REGRESSION ANALYSIS

• REGRESSION ANALYSIS

• BUSINESS STATISTICS/NET COMMERCE/NET MANAGEMENT
• DR. SHASHI AGGARWAL
• INTRODUCTION
• THE DICTIONARY MEANING OF THE TERM REGRESSION IS THE ACT OF RETURNING OR GOING BACK. THE TERM REGRESSION WAS FIRST USED IN 1877 BY FRANCIS GALTON WHILE STUDYING THE RELATIONSHIP BETWEEN THE HEIGHT OF FATHERS AND SONS.
• HIS STUDY OF HEIGHT OF ABOUT ONE THOUSAND FATHERS AND SONS REVEALED A VERY INTERESTING RELATIONSHIP.
• PUBLISHED THE RESULT IN A PAPER “ REGRESSION TOWARDS MEDIOCRITY IN HEREDITARY STATURE.

• RESULTS OF GALTON STUDY

1. TALL FATHERS TEND TO HAVE TALL SONS AND SHORT FATHER SHORT SONS
2. BUT THE AVERAGE HEIGHT OF SONS OF THE TALL FATHERS IS LESS THAN THAT OF TALL FATHERS AND THE AVERAGE HEIGHT OF THE SONS OF SHORT FATHERS IS GREATER THAN THAT OF THE SHORT FATHERS.
3. GALTON’S STUDIES REVEALED THAT OFFSPRING OF ABNORMALLY TALL OR SHORT PARENTS TEND TO REVERT OR STEP BACK TO THE AVERAGE HEIGHT OF THE POPULATION : REGRESSION TO MEDIOCRITY
4. THE LINE DESCRIBING THIS TENDENCY TO REGRESS GOING BACK WAS CALLED BY GALTON REGRESSION LINE.

• CONCLUSION
• IF AVAERAGE HEIGHT OF A CERTAIN GROUP OF FATHERS IS a CMS ABOVE ( BELOW) THE GENERAL AVERAGE HEIGHT THEN AVERAGE HEIGHT OF THEIR SONS WILL BE aXr cms ABOVE OR (BELOW ) THE GENERAL AVERAGE HEIGHT WHERE r is THE CORRELATION COEFFICIENT BETWEEN THE HEIGHT OF GIVEN GROUP OF FATHERS AND THEIR SONS. HERE CORRELATION IS POSITIVE SINCE r’s values lies between +1 to -1  WE HAVE axr<=a
• REGRESSION MEANING

1.  HAS MUCH WIDER PERSPECTIVE WITHOUT ANY REFERENCE TO BIOMETRY
2. MEANS THE ESTIMATION OR PREDICTION OF THE UNKNOWN VALUE OF ONE VARIABLE FROM THE KNOWN VALUE OF THE OTHER VARIABLES
3. TO FIND OUT THE AVERAGE PROBABLE CHANGE IN ONE VARIABLE GIVEN A CERTAIN AMOUNT OF CHANGE IN ANOTHER.
4. IN ECONOMICS IT IS THE BASIC TECHNIQUE FOR MEASURING OR ESTIMATING THE RELATIONSHIP AMONG ECONOMIC VARIABLES

5. THE STATISTICAL  TOOL WHICH IS EXTENSIVELY USED IN ALMOST ALL NATURAL SCIENCES ---NATURAL,SOCIAL AND PHYSICAL
6. SPECIALLY USED IN BUSINESS AND ECONOMICS
7. THE ESTIMATION OF PREDICTION OF FUTURE PRODUCTION,CONSUMPTION,PRICES,INVESTMENTS,SALES,PROFITS AND INCOME ARE OF PARAMOUNT IMPORTANCE
8. POPULATION ESTIMATES AND POPULATION PROJECTIONS ARE VERY MUCH IMPORTANT FOR EFFICIENT PLANNING

