parametric range estimating of building costs using regression models and bootstrap

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Parametric Range Estimating of Building CostsUsing Regression Models and Bootstrap

Rifat Sonmez1

Abstract: This paper presents a bootstrap approach for integration of parametric and probabilistic cost estimation techniques. In theproposed method, a combination of regression analysis and bootstrap resampling technique is used to develop range estimates forconstruction costs. The method is applied to parametric range estimation of building projects as an example. The bootstrap approachincludes advantages of probabilistic and parametric estimation methods, at the same time it requires fewer assumptions compared toclassical statistical techniques. This study is of relevance to practitioners and researchers, as it provides a robust method for conceptualestimation of construction costs.

DOI: 10.1061/�ASCE�0733-9364�2008�134:12�1011�

CE Database subject headings: Construction costs; Estimations; Probability; Regression analysis; Parameters.

Introduction

Key budgeting and feasibility decisions are based on the costestimates prepared at the early stages of construction projects. Asproject scope is not finalized during these conceptual phases,quantity take-off cannot be performed to form a detailed costestimate. Although conceptual cost estimates are not expected tobe precise, inaccurate estimates may lead to lost opportunities,and lower than expected returns �Oberlender and Trost 2001�.Parametric estimation methods have been suggested in severalstudies to improve accuracy of conceptual cost estimates.

In parametric estimating, a model including the important pa-rameters is developed to predict construction costs, using data ofprevious projects. Regression analysis and neural networks tech-niques are commonly used to determine an adequate cost model.In previous studies regression analysis was implemented to esti-mate construction costs for building projects �Kouskoulas andKoehn 1974; Karshenas 1984; Lowe et al. 2006�, and offshoredecommissioning operations �Kaiser 2006�. Regression techniqueallows relatively simple analysis to sort out the impacts of theparameters on the project cost. Neural networks have been pro-posed as an alternative to regression analysis for parametric costmodeling. Neural networks have been used successfully to esti-mate costs of building projects �Kim et al. 2005�, highwayprojects �Hegazy and Ayed 1998�, and concrete pavements �Adeliand Wu 1998�.

Parametric models, in general, make a point estimate for theproject cost. A single cost estimate does not provide sufficientinformation about the level of uncertainty included in the esti-mate. This is especially crucial during contingency setting, orgo/no-go decisions, as conceptual cost estimates, in general, in-

1Assistant Professor, Dept. of Civil Engineering, Middle East Tech.Univ., Ankara, 06531, Turkey. Email: [email protected]

Note. Discussion open until May 1, 2009. Separate discussions mustbe submitted for individual papers. The manuscript for this paper wassubmitted for review and possible publication on August 25, 2006; ap-proved on July 1, 2008. This paper is part of the Journal of ConstructionEngineering and Management, Vol. 134, No. 12, December 1, 2008.
©ASCE, ISSN 0733-9364/2008/12-1011–1016/$25.00.
JOURNAL OF CONSTRUCTION ENG

J. Constr. Eng. Manage. 20

clude high levels of uncertainty. The level of uncertainty includedin the estimates can be quantified by probabilistic estimating tech-niques. In probabilistic estimating, project cost is considered as arandom variable and a probability distribution function for thecost is generally established by simulation techniques �Curran1989; Touran and Wiser 1992; Wang 2002; Isidore and Back2002�. The probability distribution function developed can beused to predict the probability of a budget overrun, or to deter-mine a range of estimates for a desired level of certainty. Selec-tion of an adequate theoretical probability distribution functionfor the cost items, and inclusion of the correlations amongthe cost items are the main challenges of simulation techniques.The two major difficulties in including the correlations are �1�estimating correlation coefficients; and �2� providing an accuratetheoretical analysis approach to account for these correlations�Hudak and Maxwell 2007�. Analytical methods provide an al-ternative to simulation for range estimating of project costs�Diekmann 1983; Moselhi and Dimitrov 1993; Skitmore and Ng2002�.

Although simulation and analytical methods allow rangeestimation, the impacts of parameters on project cost are notgenerally included in these techniques. On the other hand,the parametric estimation techniques include the information ofthe parameters but, usually do not include range estimates forthe project costs. The main objective of this study is to developan integrated method, which will include the advantages of para-metric and probabilistic estimating techniques simultaneously.

