Question

Do four regression analyses using the Anscombe Quartet (1973) Data Analysis Tool in Excel Any Statistical package Summarize the results and compare those using the original data. How would you run regression analysis differently? Remove some data, if necessary Linear or curvilinear (quadratic or cube)? Re-run regression analysis Summarize the results

Answer 1

Do four regression analyses using the Anscombe Quartet (1973) Data Analysis Tool in Excel Any Statistical package Summarize the results and compare those using the original data. How would you run regression analysis differently? Remove some data, if necessary Linear or curvilinear (quadratic or cube)? Re-run regression analysis Summarize the results

Answer:

R RGui (64-bit) File History Resize Windows R R Graphics: Device 2 (ACTIVE) tled-R Editor R xl-c (10, 8, 13, 9, ll, 14, 6, 4, 12, 7, 5) .04,6.95, 7.58,8.81.8.33. , 7.24,4.26,10.84,4.82,s.68) 2=c ( 9.14,8.14 x3= x1 .3, 3 . 1, 9. 13, 7.26,4 .74) , 81.8.84.6.08.5.39,8.15,6.42,5.73) x4-c (rep (8 , 7) , 19, rep (8,3) ) 4-c (6.58, 5.76, 7 .71, 8.84,8.47, 7.04,5.25,12.5,5.56, 7.91, 6.89) -"p") abline (1m (yl~xl) ) plot (x2, y2, type="p") 4 6 10 12 4 10 12 14 x1 "p") abline (1m (y3-x3) ) plot (x4, y4, type= "p") abline (1m (y4-x4) regi-sunmary (lm (yl-xl)) reg2-summary (1m (y2-x2)) reg3-summary (1m (y3 x3)) reg4-summary (1m (y4 x4) 6 4 10 12 12 14 16 18 X4

Then run the linear regression.

regl Call: lm (formula yl xl) Residuals: Min 10 Median 3Q Max -1.92127-0.45577 -0.04136 0.70941 1.83882 Coefficients: Estimate Std. Error t value Pr (>I t |) 1.1247 (Intercept) 3.0001 2.667 0.02573 xl 0.5001 0.1179 4.241 0.00217 1 0.1 Signif. codes: 0 0.001 0.01 0.05 Residual standard error: 1.237 on 9 degrees of freedom 0.6665, Multiple R-squared: F-statistic: 17.99 on 1 and 9 DF Adjusted R-squared: 0.6295 p-value: 0.00217

> reg2 Call: lm (formula y2 x2) Residuals: Min Median 30 Max -1.9009 -0.7609 0.1291 0.9491 1.2691 Coefficients Estimate Std. Error t value Pr (>I t |) (Intercept) 3.001 1.125 2.667 0.02576 x2 0.500 0.118 4.239 0.00218 1 0.1 Signif. codes: 0 0.001 0.01 0.05 Residual standard error: 1.237 on 9 degrees of freedom Multiple R-squared 0.6662, Adjusted R-squared: 0.6292 F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002179

> reg3 Call: lm (formula = v3 x3) Residuals: Min Median 3Q Max -1.1586 -0.6146 -0.2303 0.1540 3.2411 Coefficients: Estimate Std. Error t value Pr (> tl) (Intercept) 3.0025 1.1245 2.670 0.02562 x3 0.4997 0.1179 4.239 0.00218 0.05 . Signif. codes: 0 1*** 0.001 0.01 0.1 Residual standard error: 1.236 on 9 degrees of freedom Multiple R-squared 0.6663, Adjusted R-squared: 0.6292 F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002176

reg4 Call: = v4 x4) lm (formula Residuals: 30 Min 10 Median Max 0.000 1.839 -1.751 -0.831 0.809 Coefficients: Estimate Std. Error t value Pr (>It|) 3.0017 1.1239 2.671 (Intercept) 0.02559 * 0.1178 0.00216** x4 0.4999 4.243 1 Signif. codes: 0 0.001 0.01 0.05 . 0.1 Residual standard error: 1.236 on 9 degrees of freedom Adjusted R-squared 0.6667, Multiple R-squared: 0.6297 18 on 1 and 9 DF, F-statistic: p-value: 0.002165

For all the 4 cases we see that the Multiple R-squared values are 0.66 and around it. This is because of the presence of outlier in the data set which can be clearly seen from the scatter plots.

So, now we remove the outlier and re-run the regression.

Data Set 2

Now, the multiple R-squared value is 1. So there is a perfect relationship exists between x2and y2.

