research methodology-chapter 14
DESCRIPTION
TRANSCRIPT
![Page 1: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/1.jpg)
CHAPTER-14
INTRODUCTION TO CORRELATION & REGRESSION ANALYSIS
ByDR. PRASANT SARANGI
![Page 2: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/2.jpg)
Key concepts:
Introduction to Correlation Analysis Rank Correlation Linear Regression Analysis Multiple Regression Analysis
![Page 3: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/3.jpg)
CORRELATION ANALYSIS
• Positive Correlation• Negative Correlation• Linear Correlation and • Non-linear Correlation
![Page 4: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/4.jpg)
Positive Correlation
• Two variables are said to be positively correlated when the movement of the one variable leads the movement of the other variable in the same direction.
• There exists direct relationship between the two variables.
![Page 5: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/5.jpg)
Negative Correlation
• Correlation between two variables is said to be negative when the movement of one variable leads to the movement in the other variable in the opposite direction.
• Here there exists inverse relationship between the two variables.
![Page 6: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/6.jpg)
Linear Correlation
• The correlation between two variables is said to be linear where the points when drawn is a graph represents a straight line.
• Non-linear Correlation A relationship between two variables is said to be non-linear if a unit change
in one variable causes the other variable to change in fluctuations. If X is changed then corresponding values of Y will not change in the same
proportion.
![Page 7: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/7.jpg)
Methods of Measuring Correlation
• The Graphical MethodThe correlation can be graphically shown by using scatter diagrams. Scatter diagram reveals two important useful information. Firstly, through this diagram, one can observe the patterns between two
variables which indicate whether there exists some association between the variables or not.
Secondly, if an association between the variables is found, then it can be easily identified regarding the nature of relationship between the two (whether two variables are linearly related or non-linearly related).
![Page 8: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/8.jpg)
• Karl Pearson’s Coefficient of Correlation Karl Pearson’s coefficient of correlation (developed in 1986) measures
linear relationship between two variables under study. Since, the relationship is expressed is linear, hence, two variables change in a fixed proportion. This measure provides the answer of the degree of relationship in real number, independent of the units in which the variables have been expressed, and also indicates the direction of the correlation.
![Page 9: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/9.jpg)
• Direct method
22ii
iiXY
yx
yxr
Assumed Mean Method
2222 )()(
))((
YYXX
YXYXXY
ddnddn
ddddnr
![Page 10: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/10.jpg)
• Grouped Data
2222 )()(
))((
YYXX
YXYXXY
fdfdnfdfdn
fdfddfdnr
![Page 11: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/11.jpg)
Assumptions of Coefficient of Correlation
1. The Value of the Coefficient of Correlation Lies between -1 (minus one) to +1 (plus one).
2. The Value of the Coefficient of Correlation is Independent of the Change of Origin and Change of Scale of Measurement
2222 )()(
)()(
iiii
iiiiXY
kknhhn
khkhnr
![Page 12: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/12.jpg)
Rank Correlation Coefficient
There are three different situations of applying the Spearman’s rank correlation coefficient.
• When ranks of both the variables are given• When ranks of both the variables are not given and • When ranks between two or more observations in a series are
equal
![Page 13: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/13.jpg)
• When Ranks of Both the Variables are Given
)(
61
61
2
2
3
2
nnn
dor
nn
dRXY
When Ranks of both the Variables are not Given
•In such cases, each observation in the series is to be ranked first.
•The selection of highest value depends on the researcher.
• In other words, either the highest value or the lowest value will be ranked 1 (one) depends upon the decision of the researcher.
![Page 14: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/14.jpg)
• When Ranks between Two or More Observations in a Series are Equal• The ranks to be assigned to each observation are an average of the ranks
which these observations would have got, if they differed from each other.
)1(
......)(12
1)(
12
1)(
12
16
12
3332
321
31
2
nn
mmmmmmd
RXY
![Page 15: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/15.jpg)
Simple Linear Regression Model
![Page 16: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/16.jpg)
What do we use regression models for:
1. Estimate a relationship among economic variables, such as y = f(x).
2. Test hypotheses
3. Forecast or predict the value of one variable, y, based on the value of another variable, x.
![Page 17: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/17.jpg)
Dependent and Independent Variables
Dependent variable - the variable we are trying to explain
Independent (or explanatory) variables - variables that we think cause movements in the dependent variable
![Page 18: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/18.jpg)
Simple Regression Model
Y = dependent variableX = independent variable
Model is: Y = α + Xα is the intercept or constant is the slope coefficient
![Page 19: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/19.jpg)
Linearity
Models that are linear in the variables and in the coefficients:Y = α + X
Models that are nonlinear in the variables but linear in the coefficients: Y = α + X2
![Page 20: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/20.jpg)
Models that are nonlinear in the variables and in the coefficients:Y = α + X
Some models that are nonlinear can be made linear in the coefficients:
Y = e α X
take logs:ln Y = α + ln X
![Page 21: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/21.jpg)
r
{α
E(Y|X)
E(Y|X)
AverageExpenditure
X (income)
E(Y|X)= α + X
=E(Y|X)
X
An Example showing income and average expenditure
![Page 22: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/22.jpg)
Error Term
Y is a random variable composed of two parts:
I. Systematic component: E(Y) = α + X This is the mean of Y.
II. Random component: u = Y - E(Y | X) = Y - α - X
u is called the stochastic or random error.
Together E(Y) and u form the model: Y = α + X + u
![Page 23: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/23.jpg)
Sources of error term
• Dependent variable measured with error• Model left out relevant variables• Wrong functional form• Inherent randomness of behaviour
![Page 24: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/24.jpg)
True Relationship
u4
Y
X
E(Y)= α + X
Y4
Y1
Y3
Y2
X1 X2 X3 X4
u1
u2
u3
![Page 25: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/25.jpg)
The Estimated Model
We use the data on Y and X to come up with guesses for α and . These estimated parameters or coefficients are
α and cap
^ ^
![Page 26: Research Methodology-Chapter 14](https://reader035.vdocuments.us/reader035/viewer/2022081715/547f1811b4af9ff2498b4732/html5/thumbnails/26.jpg)
Our estimated, or “fitted”, model gives the predicted value for Y for any given X:
Yi = α + Xi
The residual is the difference between the actual or observed value of Y and the predicted value:
ui = Yi - Yi = Yi - α - Xi
^ ^ ^
^ ^ ^ ^