Bootstrap Method

Bootstrap is a resampling method in which r new samples, eachthe same size as the observed data, are randomly drawn withreplacement from the observed data. The purpose of bootstrap isto mimic the process of sampling observations from the popula-tion by resampling data from the observed sample. Bootstrapmethods are more flexible than classical statistical methods asthey require fewer assumptions �Johnson 2001�.

Bootstrap method has been applied to a wide variety of statis-
tical procedures such as estimation of standard error for a mean,
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estimation of empirical probability distribution function of apopulation, regression and time series analysis, determination ofconfidence and prediction intervals, hypothesis testing, and esti-mation of prediction error �Efron and Tibshirani 1993; Davisonand Hinkley 1997�. Examples of bootstrap applications includefinancial risk management �Christoffersen and Goncalves 2005;Marsala et al. 2004�, water supply storage and delivery systemmodeling �Tasker and Dunne 1997�, and pavement structuralevaluation �Damnjanovic and Zhang 2006�.

In this study a combination of regression and bootstrap tech-niques is used to integrate parametric and probabilistic estimationtechniques. A simple linear model with one parameter p, for costitem m can be shown as

cm = �0 + �1p + � �1�

where cm=predicted cost, �0 and �1=regression coefficients, and�=random error with an expected value of 0. The random errorterm � takes into account all unknown factors that are not in-cluded in the model. The regression parameters �0 and �1 areestimated by using the observed data x= �x1 ,x2 , . . . ,xn�.

In the proposed method, the observed data pairs x= ��cm1 , p1� , �cm2 , p2� , . . . , �cmn , pn��, compiled from previousprojects for cost item m, with one parameter p, are resampledby bootstrap method, to form a data set x�. Integers i1 , i2 , . . . , in,each of which has a value between 1 and n, with a probability of1 /n are selected randomly to perform resampling. The bootstrapdata set x* = ��cmi1 , pi1� , �cmi2 , pi2� , . . . , �cmin , pin�� is formed bycorresponding members of x

x1* = xi1, x

2* = xi2, . . . , x

n* = xin �2�

The star notation indicates that x� is not the actual data set x,but rather a resampled version of x. The bootstrap dataset �x1

� ,x2� , . . . ,xn

�� consists of members of the original data set�x1 ,x2 , . . . ,xn�, some appearing zero times, some appearing once,some appearing twice or, more.

The bootstrap data set x� is used to obtain the regression co-efficients for the model �1�. A probability distribution function forthe predicted cost item m is obtained by using several bootstrapreplications. Probability distribution functions for all of the pre-dicted cost items with one or more parameters are obtained simi-larly, using the previously selected integers i1 , i2 , . . . , in, todetermine bootstrap project data pairs. Finally, predictions for thecost items are added to determine the probability distributionfunction for the total cost.

The main advantage of the nonparametric bootstrap approachpresented in this study is that it does not require any assumptionsregarding the distribution of the error term �, or the distributionsof the cost items. In an alternative approach, range estimates forthe cost items can be developed by assuming that the error term �has a normal distribution �Sonmez 2004�. This alternative ap-proach becomes complicated when several models are developedas the normality assumption of the error term must be checked foreach cost model, and transformations may be required when re-siduals are not normally distributed. The bootstrap technique alsoenables development of a robust method to integrate the informa-tion of the cost items and parameters for range estimating of the
total project cost.
1012 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMEN


Regression Models

Case Example

Data compiled from 20 U.S. building projects were used to dem-onstrate the proposed bootstrap approach. The projects consist ofcontinuous care retirement communities built by a single contrac-tor in 10 different locations, between 1983 and 1995. The costbreakdown system of the contractor included 11 cost items in-cluding; site development �Sd�, foundations and slab on grade�Fs�, structure �St�, enclosure �En�, interior finishes �If�, equip-ment and special construction �Eq�, conveying systems �Cs�, me-chanical �Me�, fire protection �Fp�, electrical �El�, and generalrequirements �Gr�. The detailed cost estimates for the items werecompiled along with the data of 20 parameters. The parametersincluded information of time and location of construction, projectduration, building characteristics, site conditions, structural frameand exterior finish types.

