An Evaluation of a 

Proposed Revision 

of the ASTM D 1990 

Grouping Procedure 

Steve P. Verrill

James W. Evans

David E. Kretschmann

Cherilyn A. Hatfield

United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL–RP–671

4 6 8 10 12

−5 −4 −3 −2 −1 0

Shape parameter

log10(probability of breakage at allowable property)

12 y

.001

.001455766

.005

current
February 2013

Verrill, Steve P.; Evans, James W.; Kretschmann, David E.; Hatfield, 

Cherilyn A. 2013. An evaluation of a proposed revision of the ASTM D 

1990 grouping procedure. Research Paper FPL-RP-671. Madison, WI: 

U.S. Department of Agriculture, Forest Service, Forest Products 

Laboratory. 34 p.

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and employer.

Abstract

Lum, Taylor, and Zidek have proposed a revised procedure 

for wood species grouping in ASTM standard D 1990. 

We applaud the authors’ recognition of the importance of 

considering a strength distribution’s variability as well as 

its fifth percentile. However, we have concerns about their 

proposed method of incorporating this information into a 

standard. We detail these concerns in this paper. We also 

provide both theoretical and empirical arguments that suggest that revisions of the standard should not be based on a 

two-parameter Weibull assumption.

Keywords: Reliability, modulus of rupture, modulus of 

elasticity, bivariate Gaussian−Weibull, pseudo-truncated 

Weibull, Weibull

Contents

1 Introduction....................................................................... 1

2 Aside ................................................................................. 3

3 The Proposal of Lum, Taylor, and Zidek (2010)............... 4

4 Performance of the Proposed Procedure........................... 5

5 Are Strength Distributions of Grades of Lumber Two- 

 Parameter Weibull? ........................................................... 8

6 Conclusions......................................................................13

7 References........................................................................14

8 Appendix..........................................................................15
An Evaluation of a Proposed Revision of the ASTM D

1990 Grouping Procedure

Steve P. Verrill, Mathematical Statistician

James W. Evans, Mathematical Statistician

David E. Kretschmann, Research General Engineer

Cherilyn A. Hatfield, Statistician

Forest Products Laboratory, Madison, Wisconsin

1 Introduction

In 2008, the American Lumber Standard Committee Board of Review formed a North American

Lumber Properties Task Group to review the application and interpretation of ASTM structural

lumber standards. The subjects under review included species grouping, withdrawal of species from

existing species groups, the appropriateness of using parametric fits to censored tails of lumber

data, and issues with sampling representativeness. The charge of one of the task subgroups was to

investigate procedures that might be improvements over the currently accepted grouping procedure

that is based on nonparametric estimates of fifth percentiles. In this paper, we evaluate a proposed

procedure that is based on two-parameter Weibull fits to lumber strength data.

Two important wood properties are bending strength (modulus of rupture or MOR) and stiffness

(modulus of elasticity or MOE). In the past, MOR populations have been modeled as lognormals or

as two- or three-parameter Weibulls. MOE populations have generally been modeled as Gaussians.

(See, for example, ASTM 2010a, Evans and Green 1988, and Green and Evans 1988.)

Design engineers must ensure that the loads to which wood systems are subjected rarely exceed

the systems’ strengths. To this end, ASTM D 2915 (ASTM 2010a), and ASTM D 245 or ASTM D

1990 (ASTM 2010b,c) describe the manner in which “allowable properties” are assigned to populations of structural lumber. In essence, an allowable strength property is calculated by estimating a

fifth percentile of a population (actually a 95% content, one-sided lower 75% tolerance bound) and

then dividing that value by duration of load and safety factors. The intent is that the population

can only be used in applications in which the load does not exceed the allowable property.

It is often the case that lumber marketers want to assign a single allowable property to a

grouping of multiple species. Currently, ASTM D 1990 details procedures for determining this

shared allowable property. For strength properties, these procedures are currently solely focused on

the fifth percentile of an individual species or group of species. Lum, Taylor, and Zidek (2010a,b,c)

and others (see, for example, Verrill and Kretschmann 2009) have noted that under the current

procedures, two strength distributions could share a fifth percentile and thus an allowable property,

and yet, if the populations had different variabilities, the associated probabilities of breakage at the

allowable property would differ. This is illustrated in Figure 1. The wide distribution has mean 1

and standard deviation .25. The narrow distribution has mean 1 − 1.645 × .25 + 1.645 × .05 and

standard deviation .05. The two distributions share a fifth percentile (1 − 1.645 × .25), and yet

it is clear visually that a value drawn from the wide distribution is more likely to fall below the

allowable property than a value drawn from the narrow distribution. (In fact, it can be shown that

a value from the wide distribution falls below the allowable property with probability .002 while a

value from the narrow distribution falls below the allowable property with probability 2.8E-15.)

1
Thus, an improved grouping procedure should take differing variabilities into account. In fact,

ideally, an improved procedure would focus on grouping based on similar probabilities of failure for

a given load distribution. Lum et al. (2010a,b,c) have proposed a grouping procedure that does

take variability into account. (In their papers they emphasize that the proposal is a draft proposal.)

However, it is not based on probabilities of failure. Apparently, the authors felt that for practical

reasons, it was still necessary to base groupings on something like an allowable property based on

a fifth percentile divided by a duration of load and safety factor. As we will detail below, they

proposed to incorporate information about a population’s variability by calculating an “adjusted

fifth percentile.” For populations with a coefficient of variation (CV) above some standard value

(more variable populations), the adjusted fifth percentile would lie below the actual fifth percentile.

For populations with a coefficient of variation below the standard value (less variable populations),

the adjusted fifth percentile would lie above the actual fifth percentile.

We applaud the authors for their insight, but believe that their current proposal needs further

work (as expected given its draft status). We believe that their proposal is unsatisfactory for four

reasons:

1. Their current proposal still permits probabilities of breakage at allowable properties to differ

among species and among groups. In Section 4.1, we do describe a particular implementation

of their proposal that would address this problem. (This particular implementation is simpler

in conception and execution than their more general proposal.) However, in Section 4.2, we

note that equal probabilities of breakage cannot be guaranteed under all duration of load

conditions.

2. In Section 4.3, we discuss the fact that their proposal needs to be investigated for safety

consequences.

3. In Section 4.4, we note that the proposed procedure can still lead to unexpected results (the

probability of breakage of the “controlling” species at the allowable property can be less than

that of a non-controlling species).

4. The current proposal is based on an assumption of a two-parameter Weibull strength distribution. Verrill, Evans, Kretschmann, and Hatfield (2012) demonstrate that if the full MOE,

MOR distribution is bivariate Gaussian–Weibull, then a subcategory based on machine stress

rated (MSR) values (or a normally distributed quality characteristic that is implicit in visually graded categories) will not be distributed as a Weibull. Instead, it will be distributed as a

pseudo-truncated Weibull (PTW). In Sections 5.2 and 5.3 we discuss empirical evidence that

actual MOR distributions are indeed PTW rather than Weibull. In Section 5.4 we discuss

simulations that we have conducted that demonstrate that breakage rate calculations that

are based on Weibull fits are susceptible to significant error if the strength distribution is

actually a pseudo-truncated Weibull.

Before proceeding with our analysis, we would like to clarify one point. We have spoken about

“probabilities of breakage at the allowable property” and about “similar probabilities of failure

for a given load distribution.” Ultimately, we would like to see (if possible) standards that assure

us that if a wood assembly’s environment (heat, moisture, biological . . . ) is drawn from a given

“distribution” and its load history is drawn from a given “distribution,” then if the members of

the assembly are drawn from a given species grouping, the probability that the assembly “fails”

(in terms of safety or, alternatively, serviceability) prior to a given time is less than a given value.

However, in the current paper, we are focusing on the probability that a piece of lumber breaks

when it is subjected to a load equal to the allowable strength property of the population from which

2
the piece is drawn. We will denote this probability by pBr and refer to it as “the probability of

breakage.”

We recognize that our “probabilities” cannot be taken entirely literally. Our probability calculations are mathematically correct, but they rely on simplified models that may not adequately

reflect the real world. Our paper is primarily an attempt to evaluate Lum et al.’s (2010a,b,c) proposed fifth percentile adjustment based on currently available data and models. As our data and

models improve, our probability estimates will also improve.

2 Aside

In connection to Figure 1, a reviewer of this paper noted 1) that the load is, of course, variable,

2) that strength populations are unlikely to have Gaussian distributions (the reviewer suggested

doing some modeling with lognormal strength and load distributions), and 3) that when variable

loads are included, the “unfair” advantage that a wide distribution is given when we group solely

on the basis of fifth percentiles is lessened. The reviewer felt that if we did some calculations

with lognormal loads and strengths (which are mathematically tractable), “you will conclude that

‘messing’ with the fifth/2.1 can’t be justified in the context of structural reliability or safety.”