• DEFINITION OF REGRESSION
• M.M BLAIR,” REGRESSION ANALYSIS IS A MATHEMATICAL MEASURE OF THE AVERAGE RELATIONSHIP BETWEEN TWO OR MORE VARIABLES IN TERMS OF THE ORIGINAL UNITS OF THE DATA
• TWO TYPES OF VARIABLES :

1. DEPENDENT VARIABLE : WHOSE VALUE IS INFLUENCED OR IS TO BE PREDICTED
2. INDEPENDENT VARIABLE : VARIABLE WHICH INFLUENCES THE VALUE OR IS USED FOR  PREDICTION

·         INDEPENDENT VARIABLE IS ALSO KNOWN AS REGRESSOR/PREDICTOR/EXPLANATORY

·         DEPENDENT VARIABLE IS KNOWN AS REGRESSES OR EXPLAINED VARIABLE

• SIMPLE REGRESSION
  SIMPLE ANALYSIS : CONFINED TO THE STUDY OF ONLY TWO VARIABLES AT A TIME IS TERMED AS SIMPLE REGRESSION
• MULTIPLE REGRESSION : FOR STUDYING MORE THAN TWO VARIABLES AT A TIME IS TERMED AS MULTIPLE REGRESSION
• LIKE YIELD OF CROP DEPENDS ON THE RAINFALL, FERTILITY OF THE LAND,MANURE USED
• THE COST OR PRICE OF A PRODUCT DEPENDS ON THE PRODUCTION AND ADVERTISING EXPENDITURE
• LINEAR REGRESSION

1. THE MATHEMATICAL EQUATION OF THE REGRESSION CURVE

2.       ( REGRESSION EQUATION) ENABLE US TO STUDY THE AVERAGE CHANGE IN THE VALUE OF THE DEPENDENT VARIABLE FOR ANY GIVEN VALUE OF THE INDEPENDENT VARIABLE

3. IF THE REGRESSION CURVE IS STRAIGHT LINE WE SAY LINEAR REGRESSION BETWEEN THE VARIABLES UNDER STUDY
4. THE EQUATION OF SUCH CURVE IS THE EQUATION OF A STRAIGHT LINE THAT IS FIRST DEGREE EQUATION IN THE VARIABLE X AND Y
5. IN CASE OF LINEAR REGRESSION THE VALUES OF THE DEPENDENT VARIABLE INCREASE BY A CONSTANT ABSOLUTE AMOUNT FOR UNIT CHANGE IN THE VALUE OF THE INDEPENDENT VARIABLE

• NON LINEAR REGRESSION
• IF THE CURVE OF REGRESSION IS NOT A STRAIGHT LINE : KNOWN AS CURVED  OR NON LINEAR REGRESSION
• THE REGRESSION EQUATION WILL BE A FUNCTIONAL RELATION BETWEEN x and Y  INVOLVING THE TERMS IN x AND Y OF DEGREE HIGHER THAN ONE LIKE
• X², Y², XY

• IMPORTANCE

1. ESTIMATES OF VALUES OF THE DEPENDENT VARIABLES FROM THE VALUES OF INDEPENDENT VARIABLES.
2. MEASURE OF THE ERROR INVOLVED IN USING THE REGRESSION LINE AS A BASIS FOR ESTIMATION.
3. CAN OBTAIN A MEASURE OF THE DEGREE OF ASSOCIATION/CORRELATION THAT EXIST BETWEEN THE TWO VARIABLES
4. THE COEFFICIENT OF DETERMINATION MEASURES THE STRENGTH OF RELATION EXIST BETWEEN THE TWO VARIABLES

• DIFFERENCE BETWEEN CORRELATION AND REGRESSION
• CORRELATION
• MEASURE OF DEGREE OF RELATIONSHIP BETWEEN X AND Y
• TOOL OF ASCERTAINING THE DEGREE OF RELATIONSHIP
• REGRESSION
• STUDY OF NATURE OF RELATIONSHIP