Selection of Parameters

Regression analysis was performed to develop a parametricmodel for each cost item. Candidate parameters for the initialregression models were determined by the senior estimators of thecontractor. Historical construction cost index, and city cost index�R. S. Means 1996� were included in all of the models as candi-date parameters, to determine the significance of inflation andlocation factors on the cost items. Similarly, total gross buildingarea was included in all of the initial cost models. The remainingcandidate parameters included in the initial models were specificto the cost item, and are given in Table 1. The candidate param-eters that did not have a significant impact on the cost item weredropped form the model, one at a time, by backward eliminationtechnique. The parameters that had a regression coefficient sig-nificant at a 0.2 significance level were included in the final costmodels. The parameters included in the final models are given inTable 2.

Building area �P1�, construction cost index �P2�, site area �P8�,major demolition on site multiplied by the site area �P9�, and sitewaste treatment multiplied by the site area �P10�, were the param-eters included in the final cost model for site development. Theparameters P9 and P10 were used to represent additional sitedevelopment costs per 1 m2 of site area, when there was a majordemolition or waste treatment on site. In the foundations and slabon grade and equipment and special construction cost models;building area, construction cost index, and city index �P3� weredetermined as the significant parameters. The parameters steelframe multiplied by building area �P12�, concrete frame multi-plied by building area �P13�, and steel and concrete frame multi-plied by building area �P14� of the structure cost model, indicatedthe additional cost of these frame types for 1 m2 of building area,compared to the wood frame. Similarly, the parameters vinyl ex-terior finish multiplied by building area �P17�, and masonry exte-rior finish multiplied by building area �P18� were included in thefinal enclosure model to represent the cost difference of theseexterior finishes for 1 m2 of building area, compared to plaster/stucco finish.

Results of the regression analysis revealed that the parameterbuilding area per residential unit �P7� had a significant impact onthe interior finish, mechanical, and electrical costs. The modelresults indicate that for a given total building area as the numberof residential units increases, interior finish, mechanical, and elec-
trical costs rise mainly due to increased number appliances, such
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as interior doors, kitchen cabinets, sinks, bathroom accessories,and mechanical and electrical fixtures. The parameter percentcommons and nursing facilities �P5� significantly impacted inte-rior finish costs, indicating a higher interior finish cost for thecommons and nursing facilities compared to the residential units.Project duration �P20� had an impact on the general requirementscosts, along with the parameters construction cost index, and citycost index. In the final cost model for the conveying systems,number of elevator stops �P19� was determined as the only

Table 1. Candidate Parameters for the Cost Models

No. Parameter description

P1 Total gross residential, commons, nursing facilities, and str

P2 Construction cost index �1995=10

P3 City cost index

P4 Number of stories

P5 Percent area of commons and nursing facilities in th

P6 Percent structured parking area in tota

P7 Total gross building area per residentia

P8 Site area in m2

P9 Major demolition on site �dummy variable� mu

P10 Site waste treatment �dummy� multiplied

P11 Wood frame �dummy� multiplied by

P12 Steel frame �dummy� multiplied by

P13 Concrete frame �dummy� multiplied b

P14 Steel and concrete frame �dummy� multipl

P15 Masonry structure �dummy� multiplied

P16 Wood exterior finish �dummy� multiplie

P17 Vinyl exterior finish �dummy� multiplied

P18 Masonry exterior finish �dummy� multipli

P19 Number of elevator stops

P20 Project duration in months

Table 2. Significance Levels of Coefficients and R2 Values for the Final

No. Sd Fs St En If

P1 0.10 0.00 0.00 0.00 0.00

P2 0.02 0.00 0.00 0.13 0.00

P3 0.00 0.02 0.14

P5 0.17

P7 0.15a

P8 0.03

P9 0.16

P10 0.03

P12 0.02

P13 0.01

P14 0.00

P17 0.16a

P18 0.02

P19

P20

R2 0.87 0.84 0.96 0.89 0.96a
Indicates a negative regression coefficient for the parameter.


significant parameter. Regression models developed indicated thatconstruction cost index was not a significant parameter for theconveying system costs. The insignificance of construction costindex for conveying system costs may be due to characteristics ofthe project data used to develop the models, or due to the possi-bility that conveying system costs did not increase significantlyfrom 1983 to 1995. The case example was limited to data com-piled from 20 projects; more data are required to make a conclu-sion for the conveying system costs.