One of our responses was that we were simply using Figure 1 to illustrate the obvious, but

sometimes overlooked point that a fifth percentile is not sufficient to characterize the behavior of a

population, and that if we group based solely on fifth percentiles, we are grouping on a basis that is

not justified from a reliability perspective. A second response was that we were evaluating a proposal

that made a Weibull strength assumption so, indeed, we would not be making a Gaussian strength

assumption in the remainder of the paper. A third response was that although full reliability

calculations must include a load distribution, we could gain insights into the proposed method

by evaluating the probability that a Weibull strength distribution would lie below an allowable

property. This makes the Weibull math tractable and leads to some elegant simplifications (see,

for example, Section 4) that permit us to gain a better understanding of the proposed method. A

fourth response is that the focus of this report is the proposed method for adjusting fifth percentiles

prior to grouping. Our purpose here is not to present a full-throated defense of the argument

that grouping should be based on similar probabilities of failure given similar load distributions

rather than on similar 5th percentiles. However, in response to the reviewer, we did perform some

calculations based on the lognormal distribution. The details of the calculations are provided

in the Appendix. We wanted to check whether significant “unfairness” (significant differences in

reliabilities) continued to exist for “narrow” and “wide” distributions that shared the same fifth

percentile when we calculate probabilities of failure based on a load distribution rather than at a

fixed point (treating the allowable property as a “worst case”). The results from these calculations

are listed in Table 1, and the second row of Table 1 is illustrated in Figure 2. In line with the

suggestions of the reviewer, we assumed lognormal strength and load distributions. The reviewer

also suggested that in our calculations, we let the load distribution have a CV approximately equal

to 0.3, and let the probability that the load exceeds the allowable property be approximately 0.02.

We considered 15 cases. In each case, the CV of the load distribution was taken to be 0.3. We

let the probability that the load exceeded the allowable property equal 0.01, 0.02, 0.05, 0.10, or

0.20. We let the CV’s of the narrow strength populations be 0.05, 0.1, or 0.2 and the CV’s of

the corresponding wide strength populations be 0.25, 0.3, or 0.4. From Table 1, we can see that

for the conditions considered, the reviewer’s intuition is at least partially validated. That is, the

pF narr and pF wide values for the cases considered in Table 1 are closer than the 2.8E-15 and .002

associated with Figure 1. However, the results continue to suggest that equal fifth percentiles do

3
not in general lead to probabilities of failure that are so close that differences can be neglected.

Of course, this is hardly a comprehensive study of the issue. It could be argued, for example,

that strength distributions are not lognormal or Weibull but pseudo-truncated versions of standard

strength distributions (see Section 5). This might lead to small probabilities of failure for both

“narrow” and “wide” strength distributions and differences in these probabilities might not be

of practical importance. Also, we have compared distributions with highly diverging CV’s (for

example, 0.05 and 0.25). In practice, differences in strength distribution CV’s might be slight

enough so that differences in probabilities of failure will again not be of practical importance. We

leave further exploration of these issues to future papers.

3 The Proposal of Lum, Taylor, and Zidek (2010)

In essence, their proposal was:

1. Strength distributions would be modeled as two-parameter Weibulls.

2. The controlling species in a group of species would be the one with the lowest pth quantile

for some pre-specified p.

3. The allowable strength property for a group would be x.05,adj/2.1 where x.05,adj is the adjusted

fifth percentile of the controlling species. This adjusted fifth percentile would be a function of

the controlling population’s coefficient of variation and some standard coefficient of variation.

If a controlling population’s coefficient of variation were smaller than the standard coefficient

of variation, then the adjusted fifth percentile would be boosted up. If a controlling population’s coefficient of variation were larger than the standard coefficient of variation, then its

fifth percentile would be adjusted down.

Here is a detailed description of their algorithm for calculating x.05,adj.

Let β be the shape and 1/γ be the scale of the assumed Weibull strength distribution of the

controlling species. The pth quantile, xp, of this distribution is given by

p = 1 − exp −(γ × xp)

β



or

xp = (− ln(1 − p))1/β/γ (1)

Let β0 be the shape parameter corresponding to the standard coefficient of variation. (For

two-parameter Weibull distributions, the coefficient of variation is solely a function of the shape

parameter.)

Find the γ0 that satisfies

p = 1 − exp −(γ0 × xp)

β0



or

γ0 = (− ln(1 − p))1/β0 /xp (2)

The adjusted fifth percentile is then given by

.05 = 1 − exp −(γ0 × x.05,adj)

β0



4
or, using (2) and (1),

x.05,adj = (− ln(1 − .05))1/β0 /γ0

=



ln(1 − .05)

ln(1 − p)

1/β0

× xp

=



ln(1 − .05)

ln(1 − p)

1/β0

×

(− ln(1 − p))1/β

γ

(3)

and the corresponding adjusted allowable property is x.05,adj/2.1. (Note that the unadjusted fifth

percentile, x.05, is given by replacing p in (3) by .05.)

4 Performance of the Proposed Procedure

To evaluate the performance of this procedure, we calculate the probability that a piece of lumber

drawn from the controlling species population breaks when it is subjected to a load equal to the

allowable strength property of the species.

From (3) we have

pBr = 1 − exp −(γ × x.05,adj/2.1)β



(4)

= 1 − exp −(arg)β



where

argβ =

"

1

2.1

×



ln(1 − .05)

ln(1 − p)

1/β0

#β

× (− ln(1 − p)) (5)

In the next subsection, we will discuss a figure in which we plot log(pBr) versus β for a variety of

p’s. Somewhat surprisingly, these plots will appear to be linear. This is due to the fact that, for

the p’s and β’s that we consider, argβis “small” so

exp 

−(arg)β



≈ 1 − (arg)β

or

1 − exp −(arg)β



≈ (arg)β

and from (5),

ln(pBr) = ln 1 − exp −(arg)β



≈ β × ln "

1

2.1

×



ln(1 − .05)

ln(1 − p)

1/β0

#

+ ln(− ln(1 − p)) (6)

which is a linear function of β. (For unadjusted fifth percentiles, (6) becomes ln(pBr) ≈ − ln(2.1) ×

β + ln(− ln(1 − .05)).)

5
4.1 Is the proposed procedure “fairer” than one based on unadjusted fifth percentiles?

In Figure 3, we plot the log base 10 of the probability of breakage of a (controlling) species at its

allowable load versus shape. The solid line represents the “current situation”—that is, the allowable

property is taken to be an unadjusted fifth percentile of a two-parameter Weibull divided by 2.1.

The long-dashed line represents the situation under the Canadian proposal when the quantile p

being considered is .005 and the baseline shape is 4.8. The short-dashed line represents the situation

under the Canadian proposal when the quantile p being considered is .001455766 and the baseline

shape is 4.8. The dotted line represents the situation under the Canadian proposal when the

quantile p being considered is .001 and the baseline shape is 4.8.

From Figure 3, we can see that a procedure based on an unadjusted fifth is unfair in that, under

it, populations with lower shapes (higher coefficients of variation) are permitted higher probabilities

of breakage at their allowable properties. The Canadian procedure is fairer as it jacks down the

permissible probabilities of breakage for shapes below 4.8 and increases them for shapes above 4.8.

As the quantile used in the procedure decreases, the slope of the line increases.

From Figure 3, we can also see that, under the proposed approach, it is possible to develop a “fair

procedure” that yields the same probability of breakage at the allowable property for all controlling

species. That is, given any baseline shape, β0, we can specify a corresponding p (0.001455766 for

β0 = 4.8) for which the pBr versus β curve is flat (the short-dashed line in Figure 3).

This can be demonstrated as follows. From results (4) and (5), we know that pBr will be a

constant function of β if



1

2.1

β0

×



ln(1 − .05)

ln(1 − p)



= 1 (7)

or

ln(1 − p) = ln(1 − .05)/2.1

β0

or

1 − p = exp ln(1 − .05)/2.1

β0



or

p = 1 − exp 

ln(1 − .05)/2.1

β0



(8)

(As noted above, for baseline β0 = 4.8, this p equals .001455766.)

In this constant pBr case, from (4), (5), and (7), we have

pBr = 1 − exp −(arg)β



(9)

= 1 − exp(ln(1 − p)) = p

That is, the probability of breakage in this “fair case” is just the p value that was used in the

proposed procedure. This tells us that in the “fair case,” the proposed procedure simply amounts

to choosing a p value and then stating that the allowable property associated with any controlling

species will be its pth quantile (so that the probability that any controlling species will break when

subjected to a load equal to its allowable property will be p).

4.2 Duration of load and inherent unfairness

Any procedure is inherently unfair. To understand this central point, consider the current situation.

Why is it “unfair” to narrower distributions? As illustrated in Figure 1, in a probability sense a

.05 quantile (fifth percentile) divided by 2.1 is farther down in the tail (“more standard deviations

6
below the mean” for a normal distribution) for a narrow distribution than for a broad distribution.

