Study of Long-Term Pavement Performance (LTPP): Pavement Deflections

Appendix M. The Generalized Likelihood Ratio For H48 Over H60

If two hypotheses are available for a model for data, a standard way to compare them is the generalized likelihood ratio. Briefly, this is a computation of the highest possible probability for the data under each hypothesis, or the maximum likelihood estimates of the parameters for the data within each hypothesis. The H48 hypothesis is that sensor 7 was offset 121.9 cm (48 inches); the H60 hypothesis is that sensor 7 was offset 152.4 cm (60 inches).

In each of the three cases examined below, the data to be modeled are the predicted positions of sensor 7, using all of the nonjoint associated deflection data for a particular site and day. These data fail tests of the hypothesis that they come from a normal distribution, but are approximately symmetric. The central limit theorem would then imply that the sample means have approximate normal distributions, with the approximation improving as the sample size (i.e., the number of dates) increases.

Thus the sample mean, x bar, is approximately normally distributed with mean µ_H and unknown standard deviation s divided by the square root of n . Here µ_H is the appropriate value dictated by the hypothesis; if H is H48, then µ_H is 48, and if H is H60, then µ_H is 60. The nuisance parameter s = s_H is estimated by its maximum likelihood estimate shown in the equation given in figure 57.

Figure 57. Equation. Maximum likelihood estimate.

The likelihood under either hypothesis is shown in the equation in figure 58.

Figure 58. Equation. The likelihood.

For example, if the hypothesis is H48, then the value of µ_H is replaced by 48, and the value of s_H is replaced as shown in the equation in figure 59.

Figure 59. Equation. Maximum likelihood estimate, s_H = 48.

This likelihood value is computed for each hypothesis, and the ratio of these two numbers is called the likelihood ratio, a computation of the relative likelihood of the data under the two competing hypotheses.

Table 12 captures all of the relevant information for the three datasets presented in the report. MLE of SD is the maximum likelihood estimate of s, the standard deviation, referenced above.

Table 12. Likelihood ratio stats for protocol versus nonprotocol d7 sensor positions.

FWD Serial Number	Sample Mean,	MLE of SD under H48	MLE of SD under H60	Sample Size, n	Likelihood under H48	Likelihood under H60	Likelihood Ratio
8002–129	47.99	1.88	12.16	12	0.7350095	0.0003252	2,260
8002–132	48.12	2.33	12.10	29	0.8840417	1.531E–07	5,773,093
8002–061	47.20	1.44	12.87	65	9.4002E–05	2.789E–15	33,706,837,035