Initial models in whichthe parameter is included

parking area in m2 All

All

All

Fs, St, En

building area If, Eq, Me, Fp, El

If, Me

If, Eq, Me, Fp, El

Sd

by P8 Sd

Sd

Fs

St

St

P1 St

St

1 En

En

P1 En

Cs

Gr

ls

Eq Cs Me Fp El Gr

0.00 0.00 0.00 0.00 0.01

0.00 0.00 0.00 0.00 0.00

0.04 0.14

0.07a 0.00a

0.00

0.09

0.80 0.95 0.86 0.75 0.91 0.83

uctured

0�

e total

l area

l unit

ltiplied

by P8

P1

P1

y P1

ied by

by P1

d by P

by P1

ed by

Mode


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Collinearity Diagnostic

Collinearity can be defined as existence of high correlation, whenone of the regression independent variables is regressed on theothers. When collinearity is significant, regression analysis maygive misleading results for hypothesis testing, estimation, andforecasting �Belsley et al. 1980�. Several diagnostic techniqueshave been developed to detect collinearity. Variance inflation fac-tor �VIF� is commonly used to evaluate the level of collinearity.In general, a VIF of 10 or larger indicates a problem due tocollinearity �Chatterjee and Price 1991�. VIF is calculated usingthe following formula:

VIFi =1

1 − Ri2 , i = 1,2, . . . ,k − 1 �3�

where Ri2=correlation coefficient of independent variable i, re-

gressed on the other independent variables, and k=number ofindependent variables.

Collinearity diagnostic was performed for the final regressionmodels, as there was a possibility of correlation between some ofthe independent variables. As an example, in the final site devel-opment model, existence of correlation between the variablesbuilding area �P1�, and site area �P8� was a possibility. Similarly,in the final general requirements model, there was a possibility ofcorrelation between the variables building area �P1�, and projectduration �P20�. VIF was used to evaluate the level of collinearity.VIF values for models including two or more independent vari-ables are presented in Table 3. The variable building area �P1� ofthe site development model had the highest VIF, with a value of3.77. The results indicated that collinearity did not cause a prob-lem for the models, as the calculated VIF values were smallerthan the suggested value of 10.

Validation of the Models

The coefficient of determination �R2� values of the regressionmodels were between 0.75 and 0.96. The R2 values show thatlinear models in general provided a sufficient fit to the data. Re-sidual plots of final regression models were used to investigatethe possible need for higher order models for the cost items.The residual plots indicated that linear models were sufficient;therefore higher order models were not developed. However, a

Table 3. VIF Values for the Final Models

No. Sd Fs St En

P1 3.77 1.12 3.57 2.04

P2 1.11 1.11 1.58 1.34

P3 1.08 1.25 1.18

P5

P7

P8 3.01

P9 1.71

P10 1.39

P12 2.45

P13 2.67

P14 3.55

P17 1.17

P18 2.25

P20

bootstrap approach is not limited to linear regression models only.



The approach can also be used to develop range estimates forhigher order regression or neural network models, when linearmodels are not sufficient.

A good fit for a model does not always guarantee accuratepredictions. Cross-validation techniques are commonly imple-mented to evaluate prediction performance of the models. In“leave-one-out” cross validation, one data point is not used duringmodeling, and the model developed is used to predict the previ-ously selected data point. The procedure is repeated for all datapoints, and predicted data values are compared with the observedvalues to assess the prediction performance.

Leave-one-out cross validation was performed to evaluate theprediction performance of the final models developed. Mean ab-solute percent error �MAPE� was used as an error measure toevaluate the prediction performance. MAPE value for a costmodel is the average of deviations between predicted cost andactual detailed cost estimate in absolute values, expressed as pro-portion of the actual detailed cost estimate. MAPE for the totalcost was calculated as 12%. The models were able to predict theactual detailed cost estimates with an average error of �12%. Theaverage prediction error of �12% before contingency was con-sidered as acceptable, as it was within the suggested accuracyrange from −15 to +25 for conceptual cost estimating of buildingprojects �AbouRizk et al. 2002�.

Range Estimates

Models developed were used to develop range estimates for acase project, which was not included in the regression analysis.The values of the parameters for the case example are given inTable 4. The site development activities for the project did notinclude demolition or waste treatment, so the parameters P9 andP10 were equal to zero. The project had both steel and concreteframes, as a result P14 was equal to building area, and both P12and P13 were equal to zero. The exterior finish of the buildingswas plaster finish, so the parameters P17 and P18 were also equalto zero.