Thus the pBr (probability of breakage at the allowable property) for the narrower distribution at

its fifth/2.1 is smaller than the pBr for the broader distribution at its fifth/2.1. If instead, we use

the pth quantile of a distribution as its allowable property, then pBr would be the same (p) for all

distributions.

However, as soon as specimens are subjected to duration of load and their strengths are reduced,

this “fairness” is likely gone. Now narrower distributions are likely to be favored. Gerhards (2000)

found that after 12.27 years under load, 16 of 50 No. 2 two by fours failed. The load was intended

to be the “10-year design load” of the distribution from which the two by fours were drawn, that

is x.05/1.62. (The 1.3 factor for safety was not included, but this does not affect our conclusions.

See below.) Gerhards later concluded that the load was actually x.08/1.62. (Again, this does not

affect our conclusions. See below.) He also calculated (by assuming that the original strength of

the strongest piece of lumber to fail was equal to the expected strength of the 16th order statistic

of 50 from the strength population) that the approximate load to strength value for the strongest

piece of lumber to fail was 0.442. If we assume that all pieces with load to strength ratios greater

than 0.442 will fail by 12.27 years, then we can calculate the probability of breakage after 12.27

years for loads set to “fair” adjusted allowable properties at day 0:

pBr = Prob Initial Strength <

x.05,adj

2.1

×

1

.442

= 1 − exp −



γ ×

x.05,adj

2.1

×

1

.442β

!

= 1 − exp −(arg)β



(10)

where

argβ =

"

1

(2.1)(.442) ×



ln(1 − .05)

ln(1 − p)

1/β0

#β

× (− ln(1 − p))

and, for β0 = 4.8 and a flat initial pBr versus β curve,

p = 0.001455766

In Figure 4, we replicate Figure 3 but also add the pBr versus β curve where pBr is calculated via

(10). Figure 4 demonstrates that after 12 years of load, the probability of breakage is no longer

“fair” — PBr is no longer a flat function of the CV. Instead, it increases as CV decreases (as β

increases).

4.3 Is the proposed procedure “as safe” as one based on unadjusted fifth percentiles?

We can see from Figure 3 that implementing the proposed approach would (for at least some p

values) decrease the probability of breakage at the allowable load for species with shape values less

than the baseline shape. However, it would increase the probability of breakage at the allowable

load for species with shape values greater than the baseline shape. It is not immediately clear that

this would translate into a situation that is as safe as the current situation. (The species associated

with the highest probabilities of breakage at their allowable properties under the current system

would have these probabilities reduced under the proposal. However, are these species as common

as those whose probabilities of breakage would increase under the proposal?)

7
In Figure 5 we plot log base 10 of the ratio of the probability of breakage under the proposed

procedure to the probability of breakage under the current procedure versus controlling species

shape for a baseline of 4.8. As a reference, we also plot the log 10(ratio) = 0 line. (pBr under the

current procedure is just 1 − exp(ln(1 − .05)/2.1

β

) and pBr under the proposed procedure is given

by (4).)

Obviously, the ratios begin below 1, increase to 1 at the baseline shape, and then continue to

increase. The increases can be quite large. (The minimum ratio in Figure 5 is .210 at controlling

species β = 2.6956 (corresponding to a CV of 40%). The maximum ratio in Figure 5 is 226 at

controlling species β = 12.154 (corresponding to a CV of 10%).) Thus, again, it is not immediately

clear that we would not be degrading our reliability by adopting the proposed procedure.

4.4 Anomalous behavior of the proposed procedure

Suppose that there are two species in the group, species A and species B. Further suppose that it

has been decided to base the procedure on the pth quantile, and that the pth quantile of species

A, xp,A, lies below the pth quantile, xp,B, of species B. Then species A is the controlling species.

Now suppose that p and the baseline shape, β0, are chosen so that the pBr versus β curve is

flat. In this case, as demonstrated in Subsection 4.1 (see equation (9)), pBr,A = p and the allowable

property is just xp,A. Since

xp,B > xp,A = the allowable property

pBr,B = prob(XB < the allowable property)

= prob(XB < xp,A)

≤ prob(XB < xp,B) = p = pBr,A

Thus, species B is at least as safe at the allowable property as the controlling species A.

However, suppose, instead, that p and the baseline shape, β0, are not chosen so that the pBr

versus β curve is flat. In this case, it is possible for the allowable property to lie below or above

xp,A, and undesirable behavior can result.

If the allowable property lies below xp,A and species B has a broader distribution than species

A, it will be possible for pBr,B to exceed pBr,A even though species A is the “controlling species.”

For example, for β0 = 4.8, p = 0.01, βA = 6, γA = 0.00941, βB = 3, γB = 0.004, we have

allowable property = 33.0, xp,A = 49.4, xp,B = 54.0, pBr,A = 0.000899, and pBr,B = 0.00230. That

is, pBr,B > pBr,A even though xp,A < xp,B. The situation is depicted in Figure 6.

Similarly, if the allowable property lies above xp,A and species B has a narrower distribution

than species A, it will be possible for pBr,B to exceed pBr,A even though species A is the “controlling

species.” For example, for β0 = 4.8, p = 0.0005, βA = 5, γA = 0.00929, βB = 9, γB = 0.017, we

have allowable property = 29.4, xp,A = 23.5, xp,B = 25.3, pBr,A = 0.00152, and pBr,B = 0.00195.

That is, pBr,B > pBr,A even though xp,A < xp,B. The situation is depicted in Figure 7.

5 Are Strength Distributions of Grades of Lumber Two-parameter

Weibull?

The answer, both theoretically and empirically, is no.

8
5.1 The theoretical argument

This argument is based upon the process by which “bins” of lumber are produced. For example,

in the United States, construction grade two by fours are often classified into visual categories—

select structural, number 1, number 2—or into machine stress-rated (MSR) grades. In the case

of MSR grades, modulus of elasticity (MOE) boundaries are selected, the MOE of a piece of

lumber is measured non-destructively, and pieces are placed into categories based upon the MOE

bins into which the pieces fall. Because MOE and MOR are correlated, bins with higher MOE

boundaries also tend to contain lumber populations with higher MOR values. However, because

the correlation between MOE and MOR is not 1, the MOR population corresponding to a MOE

bin is not a truncated Weibull. In Verrill et al. (2012), we show that if the joint distribution of the

full MOE and MOR populations is bivariate Gaussian–Weibull, then the strength distribution of

the specimens in a MSR grade is a pseudo-truncated Weibull (PTW), not a Weibull. In that paper,

we obtain the density function of the PTW and develop asymptotically efficient methods for fitting

the underlying bivariate Gaussian–Weibull. We note that the arguments that apply to MSR lumber

also apply to visually graded lumber assuming that there is an implicit quality measurement in the

visual grading process that is analogous to the MOE measurement in the MSR process.

5.2 The empirical argument

Lum et al. (2010a,b,c) reported that for species A (not revealed), number 2 lumber, the estimated

shape parameter increased from 3.1 to 4.1 to 4.9 as the data fitted went from full to bottom 20%

to bottom 10%. For species B (not revealed), number 2 lumber, the estimated shape parameter

increased from 2.9 to 3.7 to 4.5. For species C (not revealed), number 2 lumber, the estimated shape

parameter increased from 2.4 to 4.1 to 4.9. That is, in all three cases the estimated shape parameter

increased as the censoring increased. Similarly, we fit 19 cells of select structural and number 2 data

that were obtained in the Ingrade Program (Evans and Green (1988), Green and Evans (1988)).

See Table 2. In 17 of the 19 cases considered, the estimated shape parameter increased as the

censoring increased (and the estimated scale parameter decreased as the censoring increased). (We

note that we were using correct censored data estimation techniques in the censored data cases.

See Cohen (1965) or ASTM D 5457 (ASTM 2011).)

We claim that these increases are evidence that these real strength distributions are not twoparameter Weibull. Given the increase in slope estimates with increased censoring, statisticians

should immediately question the two-parameter Weibull assumption because if this assumption were

correct, maximum likelihood estimates of the shape parameter would be asymptotically unbiased

regardless of whether we were using censored estimates or full sample estimates. Thus, there should

be no systematic differences in the means of the shape estimates. We checked this intuition via simulation. We generated 20 samples of size 1200 from a two-parameter Weibull with shape parameter

3.701 (CV approximately equal to 0.30). We then generated maximum likelihood estimates of the

shape parameter for the full data, the bottom 20% of the data, and the bottom 10% of the data for

each of the 20 samples. The 20 triples of estimates are reported in Table 3. The means of the three

estimates (full, bottom 20%, bottom 10%) were 3.72, 3.70, and 3.76. Their standard deviations

were .09, .19, and .27. This simulation does not suggest that we would consistently see shape values

increasing as censoring increases if we were truly fitting a two-parameter Weibull. A listing of the

program that produced Table 3 can be found at http://www1.fpl.fs.fed.us/bivar3.html. See

the testcens.f link.

