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Regression is a concept in Statistics used to measure the relationship between two variables, a response variable and predictor variable. The predictor variable has the influence over the response variable. Usually, it is the study of cause and effect of one variable over the other variable. A rise in price and the effect of it on demand is one example where the concept of regression can be applied. The quantitative effect that one variable exerts over the other is studied in regression.  

Interestingly, the first study on regression was about the stature of parents and their children, conducted by Sir Francis Galton during the late 19th century. The heights parents were compared. The study showed that a tall father had sons who were shorter than the father himself and the short father tended to have sons who were taller than him. The heights of the sons regressed to the mean. This means that the variables are imperfectly correlated. 

Regression analysis includes the following steps:

  • Statement of  problem
  • Selection of potentially relevant variables
  • Data collection
  • Model specification
  • Choice of fitting method
  • Mode fitting
  • Model validation and criticism
  • Using the chosen model(s) for the selection of the posed problem. 

     Linear Regression Graphs

Regression Formula:
Regression Equation(y) = a + bx
Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
Intercept(a) = (ΣY - b(ΣX)) / N
where
              x and y are the variables.
              b = the slope of the regression line
              a = the intercept point of the regression line and the y axis.
              N = Number of values or elements
              X = First Score
              Y = Second Score
              ΣXY = Sum of the product of first and Second Scores
              ΣX = Sum of First Scores
              ΣY = Sum of Second Scores
              ΣX2 = Sum of square First Scores 

 Regression Example: To find the Simple/Linear Regression of

 

X Values

Y Values

60

3.1

61

3.6

62

3.8

63

4

65

4.1

 To find regression equation, we will first find slope, intercept and use it to form regression equation..

Step 1: Count of the value is 5
            N = 5

Step 2: Find XY, X2
            See the below table 

X Value

Y Value

X*Y

X*X

60

3.1

60 * 3.1 = 186

60 * 60 = 3600

61

3.6

61 * 3.6 = 219.6

61 * 61 = 3721

62

3.8

62 * 3.8 = 235.6

62 * 62 = 3844

63

4

63 * 4 = 252

63 * 63 = 3969

65

4.1

65 * 4.1 = 266.5

65 * 65 = 4225
 

    Step 3: Find ΣX, ΣY, ΣXY, ΣX2.

            ΣX = 311
            ΣY = 18.6
            ΣXY = 1159.7
            ΣX2 = 19359

  Step 4: Substitute in the above slope formula given.
            Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
            = ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311)2)
            = (5798.5 - 5784.6)/(96795 - 96721)
            = 13.9/74
            = 0.19

   Step 5: Now, again substitute in the above intercept formula given.
            Intercept(a) = (ΣY - b(ΣX)) / N
            = (18.6 - 0.19(311))/5
            = (18.6 - 59.09)/5
            = -40.49/5
            = -8.098

  Step 6: Then substitute these values in regression equation formula
            Regression Equation(y) = a + bx
            = -8.098 + 0.19x.

Suppose if we want to know the approximate y value for the variable x = 64. Then we can substitute the value in the above equation.

            Regression Equation(y) = a + bx
            = -8.098 + 0.19(64).
            = -8.098 + 12.16
            = 4.06 

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