Question

4. The anscombe data set in the datasets R package (should automatically be loaded) contains 4 pairs of response-explanatory variables. The pairs are xl-yl, x2-y2, x3-y3, and x4-y4 where x is the explanatory variable and y is the response variable. (a) Run 4 simple linear regression analyses (one on each of the 4 pairs) to verify that the regression output is exactly the same (up to numerical accuracy) b) For each pair, describe what is wrong (if anything) and use a diagnostic plot to illustrate your point i. x1-yl ii. x2-y2 iii. x3-y3 iv. x4-y4

Answer 1

### dataset : ans combe » librar.(datasets) > data-"anscombe) > yl-anscombeSy1 x1-anscombe$x1 > y2-anscombeSy2 x2-anscombex2 > y3-anscombeSy3 >x3-anscombex3 > y4-anscombeSy4 > x4=anscombe5x4 > ## a) simple linear regression line on pairs > regi-lmay1-x1) ## first regression model > regi call: ImCfor mula yl x1) coefficients: (Intercept) 3.0001 0. 5001 ## yhat-3.000ltao. 5001 *x1) > > reg2-1m(y2-x2) ## second regression model > reg2 call: 1m(formula = y2 ~ x2) coefficients: (Intercept) 3.001 0. 500

> ## yhat-3.001+ (0.500*x2) regs-1mLY3-x3) ## third regression model > > reg3 Cal1; In (formula = y3 ~ x3) coefficients: (Intercept) 3.0025 0.4997 > ## yhat-3.002 54(0.4997*x3) reg4-1mCy4-X4) reg4 > ## fourth regression model > call: coefficients: (Intercept) x4. 0.4999 3.0017 > ## yhat-3.001 74(0.4999*x3) >from all regression models intercept value is approximately same round 1s 3 and slope is 0.500 approximately. but we take upto numeric accuracy then there is slight difference values

### b) Diagnostic plot : For model 1 librarvlggfortify) autoplot(regi) Residuals vs Fitted Normal Q-Q 10 10 -2 5678910あ Fitted values Theoretical Quantiles Sale-Location uals vs Leverage 10 05. 10 2 5 6 7 8 9 10 0.0 0.1 0.2 0.3 Fitted values Leverage ## normal qq plot is linear

### b) Diagnostic plot : For model 2 librarvlggfortify) autoplot(reg2) Residuals vs Fitted Normal Q-Q 1- UD 2 -1 10 Fitted values Theoretical Quantiles OScale-Location F Residuals vs Leverage UD UD 0.9 O-1 0.3 0.1 5 6 78 910 Fitted values 0.2 Leverage 0.0 0.3 ## normal qq plot is linear

### b) Diagnostic plot : For model 3 librarvlggfortify) autoplot(reg3) Residuals vs Fitted Normal Q-Q UD Fitted values Theoretical Quantiles OScale-Location F Residuals vs Leverage N 1.0 0.5 5 6 7 8 9 10 Fitted values 0.0 0.2 Leverage 0.3 ## normal qq plot is linear

### b) Diagnostic plot : For model 4 librarvlggfortify) autoplot(reg4) Residuals vs Fitted Normal Q-Q 7 8 9 10 1 12 CD Fitted values Theoretical Quantiles Scale-Location Residuals vs Leverage 1.25 1.00 UD 0.75- 0.50- 0.25- . 7 8910 11 12 D 0.00 0.25 0.50 0.75 1.00 Fitted values Leverage ## normal qq plot is linear ## all model are linear in trend