Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home   |   Pavements Home
Report
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-RD-03-041

Evaluation and Analysis of LTPP Pavement Layer Thickness Data

Procedures for the Kolmogorov-Smirnov Goodness-of-fit Test

The Kolmogorov-Smirnov test procedure involves the comparison between the experimental cumulative frequency and an assumed theoretical distribution function. If the discrepancy is large compared to what is normally expected from a given sample size, the theoretical model is rejected.

The Kolmogorov-Smirnov test procedure involves the following steps:

  1. Sort layer thickness measurements in the ascending order.
  2. Compute cumulative frequencies of each layer thickness observation Sn(x) using the following formula:
Figure 94 in page 137 shows the cumulative frequencies (S sub n of x) definition equation. S sub n of x is equal to one of the three values depending on the interval that x falls into. S sub n of x is equal to 0 if x is less than x sub k equals 1; k/n if x is less than or equal to x sub k+1 and greater than or equal to x sub k; 1 if x is greater than or equal to x sub k equals to n, where n is the number of layer thickness measurements for the layer; x sub k is the k-th layer thickness measurement in the ascending order.

Figure 94: Equation. Cumulative frequencies definition.

Where xk is a layer thickness value from sample of n layer thickness measurements sorted in the ascending order by thickness value. The k - index indicates the order of layer thickness observation in the sorted layer thickness array.

  1. Select a candidate theoretical distribution function (for example, normal distribution).
  2. Using the layer thickness measurements data, compute descriptive statistic values necessary for definition of the selected theoretical distribution (for example, mean and standard deviation).
  3. Using selected theoretical distribution function and computed descriptive statistics, compute theoretical cumulative frequency values F(xk) for each thickness value xk.
  4. Find the difference between the observed cumulative frequency value Sn(xk) and the theoretically predicted cumulative frequency values F(xk) for each xk from the sample of n thickness measurements.
  5. Select the maximum difference between the observed cumulative frequency value Sn(xk) and the theoretically predicted cumulative frequency values F(xk) called the observed maximum difference Dn or D-max statistic. This value is a measure of discrepancy between the theoretical model and the observed data.
Figure 95 in page 138 shows the D-max (D sub n) statistic definition equation. D sub n is equal to the maximum of the absolute differences between the theoretical cumulative frequency value of x sub k, denoted by F of x sub k, and the actual cumulative frequencies, denoted by S sub n of x sub k and defined in Figure 94 in page 137.

Figure 95: Equation. D-max statistic definition.

8. Select level of significance α = 1 percent

9. Compute the critical value based on selected value of α. Based on value of n, is found as following

Figure 96 in page 138 shows the 95th percentile critical value of D sub n (D superscript alpha=95 subscript n) statistic definition equation. The 95th percentile critical value of D sub n is equal to one of the two values depending on the interval that n falls into. The 95th percentile critical value of D sub n is equal to 0.7688 times n to the power of -0.4088 if n is less than or equal to 50 and greater than or equal to 5; 1.031 times n to the power of -0.5 if n is greater than 50.

Figure 96: Equation. Critical value definition.

10. The Kolmogorov-Smirnov test determines whether, for specified level of significance α, the proposed distribution is an acceptable representation of the field data.

Figure 97 in page 138 shows the Kolmogorov-Simirnov goodness-of-fit test equation. If D sub n is less than the alpha-th percentile critical value of D sub n as defined in Figure 96 in page 138, then the theoretical distribution is acceptable; otherwise, the theoretical distribution is rejected.

Figure 97: Equation. Kolmogorov-Smirnov test evaluation criteria.

The following figure 98 demonstrates the results of the Kolmogorov-Smirnov test for a layer that did not pass the test of normality.

Figure 98 in page 139 shows an example chart for the Kolmogorov-Simirnov normal distribution goodness-of-fit test for the DGAB layer thickness data for the SPS-1 Section 01-0101.  The horizontal axis is the layer thickness measurement ranging from 185 mm to 220 mm.  The vertical axis the percentile of the theoretical or the actual cumulative frequency value of the k-th ordered layer thickness measurement, denoted by F of x sub k and by S sub n of x sub k as defined in Figure 94 in page 137.  The actual cumulative frequency values of the ordered layer thickness measurements are scattered around the curved increasing theoretical cumulative frequency line starting from the origin of the coordinate system to the upper right corner.  In this case, the null hypothesis of normal distribution is rejected because D-max with the value of 0.195, as defined in Figure 95 in page 138, is greater than the alpha-th (e.g., 95th) percentile critical value of D sub n with the value of 0.139, as defined in Figure 96 in page 138.

