# The Relationship Between More Than One Predictor Or Independent Variables Accounting Essay

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## Introduction:

As proposed by Pearson in 1908, the term multiple regression is used to study the relationship between more than one predictor or independent variables and a dependant variable or criterion. Multiple regression provides the estimate of the concerned variable based on its dependent variables. The independent variables are found out based on the project and a detailed research study to estimate the dependent variable. Multiple regression is more practical approach towards estimating a variable since, there would not be a single factor affecting the estimation of criterion and hence linear regression might not always help for practical studies. Multiple regression is more beneficial especially in construction projects, where estimated values would help in assessing the performance of the project by comparing with the actual values. This regression can be applied to determine various details in the project mainly in construction which includes duration of the project, cost involved, resource availability and many other factors. Construction projects involve many constraints related to budget, time, and resources. To effectively and efficiently complete a project, multiple regression technique helps in prediction of these constraints so that project manager can manage things optimally and lead to project success.

This assignment aims at studying the multiple regression technique, observing and collecting details from many projects and applying it to predict the estimated value of duration of office projects based on the key factors that affect this variable. With the help of a regression model obtained by applying the multiple regression technique the duration can be predicted. So our dependent variable is duration of office projects. Through out the rest of the report it is denoted by Y and is obtained in weeks. A detailed study about construction projects, their durations, scheduling and estimations have been studied and the independent variables on which Y depends are observed. These variables identified include experience of project manager, his reputation, and type of contract, his previous record, financial constraints, and location. They are obtained from the various sources which are as described below:

## Literature Review:

Mohan M Kumaraswamy and Daniel in 1995 conducted a survey and published a research paper in Construction Management and Economics journal about the determinants of construction duration. They illustrated a hierarchy of factors that affect the construction duration as shown below in the fig1. They highlighted the key factors as the construction value or financial constraints, location, type of the contract, productivity and previous developments. They have done an excellent research in Hong Kong to find out these relevant variables and hence these variables are taken into consideration for our project as well.

Ms. Theresa Keoughan Burrows, Mr. Ian Pegg, and Mr. Joe Martin in their work of predicting construction duration of buildings have considered phasing, market constraints or opportunities, design, complexity, site access, availability of resources, site conditions, and availability of finances.

Greg J. Hoffman, Alfred E. Thal Jr, Timothy S. Webb, and Jeffery D. Weir conducted research to estimate performance time in construction projects. They considered various factors under project scope, project complexity, project environment and management attributes. Based on these factors and with the help of statistical tools they predicted the performance time. They also mentioned various authors and their contributions in determining the same.

All these research studies and through extensive search on website regarding construction projects, duration and factors that affect the duration of project are studied and determined to proceed further in regression. Next section handles the details about these variables and data collection.

## Fig1: Factors affecting construction duration

## Project Study:

As mentioned above, the factors have been identified to predict the duration of the office projects. Data for these dependent variable and independent variables are obtained from various sources through questionnaires, interviews and from the secondary data available from the Human resources Department of various construction industries. The data thus collected is refined and checked for the correlation and those variables which are not correlated are finally used for the regression analysis.

Of all the variables, the main variables with less correlation are found out to be:

## Y: Duration of office projects (Dependent variable)

Exp: Experience of the Project Manager (years)

TR: Track record of contractor in building office towers (years)

Type of the contract (Lump sum, Design Build, Others)

Size: floor area (sq m)

Loc: location of site (CBD, outside CBD)

Int: interest rates, a proxy for financial constraints (percent)

Dev: Reputation of developer

The data values for these variables obtained from different sources mentioned above are as:

Y, duration for the completion of office projects are obtained in terms of number of weeks. The average value of this variable is found out to be 133.34 weeks with minimum value of 79 weeks and maximum of 201 weeks (Table1: Descriptive Statistics).

Exp, the experience of the project manager; this gives the information about the project manager's previous experience in dealing with similar projects. It is measured in terms of the number of years. The average experience was found to be 6 years with minimum of 3years and maximum of 10years experience in the sample obtained (Table1: Descriptive Statistics).

TR, track record of contractor in building towers which is also measured years gives us the information about the contractors previous duration in building office towers. Average was found out to be 2.1 years with minimum of 1year and maximum of 3years (Table1: Descriptive Statistics).

Type of Contract has got three choices of Lump sum, Design Build or others. 2 Dummy variables LSC and DB are used to measure this factor. It is considered that 0, 0 would represent other contracts and LSC=1 represents Lump sum project and DB=1 represents Design build type.

Size, which is considered to be the floor area measured in square meters. Average was found to be 31sq.m, with minimum of 25sq.m and maximum of 40 sq.m (Table1: Descriptive Statistics). Larger the floor area, larger should be the duration but depends on the skill and performance of the project manager.

Loc, which represents the location of the construction site, has got two choices of within the CBD region and outside the CBD. Thus this is again a single dummy variable with 0 within CBD and 1 for outside CBD.

Int, interest rate which represents the financial constraints involved during the project. It includes the cost, rate of return or cost of capital involved in the project. It shows an average value of 11% with minimum of 7% and maximum of 15% (Table1: Descriptive Statistics).

