【正文】 tested and confirmed IMPROVE 3 You Are Here Develop a focused problem statement Process Door Versus Data Door Organize potential causes Hypothesis Testing and Regression Analysis Design of Experiments and Response Surface Green Belt MaterialsReview Green Belt Materials Simple Regression –Theory Review Bridge Materials Transformed Data MLR Curvilinear Discrete Xs Logistic 4 Table of Contents Introduction to Linear Regression. Simple Linear Regression Regression Analysis With Transformed Data Introduction to Multiple Linear Regression Working With Multiple Regression General Procedure for Multiple Regression Introduction to Curvilinear Regression Review of Regression Appendix Regression With Discrete Xs Logistic Regression (Discrete Ys) Introduction to Linear Regression 6 Regression: Quantifies the Relationship Between X and Y Regression analysis generates a line that quantifies the relationship between X and Y. 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 X (input) Y (output) Appropriate Data for X or Y In Regression Data Type Minitab Format DiscreteOrdinal ranks 1, 2, ..., 5 Numerical DiscreteCount or Percents of defects, % defective Numerical Continuous Amounts Cycle time Numerical The line, or regression equation, is represented by mathematical equation of a straight line as: Y= bo+ b1X bo = intercept (where the line crosses X= 0) b1 = slope (rise over run, or change in Y per unit increase in X) 7 Caution! Extrapolating Beyond the Data Is Risky 20 30 40 250 300 350 Feed Screw (rpm) Carton Weight (grams) ? ? ? ? ? What is the relationship between X and Y for X 30? Range of data ? Extrapolation is making predictions outside the range of the X data – It’s a natural desire but it’s like walking from solid ground onto thin ice ? Predictions from regression equations are more reliable for Xs within the range of the observed data ? Extrapolation is less risky if you have a theory, process knowledge, or other data to guide you 8 Fitting a Line Through Data: The Residuals X 0 5 10 15 0 5 10 15 Y residual2 residual7 ? Observed Y (actual Y) o Predicted Y (fitted or expected Y— on the line) ? A residual Is the vertical distance from each data point to the regression line Equals (Observed Y – Predicted Y) Represents mon cause (= random = unexplained) variation 9 How the Regression Equation Is Determined The least squares method The regression equation is determined by a procedure that minimizes the total squared distance of all points to the line. ? Finds the line where the squared vertical distance from each data point to the line is as small as possible (or the ―least‖) ? Restated…minimizes the ―square‖ of all the residuals ? Regression uses the least squares method to determine the ―best line‖: ? Data (both X and Y values) are used to obtain b0 and b1 values ? The b0 and b1 values establish the equation ? We will use Minitab to do this Least squares method 1. Measure vertical distance from points to line 2. Square the figures 3. Sum the total squared distance 4. Find the line that minimizes that sum 10 A Note on Terminology These terms are often used interchangeably: ?Regression equation* ?Regression line ?Prediction equation ?Prediction line ?Fitted line or fits* ?Model 11 RSquared (RSq or R2): The % Explained Variation RSquared = Rsq ? Measures the percent of variation in the Yvalues that is explained by the linear relationship with X ? Ranges from 0 to 1 (= 0% to 100%) V ari ati on Ex pl ai ned % 100 x n d)V ari ati oUnex pl ai ne(E x pl ai ned T otal V ari ati on Ex pl ai ned sqR ???Y 15 Total Variation in Y X 0 5 10 15 0 5 10 Think of this distance conceptually as the explained variation*。 you39。 range = 2 to 15 ? Correlation, r = .634 ? Residuals seem OK。t really control the number of setups, but you could manage it for improvement and you could definitely use it for prediction 2c, d, e (interpretation of plots and output): Time plots of both variables showed no obvious trends. (Control charts were also checked because the mean and control limits make it easier to look for trends and other special causes。 it increased as staff level (X) increased. Do a square root transformation on Y to overe this problem. 55 Practice: Use Other Transformations with Regression, cont. Instructions: 1. Use Minitab to create a column of transformed data: Calc Calculator Sqrt(CallsAnswd) 2. Include the column of transformed data in the regression. 56 Summary of Regression Analysis with Transformed Data ?Transforming X or Y or both may be used to: ? Straighten out relationships between X amp。 remaining variation is unexplained, and presumed to result from mon causes 12 Correlation (r): The Strength of the Relationship The correlation, r: ?Ranges from 1 to 1 r = –1 = perfect negative or inverse relationship r = 0 = no linear relationship r = +1 = perfect positive relationship ?Measures the ―strength‖ of the relationship ?R2 is equal to square of r ?Known as Pearson’s correlation coefficient 13 Correlation (r): The Strength of the Relationship, cont. Note: If the slope b1 = 0, then r = 0. Otherwise there is no relationship between the slope value b1 and the correlation value, r. XYXYXYXYXYXYStrong Positive Correlation r = .95 R2 = 90% Moderate Positive Correlation r = .70 R2 = 49% No Correlation r = .006 R2 = .0036% Other Pattern No Line