• CAUSE AND EFFECT RELATION
• LINES OF REGRESSION
• LINE OF REGRESSION IS THE LINE WHICH GIVES THE BEST ESTIMATE OF ONE VARIABLE FOR ANY GIVEN VALUE OF THE OTHER VARIABLE
• TWO VARIABLES x and y TWO LINES OF REGRESSION
• GRAPHIC TECHNIQUE TO SHOW THE FUNCTIONAL RELATION BETEWEEN TWO VARIBALES X AND Y IE . DEPENDENT AND INDEPENDENT VARIABLES. IT SHOWS THE AVERAGE RELATIONSHIP BETWEEN TWO VARIABLES. LINE OF AVERAGE
• REGRESSION LINES ARE BASED ON REGRESSION EQUATIONS. THESE ARE ALSO KNOWN AS ESTIMATING EQUATIONS.
• TWO REGRESSION EQUATION

1. X ON Y
2. Y ON X

• REGRESSION  LINES
• DEFINITION:
• LINE OF REGRESSION OF Y ON X IS THE LINE WHICH GIVES THE BEST ESTIMATE FOR THE VALUE OF Y FOR ANY SPECIFIED VALUE OF X
• SIMILARLY LINE OF REGRESSION OF X ON Y IS THE LINE WHICH GIVES THE BEST ESTIMATE FOR THE VALUE OF X FOR ANY SPECIFIED ANY VALUE OF Y
• MEANING OF BEST FIT
• THE TERM BEST FIT IS INTERPRETED IN ACCORDANCE WITH THE PRINCIPLE OF LEAST SQUARE WHICH CONSISTS IN MINIMIZING THE SUM OF THE SQUARE OF THE RESIDUALS OR THE ERRORS OF ESTIMATES .THAT IS THE DEVIATION BETWEEN THE GIVEN OBSERVED VALUES OF THE VARIABLES AND THEIR CORRESPONDING ESTIMATING VALUES AS GIVEN BY THE LINE OF BEST FIT

• METHODS OF CALCULATING
• THROUGH NORMAL EQUATION
• X ON Y
• SO THE EQUATION IS :-
• X_C = a +bY 
• ∑ X =Na +b ∑ Y
• ∑ XY =a ∑ Y + b∑ Y²       
• Y ON X    
• Y_C = a +bX
• ∑ Y=Na +b ∑ X
• ∑ XY =a ∑ X + b∑ X²
•  

• SOLUTION
• X_C = a +bY 
• ∑ X =Na +b ∑ Y   , = 31 =8a+24b  --------1
• ∑ XY =a ∑ Y + b∑ Y²       =,= 83=24 a +108 b----2
•   

•     MULTIPLY BY 1 by 3
• 93=24a + 72 b------iii
• 83 =24a +108b  -----ii
• SUBTRACT iii out of ii
• SOLUTION
• 10  = -36b
• b = -10/36, =-5/18=-.28
• 8 a + 24 b =31
• 8 a +24 x(-.28) = 31
• 8a -6.72 =31
• 8a=31 +6.72
• 8a=37.72
• a= 37.72=8=4.715 ,REGRESSION EQUATION X ON Y=4.715-.028 Y
• Y_C = a +bX
  SOLUTION Y ON X
• Y_C = a +bX
• ∑ Y=Na +b ∑ Y   ,   24 =8a +31b---i

• ∑ XY =a ∑ X + b∑ Y²          
•     83=31 a + 153 b-------ii   
•  MULTIPLY i by 31 AND ii BY 8
• SOLUTION OF Y ON X
• 248 a +1224b =664
• 248 a + 961 b =744
• 263 b =-80
• b =-80/263 =-.30,
• 24= 8a +31 b
• 8a + 31X-.30 =24, 24=8a -9.3
• 8a= 24+9.3=33.3 =8a. , a= 33.3/8=4.162
• Y =4.1625 -.30 X