Regression and Correlation

Transcription

1 Chapter 1-6 Regression and Correlation 1-6-1

2 Linear Regression and Correlation Analysis Not enough to know what impacts things but need to know how they impact. Correlation establishes if something impacts, regression establishes how it impacts. 1-6-

3 Topics 1 Linear Regression Correlation Curvilinear Regression Multiple Linear Regression 1-6-3

4 Linear Regression The regression equation is determined mathematically from data collected on a process. The regression equation predicts a value for the dependent variable, y, from the independent variable

5 Least Squares Regression Model 1-6-5

6 Linear Regression If there is a correlation the equation for that linear relationship can be determined from the data. y b b 0 1 In the equation above b 0 is the intercept and b 1 is the slope. The intercept is where the curve crosses the y ais. The slope is the change in y divided by the change in The values are calculated from the normal equations: 1-6-6

7 Normal Equations Determine slope (b 1 ) and intercept (b 0 ) Developed from data Solved simultaneously y nb 0 b 1 y b b

8 Lean Si Sigma Black Belt Slope and Intercept Equations Determine slope (b 1 ) and intercept (b 0 ) Developed from data n b n y b n y y n b

9 Regression Study Collect Data. Determine independent and dependent variables. Graph the data in a scatter diagram to determine if the data appears to be a straight line. (Not an obvious curve.) Proceed to analysis if the data is linear. Consider transforming data if not. Always be aware of outliers

10 Eample It is thought that abrasion loss in microns over time is a function of Rockwell hardness. Eight samples were taken. The data is shown in the table on the net page. Note that abrasion loss is the dependent variable

11 Data for Eample X Y Data Set

12 1-6-1

13 1-6-13

14 1-6-14

15 Eplanation R square is the coefficient of determination. It is the percent of variation that can be determined by the regression equation. Adjusted R square is an attempt to compensate for the sample size. (n -1) (1 r ) [n - (k 1)] Where k is n is the number of the sample size and independent variables Standard error is a measure of the variability

16 Eample Data Set X Y

17 Using Regression Equations Make sure there is a cause-effect relationship between the dependent and independent variables. If there is no significant linear correlation don t use the regression equation to make predictions. When using the regression equation for predictions stay within the scope of the available sample data. A regression equation based on old data is not necessarily valid now. Don t make predictions about a population that is different from the population from which the sample data were drawn

18 Outliers In a scatter plot an outlier is a point lying far away from the other data points. When one is noted it should be investigated. If there is an identifiable special cause of variation it may be discarded. (We know why it was different.)

19 Confidence The regression equation gives the best estimate of the predicted value. A confidence interval can be determined for the true value using the equation shown. Calling ŷ the predicted value and ŷ - E y ŷ E Where y thetrue value : E t / s e 1 1 n n( n( 0 ) ) ( ) The value of S e is given by Ecel

20 Interval Estimates for Different Values of y Prediction Interval for an individual y, given p Confidence Interval for the mean of y, given p p 1-6-0

21 Correlation Definition The coefficient of correlation, r, measures the strength of the relationship between two variables. High correlation indicates a strong relationship. High correlation does not indicate a cause-effect relationship

22 Correlation A correlation eists between two variables when one of them is related to the other in some way. 1-6-

23 Correlation Values r = +1 means a perfect direct relationship r = -1 means a perfect indirect relationship r = 0 means no relationship 1-6-3

24 Coefficient of Determination Indicates the proportion of the variation of y which is accounted for by Calculation: r 1-6-4

25 Tip Amount Positive Correlation Meal Amount 1-6-5

26 Properties of Linear Correlation Coefficient, r 1) The value is always between 1 and 1. ) The value of r does not change if all values of either variable are converted to a different scale. 3) The value of r is not affected by the choice of or y. 4) It measures the strength of a linear relationship. It is not designed to measure the strength of a relationship that is not linear

27 Linear Correlation Coefficient The linear correlation coefficient r measures the strength of the linear relationship between the paired and y values in a sample. It is calculated as shown below. The actual calculation is performed using these calculating devices. S S S y yy n n n y ( )( y ( ( ) y) y) r Sy ( S)( Syy) 1-6-7

28 Ecel 1-6-8

29 1-6-9

30 Eample Determine the coefficient of correlation between and y in the table on the net page

31 Eample X Y Data Set

32 Calculating r r Sy ( S)( Syy ) 9.86 (4.36)(1.86)

33 Significance of Coefficient of Correlation In order to answer whether or not the value of r that is calculated is significant, a test of hypothesis must be performed. H H 0 1 : r : r 0 0 z n 3 (1 r) ( )(ln ) (1 ) r

34 Eample Refer back to the data that gave us an r of -.56 based on 14 pairs of data. Can we say, with 95 percent confidence, that the r is significant?