Two additional checks of the two-parameter Weibull assumption can be made via probability

plots and goodness-of-fit tests. (See, for example, D’Agostino and Stephens, 1986.) In the upper

9
halves of Figures 8 and 9, we provide two-parameter Weibull probability plots of two of the 19 data

sets described in Table 2. In the lower halves of Figures 8 and 9, we provide Weibull probability

plots of generated two-parameter Weibull data. The generating scales and shapes were the fitted

values for the data sets plotted in the upper halves of the figures.

Notice the differences between the paired upper and lower plots. In each upper plot, the

measured data set has a left-hand tail that is “short” compared to what one would expect if the

population truly were Weibull (hence the points lying above the line at the left side of the plot).

This left tail shortness is what one would expect from a PTW. This left side shortness does not

appear in the generated two-parameter Weibull data displayed in the lower plots. We produced

similar plots for the other 17 data sets listed in Table 2. Fourteen of the 19 data sets displayed the

left tail shortness for the observed data. Only two of the 19 generated data sets displayed any left

tail shortness.

There was no corresponding right-tail shortness in the observed strength data. At first, this

might suggest pseudo-truncation at the left and none at the right. (This is correct for the SS data.)

However, while a two-parameter Weibull must have support that extends down to 0, and thus below

a PTW, there is no required upper extension, and it is possible that a best fit of a Weibull model

to PTW data will involve a pulled-in Weibull right tail which can lead to data points at or above

the y = x line at the right side of the probability plot.

We also performed Cram ́er–Von Mises and Anderson–Darling goodness-of-fit tests of both the

observed and generated strength data. The results from these tests are displayed in Table 4. “NS”

denotes not statistically significant at a .25 level. “.25”, “.10”, “.05”, “.025”, and “.01” indicate

statistically significant at the listed levels. From these results, it is clear that the observed strength

data is not distributed as a two-parameter Weibull.

A listing of the program that produced the probability plots and performed the goodness-of-fit

tests can be found at http://www1.fpl.fs.fed.us/bivar3.html. See the weibpp.f link.

5.3 Empirical evidence of pseudo-truncated Weibull behavior

In Section 5.1 we noted that, provided that the full MOE/MOR population has a particular form of a

bivariate Gaussian–Weibull distribution, we can mathematically prove that the marginal MOR distribution for MSR lumber is not Weibull. Instead, the marginal distribution is a pseudo-truncated

Weibull (PTW) where the distribution function of a PTW is derived in Verrill et al. (2012). We

also noted that this theoretical argument also applies to visually graded lumber.

In Section 5.2, we demonstrated that there is empirical evidence that the MOR distributions of

visual grades are not Weibull.

In this section, we establish that there is empirical evidence that the distributions of visual

grades of lumber share at least one characteristic of PTW distributions. In particular, here we

report results from simulations of PTW distributions that demonstrate that fits of two-parameter

Weibulls to PTW data yield the same pattern of increasing shape estimates (and decreasing scale

estimates) with increasing censoring that was observed (see Table 2) with actual visually graded

lumber.

We created simulated number 2 populations as follows:

1. Randomly generate 1,000,000 N(0,1) x’s (MOE’s).

2. Randomly generate 1,000,000 correlated normally distributed y’s via

yi = ρ × xi +

p

1 − ρ

2 × N(0,1)

where the N(0,1)’s in the equation are statistically independent of the xi’s.

10
3. Sort the x’s and bring along the corresponding y’s. (This models machine stress rating of

lumber.)

4. Treat the y’s that correspond to the top 600,000 x’s as number 2 and betters. Treat the

bottom 400,000 of this 600,000 as number 2’s.

5. Transform these normals to Weibulls by first transforming them to U(0,1)’s and then to

Weibulls. (We have done simulation studies that indicate that the correlation value between

the MOEs and the Weibulls remains essentially the same as the correlation between the two

sets of normals.) (Shapes were determined by CV’s. Scales were chosen so that median values

were 100.)

Listings of the simulation programs are available at http://www1.fpl.fs.fed.us/bivar3.html.

See the num2sim.1m.2.f link.

We generated number 2 populations for four correlations between the underlying normals—

.50, .60, .70, .80—and three coefficient of variation (CV) values—20, 30, and 40. From each

generated population, we randomly drew (with replacement) 100 samples of size 400. For each of

these samples, we used linear regression and maximum likelihood techniques to fit two-parameter

Weibulls to the full sample, and to the bottom 20 and 10 percent of the sample. (Correct censored

data estimation techniques were used to fit the censored data. See Cohen (1965) or ASTM D

5457 (ASTM 2011).) We averaged the parameter estimates over the 100 fits. These averages are

reported in Table 5 and show the same pattern of increasing shape estimates and decreasing scale

estimates with increasing censoring that was reported for the MOR data discussed in Subsection

5.2.

5.4 If the strength populations are not really two-parameter Weibull, does it

matter?

In the subsections above, we have provided theoretical and empirical evidence that MOR subpopulations generated via MOE or visual-grading binning do not have a two-parameter Weibull

distribution. Suppose, however, that we ignore this fact and proceed to fit such data with Weibull

distributions. Does this hurt us? After all, ultimately we are not interested in estimates of shape

and scale. Instead, we are interested in estimates of probabilities of breakage. If we fit a Weibull

distribution to PTW data, are we seriously misled about probabilities of breakage? In Tables 6–9

we report results from simulations performed to investigate this question. (A listing of the program

that produced these tables can be found at http://www1.fpl.fs.fed.us/bivar3.html. See the

pfsim4.f link.)

For four generating correlations—.50, .60, .70, .80—and three coefficient of variation (CV)

values—20, 30, and 40, we generated 10,000 bivariate Gaussian–Weibull data sets. These data data

sets were generated so that each yielded 400 specimens with a Gaussian value that lay between the

40th and 80th percentiles of the Gaussian distribution. This required approximately 1000 bivariate

Gaussian–Weibull pairs in each data set. For each of these 10,000 data sets we calculated:

1. (W/Reg/All) A (theoretically incorrect) Weibull regression fit based on all 400 simulated

pseudo-truncated MOR data values.

2. (W/Reg/20) A (theoretically incorrect) Weibull regression fit based on the bottom 20% of

the 400 simulated pseudo-truncated MOR data values.

3. (W/Reg/10) A (theoretically incorrect) Weibull regression fit based on the bottom 10% of

the 400 simulated pseudo-truncated MOR data values.

11
4. (W/MLE/All) A (theoretically incorrect) Weibull maximum likelihood (ML) fit based on all

400 simulated pseudo-truncated MOR data values.

5. (W/MLE/20) A (theoretically incorrect) Weibull ML fit based on the bottom 20% of the 400

simulated pseudo-truncated MOR data values.

6. (W/MLE/10) A (theoretically incorrect) Weibull ML fit based on the bottom 10% of the 400

simulated pseudo-truncated MOR data values.

7. (PTW trunc) A (theoretically correct) fit of the 400 bivariate Gaussian–Weibull data values

binned by the Gaussian value (the Gaussian member of the pair lies between the 40th and

80th percentiles of the Gaussian distribution). (Bivariate data that has been truncated based

on one of its components is used to fit a bivariate Gaussian–Weibull which permits one to

characterize the corresponding univariate PTW.)

8. (PTW 400) A (theoretically correct) fit of the first (not the lowest) 400 of the approximately

1000 unbinned Gaussian-Weibull values. (Bivariate data that has not been truncated based

on one of its components is used to fit a bivariate Gaussian–Weibull which permits one to

characterize the corresponding univariate PTW.)

9. (PTW 1000) A (theoretically correct) fit of all of the approximately 1000 unbinned GaussianWeibull values. (Bivariate data that has not been truncated based on one of its components is

used to fit a bivariate Gaussian–Weibull which permits one to characterize the corresponding

univariate PTW.)

(The first six fits use correct methods. That is, they use censored data methods for censored data

and full sample methods for full sample data. They are “theoretically incorrect” fits because they

are based on the incorrect assumption that the data come from a Weibull population.) Each of

these fits yielded an allowable property (estimated fifth percentile divided by 2.1). In each case

we calculated the predicted probability, pBr,est, that an MOR would lie beneath the allowable

property. This value was calculated from the fitted distribution. In each case we also calculated

the true probability, pBr,true, that an MOR would lie beneath the allowable property. This value

was calculated from the (known) generating bivariate Gaussian–Weibull. Information about the

ratios of the true “probabilities of breakage” (probability that a specimen has strength lower than

the allowable property) to the estimated probabilities of breakage is presented in Tables 6–9.

