Six Sigma: Regression Modelling

Six Sigma: Regression Modelling Linear regression Linear regression investigates and models the linear relationship between a response (Y) and predictor(s) (X). Both the response and predictors are continuous variables.

Quantifies the relationship between Y and X Y = a + bX Y = Dependent Variable or Response a = Intercept (Value of Y when X = 0) b = Slope X = Independent Variable or Predictor

Hour Studied (X) 20 24 46 62 22 37 45 27 65 23 371

Test Score % (Y) 40 55 69 83 27 44 61 33 71 37 520

XY

X2

Y2

800 400 1600 1320 576 3025 3174 2116 4761 5146 3844 6889 594 484 729 1628 1369 1936 2745 2025 3721 891 729 1089 4615 4225 5041 851 529 1369 21764 16297 30160

Inserting it into the formula will give a = 15.792 and b = 0.976 Now we can find value for Y in using value of a , b and any value for X Assume If someone studies for 50 hours then Test score will be 15.972 + 0.976(50) = 64.59

Scatterplot of Test Score vs Hour Studied 90 80

Test Score

70 60 50 40 30 20 20

30

40 50 Hour Studied

60

70

You will ask if someone studied for 20 hours as in the table then he should score 40 marks but if we apply on above formula it will give 15.792 + 0.976(20) = 35.31 which is not equal to 40.

This difference is called Residual.

Hour Studied (X) 20 24 46 62 22 37 45 27 65 23 371

Test Score % (Y) 40 55 69 83 27 44 61 33 71 37 520

Y=15.792+0.976(X) Residual 35.312 39.216 60.688 76.304 37.264 51.904 59.712 42.144 79.232 38.24 520.016

4.688 15.784 8.312 6.696 -10.264 -7.904 1.288 -9.144 -8.232 -1.24 -0.016

SS 21.977 249.135 69.089 44.836 105.350 62.473 1.659 83.613 67.766 1.538 707.436

Let us check the calculations using Minitab: Regression Analysis: Test Score versus Hour Studied The regression equation is Test Score = 15.8 + 0.976 Hour Studied Predictor Constant Hour Studied S = 9.40369

Coef 15.792 0.9760

SECoef 7.543 0.1868

R-Sq = 77.3%

T 2.09 5.22

P 0.070 0.001

R-Sq(adj) = 74.5%

Analysis of Variance Source Regression Residual Error Total

DF 1 8 9

SS 2412.6 707.4 3120.0

MS 2412.6 88.4

We found the same results using Minitab.

F 27.28

P 0.001

Multiple linear regression

Y = a +bX1 + cX2 + dX3 +eX4‌‌..

Thirty-two pieces of cotton cellulose were produced at different settings of HCHO concentration (Conc), catalyst ratio (Ratio), curing temperature (Temp), and curing time (Time). The durable press rating (Rating), a measure of wrinkle resistance, was recorded for each product. The linear associations between the response, Rating, and the predictors, Conc, Ratio, Temp, and Time, were assessed by multiple linear regression.

Conc Ratio Temp Time Rating (X1) (X2) (X3) (X4) (Y) 8 4 100 1 1.40 2 4 180 7 2.20 7 4 180 1 4.60 10 7 120 5 4.90 7 4 180 5 4.60 7 7 180 1 4.70 7 13 140 1 4.60 5 4 160 7 4.50 4 7 140 3 4.80 5 1 100 7 1.40 8 10 140 3 4.70 2 4 100 3 1.60 4 10 180 3 4.50 6 7 120 7 4.70 10 13 180 3 4.80 4 10 160 5 4.60 4 13 100 7 4.30 10 10 120 7 4.90 5 4 100 1 1.70 8 13 140 1 4.60 10 1 180 1 2.60 2 13 140 1 3.10 6 13 180 7 4.70 7 1 120 7 2.50 5 13 140 1 4.50 8 1 160 7 2.10 4 1 180 7 1.80 6 1 160 1 1.50 4 1 100 1 1.30 7 10 100 7 4.60 4 1 100 1 1.40 4 1 100 1 1.45

