Regression Analysis for Cost Modelling

8/6/2019 Regression Analysis for Cost Modelling

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Regression Cost Model



Introduction Regression analysis – A statistical method by

which estimates are made of the value of avariable from a knowledge of the values of one or more other variables, and the errorsinvolved in this estimating processmeasured

Normally used in situations where therelationships between variables is notunique

Main types-

§ Simple Linear Regression Analysis

§ Multiple Linear Regression Analysis



Assumptions

§ The standard deviation in the error associated withthe dependent variable cost remains constantthroughout the domain

§ This error is normally distributed

§ The effect of any variable is always expressed interms of a fixed cost increase or decrease,irrespective of project size or type



Simple Linear RegressionAnalysis

Two-variable linear regression – describesthe relationship between two variables bycomputing a straight line through the dataobtained

Dependent variable (y) - the value to be estimated

Independent variable (x) – the factor from which theestimates are made

Constant (a)- the value of y when the independent

variable is zero

The coefficient of x (b)- The slope of the line

Expression for straight line

y = a +bx



§ Prediction within the range of values in the dataset is knownas interpolation

§ Prediction outside this range of the data is knownas extrapolation

a D e p e n d e n

t v a r i a b

l e

Independent variable

θ

b = tan θ



Steps of SLR Model



Specification

§ Begins with theoretical reasoning on the relationshipbetween variables

§ Form equations to represent the relationships betweenvariables

§Since the population parameters are unknown, sample isconsidered and the model is built with estimated values

Estimation

§ Lean squares estimation procedure is used most of the

time

§ Include a series of statistical tests to make sure that theestimated model is a good representation of thepostulated relationship

Steps of SLR Model



Validation

§ Evaluate the quality of the model

§ Evaluated on the basis of following statistics

o Coefficient of determination

o Standard error

o F ratio test- The ratio of the regression mean square to theresidual mean square

o T ratio test- The ratio of the coefficient to its standard error

Forecasting§ Forecasting should be satisfactory to the users

§ Accuracy depends on the acceptable error amount of themodel

Steps of SLR Model



Multiple Linear RegressionAnalysis

This aims to create a relationship with thedependant variable with several otherindependent variables.

§ Independent / Response variable - y§ Dependant/ Explanatory variables- x 1, x 2, x 3….. x n

y = a + b1 x1 + b 2 x 2 + b 3 x 3 + … bn xn + e



Steps of MLR Model



Specification

§ Begins with theoretical reasoning on the relationship

between variables§ Selecting a full set of explanatory(Independent) variables

Estimation

§ Determining the correlation coefficients between allpossible pairs.

§ Resolving multicollinearity

§ Eliminating non-significant variables one at a time until allthe remaining variables are significant.

o Use of the t-ratio- a large t-ratio is desirable.o Use of the F-ratio -Test for the significance of the overall

dependence of y on the variables (x1, x2, . . . , xn )

§

STEPS OF MLR MODEL



Estimation Cont’

§ Constructing a multiple linear regression modelo Making estimates of the coefficients in the regression model,

the method of least squares is used due to its simplicity.

Validation

§ Validation done before practical use in constructionindustry using another actual project.

§

Forecasting

§ If validation is a success practical use on constructionprojects

STEPS OF MLR MODEL



Application in ConstructionIndustry

Simple Linear Regression Analysis§ Case Study: Consider the possible sample values of

bricklayer hours and areas of brickwork from 10fictitious contracts in the following table

§



§Plotted scatter diagram



§ Scatter is caused by the factors other than area whichaffect the hours required

o Bricklayer-hours : Independent / Response variable

o Areas of brickwork : Dependent / Regression variable

§ To avoid individual judgement in constructing the line –method of least squares used

§ In fitting the regression line to a set of data, severalparameters are estimated which need to be tested forthe significance before being accepted

§ As an overall guide to the strength of association betweenthe two variables the correlation coefficient is calculated

Perfect correlation = 1

Calculated coefficient correlation = 0.998



§ Shows an excellent degree of correlation which cannot befound by using one variable only

§ Standard error of estimate ; anticipated differencebetween the actual values and what the regression linepredicts, should be calculated

A li i i C i



Multiple Linear Regression Analysis

Case Study : Obtaining a model that will estimateproductivity rates of concrete operations. For this awastewater project was observed in the North-Eastof Scotland (Project A).

The regression analysis methodology used in this study isbackward elimination, stepwise regression.

§

§ Firstly listing out the independent variables of Project Ao Type of pouro Total volume(m3)

o Number of trucks on job

o Average volume of load(m3)

o Start time

Application in ConstructionIndustry

o Average truck time(minutes)

o Number of loads

o Weather

o Concrete mix



§Calculating the correlation coefficients between all possible

pairs by using the in built functions of Microsoft Excel

§

Resolving multicollinearity by removing one variable (Totalvolume) out of the highly co rerated two variables ( i.e Total

volume and No. of Loads).



§ Estimating partial regression coefficients and thecorresponding t-statistics from the regression on actualproductivity for all eight explanatory variables.

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§ Insignificant variables have small absolute values-Shouldbe eliminated

§ Carrying out two further runs, eliminating the insignificantvariables: concrete mix (t-statistic=0.97) and the starttime (t-statistic=1.72) from the regression model.



§ An important assumption made is the variability of thedata does not change for different levels of theresponse or explanatory variables.o This is checked by carrying out residual plots.

§

§ Constructing a multiple linear regression model for actualproductivity for a single server concrete system.

Pactual =1.31T p + 1.75V a + 0.56T n + 0.59W 0.01C t þ 0.37Ln – 6.95

Tp = Type of pour

Va = Average volume of concrete

Tn = Number of trucks on job

W = Weather

Ct = Average cycle time

Ln = Number of loads



§ Validation is done by using an actual concrete pours from

another wastewater project in Scotland by a different

contractor (Project B).

§

The actual productivities achieved on 32 operations observed onProject B are compared to the predicted productivities using

the derived regression model



Drawbacks in regression Multicollinearity§ If the explanatory variables in multiple regression are

correlated, and if the correlation coefficient (positive/negative) is high it is difficult to get their separateeffects on the dependent variable.

§ Leads to a poorly estimated partial regression coefficient.

Omitted variables§ If independent variables that have significant relationships

with the dependent variable are left out of the model,the results will not be satisfactory. E.g location, qualityetc cannot be quantified

§

Biasness of selecting independent variables§

Endogeneity§ Changes in the dependent variable cause changes in the

independent variable.

§



Development of RegressionModel

Development of technologies in computing,accessing, processing and storing data

§ All major statistical software packages perform leastsquares regression analysis.

§ Simple linear and multiple regression using least squares canbe done in some spreadsheet applications and on somecalculators.

§ Specialized regression software has been developed for use

in fields (survey analysis, neuro imaging).§ The Constructive Cost Model (COCOMO)- An example of an

algorithmic software cost estimation model developedusing basic regression formula.

§



Conclusions Regression Analysis falls under the Algorithmic Cost Model

which uses mathematical formulae linking costs/inputswith metrics to produce an estimated output.

It is used not only for estimating costs but also forforecasting productivity, time and any other parameter.

A widely used method not just in the construction Industry. When there is only one major factor affecting the response

SLR can be used When there are more than 1 major factor affecting the

respone MLR can be used There are several drawbacks and limitations in this method.

The knowledge of using Regression analysis in a specializedcost estimation software, in spread sheets and in acalculator is beneficial for the Quantity Surveyor

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Regression Analysis for Cost Modelling