In column 1 of these tables, we provide the generating coefficient of variation. In column 2 we

describe the type of fit. (These correspond to the nine fitting techniques listed above.) In column 3

we provide the median of the 10,000 ratios of true to estimated probabilities of breakage. A value

below 1 indicates an approach that is conservative at the median. However, as can be seen from

the remainder of the table, the median represents an insufficient summary. The remaining columns

list the fraction of the time that a particular technique had ratios of true to estimated probabilities

of breakage that lay in the intervals [0, .01], (.01, .02], (.02, .1], (.1, .2], (.2, .5], (.5, 1), (1, 2), [2, 5),

[5, 10), [10, 50), [50, 100), [100, ∞). Obviously, we could have chosen other intervals. However, our

main points are clear from the current tables:

• The Weibull fits (regression and ML) tend to be overly conservative. That is, true probabilities

of breakage can be much less than estimated probabilities of breakage.

• At the same time, the censored (20% and 10%) Weibull fits can be much more variable than

the correct bivariate Gaussian–Weibull fits with the result that they can occasionally yield

highly non-conservative fits. That is, the actual probabilities of breakage can be much greater

than estimated probabilities of breakage.

12
• If the joint MOE/MOR distribution is truly bivariate Gaussian–Weibull, we can obtain better

estimates of the probability of breakage by taking the same number of specimens from the

full distribution than by restricting ourselves to binned values. (Compare the PTW 400 and

PTW trunc results.)

6 Conclusions

We have evaluated a proposed method for combining wood species into groups with similar strength

properties. We commend the authors of the proposal for recognizing the importance of considering

a strength distribution’s variability as well as its fifth percentile. However, we have concerns about

the proposed implementation:

• The proposed procedure will reduce the allowable properties of more variable populations

and increase the allowable properties of less variable populations. However, unless we choose

the p and β0 tuning parameters to yield a flat probability of breakage versus β curve, there

will still be “unfairness” in the allowable property calculations. That is, the probability of

breakage at the allowable property will depend on a population’s CV.

• If we do tune the procedure to ensure a flat probability of breakage versus β curve, then the

procedure simply amounts to choosing a p value (e.g., .01 or .001) and taking all controlling

species’ allowable properties to be their pth quantiles.

• However, as we demonstrated in Section 4.2, if we achieve “fairness” via this method at one

time, the “fairness” will disappear as the lumber is subjected to duration of load.

• We also noted that although the proposed procedure could reduce the probabilities of breakage

associated with more variable species, it will increase the probabilities of breakage associated

with less variable species. Thus, we are not assured (without further work) that an implementation of the procedure would preserve current safety levels.

• Currently, the proposed procedure makes use of the Weibull distribution to obtain estimates

of allowable properties. We have demonstrated that MOR grade populations are not Weibull

distributed and that we can obtain poor estimates of probabilities of breakage at allowable

properties if we make a Weibull assumption.

Given these issues, a great deal of additional work would need to be done before the proposed

procedure could be accepted for implementation as part of an ASTM standard. Having said this,

we do support continued work by Lum, Taylor, and Zidek (and others) in this area. Ultimately,

if we want to take a reliability based approach to species grouping, the grouping cannot be based

on a single parameter of a strength distribution (the fifth percentile) given the simple fact that

strength distributions cannot generally be characterized by single parameters.

We note that our work also suggests that it would be worthwhile to investigate whether full

joint MOE/MOR populations truly have a bivariate Gaussian–Weibull distribution. If so, we could

exploit this fact to obtain more accurate and precise estimates of probabilities of failure. (However,

we fear that given the mixture nature of lumber populations, it is likely that no “simple” joint

distribution is appropriate for modeling a full population of MOE/MOR pairs.)

13
7 References

ASTM. 2010a. Standard practice for evaluating allowable properties for grades of structural

lumber. D 2915-10. West Conshohocken, PA: ASTM International.

ASTM. 2010b. Standard practice for establishing structural grades and related allowable properties for visually graded lumber. D 245-06. West Conshohocken, PA: ASTM International.

ASTM. 2010c. Standard practice for establishing allowable properties for visually-graded dimension lumber from in-grade tests of full-size specimens. D 1990-07. West Conshohocken, PA:

ASTM International.

ASTM. 2011. Standard Specification for Computing Reference Resistance of Wood-based Materials and Structural Connections for Load and Resistance Factor Design. D 5457-10. West

Conshohocken, PA: ASTM International.

Cohen, A.C. (1965), “Maximum Likelihood Estimation in the Weibull Distribution Based on

Complete and on Censored samples,” Technometrics, 7, 579–588.

D’Agostino, R.B. and Stephens, M.A. (1986), Goodness-Of-Fit Techniques, New York: Marcel

Dekker.

Evans, J.W. and Green, D.W. (1988), Mechanical Properties of Visually Graded Dimension Lumber: Vol 2. Douglas Fir–Larch., Pub. PB-88-159-397, Springfield, VA: National Technical

Information Service, 476 pages.

Gerhards, C.C. (2000), “Bending Creep and Load Duration of Douglas-Fir 2 by 4s under Constant

Load for up to 12-Plus Years,” Wood and Fiber Science, 32(4), 489–501.

Green, D.W. and Evans, J.W. (1988), Mechanical Properties of Visually Graded Dimension Lumber: Vol 4. Southern Pine., Pub. PB-88-159-413, Springfield, VA: National Technical Information Service, 412 pages.

Lum, Conroy (FPInnovations), Taylor, Carolyn (University of British Columbia), and Zidek,

James (University of British Columbia) (2010a), representing the Canadian Wood Council,

personal communications.

Lum, C., Taylor, C., and Zidek, J. (2010b), “Comments and Questions on Grouping Procedures

2.0,” unpublished working paper prepared for the Lumber Properties Task Group of the

American Lumber Standards Committee.

Lum, C., Taylor, C., and Zidek, J. (2010c), “Using Reliability Concepts to Revise the D1990

Species Grouping,” presentated to the American Lumber Standards Committee Lumber Properties Task Group in St. Louis, Missouri on April 20, 2010.

Verrill, S.P. and Kretschmann, D.E. (2009), “Material Variability and Repetitive Member Factors for the Allowable Properties of Engineered Wood Products,” Journal of Testing and

Evaluation, 37(6), 607–615.

Verrill, S.P., Evans, J.W., Kretschmann, D.E., and Hatfield, C.A. (2012), “Asymptotically Efficient Estimation of a Bivariate Gaussian–Weibull Distribution and an Introduction to the

Associated Pseudo-truncated Weibull,” USDA Forest Service Forest Products Laboratory

Research Paper RP-666 (http://www1.fpl.fs.fed.us/fpl rp666.pdf).

14
8 Appendix

In this Appendix, we describe the manner in which we created Table 1 and Figure 2. None of the

mathematics is novel. We describe it in some detail simply because this permits easy replication of

our work, and because this highlights the facts that we need two constraints to characterize a two

parameter strength or load distribution, that specifying a fifth percentile of a strength distribution

does not permit us to calculate a probability of failure, and thus that equality or similarity of fifth

percentiles alone should not be a basis for grouping.

8.1 Characterizing a lognormal distribution from its CV and its fifth percentile

By definition, a random variable X is distributed as a lognormal(μ, σ2) if ln(X) is distributed as a

normal(μ, σ2). Thus, characterizing the lognormal is equivalent to determining the two parameters

μ and σ. For a lognormal distribution, it can be shown that

CV = pexp(σ

2) − 1

or

ln 1 + CV2



= σ

2

(11)

Thus, for lognormals, the parameter σ can be calculated from a knowledge of the (alternative

parameter) CV.

Now let x.05 denote the fifth percentile of the lognormal. Then

Prob LN(μ, σ2) ≤ x.05= 0.05

or

Prob N(μ, σ2) ≤ ln(x.05)



= 0.05

or

Prob N(0, 1) ≤

ln(x.05) − μ

σ



= 0.05

or

ln(x.05) − μ

σ

= Φ−1(.05)

where Φ denotes the N(0,1) cumulative distribution function. Thus,

μ = ln(x.05) − σ × Φ

−1

(.05) (12)

From results (11) and (12), we see that we can calculate the two parameters μ and σ of a

lognormal distribution from a specification of its fifth percentile and its coefficient of variation.

Thus, in our calculations for Table 1, we were able to fully characterize the “narrow” and “wide”

lognormal strength distributions by specifying a shared fifth percentile and the CV’s of the narrow

and wide strength distributions. (For the shared fifth percentile, we arbitrarily choose x.05 = 50.

The specific shared value does not matter because different values simply amount to changes in

scale.)

15
8.2 Characterizing a lognormal distribution from its CV and the probability

that a value randomly drawn from the distribution exceeds a particular

value

Next, we claim that we can characterize a lognormal load distribution by specifying its CV and

the probability, q, that it lies above the allowable property (given that we have previously specified

x.05).

From (11) we know that we can obtain a value for its σ parameter from its CV. We calculate

its μ parameter as follows.

