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| How the LogLikelihood Ratio Criterion Works – Constructing Confidence Bounds on Probability of Detection Curves
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| Likelihood is "the
probability of the data." It is proportional to the probability that the
experiment turned out the way it did. So some POD model parameters are
more likely than others because they explain the inspection outcome
better than other values. We choose the "best" parameter
values, i.e. those
that maximize the likelihood, called not surprisingly, "maximum likelihood
parameter estimates." The most likely parameter values are the "+" but other values are also plausible, although less likely. That (x, y) pair produces the best GLM (Generalized Linear Model) fit of the data, shown as the black line in Figures 2 and 3. Each point along the 95% confidence contour of the loglikelihood surface produces a GLM line (Figure 2) and its corresponding POD(a) fit (Figure 3). The triangle identifies the (x, y) pair (location and shape, m and s) whose line is associated with a90/95.
Figure 1 - LogLikelihood Ratio Surface |
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Figure 2 - Logit (and POD) vs. Size |
Figure 3 - POD vs. Size |
| If we choose slightly different values, the resulting likelihood diminishes. As a consequence of the Central Limit Theorem, the ratio of the logs of the new values to their maximum values, the loglikelihood ratio, L, has an asymptotic chi-square (c2) density. That provides a means for constructing likelihood ratio confidence bounds: Move the POD(a) model parameters away from their maximum values but not too far – only until the criterion is reached. In other words, values of the parameters that are "close" to the best estimates are plausible, but values that are "far" are unlikely to describe the data. The asymptotic behavior of provides a way of determining what is meant by "close." Thus the confidence bounds are formed by the locus of POD(a) curves that are "close" to the best (MLE) curve Figure 1 has some additional interesting features. Notice that the maximum likelihood estimates (the big +) are not in the precise center of the loglikelihood contours, and that the contours are not symmetrical. As the sample size is increased the resulting contours contract toward the MLEs and the contours become symmetrically centered asymptotically, but for this smaller sample (n=92) the contour is decidedly not symmetric. There is another ellipse (dotted line) that is centered at the MLE values. That is the Cheng and Iles approximation to the confidence contour (Cheng and Iles, 1983). For small sample sizes it is a poor approximation, as is evident here.
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Mail to Charles.Annis@StatisticalEngineering.com |
