terms (e.g., squared values) or interactions (e.g., between
material and discontinuity size), we recommend centering
the continuous predictor to reduce the potential for multicol-
linearity [17, 23]. When predictor variables are highly correlated
(such as and 2 ) the variances associated with the param-
eters may be inflated, which can affect variance estimation in
the POD curve. Although other more complex methods exist
to reduce multicollinearity, a simple centering of the variables
often resolves any issues and requires no structural changes to
the 90/95 estimation methods in Section 3. The need for center-
ing can be determined using the variance inflation factor (VIF)
[17, 23].
Simulation 1 considers a single random instance from
Equation 42, and Simulation 2 considers 10 000 random
instances. Figure 2 shows the fitted linear models for all
approaches from Simulation 1 plotted against the data, with
the top graph showing Material A and the bottom showing
Material B. The combined models ( =x and = x 2 +x)
are the same in both graphs because they do not account for
material. The by-case models ( =x for Material A, =x
for Material B, = x 2 +x for Material A, and = x 2 +x for
Material B) and those that include material ( =x +m and =
x 2 +x +m +mx +m x 2 are different between Material A and
Material B. Models including 2 naturally show more curva-
ture, and models fit separately to A and B demonstrate steeper
increases for Material B and less for Material A.
For Material A, the line =x is identical to the =x +m
line, and similarly for Material B. The line = x 2 +x for
Material A is identical to the = x 2 +x +m +mx +m x 2 line,
and similarly for Material B. Since these models have the same
mean, the latter lines are overlaid on the former in Figure 2
and may appear to be missing. However, the variances are dif-
ferent between these models, so they will have different POD
curves (see Figure 3).
The methods in Section 3.1 were used to create the models
in Tables 1, 2, 4, and 5. Tables 4 and 5 list the ˆ μpod and ˆ σpod =
σx/α1 values for Materials A and B. Variable 1 differs for the
materials, so each material has a different ˆ σpod The values for
ˆ μpod and σpod ˆ in Table 4 are in terms of while Table 5 values
are in terms of so the numbers differ because they are on
different scales.
When the data is split into subsets (model =x for each
material), the variance for Material A is estimated as 1.072 =
1.1449 and for Material B as 1.862 =3.4596. However, when the
materials are modeled together ( =x +m +mx a single
variance is estimated for both materials simultaneously:
1.522 =2.3104. This is different from the =x model where
material is ignored, which yields 2.382 =5.6644 this larger
variance occurs because the important material variable is
not included in the mean estimate. This applies to the qua-
dratic models as well, though estimated in -space rather
than -space.
Table 1 provides each fitted model resulting from the
maximum likelihood estimation (MLE). The MLE fit informa-
tion is provided in Table 2, in which smaller values of –2 log
likelihood (-2LL), AIC, and BIC indicate better models when
comparing the same dataset. These statistics are useful even
for comparing nonlinear models [16, 20]. Because Table 2
contains information from linear models, it is possible to
20
15
10
5
0
0.00 0.25 0.50 0.75 1.00
Discontinuity size
y =x
y =x for A
y =x for B
y =x2 +x
y =x2 +x for A
y =x2 +x for B
y =x2 +x +material
y =x +material
20
15
10
5
0
0.00 0.25 0.50 0.75 1.00
Discontinuity size
y =x
y =x for A
y =x for B
y =x2 +x
y =x2 +x for A
y =x2 +x for B
y =x2 +x +material
y =x +material
Figure 2. Linear model fits for each case and material: (a) Material A
(b) Material B.
y =x
y =x for A
y =x for B
y =x2 +x
y =x2 +x for A
y =x2 +x for B
y =x2 +x +material for A
y =x2 +x +material for B
y =x +material for A
y =x +material for B
0.00 0.25 0.50
Discontinuity size
1.00
0.75
0.50
0.25
0.00
1.00
0.75
0.50
0.25
0.00
0.00 0.25 0.50
Discontinuity size
Figure 3. Probability of detection (POD) curves for each model:
(a) Material A (b) Material B.