Regression Analysis: Rating versus Conc, Ratio, Temp, Time The regression equation is Rating = - 0.759 + 0.157 Conc + 0.216 Ratio + 0.0107 Temp + 0.0960 Time Predictor Constant Conc Ratio Temp Time

Coef -0.7591 0.15691 0.21633 0.01072 0.09601

S = 0.806964

SE Coef 0.7317 0.06296 0.03145 0.004594 0.05422

R-Sq = 73.1%

T -1.04 2.49 6.88 2.33 1.77

P 0.309 0.019 0.000 0.027 0.088

R-Sq (adj) = 69.1%

The interpretation of the regression equation follows:  The slope (X1 = 0.157) is the change in Rating when Concentration increases by 1. That is, when the Concentration increases by one unit, the Rating increases by 0.157 units.  The slope (X2 = 0.216) is the change in Rating when Ratio increases by 1. That is, when the Ratio increases by one unit, the Rating increases by 0.216 units.  The slope (X3 = 0.0107) is the change in Rating when Temperature increases by 1. That is, when Temperature increases by one unit, the Rating increases by 0.0107 units.  The slope (X4 = 0.0960) is the change in Rating when Time increases by 1. That is, when Time increases by one unit, the Rating increases by 0.0960 units.  The constant (intercept) value (a =  0.759) is the predicted value of Rating when each predictor (Conc, Ratio, Temp, Time) is zero. That is, when the predictors are zero the Rating is  0.759.

Scatterplot of Rating vs Conc, Ratio, Temp, Time 5

Variable Conc Ratio Temp Time

Rating

4

3

2

1 0

50

100 X-Data

150

200

We can see that temp has the least slope and is different from the rest of the factors.

General regression General regression investigates and models the relationship between a response (Y) and predictor(s) (X). The response must be continuous, but you can have both continuous and categorical predictors. A pharmaceutical company wants to model the relationship between the salaries of research scientists with their gender, years of experience, and an index of publication quality. The company wants to test the interactions between all predictors.

Salary Gender Publication Years 88 F 4.5 11 77.3 M 7.8 6 75.3 M 4.3 7 96.4 M 8.8 20 87.7 M 5.1 8 58.1 F 2.1 3 63.1 F 2.4 4 58.5 M 2.6 2 95.4 F 8.2 16 92 F 8.8 14 94.5 M 6.6 12 103 M 8.4 18 94 F 5.8 14 64 M 3.2 5 77.3 M 3.7 10 83.7 M 5.3 8 74.8 F 6.4 7 92.5 F 5.6 17 79.7 M 5.9 8 64 F 2.5 5 94.3 M 6.8 14 100.2 M 7.9 19 87.7 M 4.3 13 71.3 F 3.3 6 97 M 4.7 21 83.4 F 7.9 8 90.8 M 5.5 12 78.5 F 5.8 5 97.3 M 4.7 19 61.6 F 4.3 3 65.3 F 5.5 2 89.7 F 6.6 10 83.8 M 5.8 7 104.5 M 8.5 18 78.6 M 5.6 7

Regression Equation Gender: F

Salary = 39.9669 + 4.17447 Publication + 3.68777 Years - 0.293194 Publication*Years

M

Salary = 43.6179 + 4.50745 Publication + 2.60465 Years - 0.18553 Publication*Years

You can interpret each slope value as the change in salary when the predictor increases by 1. For example, for females, when publication quality increases by one unit, salary increases by 4.17447.

You can interpret each intercept value as the predicted value of salary when each predictor is zero. For example, for males, when each predictor is zero, salary is 43.6179

Scatterplot of Salary vs Publication, Years 110

Variable Publication Publication Years Years

100

Gender F M F M

Salary

90

80

70

60 0

5

10 X-Data

15

20

Gender is not an impacting factor.

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