Range estimates for the case example were developed by boot-strap resampling method. One thousand bootstrap data sets witheach containing 20 data points, were drawn randomly with re-placement form the original project data. One thousand regression

Eq Me Fp El Gr

1.12 2.01 1.08 2.01 1.99

1.11 1.14 1.08 1.03 1.23

1.08 1.10

2.13 2.09

2.25

If

2.75

1.13

1.54

2.13

coefficient sets were determined for each final cost item model by


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using the bootstrap data sets. The regression coefficients sets wereused to make 1,000 predictions for the example project costitems. The predictions were then used to obtain the probabilitydistribution function for each cost item, and for the total cost.Empirical distribution function for the predicted total project costis presented in Fig. 1. Once probability distribution functions areobtained, range estimates for a desired level of certainty can bemade. Table 5 provides range estimates for 90% probability level.There is a 90% chance that the detailed cost estimate for the caseproject would be between $16,900,782 and $21,153,484. Actualdetailed cost estimate for the case project was $20,800,000. Theactual detailed cost estimate was larger than the 50% predictionvalue of $18,593,888, and was close to the upper end of theprediction range.

The range estimates for the cost items provide an indicationfor the uncertainties included in the estimates. As an example, thedifference of 95 and 5% range estimates for the enclosure cost is$1,658,940, which is larger than the 50% estimate of $1,398,060.But, the difference of upper and lower ranges for the fire protec-tion is $105,566, which is less than one third of the 50% estimateof $366,711. Comparison of the range estimates indicates that, theuncertainties included in the cost estimate for enclosure is largerthan the uncertainties included in the fire protection estimate. Thelimitations of the parameters included in the models, and thevariation levels of the cost items can have an impact on the un-certainty levels of the cost estimates.

Table 4. Parameter Values for the Case Example

Parameter Value Unit

P1 22,779 m2

P2 100.6 —

P3 87.8 —

P5 41.8 —

P7 201.6 m2 /unit

P8 4,047 m2

P14 22,779 m2

P19 61 Each

P20 16 Months

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5

Predicted total project cost ($Million)

Probabilityofdetailedestimatebeinglessthanthe

predictedcost

Fig. 1. Empirical probability distribution function for the predictedcost



Conclusions

Parametric and probabilistic estimating techniques were inte-grated using regression analysis and bootstrap methods. The boot-strap method does not require any assumptions regarding theprobability distribution of the model errors, or the distributions ofthe cost items, and can be implemented by using a spreadsheetprogram. Integration of parametric and probabilistic estimatingtechniques will hopefully lead to more realistic cost expectationsduring early project stages.

The models presented in this study were developed using datacompiled from 20 projects. Therefore, the revealed relations be-tween the parameters and cost items were determined with a lim-ited data set. Future studies including larger data sets, andadditional parameters can help to improve the understanding ofbuilding costs.

This study presented implementation of bootstrap method forconceptual cost estimating. Bootstrap method offers several ad-vantages over the classical statistical techniques. The nonparamet-ric bootstrap avoids restrictive and sometimes dangerousassumptions about the form of the underlying populations. In ad-dition to cost estimating, potential implementation areas of boot-strap in construction management include risk analysis,probabilistic scheduling, simulation of construction operations,and modeling and hypothesis testing applications.

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Table 5. Range Estimates �$� for the Case Example

Example

Probability level

5% 50% 95%

Site development 762,743 1,189,386 1,741,066

Foundations and slab on grade 654,092 896,173 1,134,172

Structure 2,290,073 2,659,794 3,329,636

Enclosure 314,165 1,398,060 1,973,105

Interior finishes 4,019,519 4,860,802 5,800,377

Equipment and specialconstruction

232,315 295,305 452,018

Conveying systems 700,761 739,380 804,205

Mechanical 1,613,006 2,177,185 3,163,362

Fire protection 321,291 366,711 426,857

Electrical 1,333,400 1,586,136 1,841,553

General conditions 2,060,346 2,431,586 3,021,058

Total project cost 16,900,782 18,593,888 21,153,484

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08.134:1011-1016.

parametric range estimating of building costs using regression models and bootstrap

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