35 Solution Methodology State Hypothesis Identify Test Statistic Specify Confidence Calculate Test Statistic Identify Table Value Compare Test and Table Statistics

36 Calculating r Hypothesis: H H 0 1 : r : r 0 0 Test Statistic: z ( n 3 )(ln (1 r) ) (1 r) z z 14 3 ( )(ln.10 (1.56) ) (1.56)

37 Calculating r 95% Table Z Value: Comparison and Conclusion: Since the calculated z (.10) is greater than 1.96, the test rejects H 0 and accepts H 1. Therefore, r is not equal to zero and correlation is significant

38 Eample An analyst observes a kitting operation and collects data on package volume and time required per unit for the operation. Determine if the correlation is significant at the.01 level. The data is on the net page

39 Data for Eample Data Set Operation Time/Unit Kit Volume

40 1-6-40

41 Curvilinear Regression Determines the relationship between one dependent and one independent variables when the relationship is not linear Transform data Proceed as if linear High correlation does not necessarily imply a cause effect relationship

42 Typical Curvilinear Models 1-6-4

43 Curvilinear Regression Normal Equations y b b b 0 1 y nb 0 b1 b y b 0 b1 b 3 y b 0 b1 b

44 Eample - Curvilinear X Y Data Set

45 Practice Data Set Y X

46 Multiple Linear Regression Determines the relationship between one dependent and two or more independent variables Methodology is similar Best done using appropriate statistical software (EXCEL)

47 Lean Si Sigma Black Belt Multiple Linear Regression Normal Equations b b b b y b b b b y b b b b y b b b nb y b b b b y

48 Graph of a Two-Variable Model Three dimension Y Y b 0 b1x1 bx X X

49 Eample Downtime/ Month Machine Speed Machine Age Data Set

50 Improving the Regression Evaluate the correlation of the independent variables

51 Eample Sum of Demand through t Population (DPopulation (Dayepartures (ChecArrivals (CheckOc Sum of Demand through 11:00am 1 Resort Population (Day of) Resort Population (Day before) Room Departures (Check-outs) Room Arrivals (Check-Ins) Rooms Occupied A high correlation between two independent variables indicates colinearity. Using both variables in the multiple regression equation may dilute the overall results of the regression equation. Colinearity is a situation where there is close to a near perfect linear relationship among some or all of the independent variables in a regression model. In practical terms, this means there is some degree of redundancy or overlap among your variables

52 Using all the Variables SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 7 Coefficients Standard Error Intercept Resort Population (Day of) Resort Population (Day before) Room Departures (Check-outs) Room Arrivals (Check-Ins) Rooms Occupied

53 Dropping one of the Variables Rooms Occupied SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 7 Coefficients Standard Error Intercept Resort Population (Day of) Resort Population (Day before) Room Departures (Check-outs) Room Arrivals (Check-Ins) This improves the adjusted r square and makes for a better fit

54 Multiple Regression Guidelines Use common sense and practical considerations to include or eclude variables. Include as few variables as possible. Use adjusted R to guide

55 Practice Problems (Data Set 1-6-6) An industrial engineer must use a regression based standard data system of work measurement to estimate the time required to cut various sizes of boards. Representative time studies were performed on sample sizes. The results of the studies are shown in the table on the right. The dimensions are in inches. The times are in minutes. What is the relationship? How good is it? Width Thick Time

56 Practice Problems The State Agricultural Etension Service has hired you, as a Si Sigma Black Belt to help improve a process. They desire to help soybean farmers increase the yield (in bushels per acre) from their fields based on certain easily measurable and controllable factors. We know that fertilizer sells for $5 per 100 pounds. Water costs $.18 per gallon. Lime costs $1.75 per 100 pounds. Soybeans sell for $6.85 a bushel. What would be the best combination of fertilizer, water, and lime to maimize the profit for the farmer? Don t limit the analysis to the eperimental values. Use the data collected on the following page

57 Soybean Data Bushels per Acre Pounds Fertilizer Gallons Water Pounds Lime Bushels per Acre Pounds Fertilizer Gallons Water Pounds Lime Data Set