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 65
Signal
response
Signal
response
POD
POD
material and discontinuity size), we recommend centering
the continuous predictor to reduce the potential for multicol-
linearity [17, 23]. When predictor variables are highly correlated
(such as and 2 ) the variances associated with the param-
eters may be inflated, which can affect variance estimation in
the POD curve. Although other more complex methods exist
to reduce multicollinearity, a simple centering of the variables
often resolves any issues and requires no structural changes to
the 90/95 estimation methods in Section 3. The need for center-
ing can be determined using the variance inflation factor (VIF)
[17, 23].
Simulation 1 considers a single random instance from
Equation 42, and Simulation 2 considers 10 000 random
instances. Figure 2 shows the fitted linear models for all
approaches from Simulation 1 plotted against the data, with
the top graph showing Material A and the bottom showing
Material B. The combined models ( =x and = x 2 +x)
are the same in both graphs because they do not account for
material. The by-case models ( =x for Material A, =x
for Material B, = x 2 +x for Material A, and = x 2 +x for
Material B) and those that include material ( =x +m and =
x 2 +x +m +mx +m x 2 are different between Material A and
Material B. Models including 2 naturally show more curva-
ture, and models fit separately to A and B demonstrate steeper
increases for Material B and less for Material A.
For Material A, the line =x is identical to the =x +m
line, and similarly for Material B. The line = x 2 +x for
Material A is identical to the = x 2 +x +m +mx +m x 2 line,
and similarly for Material B. Since these models have the same
mean, the latter lines are overlaid on the former in Figure 2
and may appear to be missing. However, the variances are dif-
ferent between these models, so they will have different POD
curves (see Figure 3).
The methods in Section 3.1 were used to create the models
in Tables 1, 2, 4, and 5. Tables 4 and 5 list the ˆ μpod and ˆ σpod =
σx/α1 values for Materials A and B. Variable 1 differs for the
materials, so each material has a different ˆ σpod The values for
ˆ μpod and σpod ˆ in Table 4 are in terms of while Table 5 values
are in terms of so the numbers differ because they are on
different scales.
When the data is split into subsets (model =x for each
material), the variance for Material A is estimated as 1.072 =
1.1449 and for Material B as 1.862 =3.4596. However, when the
materials are modeled together ( =x +m +mx a single
variance is estimated for both materials simultaneously:
1.522 =2.3104. This is different from the =x model where
material is ignored, which yields 2.382 =5.6644 this larger
variance occurs because the important material variable is
not included in the mean estimate. This applies to the qua-
dratic models as well, though estimated in -space rather
than -space.
Table 1 provides each fitted model resulting from the
maximum likelihood estimation (MLE). The MLE fit informa-
tion is provided in Table 2, in which smaller values of –2 log
likelihood (-2LL), AIC, and BIC indicate better models when
comparing the same dataset. These statistics are useful even
for comparing nonlinear models [16, 20]. Because Table 2
contains information from linear models, it is possible to
20
15
10
5
0
0.00 0.25 0.50 0.75 1.00
Discontinuity size
y =x
y =x for A
y =x for B
y =x2 +x
y =x2 +x for A
y =x2 +x for B
y =x2 +x +material
y =x +material
20
15
10
5
0
0.00 0.25 0.50 0.75 1.00
Discontinuity size
y =x
y =x for A
y =x for B
y =x2 +x
y =x2 +x for A
y =x2 +x for B
y =x2 +x +material
y =x +material
Figure 2. Linear model fits for each case and material: (a) Material A
(b) Material B.
y =x
y =x for A
y =x for B
y =x2 +x
y =x2 +x for A
y =x2 +x for B
y =x2 +x +material for A
y =x2 +x +material for B
y =x +material for A
y =x +material for B
0.00 0.25 0.50
Discontinuity size
1.00
0.75
0.50
0.25
0.00
1.00
0.75
0.50
0.25
0.00
0.00 0.25 0.50
Discontinuity size
Figure 3. Probability of detection (POD) curves for each model:
(a) Material A (b) Material B.
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 65
Signal
response
Signal
response
POD
POD
