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## Introduction to Regression Lecture 2.1

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**Introduction to RegressionLecture 2.1**• Review of Lecture 1.1 • Correlation • Pitfalls with Regression and Correlation • Introducing Multiple Linear Regression • Job times case study • Stamp sales case study • Homework Diploma in Statistics Introduction to Regression**Review of Lecture 1.1**Scatter plot of US mail handling data,exceptions deleted Diploma in Statistics Introduction to Regression**Always look ar your data!**"Although regression can be done without ever looking at a scatter plot, that is the statistical equivalent of flying blind" Amy Lap Mui Choi, JF MSISS, 1993/94. "Decision-making under risk is when you know what will probably happen and decision-making under uncertainty is when you probably know what will happen." Anon., JF MSISS 1995/96 Diploma in Statistics Introduction to Regression**Simple linear regression modelwith Normal model for chance**variation Y = α + βX + Diploma in Statistics Introduction to Regression**The prediction formula**Prediction equation: Prediction equation allowing for chance variation: Diploma in Statistics Introduction to Regression**Homework**Use the prediction formula to predict the extra manpower requirement during Christmas period, based on the experience of Period 7, Fiscal 1963, when Y was 1,070 and X was 270. Compare with actual. Comment. Diploma in Statistics Introduction to Regression**Application 1Confidence interval for marginal change**Recall confidence interval for m: or Confidence interval for b: Small sample: Diploma in Statistics Introduction to Regression**Diploma in Statistics**Introduction to Regression**Application 2Testing the statistical significance of the**intercept Formal test: H0: a = 0 Test statistic: Critical value: 2 (or t21, .05 = 2.08) Calculated value: 0.848 Comparison: Z < 2 (or t < 2.08) Conclusion: Accept H0 Diploma in Statistics Introduction to Regression**Testing the statistical significance of the intercept**Informal test: is less than its standard error, Draw a picture! Diploma in Statistics Introduction to Regression**More on Minitab results**Regression Analysis: Manhours versus Volume The regression equation is Manhours = 50.4 + 3.35 Volume Predictor Coef SE Coef T P Constant 50.44 59.46 0.85 0.406 Volume 3.3454 0.3401 9.84 0.000 S = 18.9300 Diploma in Statistics Introduction to Regression**Homework**In a study of a wholesaler's distribution costs, undertaken with a view to cost control, the volume of goods handled and the overall costs were recorded for one month in each of ten depots in a distribution network. The results are presented in the following table. Diploma in Statistics Introduction to Regression**Homework**The simple linear regression of costs (Y) on volume (X) was calculated, and resulted in the following numerical summary. Regression Analysis: Costs versus Volume The regression equation is Costs = 2.98 + 0.332 Volume Predictor Coef SE Coef T P Constant 2.982 1.646 1.81 0.108 Volume 0.33174 0.03182 10.42 0.000 S = 0.667603 Diploma in Statistics Introduction to Regression**Homework**(i) Draw a scatter plot for these data. Comment. Interpret the numerical summary in context. (ii) Calculate a prediction interval for costs next month when Volume in Depot 1 is planned to be £40,000, and Volume in Depot 2 is planned to be £51,000. (iii) Next month, when the two depots recorded volumes of £40,000 and £51,000 as planned, costs were £1,700 and £2,300 respectively. Comment on each case. Illustrate with an enhancement of your scatter plot. Diploma in Statistics Introduction to Regression**Homework Solution (i)**There appears to be a strong positive relationship between Costs and Volume. Diploma in Statistics Introduction to Regression**Homework Solution (i)**Costs increase approximately linearly with Volume, by around £33.20 for every £1,000 increase in Volume, from a base of around £300. (Costs = 2.98 + 0.332 Volume) The cost for a given volume is subject to chance variation with a standard deviation of around £67. (S = 0.667603) Diploma in Statistics Introduction to Regression**Homework Solution**(ii) Volume = £40,000, Costs (£1,491 , £1,759) Volume = £51,000, Costs (£1,857 , £2,124) (iii) £1,700 is within the corresponding prediction interval, satisfactory. £2,300 is outside the corresponding prediction interval, too high. An investigation is needed. Illustrate Diploma in Statistics Introduction to Regression**More precise formulas**Confidence interval for mean response: Prediction interval for next response: (ii) Volume = £40,000, Costs (£1,444 , £1,807) Volume = £51,000, Costs (£1,829 , £2,151) Diploma in Statistics Introduction to Regression**Standard error**• of prediction • of estimation Ref: "The Standard Error of Prediction" Extra Notes folder in mstuart/get or Diploma webpage Diploma in Statistics Introduction to Regression**Homework Solution**Diploma in Statistics Introduction to Regression**2. Correlation**• The correlation coefficient formula • r and reduction of prediction error • Positive and negative correlation • Perfect correlation • Conventional interpretations of r Diploma in Statistics Introduction to Regression**The correlation coefficient formula**Recall equivalently, Diploma in Statistics Introduction to Regression**Scatter plot showing zero correlation**Diploma in Statistics Introduction to Regression**Correlation r = 0.1 to r = 0.9**Data Desk Diploma in Statistics Introduction to Regression**r and reduction in prediction error**Diploma in Statistics Introduction to Regression**r and reduction in prediction error**Diploma in Statistics Introduction to Regression**Positive and negative correlation**Diploma in Statistics Introduction to Regression**Perfect correlation, positive