Prob LN(μ, σ2) ≤ x.05/2.1



= 1 − q

or

Prob N(μ, σ2) ≤ ln(x.05/2.1)= 1 − q

or

Prob N(0, 1) ≤

ln(x.05/2.1) − μ

σ



= 1 − q

or

ln(x.05/2.1) − μ

σ

= Φ−1(1 − q)

Thus,

μ = ln(x.05/2.1) − σ × Φ

−1

(1 − q) (13)

So, assuming lognormality of both the load and strength distributions, specifying the shared (in

our case) x.05, the CV’s of the narrow and wide strength distributions and the load distribution,

and the probability that the load distribution lies above the allowable property permits us to fully

characterize the load and strength distributions (that is, we can calculate their parameters μ and

σ).

Finally, we note that if X1 is distributed as a LN(μ1, σ2

1

) and X2 is distributed as an independent

LN(μ2, σ2

2

), then

Prob(X2 < X1) = Prob LN(μ2, σ2

2

) < LN(μ1, σ2

1

)



= Prob N(μ2, σ2

2

) < N(μ1, σ2

1

)



= Prob N(μ2, σ2

2

) − N(μ1, σ2

1

) < 0



= Prob N(μ2 − μ1, σ2

1 + σ

2

2

) < 0



= Φ 

p

μ1 − μ2

σ

2

1 + σ

2

2

!

(14)

Result (14) permits us to calculate, for example, the probability that a random value drawn from a

wide lognormal strength distribution falls below a random value drawn from the load distribution,

that is, the probability of failure for pieces of lumber drawn from the wide distribution when they

are subjected to the load distribution.

At http://www1.fpl.fs.fed.us/bivar3.html, we provide listings of FORTRAN and S versions of a program that we developed to utilize Equations (11) – (14) to produce Table 1 and Figure

2. See the narr.wide.f and narr.wide.s links.

16
CV’s of narrow and wide q pF narr pF wide pF ratio

.01 .21E-6 .89E-6 .23

.02 .82E-6 .25E-5 .33

.05, .25 .05 .57E-5 .11E-4 .54

.10 .28E-4 .35E-4 .79

.20 .16E-3 .14E-3 1.17

.01 .15E-6 .22E-5 .07

.02 .57E-6 .54E-5 .11

.10, .30 .05 .37E-5 .20E-4 .19

.10 .18E-4 .58E-4 .31

.20 .10E-3 .20E-3 .51

.01 .38E-6 .11E-4 .036

.02 .12E-5 .22E-4 .054

.20, .40 .05 .60E-5 .62E-4 .096

.10 .23E-4 .15E-3 .153

.20 .10E-3 .40E-3 .254

Table 1: q is the probability that the load lies above the allowable property. pF narr is the probability that a value drawn from the narrow strength distribution falls below a value drawn independently from the load distribution. pF wide is the probability that a value drawn from the

wide strength distribution falls below a value drawn independently from the load distribution.

pF ratio = (pF narr)/(pF wide).

17
Lumber Sample Censored Shape estimate Scale estimate

Species size Grade size size Censored Full Censored Full

DF 2 × 4 SS 414 60 5.664 4.618 10.474 11.074

DF 2 × 8 SS 493 60 4.066 3.747 8.051 8.446

DF 2 × 10 SS 414 60 4.980 3.985 7.083 7.866

DF 2 × 4 2 386 60 5.121 3.017 6.612 8.728

DF 2 × 8 2 1964 196 4.462 2.597 4.530 6.384

DF 2 × 10 2 388 60 5.188 2.436 3.712 5.682

HF 2 × 4 SS 428 60 5.902 4.571 8.908 9.420

HF 2 × 8 SS 375 60 5.495 4.242 6.715 7.508

HF 2 × 10 SS 368 60 4.858 4.251 6.295 6.425

HF 2 × 4 2 406 60 5.076 2.897 5.580 7.336

HF 2 × 8 2 372 60 4.229 2.717 4.630 5.859

HF 2 × 10 2 361 60 4.944 2.796 3.717 4.913

SP 2 × 4 SS 413 60 6.280 4.606 10.926 11.989

SP 2 × 8 SS 626 62 4.339 4.625 9.870 9.262

SP 2 × 10 SS 413 60 6.172 6.321 8.279 7.864

SP 2 × 4 2 413 60 4.722 2.783 7.135 8.888

SP 2 × 6 2 413 60 4.541 2.647 5.650 7.866

SP 2 × 8 2 1367 136 4.552 2.764 4.955 6.967

SP 2 × 10 2 412 60 4.256 3.469 5.630 6.441

Table 2: Maximum likelihood estimates of the shape and scale parameters for both censored and

full data sets.

18
Full Bottom 20% Bottom 10%

3.70 4.05 4.14

3.68 3.68 4.13

3.74 3.67 3.76

3.58 3.61 3.73

3.91 3.91 3.88

3.59 3.81 4.09

3.81 3.97 4.07

3.82 3.95 3.96

3.67 3.61 3.45

3.81 3.89 3.55

3.64 3.68 3.82

3.65 3.77 3.76

3.84 3.78 3.93

3.70 3.60 3.41

3.57 3.65 4.11

3.75 3.55 3.36

3.74 3.47 3.65

3.69 3.46 3.51

3.70 3.51 3.41

3.75 3.33 3.50

Table 3: Maximum likelihood estimates of the shape parameter from simulated two-parameter

Weibull data with shape parameter 3.701.

19
Lumber Sample Observed data Generated data

Species size Grade size CVM AD CVM AD

DF 2 × 4 SS 414 .10 .05 NS NS

DF 2 × 8 SS 493 NS NS NS NS

DF 2 × 10 SS 414 NS NS NS NS

DF 2 × 4 2 386 .25 .10 NS NS

DF 2 × 8 2 1964 .01 .01 NS NS

DF 2 × 10 2 388 .01 .01 NS NS

HF 2 × 4 SS 428 .05 .01 NS NS

HF 2 × 8 SS 375 .05 .05 NS NS

HF 2 × 10 SS 368 .10 .05 NS NS

HF 2 × 4 2 406 .01 .01 NS NS

HF 2 × 8 2 372 .01 .01 NS NS

HF 2 × 10 2 361 .01 .01 .25 .25

SP 2 × 4 SS 413 .05 .025 .25 .05

SP 2 × 8 SS 626 .05 .025 .25 NS

SP 2 × 10 SS 413 .01 .01 NS NS

SP 2 × 4 2 413 .01 .01 NS NS

SP 2 × 6 2 413 .01 .01 NS NS

SP 2 × 8 2 1367 .01 .01 .10 .25

SP 2 × 10 2 412 .10 .25 NS NS

Table 4: Cram ́er–Von Mises and Anderson–Darling goodness-of-fit tests of both observed and

generated data.

20
Shape estimate Scale estimate

ρ CV Data Reg MLE Reg MLE

all 7.040 6.861 108.5 108.6

20 20% 7.368 7.556 107.6 106.8

10% 7.414 7.691 108.2 106.5

all 4.500 4.382 113.6 113.7

.50 30 20% 4.756 4.826 112.3 111.1

10% 4.887 5.001 112.9 109.7

all 3.283 3.209 119.4 119.7

40 20% 3.505 3.504 116.4 115.9

10% 3.602 3.698 117.4 112.3

all 7.759 7.403 108.8 109.0

20 20% 8.601 8.635 106.4 106.2

10% 8.834 9.065 106.4 105.0

all 4.930 4.726 113.8 114.1

.60 30 20% 5.400 5.415 110.3 110.0

10% 5.552 5.662 110.4 107.9

all 3.610 3.448 119.7 120.2

40 20% 3.977 3.979 114.5 114.4

10% 4.099 4.164 113.7 111.6

all 8.700 8.170 108.7 109.0

20 20% 9.820 9.772 106.2 106.2

10% 10.137 10.309 106.1 105.0

all 5.597 5.255 114.1 114.5

.70 30 20% 6.452 6.341 109.3 109.7

10% 6.723 6.749 108.8 107.4

all 4.063 3.820 120.2 120.7

40 20% 4.651 4.565 113.3 114.2

10% 4.818 4.911 112.3 110.2

all 10.303 9.430 108.7 109.0

20 20% 12.392 12.222 105.2 105.5

10% 12.918 12.977 104.7 104.3

all 6.607 6.014 114.1 114.6

.80 30 20% 8.139 7.935 107.5 108.3

10% 8.490 8.533 106.3 106.0

all 4.761 4.359 119.8 120.5

40 20% 5.639 5.610 112.4 112.7

10% 5.877 5.897 110.9 110.5

Table 5. Maximum likelihood and regression Weibull parameter estimates for pseudo-truncated

Weibull data as a function of correlation, coefficient of variation, and censoring.