Figure 98: Chart. Example of Kolmogorov-Smirnov normal distribution goodness-of-fit test (DGAB layer SPS-1 LTPP section 01_0101).

Results of the Kolmogorov-Smirnov Goodness-of-fit Tests

The layer thickness measurements taken along the SPS LTPP sections for the structural layers were tested to determine how well the distribution of layer thickness measurements taken along the LTPP section follow selected theoretical distribution. The following table 69 provides the description of the layer and material types used in the SPS experiments. The table also provides information about layer thickness measurement sample sizes available in the LTPP database.

Table 68. Number of pavement layers and number of layer thickness measurements per layer grouped by material and layer type.

Layer-Material Type

Total number of samples

Number of samples with the following number of observations

1

5

10

15

20

25

30

35

40

45

50

55

60 or more

AC_SURFACE_COURSE

133

4

0

0

1

1

7

0

0

0

0

3

117

0

BINDER_COURSE

50

1

3

0

0

1

3

0

0

0

0

4

38

0

DENSE_GRADE_AGG_BASE

220

1

0

2

5

0

3

15

0

1

8

1

174

10

DENSE_GRD_ASPH_TREAT_BASE

97

0

0

1

0

0

0

0

0

0

2

2

92

0

LEAN_CONCRETE

48

0

0

0

0

0

0

8

0

0

0

0

35

5

PCC_SURFACE

178

1

0

1

0

0

2

40

1

0

2

3

112

16

PERM_ASPH_TREAT_BASE

130

1

0

2

0

0

1

9

0

0

1

1

111

4

AC_SURFACE_AND_BINDER

191

0

0

2

0

0

0

0

1

0

4

4

177

3

One data sample represents a group of measurements taken along the LTPP section for a specific layer and material type. There are 1,047 layers with thickness measurements along the LTPP section available in the LTPP database for the surface and base courses. The number of thickness measurements per layer and material type taken along the LTPP section ranges from 1 to 60. About 85 percent of all layers have at least 55 observations.

A total of 1034 pavement layers were tested to determine how well variability in layer thickness data along the LTPP section could be described using normal distribution. Kolmogorov-Smirnov goodness-of-fit test evaluated for level of significance alpha equal to 1 percent are summarized in table 70.

The results did not show as strong an indication of layer thickness distribution normality as the results of combined skewness and kurtosis test. This could be explained by lower power of Kolmogorov-Smirnov goodness-of-fit test compared to the combined skewness and kurtosis test. Low power indicates high probability of failing to reject the false null hypothesis.

Table 69. Summary of the goodness-of-fit results using Kolmogorov-Smirnov test with 1 percent level of significance.

Experiment

Number of layers

Not rejected (Normal)

Rejected (Not normal)

AC_SURFACE_COURSE

SPS-5

93

34 (36.6 %)

59 (63.4 %)

SPS-6

36

12 (33.3 %)

24 (66.7 %)

SURFACE_AND_BINDER

SPS-1

167

61 (36.5 %)

106 (63.5 %)

SPS-8

22

14 (63.6 %)

8 (36.4 %)

PERM_ASPH_TREAT_BASE

SPS-1

83

46 (55.4 %)

37 (44.6 %)

SPS-2

46

28 (60.9 %)

18 (39.1 %)

PCC_SURFACE

SPS-2

139

70 (50.4 %)

69 (49.6 %)

SPS-7

24

21 (87.5 %)

3 (12.5 %)

SPS-8

14

9 (64.3 %)

5 (35.7 %)

LEAN_CONCRETE

SPS-2

48

26 (54.2 %)

22 (45.8 %)

DENSE_GRD_ASPH_TREAT_BASE

SPS-1

97

45 (46.4 %)

52 (53.6 %)

DENSE_GRADE_AGG_BASE

SPS-1

97

63 (64.9 %)

34 (35.1 %)

SPS-2

84

53 (63.1 %)

31 (36.9 %)

SPS-8

38

30 (78.9 %)

8 (21.1 %)

BINDER_COURSE

SPS-5

33

11 (33.3 %)

22 (66.7 %)

SPS-6

13

7 (53.8 %)

6 (46.2 %)


The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration