UDC 661.677.36:661.679.6:626.72:626.042:66.016.2
Forecasting Work Conditions for Road
Construction Activities: An Application
of Alternative Probability Models
EMlL D. ATTANASI and S. R. JOHNSON-Department of Economics, University of
Missouri, Columbia, Mo.
SHARON LeDUC-Department of Atmospheric Science, University of MisKouri,
Columbia, Mo.
JAMES D. McQUIGG-Environmental Data Service, NOAA, and Department of
Atmospheric Science, University of Missouri, Columbia, Mo.
ABSTRACT-certain activities of highway construction
are particularly sensitive to such weather conditions as
soil moisture, precipitation, and daily temperature. Re-
gression analysis is used to obtain three alternative
probability models designed to translate observed weather
conditions into probabilities for carrying out construction
activities. The models were developed using generalized
least squares, normit analysis, and logit analysis. The
generalized least squares method was the most convenient
computationally, but it had severe interpretative dis-
advantages. The results obtained by logit analysis gave
the desired probabilistic interpretation most readily and
had the best predictive ability. Comparison of sample
observation and predicted work probabilities for common
excavation during wet and dry months indicated that the
logit analysis model could accurately translate weather
conditions into probabilities that work would take place.
Models for paving and asphalt work and for bridge and
drainage structure are also estimated using logit analysis.
These estimates indicate a strong sensitivity of the latter
category of work to precipitation conditions. Such models
may aid contract letting agencies in planning payment
schedules, penalty clauses, and completion dates for new
roads; construction firms may find such models valuable
in planning effective use of men and equipment. 0
1. INTRODUCTION
The road construction industry is highly sensitive to
weather conditions. This sensitivity arises from the fact
that certain types of weather conditions must exist for
the various construction activities to operate eficiently.
For example, common excavation cannot be efficiently
carried out during periods of excessive rainfall, concrete
pouring is hazardous during periods in which the tempera-
tures are below freezing, and the like. Because of this
sensitivity to ambient weather conditions, it is important
for contractors and contracting firms to have a convenient
means by which to translate weather forwapts and obser-
vations into probabilistic statements regarding the feasi-
bility of conducting the various road building activities.l
In regard to forecasts, these probabilistic statements
can aid contracting firms in deciding whether or not to
ask laborers to report to work. Such forecasts are of
increased importance if union arrangements attach a high
penalty to errors in such decisions as they currently do;
that is, if union contracts require partial or full wages
for days in which laborers are called out but work is not
feasible. Translation of observed weather conditions into
1 State Highway Departments typically use district engineers to make onsite decisions
as to whether or not conditions are suitable for work. Since penalty clauses are basedboth
on numbers of working days and numbers of calendar days, variability from one
district cngineer to another frequently causes difficulties in decisions concerning work
conditions.
probabilities for conducting construction activities is also
important in long-range planning. With such information,
one can use historical weather records to calculate the
number of days in which various construction activities
would have been possible in previous years. The historical
perspective gained from these result? could help contract-
letting agencies plan payment schedules, penalty clauses,
and opening dates for new highways; at the same time,
construction firms could more effectively plan the use of
men and equipment and, thereby more accurately de-
termine their capacity for taking on new contracts.'
One method for analytically determining work condi-
tions from sample data is based on the idea of a linear
probability model. Regression techniques are used to
relate variables reflecting actual working conditions to
available information on climatic variables a t construction
sites. In this paper, we apply the probability models to
data made available by the Missouri State Highway Com-
mission. The objective is to show how these probability
models can translate weather data into useful information
for determining the daily feasibility of alternative con-
struction activities and to indicate how such information
might be used.
2 Contractors and contracting agencies currently have methods of translating their
experience into operational and planning decisions. The proposed methods and results
are designed to complement existing procedures by facilitating the use of existing weather
data in obtaining additional information to be used in such decision-making processes.
~
March1973 / 2 2 3
2. SPECIFICATION AND ESTIMATION
OF PROBABILITY MODELS
First, it is necessary to introduce a mechanism for
quantifying working conditions. Initially, two classifica-
tions of work conditions will be used: (1) work day, where
all conditions are suitable for a particular construction
activity, and (2) no-work day, where some or all condi-
tions are unsuitable for a particular construction activity.
Given these two classifications, we can introduce an
artificial variable for quantification. The artificial variable
to be employed has the value 1 for work days and 0 for
no-work days.
As previously indicated, the probability model is
designed to estimate probabilities of work days on the
basis of observable climatic variables. These probabilities
are estimated from a random sample of size N of con-
comitant observations of the artificial variable and the
climatic variables. More formally, let y,-be the nth value
of the observed artificial variable, Xh be the nth observa-
tion on the set (e.g., k) of climatic variables, and tn be
the nth disturbance term. With this specification of the
variables, the objective of the probability models is to
determine a parameter vector p, conformable with Xn,
which will give estimated values of yn with some statisti-
cally desirable properties. In other words, we wish to
estimate p from the model,
y=XP+E. (1)
a In specifying eq (l ), we have used y to denote the vector
of N sample values for yn, X to denote the N sample values
with X i as rows, E as the associated vector of unobservable
disturbance terms, and the prime mark to denote the
transposition of a matrix.
Problems encountered in obtaining estimates of y
from eq (1) are mainly associated with peculiarities of
the disturbance term, E. These peculiarities, as we shall
subsequently demonstrate, come about as a result of J
being a qualitatively specified variable.
In the discussion t o follow, we develop three procedures
for estimating p from eq (1). These estimation procedures
are heuristically developed in connection with ordinary
least squares. The approach serves to tie the procedures
to the more customary methods of estimation; this is ac-
complished a t little expense in terms of space. It also points
up the fact that the proposed estimation procedures are
derived on the basis of the treatment of the disturbance
term, E.
Given a model of the form of eq (I), one naturally
contemplates the estimation of p by ordinary least squares.
This approach has some decided limitations, however.
Application of ordinary least squares requires E(X’6) =
0, E(E)=O, and E(EE’)=Q=a21, where I is the identity
matrix and Q is a scalar constant, if estimated parameter
values are to be unbiased and effi~ient.~ The inapplicabil-
ity of these conditions is easily shown. If we again denote
I
8 E is the expectation operation. The unbaised and efficiency conditions follow from
the Gauss-Markov theorem on least squares.
the nth samplevalue of X’ as XA, it follows that tn=yn-
X$?. Since yn is either 0 or 1, tn must be either -Xlp
or 1-X,)3. Hence, for E(E)=O, E(tn) must be l-X,!,p
when tn=-X$? and XLP when t,=l-X;p. On these
conditions, the variance of t n , E((:), is equal to E(y,
[l-E(yn)]). Thus, the disturbance varies with E(yn)
or equivalently with the observed climatic variables,
XL, and E(EE’) #Q ~I .~
The appropriate estimation procedure for situations
in which Q#uZI is the Aitken’s generalized least squares
(Goldberger 1964, p. 233), rather than the ordinary least-
squares procedure. Although the generalized least-squares
estimators have some undesirable properties in terms of
estimated values of yn, they are more efficient than the
ordinary least-squares estimators and are easy to compute.
On the basis of these latter two properties, the generalized
least-squares estimators are suggested as one of the meth-
ods appropriate for estimating work day probabilities.
Using the generalized least-squares method, we can esti-
mate p from eq (1) by (1) estimating Q from the ordinary
least-squares residuals and (2) using the estimated value,
h, in obtaining an estimation of p, as
The estimated variance-covaria\ce matrix the general-
ized least-squares estimator, b, is (X’Q-lX)-l. Then
y,=Xnb is naturally interpreted as the conditional
probability of a work day, given that the climatic con-
ditions described by X , occur.
It is in regard to the. estimated values for yn that the
limitations of the Aitken’s estimators are found. These
limitations exist because the procedure incorporates no
restriction that the in fall within the unit interval.
Therefore, although the generalized least-squares estimators
may yield good first approximations of the conditional
probabilities, one may still obtain values for f n that are
outside the 0, 1 interval.
Two similar estimation methods are designed to handle
this difficulty. As might be expected, these methods
involve the imposition of further restrictions or conditions
on t,. The proposed methods are called normit and probit
analyses (Berkson 1955, Goldberger 1964, and Tobin
1955). Since the two procedures involve similar methods
for handling the problem of J outside the 0, 1 interval,
their derivations are highly related. We will focus upon
the normit model, keeping in mind that it can be slightly
altered to obtain the probit model.
The normit estimation procedure is based on the follow-
ing considerations. Let I , be defined as a linear function
of the regressors; that is, In=Xnp. Also, let I ; be distrib-
uted N(0, 1) and assume that the value of yn is given by
A A
(3)
f.
Notice that this assumption simply involves a set of con-
ditions on the tn in eq (1). It follows from eq (3) that the
yn are now functions of the Xh and I;. The I;s, which am
4 Goldberger (1964, pp. 248-262) has a more elaborate discussion of these details.
224 / Vol. 101, No. 3 / Monthly Weather Review
in essence disturbances, are interpreted as critical values of
the index I,. It follows from eq (3) that, if we let F(z)
distribution a t z, then
weights. Specifically, the term Z(Xbb)/Xhb( 1 -Xhb) is
employed in estimating the weights, with b=(X’X)-lX’y
weights are set to 0.99 for n when X,b 2.1 and 0.01 for n
when XnblO. The estimated probability of a working
day, given that the climatic conditions Xh prevail, is then
t denote the value of the cumulative of the standard normal and the additional conditions that the arguments in the
P=Prob. (y,=lII,)=Prob. (I:lInlIn)=F(In) (4)
Q=Prob. (y,=O 11,) =Prob. (IE>In 11,) = 1 -F(Id (5 )
and
P=F(7,,)=F(X:i;)=F[X~(X’Q,’x)(X’Q,’y)] (10)
where the equivalence of these conditional statements
follows from the assumed N(0, I), distribution of I ,
(Zellner and Lee 1965, p. 384).5 Since I , (and thus the
probabilities P and &) are functions of P (through the
definition of I:), a maximum likelihood or minimum chi-
square procedure can be employed in estimating the
parameters of the linear functions defining I:.
The asymptotic properties of the maximum likelihood
and minimum chi-square estimations are the same. The
minimum chi-square procedure is developed here because
it is computationally more convenient. Following the
development of Zellner and Lee (1965, pp. 383-385), we
can combine eq (3) and (1) (recalling, of course, that we
are simply redefining 6) to get
where F-’ is the inverse of the cumulative normal distribu-
tion; the right side of eq (6) is an expansion about X n P
with R, representing higher order terms. Differentiating
eq (6) with respect to the distribution function and using
the definition of I,, we have
(7)
I: is the observed “normit” and XLP is the “true” normit
while Z(P,) is the value of the unit normal.
The variance of the newly defined error term t,/Z(P,)
is given by
Hence, P can’ be estimated by minimizing the normit x2,
where
where 7, is the sample estimate of the true normit and i;
is the sample estimate of P.
A second approach to the problem of obtaining esti-
mated values for probabilities that lie outside the unit
interval is called the logit model. The estimation proce-
dure associated with the logit model is, in principle, simi-
lar to that of the normit model. To demonstrate the
derivation of the procedure, we again write an n index
based upon fixed variables Xh; for example, J=X$,
where n=l, 2, . . ., N. We suppose that the probability
that y,=l can be written
Prob (y,= 1 IX,) = F(XbP) =V 1 +eX&
This expression can be derived on the basis of the usual
discriminant analysis formulation of the problem. I n fact,
the expression is the ratio of probability distributions
used in the initial phase of discriminant analysis. How-
ever, since a model based on the set of fixed explanatory
variables.XL, where n =l , . . ., N , is now being postulated,
the function will not be simply fitted to the discriminant
function (as in standard discriminant analysis). Instead,
we estimate the parameters or weights using a different
approach and, thus, obtain estimators with different
properties. As with the normit analysis, we can form a
likelihood function for the parameters to be estimated,
given eq (11) and a similar expression for the probability
that yn=O.
In deriving the likelihood function, we assumed that
the y, are ordered so that the first s are ones and the sec-
ond N-s are zeros. The likelihood function for the
sample can then be written
N
Normi t x2= [z(Pn)12 (I;-X$)’, (9) Taking logarithms and differentiating with respect to P ,
we obtain
n=l (P n +t n )(l -P n +t n )
(13)
with respect to the vector P [eq (l)]. The procedure for
obtaining /3 reduces to an estimator similar to the Aitken’s apt n=l n=l l+exAfl-Oj
’ estimator but with the “weights”ogiven by Qe, a diagonal
matrix composed of the terms post-multiplied by (yn-
X,p) in eq (9). For purposes of the subsequent empirical
work, the elements of Qe are approximated by using the
N X,,eXS a -- In =*-f= X,,-C --
or, utilizing eq (1 I), we get
8 N
X,,-C X,t Prob (yn=lIX;)=O. (14) z least-squares estimators of the vector @ in computing the n=l n=l
8 Specifically, it follows from our assumptions and the definitions implicit in eq (4146)
X ‘B P (I ) =(zd-VaS -a ” exp ( -:) ds.
The logit procedure produces a set of normal equations
that must be solved for the parameters by iterative
processes.
that
0
March 1973 Attanasi, Johnson, LeDuc, and McQuigg 1 225
The computational procedure may be greatly simplified
by using the approximation
I1 i>3
j <-3
The procedure used in this paper and developed is due to
Cox (1966, pp. 60-61). To obtain initial estimates of the P s ,
we used eq (15) and applied unweighted least squares to
the model,
A
(16) 1 1 " E(yf)=z+G (flOXO+"*fflkXk).
Recalling that we have k climatic variables, we define 6
to be a (k+l) X 1 vector of estimates, So to be a (k f l ) X
(k+ 1) matrix of cross-products for which the (r,s) element
is C X I ,X t ,, and To to be a (k+1) X 1 column vector for
which the rth element is C (y f -%)X i ,. This first approx;
mation 5 satisfies
f
i
(17)
We adjust for the discontinuity in the approximation
obtained by eq (17) by the following procedure. For
observations such that
we delete their contribution to So giving a new matrix S,.
Also, To is redefined to be T, for which the rth element is
where 2- and Z+ denote summations over these observa-
tions such that,
and (20)
The newly constructed S1 and T, matrices can be used to
obtain maximum likelihood estimates of p for the second
iteration (=) from
Y
1 S1i=T,. 6.
This procedure may be applied iteratively to refine
estimates of the ps to satisfy eq (14).
Note that the logit and normit models are similar. In
both, a "symmetrical sigmoid curve" is fitted to a linear
function of the observed data sample. The computational
burden of the'logit model is not as severe as that for the
normit analysis because of the degree of nonlinearity in
the first-order conditions. In particular, the normit
analysis requires the evalption of a number of normal
integrals. Since both methods require an assumption
226 / Vol. 101, No. 3 / Monthly Weather Review
about the distribution form for the two populations,
neither has much advantage in terms of statistical
generality. Although both types of estimations for the
weights on the weather variables will be subsequently
presented, applications of these approaches for operational
purposes should probably be guided by computational
feasibility. On the basis of this computational considera-
tion, approximations based on the generalized least squares
may be the most feasible. The subsequent comparisons
of results for the Missouri Highway Department data
may provide some insights into the legitimacy of this
last observation.
3. DATA FOR CONSTRUCTION
AND CLIMATIC VARIABLES
Records giving the sample values for the qualitative
variables, y,, were obtained from the Missouri State
Highway Commission. These records were taken from
two highway construction projects completed in the
vicinity of Jefferson City, Mo., during 1966-67.6 Speci-
fically, the records included information on common
excavation, finishing and grading, and paving activities.
Although more detailed information on each of these ac-
tivities was available, our data were supplied by the
engineers' reports indicating when the activities were
active and when they were inactive. For purposes of
maintaining some homogeneity in the data, the sample
days were restricted to those occurring between April 1
and October 31. The records from the two projects gave
218 observations for common excavation, 122 observations
for bridge, culvert, and drainage structures, and 80
observations for paving and asphalt. Because of the
relative number of observations on common excavation,
the importance of common excavation in road building,
and the sensitivity of common excavation to climatic
conditions, the major portion of the numerical results will
be related to this particular construction activity.
As was implied earlier, the independent variables are
related to climatic conditions. They are X,,, a soil mositure
index; Xzn, a 7-day average precipitation; X3,, 4-day
average precipitation; X,,, precipitation for the current
day plus X7,; X5,, a 0, 1 variable indicating whether or
not precipitation occurred on the day on which working
conditions were recorded; x,,, the average temperature
on the day the work conditions were recorded; and X7,,
the 3-day average precipitation. In the variables defined
as averages, the period includes the calendar days im-
mediately prior to the working day in question.
With the exception of X,,, the climatic variables are all
available in the records for cooperative weather stations.
In the case of this study, they were taken from the Na-
tional Weather Service Climatological Station located a t
Lincoln University, Jefferson City, Mo., approximately
2 mi from the construction sites. The soil moisture vari-
able is derived from precipitation and temperature data.
It is designed to reflect the moisture content in the top
e The projects are coded as C026-54(6) and F-54-3(14) by the Missouri Stato HighWaY
Commission. Each set of records encompassed the full duration of tho constructioll Of a
section of two-lane, hard surface highway.
f
12 in. of soil and is based upon technical data supplied
by the U.S. Forest Service and the US. Army Corps of
Engineers. For purposes of a cursory investigation of the
empirical results, it is sufficient to indicate that the index
is always positive and takes high values when the soil is
wet and low values when the soil is dry. Maunder et al.
(1971) describe the derivation of the soil moisture index.
As for the other variables, temperatures are measured in
degrees Fahrenheit and precipitation is measured in
hundredths of an inch.
4. ESTIMATED RELATIONS OF CLIMATIC
VARIABLES TO WORKING CONDITIONS
FOR COMMON EXCAVATION
As was indicated in the preceding section, the largest
number of sample observations existed for common
excavation. In this section, the values of the qualitative
variables reflecting work conditions for common excava-
tion are related to the climatic variables using a number of
functional forms. Since this estimation is of an exploratory
nature, coefficients for each of the functional forms are
reported. In addition to providing alternatives €or fore-
casting working days, it is hoped that the forms may
give some guidelines for further applications of such
probability models; that is, the alternative functional
forms together may give some indication as to robustness
of the underlying technical relationship.
The equations were estimated using the generalized
least squares, normit, and logit procedures. The results
are included in tables 1, 2, and 3, respectively. In each
case, the information presented consists of estimates of
the parameter values on the included variables and a
measure of the error.’ The measure of the error can be
used to evaluate the comparative forecasting accuracy
of the equations listed. As the discussion in section 2
would imply, the equations for each of the estimation
methods are linear in the parameters.
Because of the forecasting limitations of the generalized
least-squares estimations and the computational difficulties
associated with the normit estimations, tables 1 and 2
include fewer functional forms than table 3. We only
include enough equation estimates to allow some com-
parison with the logit results in table 3. With the limita-
tions discussed in section 2, the equations recorded in
tables 2 and 3 look reasonable; that is, negative co-
efficients on precipitation variables and small positive
coefficients on temperature variables. For the generalized
least-squares estimation method, eq (E) in table 1 appears to
give the best fit and, thus, the best predictive ability.
Equation (A), which has only the soil moisture index and
average temperature as arguments, also has some appeal
on the basis of its simplicity.
For the normit results, eq (A) appears to give the best
fit (table 2). Note also that eq (A) is a simple linear func-
tion in the soil moisture index. The comparative fits of the
estimated functions for the normit and generalized least-
1 The measure is just the error sum of squares. A variable for the regressions could be
calculatedsimply by dividing the number by 218,less the number of parametersestimated
in each equation.
squares methods suggest that simple equations involving
precipitation are likely to do relatively well, as compared
to more complicated polynomials in precipitation and
temperature, as predictors of working day probabilities
for common excavation. It is also apparent that some
physically based function in precipitation, such as the soil
moisture index, can be effectively employed as a forecast-
ing variable.
The logit equations (table 3) are more varied than either
the normit or generalized least-squares results. Our
preliminary results suggest that the logit method produced
the best fits, and because of its computational advantages,,
a number of alternatives not previously considered were
explored. The most interesting of these, eq (G) in table 3,
included 3-day average precipitation, the log of average
daily temperature, and the 0, 1 variable indicating whether
or not precipitation occurred on the day in which y, was
observed. The small error, correct sign pattern, and
possibilities of furnishing probabilities for the 0, 1 variable
for purposes of forecasting seem to designate it as the most
promising of any of the equations estimated by the three
methods.8
In summary, we conclude that the probability m Jdels
can be effectively applied to the physical relationships
involving climate variables and work day probabilities.
The coefficients estimated on precipitation variables were
consistently negative and those on temperatures were
positive. Generally, the success of the application suggests
that some of the work-no work decisions can possibly be
made in a quantitative and impersonal manner. We
illustrate the best of these models in the following section.
5. AN ILLUSTRATIVE APPLICATION
OF THE PROBABILITY MODELS
The equations presented in tables 1-3 may be difficult
to evaluate with respect to their actual predictive ability.
For this reason, we have elected to illustrate the prob-
ability models by comparing the predicted values to those
that actually occurred in the sample period. Two months
are selected for this purpose; one more wet than typical
and one more dry than typical. The extreme periods
were selected so that the predictions from the probability
models could be compared under the least favorable
conditions. (These are sample extremes.) In addition to
the observed values of the dichotomous dependent
variable and the concomitant values of the included
climatic variables, a number of special comments are
included. These comments are recorded on the days in
which the predictions from the probability model do not
appear to conform with the observed value of yn. These
comments may give some insight into possibilities for
more refined specifications of functional forms for such
models as well as indicating the limitations of those
presented.
8 Further exploration of the applicability of this equation would be a very useful ex-
ercise, particularly as i t relates to the decision aspects of the model. Weather information
involving precipitation is frequently given in terms of probabilities. Hence, the work-no
work probability is jointly determined by the error of the statistical relationship de-
scribing the probability model and the precipitationprobability. Although calculationof
exact results for such situations may be cumbersome, a numerical application of this
result using simulation modeling would be feasible.
March 1973 / Attanasi, Johnson, LeDuc, and McQuigg / 227 499-057 0 - 73 - 6
TABLE l.-Generalized least-squares estimates of linear equations relating climatically oriented variables to working day probabilities for common
excavation. Numbers i n columns refer to the regression coeficients of the variables heading the respective columns.
, Equation 1 xo. XI " x2 n x3. Xi. Xi" Xa n 1nX1. x2.xe. Xi, I Error
A
B
C
D
E
~~
1. 1947 -0.4296 29. 17
0.7341 -0.3492 32. 38
1. 1126 -6. 1571 6.2250 0.00004 -0. 0043 0.0323 28.05
0. 7707 -1. 3634 0.00003 -0.0002 27.40
0.8308 -3. 5582 0.0020 3. 7844 24. 63
TABLE 2.-Normil estimates of h e a r functions relating climatically oriented variables to working day probabilities for common excavation.
Numbers i n columns refer to the regression coeficients of th6 variables heading the respeclive columns.
Equation I Xa. XI. lnX1, x2 n Xs " 1nX2. InXs. x2.xa. XL Xi. 1 Error
A
B
C
D
E
4. 5685 -3. 3123 25. 98
0.9230 -3.0718 I 27.89
-0.6869 -0.0860 0.0339 31.80
7. 4842 -0.8408 2.2537 30.30
3.4260 -0. 2580 -0.0663 0. 0006 0.0007 0.0034 30. 90
TABLE 3.- Logit estimates of linear functions relating climatically oriented variables to working day probabilities for common excavation. Numbers
in columns refer to the regression coeflcients of the variables heading the respective columns.
Equation 1 Xo. XI n X2. XS. X:, Xi,, Xz.Xs. X3. X:. 1nXa. X4" X,". X6" Xi" I Error
8.352 -6.5980
3.798 -0.3285 0.0484 0.0036 0.0005 0.0016
-0.104 -0.1031 0.0374
13. E46 0.6232 0.0096 3.7840
-6.128 -0.0242 0.0037 1.9086
-13.366 3.8237 0.0640 O.ooO2
1 -7.029 2.3464 3.0658 0.0613
Xo.=l'
XI.=soil moisture index
Xzn=7-day average precipitation (10-2 in.)
Xa.=average temperature
Xa.=kday average preclpitation (10-2 in.)
X4.=Xr.+ current day's precipitation (10-2 in.)
Xa.=O, 1 variable dally precipitation
X?.=Sday average precipitation (10-2 In.)
22.28
31.12
28.65
22.83
28.69
19.67
23.78
The wet month within the sample period was April
1966. Table 4 includes the sample observations for this
month together with the comments and the estimated
work probabilities from models F and G in table 3.
A comparison of the predicted values with the observed
variable yn indicates that the models give the largest
error on days in which late afternoon rains occur. Al-
though model G does better than model F, it is obvious
that an alternative specification could produce more
precision in the forecasts. This lack of precision makes it
difficult to decide whether or not to charge for a work day.
One could, however, establish critical levels of work day
228 1 Vol. 101, No. 3 / Monthly Weather Review
probabilities to give fairly accurate decisions as to the
charging and planning of work days.
The dry month selected for discussion was July 1966.
Predictions from logit models F and G are also reproduced
for this dry month in table 5.
The only no-work day in the month occurred on July 26.
Although an ideal equation might estimate the probability
more closely, the two models we have selected do rather
well. Comparison of the dry and wet months suggests
that, given the two estimated functional forms, the thresh-
old or critical values could be advantageously set higher
for midsummer months than for early spring and late
TABLE 4-Sample observations and predicted work probabilities for common excavation during April 1966
Worklno- Estimated probabilities Avg. daily 3day avg. 0 ,l precipitation
U8l Model F Model G Xa n Xl. xs.
variable Special commonts Date I work ' temperature precipitation
1
4
6
6
7
14
16
18
19
20
22
26
26
27
28
29
1
1
1
1
1
0
1
0
0
0
0
1
0.938
.687
,687
.707
.767
.181
.425
.674
.669
.018
.070
.823
0.946
.840
.%lo
,847
.870
,263
.672
,379
,273
.122
.663
.869
67.0
40.6
40.6
41.6
46.6
46.0
48.0
63.0
68.0
67.0
60.6
61.6
0 ,810 .226 62.0
0 .310 .393 62. 6
1 .696 ,141 67.6
1 ,714 .767 60.6
0
0
0
0
0
0.497
.293
,360
,443
2.377
0.833
,137
.167
.667
,293
,233
0
0
0
0
0
1
0
1 Rain in late afternoon
Rain in late afternoon 1
1
0
0
Rain beginning after 1400 LST
Rain, previous evening
1
TABLE 5.-Sample observations and predicted work probabilities f o r common excavation during July 1966
Date
1
6
6
7
8
11
12
13
14
16
18
19
21
22
26
28
27
28
29
31
m
Work/no- Estimated probabilities -4vg. daily 3day avg. pre- 0 ,l precipitation
U" Model F Model G XS" X7. XS.3
work temperature cipitation variable . Special comments
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0.966
,973
,976
.974
.966
.976
,980
.978
.976
.976
.966
.966
.961
.9Ea
.962
.866
.w
.933
.939
.694
.632
0.962
,967
.968
.968
.961
,968
.972
.970
,969
.969
.942
,939
,964
.966
,963
.643
,419
.924
.928
.431
.603
79.0
84.0
86.0
86.0
78.6
86.6
91.0
88.6
86.6
86.6
88.0
86.0
81.0
74.6
72.0
79.6
79.6
81.0
83.0
84.0
73.6
0
0
0
0
0
0
0
0
0
0
0.1133
.0131
0
0
0
0
0.0833
,1300
.1300
.0467
.4333
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
Heavy rainfall late afternoon
fall months. In any large-scale application of this proce-
dure, however, one would probably break the sample
period into subperiods as a means of handling the difIi-
culties suggested by tables 4 and 5.
' 6. APPLICATIONS OF PROBABILITY MODELS
TO OTHER CONSTRUCTION ACTIVITIES
I n section 3, we noted that the records from which the
information on common excavation were obtained con-
tained data on other construction activities. In this section,
data from two of these activities are employed to estimate
probability functions. Our purposes in presenting these
results are to illustrate the applicability of the probability
model to other types of construction activities and to
provide a basis for evaluating the comparative effects of
climatic variables on other activities.
The two activities selected for the application are
(1) paving and asphalt, and (2) brid'ge, culvert, and
drainage structures. The regression equations that were
2
estimated are of the form
wn= - 15.5157+0.0037X4,--0.0004X$t-4. 1588 lnXan
(22)
for asphalt and paving and
wn= 14.8786-69.8991X4,+31.5216X~n+3.9810 lnX,,
(23)
for bridge, culvert, and drainage structures where
e" Prob. (.y,=lIX:)=-. 1 +ew
Both models were estimated by the logit method, and the
errors for the models are comparable to those for common
excavation.
These results appear to be acceptab1.e. For asphalt and
paving, the variable X, It, which reflwts the precipitation
for 4 days, is far less important, in terms of decreasing
work day probdbilities, than for either the common
March 1973 1 Attanasi, Johnson, LeDuc, and McQuigg 1 229
excavation equations or the bridge, culvert, and drainage
structures equation. In the latter case, the precipitation
variable is important, as would be expected. Average
temperatures, Xen, are of approximately equal importance
in both equations. This may a t fist appear to be question-
able, but when it is recalled that the data are only for
the summer season, the result is reasonable.
7. SUMMARY AND CONCLUSION
We have suggested and applied probadility models to
the problem of determining working days for road con-
struction activities. The results of the application are
encouraging. A number of functions that are simple in
terms of their arguments involving climatic variables
appear’to fit the data rather well. In fact, the accuracy
achieved with this limited amount of data indicates that
a more thorough application of these methods could
provide a substantive basis for planning by highway
departments and construction firms. These plans could
be a basis for contract penalty clauses and other technical
specifications as well as a basis for long-range projections
relating to equipment requirements and payment sched-
ules. In many instances, these types of decisions are
already being made with the aid of data provided by
simulation models. These probability functions could be
easily incorporated into such planning models as a method
of portraying climate-oriented uncertainties.
ACKNOWLEDGMENT
Research for this paper was supported by National Science
Foundation Grant GA15413. The authors also wish to acknowledge
Carl Klamm, Construction Division engineer for the Missouri
State Highway Commission, and the Missouri State Highway
Commission for their cooperation in providing the data used in the
paper and for helpful comments on various aspects of the problem.
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mum Normit Chi-Squarc With Particular Rcfercnce to Bio-
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Cox, David Roxbcc, “Some Procedures Connccted With the
Logistic Qualitative Response Curve,” Research Papers in
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Goldberger, Arthur Stanlcy, Econometric Theory, John Wiley & Sons,
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Tobin, J., “The Application of Multivariate Probit Analysis to
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Cowles Foundation, New Havcn, Conn., 1955, 48 pp.
Zellner, A,, and Lee, T. H., “Joint Estimation of Relationships
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No. 2, Econometric Society, Ncw Haven, Conn., Apr. 1965,
pp. 382-394.
[Received April 20, 1972; revised October 17, 19721
230 1 Vol. 101, No. 3 1 Monthly Weather Review