21
Median pBr ratio frequencies

COV Fit pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

W/Reg/All .62 .0000 .0000 .0002*† .0028*† .2983*† .5661 .1298 .0028 .0000 .0000 .0000 .0000W/Reg/20 .84 .0000 .0000 .0041*† .0444*† .2409*† .2880 .2344 .1418 † .0348*† .0111*† .0004*† .0001*†W/Reg/10 .86 .0000 .0001*† .0123*† .0679*† .2374*† .2344 .1952 .1495 † .0578*† .0378*† .0047*† .0029*†W/MLE/All .51 .0000 .0000 .0002*† .0021*† .4662*† .5028 .0287 .0000 .0000 .0000 .0000 .0000

0.20 W/MLE/20 .93 .0000 .0000 .0006*† .0150*† .1840*† .3378 .2903 .1453 † .0236 † .0034 † .0000 .0000

W/MLE/10 1.10 .0000 .0000 .0018*† .0232*† .1770*† .2615 .2440 .1943*† .0615*† .0338*† .0022*† .0007*†PTW trunc 1.06 .0000 .0000 .0000 .0000 .0647 † .3941 .3529 .1510 † .0285 † .0088 † .0000 .0000PTW 400 1.04 .0000 .0000 .0000 .0000 .0366 .4307 .4694 .0630 .0003 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0025 .4773 .5154 .0048 .0000 .0000 .0000 .0000W/Reg/All .74 .0000 .0001*† .0002*† .0002*† .1213*† .7013 .1758 .0011 .0000 .0000 .0000 .0000W/Reg/20 .90 .0000 .0000 .0001*† .0053*† .1716*† .3861 .3091 .1178*† .0092*† .0008*† .0000 .0000W/Reg/10 .91 .0000 .0000 .0000 .0097*† .1989*† .3351 .2663 .1483*† .0323*† .0090*† .0003*† .0001*†W/MLE/All .64 .0000 .0000 .0000 .0000 .1888*† .7678 .0432 .0002 .0000 .0000 .0000 .0000

0.30 W/MLE/20 .99 .0001*† .0000 .0001*† .0004*† .1025*† .4075 .3795 .1066*† .0030 † .0003 † .0000 .0000

W/MLE/10 1.09 .0001*† .0000 .0002*† .0019*† .1137*† .3304 .3474 .1769*† .0248*† .0045*† .0001*† .0000PTW trunc 1.05 .0000 .0000 .0000 .0000 .0199 † .4341 .4369 .1029 † .0058 † .0004 † .0000 .0000PTW 400 1.03 .0000 .0000 .0000 .0000 .0092 .4558 .5143 .0207 .0000 .0000 .0000 .0000PTW 1000 1.02 .0000 .0000 .0000 .0000 .0001 .4692 .5305 .0002 .0000 .0000 .0000 .0000W/Reg/All .80 .0000 .0000 .0001*† .0000 .0548*† .7435 .2005 .0011 .0000 .0000 .0000 .0000W/Reg/20 .92 .0000 .0000 .0000 .0007*† .1085*† .4504 .3571 .0806*† .0027*† .0000 .0000 .0000W/Reg/10 .94 .0000 .0000 .0000 .0006*† .1243*† .4121 .3309 .1183*† .0115*† .0023*† .0000 .0000W/MLE/All .70 .0000 .0000 .0000 .0000 .0827*† .8478 .0694 .0001 .0000 .0000 .0000 .0000

0.40 W/MLE/20 .99 .0000 .0000 .0001*† .0001*† .0547*† .4513 .4293 .0636*† .0009*† .0000 .0000 .0000

W/MLE/10 1.07 .0000 .0001*† .0000 .0002*† .0627*† .3892 .4032 .1357*† .0078*† .0011*† .0000 .0000PTW trunc 1.04 .0000 .0000 .0000 .0000 .0059 † .4473 .4919 .0546 † .0003 † .0000 .0000 .0000PTW 400 1.02 .0000 .0000 .0000 .0000 .0037 .4646 .5246 .0071 .0000 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0001 .4818 .5180 .0001 .0000 .0000 .0000 .0000

Table 6: Frequencies for pBr,true/pBr,est for a bivariate Gaussian–Weibull with generating correlation .50. *’s denote “worse behavior”

than PTW trunc behavior. †’s denote “worse behavior” than PTW 400 behavior.
Median pBr ratio frequencies

COV Fit pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

W/Reg/All .44 .0000 .0000 .0002*† .0348*† .5761*† .3561 .0326 .0002 .0000 .0000 .0000 .0000W/Reg/20 .73 .0000 .0000 .0152*† .0688*† .2745*† .2605 .2071 .1308 † .0297 † .0131 † .0002 † .0001 †W/Reg/10 .78 .0000 .0000 .0327*† .0900*† .2431*† .2084 .1720 .1473 † .0548*† .0447*† .0032*† .0038*†W/MLE/All .31 .0000 .0000 .0012*† .1025*† .7927*† .1027 .0009 .0000 .0000 .0000 .0000 .0000

0.20 W/MLE/20 .79 .0000 .0000 .0024*† .0336*† .2442*† .3272 .2512 .1202 † .0178 † .0032 † .0002 † .0000

W/MLE/10 .99 .0000 .0000 .0071*† .0438*† .2094*† .2427 .2194 .1785*† .0602*† .0359*† .0019*† .0011*†PTW trunc 1.05 .0000 .0000 .0000 .0011 † .1271 † .3445 .2975 .1679 † .0396 † .0214 † .0007 † .0002 †PTW 400 1.03 .0000 .0000 .0000 .0002 .0534 .4197 .4416 .0846 .0005 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0067 .4819 .4988 .0126 .0000 .0000 .0000 .0000W/Reg/All .60 .0000 .0000 .0002*† .0006*† .2987*† .6315 .0690 .0000 .0000 .0000 .0000 .0000W/Reg/20 .85 .0000 .0000 .0001*† .0125*† .2095*† .3670 .2847 .1129 † .0113 † .0020*† .0000 .0000W/Reg/10 .88 .0000 .0000 .0006*† .0200*† .2292*† .3073 .2448 .1485*† .0352*† .0137*† .0004*† .0003*†W/MLE/All .46 .0000 .0000 .0001*† .0016*† .6046*† .3895 .0042 .0000 .0000 .0000 .0000 .0000

0.30 W/MLE/20 .91 .0000 .0000 .0001*† .0031*† .1478*† .4117 .3411 .0915 † .0043 † .0004 † .0000 .0000

W/MLE/10 1.05 .0000 .0000 .0002*† .0055*† .1388*† .3329 .3103 .1737*† .0309*† .0076*† .0001*† .0000PTW trunc 1.05 .0000 .0000 .0000 .0000 .0496 † .4146 .3919 .1304 † .0124 † .0011 † .0000 .0000PTW 400 1.03 .0000 .0000 .0000 .0000 .0164 .4532 .4993 .0311 .0000 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0009 .4814 .5169 .0008 .0000 .0000 .0000 .0000W/Reg/All .69 .0000 .0000 .0000 .0005*† .1442*† .7471 .1080 .0002 .0000 .0000 .0000 .0000W/Reg/20 .90 .0000 .0000 .0001*† .0020*† .1398*† .4320 .3336 .0882*† .0040*† .0003*† .0000 .0000W/Reg/10 .92 .0000 .0000 .0001*† .0025*† .1571*† .3848 .3043 .1298*† .0168*† .0046*† .0000 .0000W/MLE/All .55 .0000 .0000 .0001*† .0001*† .3519*† .6396 .0083 .0000 .0000 .0000 .0000 .0000

0.40 W/MLE/20 .94 .0000 .0000 .0002*† .0006*† .0854*† .4646 .3810 .0672 † .0010 † .0000 .0000 .0000

W/MLE/10 1.05 .0000 .0000 .0001*† .0003*† .0883*† .3797 .3731 .1427*† .0134*† .0024*† .0000 .0000PTW trunc 1.04 .0000 .0000 .0000 .0000 .0188 † .4482 .4433 .0875 † .0022 † .0000 .0000 .0000PTW 400 1.02 .0000 .0000 .0000 .0000 .0081 .4682 .5122 .0115 .0000 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0001 .4812 .5186 .0001 .0000 .0000 .0000 .0000

Table 7: Frequencies for pBr,true/pBr,est for a bivariate Gaussian–Weibull with generating correlation .60. *’s denote “worse behavior”

than PTW trunc behavior. †’s denote “worse behavior” than PTW 400 behavior.
Median pBr ratio frequencies

COV Fit pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

W/Reg/All .24 .0000 .0000 .0355*† .3298*† .5773*† .0565 .0009 .0000 .0000 .0000 .0000 .0000W/Reg/20 .56 .0000 .0002*† .0503*† .1184*† .2899*† .2254 .1615 .1067 .0326 † .0140 † .0009 † .0001 †W/Reg/10 .64 .0000 .0013*† .0757*† .1251*† .2288*† .1791 .1420 .1265 † .0535 † .0556*† .0063*† .0061*†W/MLE/All .13 .0000 .0001*† .2359*† .6097*† .1528 † .0015 .0000 .0000 .0000 .0000 .0000 .0000

