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How the LogLikelihood Ratio Criterion Works

Weibull Topics

I'm actively working on the Weibull pages.  Please visit again soon.

Likelihood is "the probability of the data."

It is proportional to the probability that the experiment turned out the way it did.

So some Weibull model parameters are more likely than others because they explain the observed failures better than other values.  We choose the "best" parameter values, i.e. those that maximize the likelihood, which are called, not surprisingly, "maximum likelihood parameter estimates."

The most likely parameter values, given the data, for the Weibull slope, Weibull beta, and location, Weibull eta, are the "+" in Figure 1, but other values are also plausible, although less likely.  That (Weibull eta, Weibull beta) pair produces the best fit of the data, shown as the black line in Figure 2.

The dataset below is severely right-censored: 58 of the 70 components were removed from service at various times before they failed, and are indicated by red tic-marks along the top of figure 2.  12 failed, plotted as points.   The data are from Meeker and Escobar (1998), Statistical Methods for Reliability Data, Wiley, Table C-1, p630.

Notice that even after transformation the confidence "ellipse" isn't very elliptical.  It is very asymmetric along its major and minor axes, especially the major axis.  That is why we choose not to use methods that employ an ellipse to approximate the loglikeihood surface (like inverting the Fisher Information Matrix) and use the true loglikelihood instead.

Each point along the 95% confidence contour of the loglikelihood surface (Figure 1) produces a Weibull line (Figure 2).

The most likely parameter values, given the data, for the Weibull slope, Weibull beta, and location, Weibull eta,  are the "+" but other values are also plausible, although less likely.  That (Weibull eta, Weibull beta) pair produces the best fit of the data, shown as the black line in Figure 2.

Note: If the figures get out of sync, please reload the page.  To see an example with uncensored data, click here.

loglikelihood ratio

 

Weibull lines, censored

 

The animated figures illustrate how the model parameters (left figure) control the behavior of the Weibull model (right figure).  The  "+" plots the maximum likelihood estimates for (Weibull eta, Weibull beta), and is represented by the black Weibull line.  But values near the  "+" are not unreasonable and we can use the chi-square criterion to gauge "reasonableness."  (More on this later.) 

The pseudo-ellipse on the left translates to the confidence bounds on the right, which are constructed one (Weibull eta, Weibull beta) pair at a time.  For each point on the left there is a corresponding line on the right.  Watch to see how the critical isocline on the loglikelihood surface is traversed.  Each point represents a F(X) probability on the right where the Weibull fit is as far away from the MLEs as possible while still remaining "reasonable."  The locus of all such "not unreasonable" lines forms the loglikelihood ratio confidence bounds on the Weibull model.

Because the log of the ratio of likelihoods tends toward a chi-square distribution asymptotically (i.e. as the sample size increases), we have a criterion, chi-square, for constructing confidence bounds on the Weibull model.