Dev, acts as a representative for the reputation of the developer and is measured on a scale of 1 to 5 where 1 is for Very good developer and 5 is for a poor developer.

## Y (weeks)

## Exp (yrs)

## TR (yrs)

## LSC

## DB

## Size

## Loc

## Int

## Dev

## Mean

133.3400

6.1000

2.1000

0.4800

0.3200

31.9800

0.5000

0.1108

2.9800

## Standard Error

4.8961

0.3286

0.1040

0.0714

0.0666

0.6197

0.0714

0.0031

0.1862

## Median

126.5000

6.5000

2.0000

0.0000

0.0000

32.0000

0.5000

0.1100

3.0000

## Mode

89.0000

3.0000

2.0000

0.0000

0.0000

34.0000

1.0000

0.0900

2.0000

## Standard Deviation

34.6207

2.3234

0.7354

0.5047

0.4712

4.3822

0.5051

0.0218

1.3169

## Sample Variance

1198.5963

5.3980

0.5408

0.2547

0.2220

19.2037

0.2551

0.0005

1.7343

## Kurtosis

-1.0016

-1.3532

-1.0945

-2.0780

-1.4254

-1.1472

-2.0851

-0.8343

-1.0601

## Skewness

0.2189

0.0570

-0.1604

0.0826

0.7958

0.0762

0.0000

0.1508

0.1498

## Range

122.0000

7.0000

2.0000

1.0000

1.0000

15.0000

1.0000

0.0800

4.0000

## Minimum

79.0000

3.0000

1.0000

0.0000

0.0000

25.0000

0.0000

0.0700

1.0000

## Maximum

201.0000

10.0000

3.0000

1.0000

1.0000

40.0000

1.0000

0.1500

5.0000

## Sum

6667.0000

305.0000

105.0000

24.0000

16.0000

1599.0000

25.0000

5.5400

149.0000

## Count

50.0000

50.0000

50.0000

50.0000

50.0000

50.0000

50.0000

50.0000

50.0000

## Table1: Descriptive Statistics

## Regression Model:

After collection of data, with the help of MS Excel, multiple regression is being applied on the data with Y as the dependent variable and all others as the independent variables.

A regression model would look like:

## Y = b0 + b1X1 + b2X2 + … + brXr + e,

where,

X1, X2,.....Xr denotes the r variables that affect the dependent variable Y.

b0 is the intercept

e is the residual error and

b1, b2,...... br represents the coefficients of these variables.

As given above the labels for the variables our model would be like:

## Y = b0 + b1(Exp) + b2(TR) + b3(LSC) + b4(DB) + b5(Size) + b6(Loc) + b7(Int) + b8 (Dev) + e

On applying regression on the collected data with consideration of above model, it provided the following model:

Y = 61.171 -4.695 (Exp) + 0.229 (TR) + 9.308 (LSC) + 6.228 (DB) + 0.001 (Size) + 8.379 (Loc) + 579.082 (Int) + 8.552 (Dev)

Regression Statistics

Multiple R

0.9592

R Square

0.92

Adjusted R Square

0.9044

Standard Error

10.706

Observations

50

Table2: Regression statistics

The regression statistics shows the goodness fit of the model predicted. The above mentioned model is having an R2=0.92 which implies that 92% of the deviations of Y around Y-mean is explained by the above model through regressors Exp, TR, LSC, DB, Size, Loc, Int and Dev factors.

ANOVA

df

SS

MS

F

Significance F

Regression

8

54032.1394

6754.017424

58.92955191

4.8822E-20

Residual

41

4699.080604

114.6117221

Total

49

58731.22

Table3: ANalysis Of Variance

Analysis of Variance table splits the sum of squares into residual and regression sum of squares. The F-test result provides the test for the hypothesis that all the coefficients are equal to 0 to the alternative hypothesis that atleast one of them is non zero. Significance F value being much smaller than the 0.05 for 95% confidence level, the alternative hypothesis is accepted.

Further with the help of the t stats provided in the table below, significance of each individual variable can be determined.

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

61.171

23.101

2.648

0.011

14.518

107.825

Exp (yrs)

-4.695

1.786

-2.629

0.012

-8.301

-1.088

TR (yrs)

0.229

2.988

0.077

0.939

-5.805

6.264

LSC

9.308

4.491

2.073

0.045

0.238

18.377

DB

6.228

4.458

1.397

0.170

-2.776

15.231

Size

0.001

0.412

0.002

0.998

-0.832

0.834

Loc

8.379

3.514

2.385

0.022

1.284

15.475

Int

579.082

115.225

5.026

0.000

346.380

811.783

Dev

8.552

2.654

3.223

0.002

3.193

13.911

Table4: Coefficients Table

## TEST OF SIGNIFICANCE:

Significance of each individual variable or factor can be obtained with the help of t-statistics. It is determined from the values of coefficient and standard errors.

T-stat= (coefficient-test value)/standard error

If the t-stat falls in the region of critical values then one cannot reject the null hypothesis (H0).

