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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 μH and unknown standard deviation σ divided by the square root of n . Here μH is the appropriate value dictated by the hypothesis; if H is H48, then μH is 48, and if H is H60, then μH is 60. The nuisance parameter σ = σH is estimated by its maximum likelihood estimate shown in the equation given in figure 57.

Figure 57. Equation. Maximum likelihood estimate. The nuisance parameter lowercase sigma subscript uppercase H equals the square root of the quotient of the product of uppercase sigma times the square of the difference between lowercase X and lowercase mu subscript uppercase H, divided by lowercase N.

Figure 57. Equation. Maximum likelihood estimate.

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

Figure 58. Equation. The likelihood. The likelihood is equal to the quotient of the square root of lowercase N divided by the square root of the product of 2 times pi times lowercase sigma subscript uppercase H, this quotient then being multiplied by the natural number lowercase E raised to the power of the product of the quotient of negative lowercase N divided by 2, times the squared quotient of the difference between lowercase X bar and lowercase mu subscript uppercase H divided by lowercase sigma subscript uppercase H.

Figure 58. Equation. The likelihood.

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

Figure 59. Equation. Maximum likelihood estimate, lowercase sigma subscript uppercase H equals 48. The nuisance parameter lowercase sigma subscript uppercase H equals the square root of the quotient of the product of uppercase sigma times the square of the difference between lowercase X and 48, divided by lowercase N.

Figure 59. Equation. Maximum likelihood estimate, σH = 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 σ, the standard deviation, referenced above.

Table 12. Likelihood ratio stats for protocol versus nonprotocol d7 sensor positions.
FWD Serial Number Sample Mean, X with a line above it 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
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