and negative**Diploma in Statistics Introduction to Regression**Conventional interpretations of r**Science / Engineering: r > 0.9 is "interesting" Econometrics: r > 0.7 is "interesting", otherwise, r > 0.5 is "interesting" Sociology: r > 0.3 is "interesting" Recommendation: compare s to SY Diploma in Statistics Introduction to Regression**3. Pitfalls with regression and correlation**Diploma in Statistics Introduction to Regression**Anscombe's data summary**Diploma in Statistics Introduction to Regression**Anscombe's scatter plots**Diploma in Statistics Introduction to Regression**Homework**The shelf life of packaged foods depends on many factors. Dry cereal (such as corn flakes) is considered to be a moisture-sensitive product, with the shelf life determined primarily by moisture. In a study of the shelf life of one brand of cereal, packets of cereal were stored in controlled conditions (23°C and 50% relative humidity) for a range of times, and moisture content was measured. The results were as follows. Draw a scatter diagram. Comment. What action is suggested? Why? Diploma in Statistics Introduction to Regression**Following appropriate action, the following regression was**computed. The regression equation is Moisture Content = 2.86 + 0.0417 Storage Time Predictor Coef SE Coef T P Constant 2.86122 0.02488 115.01 0.000 Storage Time 0.041660 0.001177 35.40 0.000 S = 0.0493475 Calculate a 95% confidence interval for the daily change in moisture content; show details. Diploma in Statistics Introduction to Regression**Was the action you suggested on studying the scatter diagram**in part (a) justified? Explain. Predict the moisture content of a packet of cereal stored under these conditions for 3 weeks; calculate a prediction interval. What would be the effect on your interval of not taking the action you suggested on studying the scatter diagram? Why? Taste tests indicate that this brand of cereal is unacceptably soggy when the moisture content exceeds 4. Based on your prediction interval, do you think that a box of cereal that has been on the shelf for 3 weeks will be acceptable? Explain. What about 4 weeks? 5 weeks? What is acceptable? Diploma in Statistics Introduction to Regression**Reading**SA Sections 6.4, 6.5 Diploma in Statistics Introduction to Regression**4 Introducing Multiple Linear Regression**• SLR explaining variation in Y in terms of variation in X • MLR explaining variation in Y in terms of variation in several X 's Diploma in Statistics Introduction to Regression**Example 1What determines the taste of mature cheese?**• X1 = Acetic Acid • X2 = Hydrogen Sulphide • X3 = Lactic Acid • Y = Taste Score Diploma in Statistics Introduction to Regression**Example 2Explaining crime rates**Variable Description M percentage of males aged 14–24 So indicator variable for a southern state Ed mean years of schooling Po1 police expenditure in 1960 Po2 police expenditure in 1959 LF labour force participation rate M.F number of males per 1000 females Pop state population NW number of nonwhites per 1000 people U1 unemployment rate of urban males 14–24 U2 unemployment rate of urban males 35–39 GDP gross domestic product per head Ineq income inequality Prob probability of imprisonment Time average time served in state prisons Crime rate of crimes in a particular category per head of population Diploma in Statistics Introduction to Regression**Example 3Estimating tree volume / timber yield**For a sample of 31 black cherry trees in the Allegheny National Forest, Pennsylvania, measure • Y = volume (cubic feet), • X1 = height (feet) • X2 = diameter (inches) (at 54 inches above ground Diploma in Statistics Introduction to Regression**Example 4The Stamp Sales Case Study**The problem • January 1984, An Post established • New business plan; sales forecasts required • Historical sales data available bring in a consultant! Diploma in Statistics Introduction to Regression**Example 5A production prediction problem**• The problem • The data • Initial data analysis • dotplots • lineplots (time series plots) • scatterplot matrix • Model fitting / estimation • Model criticism • Application Diploma in Statistics Introduction to Regression**Erie Metal Products: The problem**Metal products fabrication: customers order varying quantities of products of varying complexity; customers demand accurate and precise order delivery times. Diploma in Statistics Introduction to Regression**Rear view**Stephan Clark Metal Products A specially designed cabinet Diploma in Statistics Introduction to Regression**Another view**Stephan Clark Metal Products Instrument casing Diploma in Statistics Introduction to Regression**Lockers**Stephan Clark Metal Products Instrument casing; oblique view Diploma in Statistics Introduction to Regression**Stephan Clark Metal Products**• "One customer is an international manufacturer of petrochemical equipment." • "Stephen Clark supplies painted metalwork components, panels and fabrications, which are used throughout the customer's product range." • "Stephen Clark plays an important part in them being able to cope with frequent scheduling changes." • "Through careful program management, we are able to offer excellent flexibility of supply, delivering finished product against weekly call-offs." Diploma in Statistics Introduction to Regression**Erie Metal Products: The data**Diploma in Statistics Introduction to Regression**The variables**• Response: • Jobtime, time (hours) to complete an order • Explanatory: • Units, the number of units ordered • Operations per Unit, the number of operations involved in manufacturing a unit, • Rushed, indicator of "rushed" priority status • Total Operations Units × Operations per Unit Diploma in Statistics Introduction to Regression**Initial data analysis, dotplots**Diploma in Statistics Introduction to Regression