Statistics - Simple Linear and Multiple Linear Regression

Introduction to simple and multiple linear regression.

https://issuu.com/arbrylledisonparmodules/docs/archi203_par_report_multiple_and_simple_linear_reg

1. PRESENTED BY |
Advanced Quantitative Research in the Designed and Built Environment
Simple Linear and Multiple Linear Regression
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
BRYLL EDISON C. PAR
A.INTRODUCTION TO SIMPLE LINEAR REGRESSION
B.HOW TO PERFORM LINEAR REGRESSION
C.MULTIPLE REGRESSION

2. PRESENTED BY |
PART 1 –Introduction to Simple Linear Regression
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
BRYLL EDISON C. PAR

3. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: https://www.scribbr.com/statistics/simple-linear-regression/
IMAGE REFERENCES: From left-analyticsvidhya.com; Nwaogazie(2017)
Regression modelsdescribe the relationship between variables by fitting a line to the observed data.
Linear regression models use a straight line, while logistic and nonlinear regression models use a
curved line. Regression allows you to estimate how adependent variablechanges as the independent
variable(s) change.
FIGURE 1 –LINEAR REGRESSION MODEL FIGURE 2 –LOGISTIC AND NONLINEAR REGRESSION MODEL

4. Advanced Quantitative Research in the Designed and Built Environment
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INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
Linear regression attempts to model the relationship between two variables by fitting a linear equation to
observed data. One variable is considered to bean explanatory variable, and the other is considered to be
a dependent variable.
SOURCE: stat.yale.edu
IMAGE REFERENCES: From left-Scribbr; Auerkari, e at.,(2017); sphweb.bumc.bu.edu

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UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Objectives of Linear Regression
To establish if there is a relationshipbetween two
variables:
-More specifically, establish if there is a statistically
significant relationship between the two.
-Examples: Income and spending, wage and gender,
and student height and exam scores.
Forecast new observations:
-Can we use what we know about the relationship to
forecast unobserved values?
-Examples: What will our sales be for the next quarter?
What will the ROI of a new store opening be contingent
on store attributes.
Variable Roles The Magic
•Dependent Variable –Denoted by “y”
•Independent Variable –Denoted by “x”
Slope-intercept form
•y = a+bx
•y = mx+b
Linear Equation in Statistics
y =β0+ β1x
where:
β0 = Intercept/constant value
β1 = slope of x

6. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Note: We call it “linear equation” because the equation represents a
straight line in a bi-dimensional plot
Change in intercept
Change in slope
Slope-intercept form
•y = a+bx
•y = mx+b

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INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Simple Linear Regression Error Term
Linear Regression Model Linear Regression Equation
y =β0 + β1x + ε
where:
y = Dependent Variable
x = Independent Variable
β0 = Intercept/constant value
β1 = Coefficient/slope of x
ε= error term
y =β0 + β1x
where:
y = Dependent Variable
x = Independent Variable
β0 = Intercept/constant value
β1 = Coefficient/slope of x
Note: There is no error term since the error is assumed to be zero

8. PRESENTED BY |
PART 2 –How to Perform Linear Regression
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
BRYLL EDISON C. PAR

9. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Step 1. Compile the observations/true value on a table in the Microsoft Excel program and
save it as a CSV. File.

10. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)

11. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Step 2. Review the linear regression model and identify the independent as well as the
dependent variable.

12. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Linear Regression Model Explanation
y =β0 + β1x + ε
where:
y = Dependent Variable
x = Independent Variable
β0 = Intercept/constant value
β1 = Coefficient/slope of x
ε= error term
CONSUMPTION = β0 + β1 INCOME + ε
where:
y = Consumption
x = Income
β0 = Intercept/constant value
β1 = Coefficient/slope of x
ε= error term
Assumption: Income explains consumption

13. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Step 3. Find the coefficients of the constant and the independent variable
In this case an open-source statistical package will be used (Gretlsoftware)
The software may be downloaded on this link: http://gretl.sourceforge.net/win32/

14. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Open Gretlsoftware Click the file tab and hover to open data and user file

15. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
A window will pop up. Locate your csv file from your pc and
then click open
After you click open, you will be redirected to this window

16. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Select both the dependent and independent variable (income
and consumption) and then click the Beta icon below
Choose consumption by clicking the blue arrow pointing right

17. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Choose income by clicking the green arrow pointing right
below the blue arrow then click OK
A window will pop up showing the summary of the data you
need

18. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Gretlis an open-source statistical
package, mainly for econometrics.
Econometrics is the science or field of
knowledge that analyses data with
statistical models to test hypothesis and
reach conclusions.
y =48.77+ 0.85 x + ε
Consumption =48.77+ 0.85 income + ε

19. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Step 4. Forecast using gretl. Please follow the instructions

20. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Hover on the analysis tab and click forecast and wait for
another window to pop up
Check the value of the forecast range and click OK

21. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
A summary of the predictions, standards error and the intervals will open as well as the “Forecast evaluation statistics using 40
observations” not shown in this figure. The graph will also pop up as shown on the next slide

22. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Estimated vs. Actual Values

23. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com; GretlTutorial 1: Simple Linear Regression by dataminingincaeFound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
Step 5. Finalize result and proceed with conclusion

24. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Example: Family’s consumption of a given product (Relationship between the family’s income and
the consumption)
y =48.77+ 0.85 x + ε
Consumption =48.77+ 0.85 income + ε
Estimated model of consumption
48.77 = Interpreted consumption of a family with 0 income
0.85 = Marginal effect of one unit increase of income on consumption
x = It doesn’t have an intuitive interpretation meaning that in most cases we will actually beignoring it.
Conclusion:
Income will grow 0.85 for every unit increase in income. Ex: A family's income is 50 dollars more.
0.85x
(0.85)(50 dollars) = 42.5 dollars
It means that for every 50 dollars of income a family earns more per week, the consumption will grow on average
an expected of 42.5 dollars.

25. PRESENTED BY |
PART 3 –Multiple Regression
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
BRYLL EDISON C. PAR

26. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: https://www.scribbr.com/statistics/multiple-linear-regression/
Multiple linear regressionis used to estimate the relationship betweentwo or more independent
variablesandone dependent variable.
IMAGE REFERENCES: From left –Jacome (2016); scribbr

27. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Objectives of Multiple Linear Regression
You can use multiple linear regression when you want to
know:
•How strong the relationship is between two or more
independent variables and one dependent variable (e.g.
how rainfall, temperature, and amount of fertilizer added
affect crop growth)
You can use multiple linear regression when you want to
know:
•The value of the dependent variable at a certain value
of the independent variables (e.g.the expected yield of
a crop at certain levels of rainfall, temperature, and
fertilizer addition).
Variable Roles The Magic
•Dependent Variable –Denoted by “y”
•Independent Variable –Denoted by “x”
Slope-intercept form
•y = a+bx
•y = mx+b
Multiple Linear Regression Model
y =β0+ β1X1 + β2X2 + … βpXp+ ε
where:
β0 = Intercept/constant value
β1 = slope of x

28. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Multiple Regression Key Concept
Simple linear regression
(One to one relationship)
DVIV
Multiple regression
(Many to one relationship)
DV
IV IV
IV IV
… or more
Note: Adding more independent variables to a multiple regression procedure does not mean the regression will be
“better” or offer better predictions; in fact, it can make things worse. This is called “Overfitting”
Theaddition of more independent variables creates more relationships among them. So not only the independent
variable, but they are also potentially related to each other. Whenthishappen, it is called “Multicollinearity”

29. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Multicollinearity–the independent variables are correlated with each other.
The idealisforalltheindependentvariables to be correlated with the dependent variable but not with each other.
DV
IV
IV
IV
IV
Check for the relationship between each
independent variable and the dependent variable.
Consider alltherelationships between each
independent variables.
Multiple regression
(Many to one relationship)
Intisexample.10 relationships should be considered.
4relationshipsbetweenIVandDVandanotherand6
relationships between IV and IV
Note: The more Independent variable added the
relationships become numerous. Some independent
variables, or set of independent variables, are better at
predicting the dependent variable than others. Some
contribute nothing.
The ideal is for all the independent variables to be
correlated with the dependent variable but not with each
other.

30. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Facts
•Multiple regression is an extension of simple linear
regression.
•Two or more independent variable is used to
predict/explain the variance in one dependent variable.
•Two problems may arise:
•1. Overfitting: is caused by adding too many
independent variables; they account for more variance
but add nothing to the model
•2. Multicollinearity: happens when some/all the
independent variables are correlated with eachother.
•In multiple regression, each coefficient is interpreted as
the estimated change in y corresponding to a one unit
change in a variable, when all other variables are held
constant.
y =β0+ β1X1 + β2X2 + … βpXp+ ε
Multiple Linear RegressionModel
Sum of Linear Parameters Error Term
Multiple Linear RegressionEquation
y =β0+ β1X1 + β2X2 + … βpXp
Error Term is assumed to be zero
Estimated Multiple Linear
RegressionEquation
ŷ=b0+ b1X1 + b2X2 + … bpXp
b1,b2,…bpare the estimates of β1, β2,…βp

31. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
EXAMPLE 1: ESTIMATED MULTIPLE REGRESSION
EQUATION
Ŷ = 5.344 + 0.025 X1+ 0.234 X2–0.529 X3
(Standard form of a multiple regression equation)
ŷ=b0+ b1X1 + b2X2 + … bpXp
(Estimated multiple regression equation)
Estimates of a multiple regression model
Variables: X1,X2,andX3
Coefficients: 0.025,0.234, and -0.529
Intercept:5.344
EXAMPLE 2: INTERPRETTING COEFFICIENTS
Ŷ = 27 + 9 X1+ 12 X2
(Standard form of a multiple regression equation)
X1= Capital Investments (1000 usd)
X2=Marketing Expenditures (1000usd)
Ŷ = Predicted Exam Score (1000 usd)
Note: In multiple regression, each coefficient is
interpreted as the estimated change in y
corresponding to a one unit change in a variable, when
all other variables are held constant.
Inthis example, 9000 usdis an estimate of the
expected increase in sales y, corresponding to a 1000
usdincrease in capital investment (X1) when
marketing expenditure (X2) are held constant.

32. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Multiple Regression Pre-work/Data Preparation
1.Generate a list of potential variables; independent(s) and dependent.
2.Collect data on the variables.
3.Check the relationships between each independent variable and the dependent variable using scatterplots and
correlations.
4.Check the relationships between independent variables using scatterplots and correlations.
5.(Optional) Conduct simple linear regression for each independent and dependent pair.
6.Use the non-redundant independent variables in the analysis to find the best fitting model.
7.Use the best fitting model to make predictions about the dependent variable.

33. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service
RDSDataandVariableNaming
Toconductyouranalysis, youtake a random sample of 10 past trips and record four pieces of information for each
trip: 10 Total miles traveled, 2) number of deliveries, 3) the daily gas price, and 4) total travel time in hours.
Miles Traveled, (X1) Number of Deliveries, (X2) Gas Price, (X3) Travel Time (Hours), (y)
89 4 3.84 7
66 1 3.19 5.4
78 3 3.78 6.6
111 6 3.89 7.4
44 1 3.57 4.8
77 3 3.57 6.4
80 3 3.03 7
66 2 3.51 5.6
109 5 3.54 7.3
76 3 3.25 6.4

34. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service (Sketching out relationships)
Independent Variables
Miles
Traveled,
(X1)
Number of
Deliveries,
(X2)
Gas Price,
(X3)
Travel
Time, (y)
Dependent Variable
6 Relationships Should be Analyzed

35. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service –Dependent Variable vs Independent Variable Scatterplots (Using
Gretlapplication)
X1vs. y X2vs. y X3vs. y
R squared=0.862
PValue(F)=0.000
R squared=0.840
PValue(F)=0.000
R squared=0.071
PValue(F)=0.455

36. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service –Multicollinearity Scatterplots (Using Gretlapplication)
X1vs. y X2vs. y X3vs. y
R squared=0.914
PValue(F)=0.000
R squared=0.127
PValue(F)=0.313
R squared=0.248
PValue(F)=0.143

37. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service –Correlation Summary
Correlationanalysis confirmsthe conclusion reached by visual examination of the scatterplots
Redundantmulticollinear variables
•Miles Travelled and Number of Deliveries are both highly correlated with each other and therefore are redundant;
only one should be used in the multiple regression analysis.
Non-contributingvariables
•Gaspriceisnotcorrelatedwiththedependentvariableandshouldbeexcluded.
Note:Foreducationpurposes, all three relationships will be retained.

38. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service –Individual Summary Output
Travel Time (y) vs. Miles Travelled (X1)
Ŷ = 3.186+ 0.0403 (Miles Travelled)
Ŷ = 3.186+ 0.0403 X1
An increase in 1 mile will increase delivery
time by 0.0403 hours

39. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service –Individual Summary Output
Travel Time (y) vs. Number of Deliveries (X2)
Ŷ = 4.845+ 0.498 (Number of Deliveries)
Ŷ = 4.845+ 0.498 X2
An increase in 1 delivery will increase
delivery time by 0.498 hours

40. Advanced Quantitative Research in the Designed and Built Environment
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
INTRODUCTION TO SIMPLE LINEAR REGRESSION | HOW TO PERFORM LINEAR REGRESSION | MULTIPLE REGRESSION
SOURCE: Introduction to Simple Linear Regression by dataminingincaefound at YouTube.com
Sample Problem: Regional Delivery Service –Individual Summary Output
Travel Time (y) vs. Gas Price (X3)
Ŷ = 3.536+ 0.811 (Gas Price)
Ŷ = 3.536+ 0.811 X3
Gas Price is not a variable that contributes
to travel time. No need to explore this value.

41. PRESENTED BY |
Advanced Quantitative Research in the Designed and Built Environment
Simple Linear and Multiple Linear Regression
UNIVERSITY OF THE PHILIPPINES –DILIMAN | INTEGRATED GRADUATE PROGRAM (IGP) | URBAN DESIGN STUDIO LAB
BRYLL EDISON C. PAR
A.INTRODUCTION TO SIMPLE LINEAR REGRESSION
B.HOW TO PERFORM LINEAR REGRESSION
C.MULTIPLE REGRESSION