If it lies out side the critical region then null hypothesis can be rejected and alternative hypothesis is accepted.

From the above table, coefficient of each regressor and its standard errors can be observed. T-statistic can be calculated from the coefficient and standard error values and are determined as given in the table. Depending on the t stat value and the critical value of 1.96 at 95% confidence, the established hypotheses as mentioned below would be either accepted or rejected.

## Hypotheses:

H0: b1=0 (Experience of the project manager does not have any significant impact on the duration of the project).

H1: b1≠0 (Experience of the project manager has significant impact on the duration of the project).

H0: b2=0 (Training Record of building office towers does not have any significant impact on the duration of the project).

H1: b2≠0 (Training Record of building office towers has significant impact on the duration of the project).

H0: b3=0 (Type of contract (LSP) does not have any significant impact on the duration of the project).

H1: b3≠0 (Type of contract (LSP) has significant impact on the duration of the project).

H0: b4=0 (Type of contract (DB) does not have any significant impact on the duration of the project).

H1: b4≠0 (Type of contract (DB) has significant impact on the duration of the project).

H0: b5=0 (Size of the project does not have any significant impact on the duration of the project).

H1: b5≠0 (Size of the project has significant impact on the duration of the project).

H0: b6=0 (Location of the site does not have any significant impact on the duration of the project).

H1: b6≠0 (Location of the site has significant impact on the duration of the project).

H0: b7=0 (Interest rate does not have any significant impact on the duration of the project).

H1: b7≠0 (Interest rate has significant impact on the duration of the project).

H0: b8=0 (Reputation of developer does not have any significant impact on the duration of the project).

H1: b8≠0 (Reputation of developer has significant impact on the duration of the project).

From the table we can observe that, the t-stat values of the variables Exp= -2.629, type of contract (LSC) =2.073, Loc =2.385, Int = 5.026, and Dev= 3.233. These statistic values are greater than the critical value at 95% confidence level= 1.96. Thus these variables reject their null hypothesis (H0) and accept the alternative that they have a significant impact on the duration of office projects.

Thus of all the eight variables chosen only five variables have met the significance criteria. So the insignificant variables TR (track record of building office towers), type of contract (DB), and the size or the floor area are dropped from the original list and model is refined with the significant variables.

## Final Model:

Thus dropping those insignificant variables, the multiple regression technique is applied on the data of significant variables and the model is re-estimated as given below:

Y = 62.316 -4.52 (Exp) + 5.813 (LSC) + 8.125 (Loc) + 603.81 (Int) + 8.335 (Dev)

Regression Statistics

Multiple R

0.957

R Square

0.916

Adjusted R Square

0.907

Standard Error

10.580

Observations

50.000

Table5: Regression statistics

The final model thus obtained from the regression as shown above with all is significant variables has got the R2 value equals to 0.916 showing that 91.6% deviations in the duration of the office projects would be determined by the established model with the significant variables of Experience of the PM, Type of contract (LSC), Location of the site, Interest Rate and the Reputation of the Developer.

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

62.316

22.294

2.795

0.008

17.385

107.247

Exp (yrs)

-4.520

1.607

-2.812

0.007

-7.759

-1.280

LSC

5.813

3.571

1.628

0.111

-1.384

13.009

Loc

8.125

3.327

2.442

0.019

1.420

14.829

Int

603.810

103.899

5.811

0.000

394.414

813.205

Dev

8.335

2.520

3.307

0.002

3.256

13.414

Table6: Coefficients table

From the coefficients table we can found out that all the variables chosen have met the significance test with a deviation in the variable type of contract (LSC) which is a dummy variable.

## INTERPRETATION:

The model established thus says that any office project would take a minimum of 62weeks to complete. This can be interpreted from the value of intercept which determines the minimum value for the duration of the office project.

The coefficient found for example, a difference of 1 year experience of the project manager might lead to a reduction of about 4.5 weeks in the duration of the project. This is well supported by the literature as well saying that an experienced project manager would be handling the project effectively and efficiently, thus completing the project in the minimum possible duration.

Similarly, interest rate measured in terms of percentage shows that a change of 1% in the interest rate i.e., 0.01 increase in the Int would lead to a increment of 6 weeks in the duration of project. That is for generating a higher rate of return from the project, it might take a longer duration.

So is the reputation of the developer. As it is measured on a scale of 1 to 5 highlighting very good to poor project manager; the increment of rating by 1 scale would lead to an increment of 8 weeks in duration of project. That is highly reputed project manager might finish the project in the least possible time and as his rating increases, correspondingly the duration also increases.

In this way, the estimated model would help in the prediction of duration of the office project based on the variables Experience of the PM, Type of contract (LSC), Location of the site, Interest Rate and the Reputation of the Developer and helps in completion of the project more effectively and efficiently.

## CONCLUSION:

Thus with the help of all the discussion above, the final model shown below can be used to predict the estimates of duration of the office projects.

Y = 62.316 -4.52 (Exp) + 5.813 (LSC) + 8.125 (Loc) + 603.81 (Int) + 8.335 (Dev)