0.20 W/MLE/20 .55 .0000 .0000 .0204*† .0989*† .3320*† .2786 .1688 .0818 .0144 † .0051 † .0000 .0000

W/MLE/10 .81 .0000 .0003*† .0284*† .0878*† .2333*† .2156 .1810 .1490 † .0578*† .0403 † .0047*† .0018 †PTW trunc 1.07 .0000 .0000 .0004 † .0158 † .1889 † .2694 .2353 .1780 † .0566 † .0481 † .0040 † .0035 †PTW 400 1.05 .0000 .0000 .0000 .0008 .0868 .3791 .4106 .1174 .0048 .0005 .0000 .0000PTW 1000 1.02 .0000 .0000 .0000 .0000 .0183 .4578 .4950 .0289 .0000 .0000 .0000 .0000W/Reg/All .41 .0000 .0000 .0002*† .0259*† .6829*† .2818 .0092 .0000 .0000 .0000 .0000 .0000W/Reg/20 .73 .0000 .0000 .0019*† .0345*† .2766*† .3376 .2294 .1018 † .0149 † .0033 † .0000 .0000W/Reg/10 .79 .0000 .0000 .0041*† .0507*† .2664*† .2691 .2080 .1383 † .0404*† .0213*† .0013*† .0004*†W/MLE/All .27 .0000 .0000 .0008*† .1846*† .7907*† .0239 .0000 .0000 .0000 .0000 .0000 .0000

0.30 W/MLE/20 .74 .0000 .0000 .0002*† .0142*† .2540*† .4076 .2484 .0697 † .0055 † .0004 † .0000 .0000

W/MLE/10 .92 .0001*† .0000 .0005*† .0182*† .2069*† .3132 .2577 .1549*† .0340*† .0139*† .0006*† .0000PTW trunc 1.04 .0000 .0000 .0000 .0005 † .1047 † .3710 .3341 .1536 † .0283 † .0077 † .0000 .0001 †PTW 400 1.03 .0000 .0000 .0000 .0000 .0343 .4380 .4710 .0567 .0000 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0023 .4807 .5133 .0037 .0000 .0000 .0000 .0000W/Reg/All .53 .0000 .0000 .0000 .0008*† .4249*† .5530 .0213 .0000 .0000 .0000 .0000 .0000W/Reg/20 .83 .0000 .0001*† .0000 .0065*† .1969*† .4150 .2852 .0896 † .0064 † .0003 † .0000 .0000W/Reg/10 .87 .0000 .0000 .0000 .0090*† .2053*† .3580 .2648 .1311*† .0250*† .0067*† .0001*† .0000W/MLE/All .37 .0000 .0000 .0001*† .0115*† .8589*† .1295 .0000 .0000 .0000 .0000 .0000 .0000

0.40 W/MLE/20 .83 .0000 .0000 .0001*† .0016*† .1582*† .4729 .3055 .0601 † .0015 † .0001 † .0000 .0000

W/MLE/10 1.00 .0000 .0000 .0001*† .0027*† .1386*† .3576 .3421 .1398*† .0160*† .0031*† .0000 .0000PTW trunc 1.03 .0000 .0000 .0000 .0000 .0480 † .4284 .4021 .1133 † .0077 † .0005 † .0000 .0000PTW 400 1.02 .0000 .0000 .0000 .0000 .0102 .4635 .5032 .0231 .0000 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0004 .4821 .5164 .0011 .0000 .0000 .0000 .0000

Table 8: Frequencies for pBr,true/pBr,est for a bivariate Gaussian–Weibull with generating correlation .70. *’s denote “worse behavior”

than PTW trunc behavior. †’s denote “worse behavior” than PTW 400 behavior.
Median pBr ratio frequencies

COV Fit pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

W/Reg/All .06 .0006*† .0223*† .8303*† .1364*† .0097 .0007 .0000 .0000 .0000 .0000 .0000 .0000W/Reg/20 .27 .0026*† .0126*† .2162*† .1836*† .2536*† .1482 .0951 .0577 .0183 .0110 † .0009 † .0002 †W/Reg/10 .35 .0081*† .0235*† .2039*† .1426*† .1993 † .1256 .0995 .0934 .0387 † .0461 † .0102 † .0091 †W/MLE/All .02 .0444*† .3862*† .5678*† .0006 .0006 .0004 .0000 .0000 .0000 .0000 .0000 .0000

0.20 W/MLE/20 .23 .0004*† .0070*† .2127*† .2368*† .2986*† .1381 .0701 .0297 .0047 .0019 .0000 .0000

W/MLE/10 .41 .0015*† .0081*† .1586*† .1527*† .2263*† .1622 .1161 .0942 .0392 † .0340 † .0037 † .0034 †PTW trunc 1.09 .0000 .0000 .0212 † .0720 † .1996 † .1868 .1691 .1555 .0780 † .0849 † .0145 † .0184 †PTW 400 1.04 .0000 .0000 .0002 .0066 .1436 .3295 .3300 .1678 .0199 .0024 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0001 .0574 .4323 .4337 .0753 .0012 .0000 .0000 .0000W/Reg/All .18 .0000 .0000 .0695*† .5183*† .4053*† .0069 .0000 .0000 .0000 .0000 .0000 .0000W/Reg/20 .51 .0000 .0000 .0308*† .1140*† .3424*† .2677 .1529 .0748 .0140 † .0032 † .0002 † .0000W/Reg/10 .60 .0000 .0001*† .0438*† .1172*† .2794*† .2157 .1566 .1146 † .0377 † .0306*† .0033*† .0010*†W/MLE/All .09 .0001*† .0000 .6420*† .3438*† .0137 .0004 .0000 .0000 .0000 .0000 .0000 .0000

0.30 W/MLE/20 .46 .0001*† .0000 .0153*† .1083*† .4175*† .2868 .1324 .0368 .0026 † .0002 † .0000 .0000

W/MLE/10 .68 .0000 .0001*† .0185*† .0777*† .2861*† .2663 .1849 .1151 † .0332 † .0168 † .0011*† .0002 †PTW trunc 1.04 .0000 .0000 .0004 † .0111 † .1754 † .2972 .2547 .1791 † .0520 † .0290 † .0009 † .0002 †PTW 400 1.03 .0000 .0000 .0000 .0003 .0652 .4134 .4284 .0911 .0016 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0086 .4763 .4985 .0166 .0000 .0000 .0000 .0000W/Reg/All .31 .0000 .0000 .0003*† .0997*† .8158*† .0836 .0006 .0000 .0000 .0000 .0000 .0000W/Reg/20 .69 .0000 .0000 .0015*† .0317*† .3013*† .3616 .2185 .0747 † .0092 † .0014 † .0001*† .0000W/Reg/10 .76 .0000 .0000 .0022*† .0420*† .2757*† .2921 .2170 .1274 † .0290*† .0134*† .0008*† .0004*†W/MLE/All .17 .0001*† .0000 .0494*† .6682*† .2817*† .0006 .0000 .0000 .0000 .0000 .0000 .0000

0.40 W/MLE/20 .64 .0000 .0000 .0011*† .0199*† .3230*† .4219 .1902 .0422 .0015 † .0002 † .0000 .0000

W/MLE/10 .84 .0000 .0000 .0006*† .0176*† .2280*† .3391 .2583 .1265 † .0229 † .0067*† .0003*† .0000PTW trunc 1.03 .0000 .0000 .0000 .0010 † .1116 † .3733 .3258 .1599 † .0229 † .0055 † .0000 .0000PTW 400 1.04 .0000 .0000 .0000 .0000 .0244 .4413 .4863 .0477 .0003 .0000 .0000 .0000PTW 1000 1.01 .0000 .0000 .0000 .0000 .0007 .4837 .5122 .0034 .0000 .0000 .0000 .0000

Table 9: Frequencies for pBr,true/pBr,est for a bivariate Gaussian–Weibull with generating correlation .80. *’s denote “worse behavior”

than PTW trunc behavior. †’s denote “worse behavior” than PTW 400 behavior.
0.0 0.5 1.0 1.5 2.0

02468

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Shape parameter

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Shape parameter

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0 50 100 150 200

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Weibull densities

0 20 40 60 80

0.0 0.01 0.02 0.03 0.04 0.05

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Weibull scores (cdf^(−1)((i − .3)/(n + .4)))

Ordered data

0 2 4 6 810 12

0 2 4 6 8 10 12

Weibull scores (cdf^(−1)((i − .3)/(n + .4)))

Ordered data

2 4 6 8 10 12

2 4 6 8 10 12

  	

 

    
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Weibull scores (cdf^(−1)((i − .3)/(n + .4)))

Ordered data

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Weibull scores (cdf^(−1)((i − .3)/(n + .4)))

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2 4 6 810

2 4 6 8 10

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