, all the cells contain samples from populations that have the same value of (the slope does not change for different stress cells). Also, the T50's are related to one another by the acceleration model; they all can be written using the acceleration model equation with the proper cell stresses put in. To form the Likelihood equation under the acceleration model assumption, simply rewrite each cell Likelihood by replacing each cell T50 by its acceleration model equation equivalent and replacing each cell sigma by the same one overall . Then, multiply all these modified cell Likelihoods together to obtain the overall Likelihood equation. Once you have the overall Likelihood equation, the maximum likelihood estimates of sigma and the acceleration model parameters are the values that maximize this Likelihood. In most cases, these values are obtained by setting partial derivatives of the log Likelihood to zero and solving the resulting (non-linear) set of equations. The method is complicated and requires specialized software As you can see, the procedure is complicated and computationally intensive, and only practical if appropriate software is available. It does have many desirable features such as: ● the method can, in theory at least, be used for any distribution model and acceleration model and type of censored data ● estimates have "optimal" statistical properties as sample sizes (i.e., numbers of failures) become large ● approximate confidence bounds can be calculated ● statistical tests of key assumptions can be made using the likelihood ratio test. Some common tests are: ❍ the life distribution model versus another simpler model with fewer parameters (i.e., a 3-parameter Weibull versus a 2-parameter Weibull, or a 2-parameter Weibull vs an exponential) ❍ the constant slope from cell to cell requirement of typical acceleration models ❍ the fit of a particular acceleration model In general, the recommendations made when comparing methods of estimating life distribution model parameters also apply here. Software incorporating acceleration model analysis capability, while rare just a few years ago, is now readily available and many companies and universities have developed their own proprietary versions. http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (1 of 12) [5/1/2006 10:42:29 AM] 8.4.2.2. Maximum likelihood Example Comparing Graphical Estimates and MLE 's Arrhenius example comparing graphical and MLE method results The data from the 3-stress-cell Arrhenius example given in the preceding section were analyzed using a proprietary MLE program that could fit individual cells and also do an overall Arrhenius fit. The tables below compare results. Graphical Estimates ln T50 Cell 1 Cell 2 Cell 3 8.17 6.42 5.32 MLE's Sigma ln T50 Sigma .91 .66 .81 8.89 6.47 5.33 1.21 .71 .81 Acceleration Model Overall Estimates H Graphical MLE .808 .863 Sigma ln A .74 .77 -18.312 -19.91 Note that when there were a lot of failures and little censoring, the two methods were in fairly close agreement. Both methods were also in close agreement on the Arrhenius model results. However, even small differences can be important when projecting reliability numbers at use conditions. In this example, the CDF at 25°C and 100,000 hours projects to .014 using the graphical estimates and only .003 using the MLE estimates. MLE method tests models and gives confidence intervals The Maximum Likelihood program also tested whether parallel lines (a single sigma) were reasonable and whether the Arrhenius model was acceptable. The three cells of data passed both of these Likelihood Ratio tests easily. In addition, the MLE program output included confidence intervals for all estimated parameters. SAS JMP™ software (previously used to find single cell Weibull MLE's) can also be used for fitting acceleration models. This is shown next. Using SAS JMP™Software To Fit Reliability Models Detailed explanation of how to use JMP software to fit an Arrhenius model If you have JMP on your computer, set up to run as a browser application, click here to load a lognormal template JMP spreadsheet named arrex.jmp. This template has the Arrhenius example data already entered. The template extends JMP's analysis capabilities beyond the standard JMP routines by making use of JMP's powerful "Nonlinear Fit" option (links to blank templates for both Weibull and lognormal data are provided at the end of this page). First, a standard JMP reliability model analysis for these data will be shown. By working with screen windows showing both JMP and the Handbook, you can try out the steps in this analysis as you read them. Most of the screens below are based on JMP 3.2 platforms, but comparable analyses can be run with JMP 4. The first part of the spreadsheet should appear as illustrated below. http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (2 of 12) [5/1/2006 10:42:29 AM] 8.4.2.2. Maximum likelihood Steps For Fitting The Arrhenius Model Using JMP's "Survival" Options 1. The "Start Time" column has all the fail and censor times and "Censor" and "Freq" were entered as shown previously. In addition, the temperatures in degrees C corresponding to each row were entered in "Temp in C". That is all that has to be entered on the template; all other columns are calculated as needed. In particular, the "1/kT" column contains the standard Arrhenius 1/kT values for the different temperature cells. 2. To obtain a plot of all three cells, along with individual cell lognormal parameter estimates, choose "Kaplan - Meier" (or "Product Limit") from the "Analysis" menu and fill in the screen as shown below. Column names are transferred to the slots on the right by highlighting them and clicking on the tab for the slot. Note that the "Temp in C" column is transferred to the "Grouping" slot in order to analyze and plot each of the three temperature cells separately. http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (3 of 12) [5/1/2006 10:42:29 AM] 8.4.2.2. Maximum likelihood Clicking "OK" brings up the analysis screen below. All plots and estimates are based on individual cell data, without the Arrhenius model assumption. Note: To obtain the lognormal plots, parameter estimates and confidence bounds, it was necessary to click on various "tabs" or "check" marks - this may depend on the software release level. http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (4 of 12) [5/1/2006 10:42:29 AM] 8.4.2.2. Maximum likelihood This screen does not give -LogLikelihood values for the cells. These are obtained from the "Parametric Model" option in the "Survival" menu (after clicking "Analyze"). 3. First we will use the "Parametric Model" option to obtain individual cell estimates. On the JMP data spreadsheet (arrex.jmp), select all rows except those corresponding to cell 1 (the 85 degree cell) and choose "Exclude" from the "Row" button options (or do "ctrl+E"). Then click "Analyze" followed by "Survival" and "Parametric Model". Enter the appropriate columns, as shown below. Make sure you use "Get Model" to select "lognormal" and click "Run Model". http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (5 of 12) [5/1/2006 10:42:29 AM] 8.4.2.2. Maximum likelihood This will generate a model fit screen for cell 1. Repeat for cells 2 and 3. The three resulting model fit screens are shown below. http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (6 of 12) [5/1/2006 10:42:29 AM] 8.4.2.2. Maximum likelihood Note that the model estimates and bounds are the same as obtained in step 2, but these screens - also give LogLikelihood values. Unfortunately, as previously noted, these values are off by the sum of the {ln(times of failure)} for each cell. These sums for the three cells are 31.7871, 213.3097 and 371.2155, respectively. So the correct cell -LogLikelihood values for comparing with other MLE programs are 53.3546, 265.2323 and 156.5250, respectively. Adding them - together yields a total LogLikelihood of 475.1119 for all the data fit with separate lognormal parameters for each cell (no Arrhenius model assumption). 4. To fit the Arrhenius model across the three cells go back to the survival model screen, this time with all the data rows included and the "1/kT" column selected and put into the "Effects in Model" box via the "Add" button. This adds the Arrhenius temperature effect to the MLE analysis of all the cell data. The screen looks like: Clicking "Run Model" produces http://www.itl.nist.gov/div898/handbook/apr/section4/apr422.htm (7 of 12) [5/1/2006 10:42:29 AM]