RRGui (64-bit) File Edit Packages Windows Help RRGraphics Device 2 (ACTIVE) R Untitled R Editor v4-c (6.58,5.76,7.71, 8.84,8.47,7.04,5.25,12.5,5.56,7.9 RR Console par (mfrow-c (2,2 )) reg.new2-m (y2poy(x, plot (xl,yl, type="p"I abline (1m (yl xl)) plot (x2,y2, type-"p") >summary (reg.new2) Call: plot (x3, v3, Eype = "p " ) abline (1m (y3-x3)) plot (x4,y4, type-"p") abline (1m (y4 x4) ) poly (x2, 2)) 1m (formula y2 Residualsa: Min 10 Median 30 Max 0.0023776 -0.0013287 -0.0011888 -0.0006294 0.0008741 regl-summary (1m (yl-xl)) reg2- summary (1m (y2~x2 )) Coefricienta: reg3-summary (lm (y3-x3) Estimate Std. Error t value Pr (>|t) reg4-summary (1m (y4-x4)) 7.5009091 0.0005043 <2e-16 (Intercept 1875 ... <2e-16 * 2) 2-3.7116396 poly (x2, 0.0016725 -2219 reg.new2-1m (y2~poly (x2 , 2)) summary (reg.new2) 0.001 0.01 0.05 .0.1 '1 0 Signir, codes: degrees of freedom Residual standard error: 0.001672 on Multiple R-squared: Adjusted R-squared: 1 P-value:< 2.2e-16 F-statistic: 7.378e+06 on 2 and 8 DF,

So, we have seen that after removing the outlier the multiple R-squared become 1 indicating a perfect linear relationship exhibited.

R RGui (64-bit) X File Edit Packages Windows Help RRGraphics Device 2 (ACTIVE R Untitled R Editor y2=c (9.14,8.14,8.74,8.77,9.26, 8.1, 6.13, 3.1,9.13,7.26, 4.74) Console x3-x1 y3 c(7.46, 6.77,12.14, 7.81, 8.84, 6.08,5.39, 8.15,6.42,5.73) nlot (x3. v3.Evpe y4=c (6.58, 5 .76,7.71, 8 .84,8.47,7.04,5.25, 12.5,5.56, 7.91,6.89) abline (1m (y3-x3) ] >plot (x4, y4, cype"p") par (mfrow c (2,2)) plot (x1,yl, type="p") abline (1m (yl~xl) svEC (10.9.9.11.14,6,4,12,7,5) > y3.new-c (7.46, 6.77, 7.11,7.81,8.84,6.08,5.39,0.15,6.42, 5.73) > reg.new3-1m (y3.newx3.new) >summary (reg.new3) ablins/n2x2)1 Call: plot (x3, y3, type="p") abline (1m (y3-x3) ) plot (x4, y4, type- "p") abline (1m (y4-x4)) lm (formula- y3.new - x3.new) Residuals: Min Median 3с Нах -0.0041558 -0.0022240 0.0000649 0.0018182 0.0050649 - summary (1m (y1~x1) ) req2=summary (1m (y2-x2)) reg3-summary ( lm (y3~x3) ) reg4-summary (1m (y4 x4)) rec Coefficients: value Pr (>It ) Estimate Std. Error 2e-16 (Intercept) 4.0056494 o.0029242 1370 x3.new 0.3453896 0.0003206 1077 <2e-16 0 0.001 0.01 0.05 .0.1 ' 1 .hew2=1m ( y2 poly (x2 , 2) ) .new2) Signif. codes: rror: 0.0030 reedom MulEinle B-mared Adiusted B-8muared: x3.new-c(10,8,9, 11, 14, 6,4, 12, 7, 5) y3.new-c (7.46,6.77,7.11,7.81,8.84,6.08,5.39, 8.15, 6.42,5.73) reg.new3=1m (y3.new ~x3.new) summary (reg.new3) F-statistic: 1.161e+06 on 1 and 8 DF, p-value: < 2.2e-16

Data Set 4

R RGui (64-bit) File Edit Packages Windows Help aphics: Device 2 (ACTIVE R Untitled REditor RR Console par (mfrow=c ( 2, 2 )) plot (x1, y1, type"p") 76.7.71.8.84.0.47,7.04,5.25,5.56, 7.91, 6.89) >reg.new4-1m (y4.new-x4.new) >summary (reg.newi) abline (1m (y2 x2)) plot (x3, y3, type="p") abline (1m (y3 ~x3) ) plot (x4, y4, type="p") abline (1m (y4-x4)) Call: 1m (formula=v4.new x4.new) Residuals 10 Median 30 Max 1.839 regi summary (lm (y1 ~x1)) reg2-summary (lm (y2 ~x2)) reg3-summary (1m (y3x3)) reg4 summary (1m (y4~x4)) -1.751 -1.036 -0.036 0.859 Coefficients: (1 not defined because of singularities) st Std 17.92 2.39-08 * o 3908 (Intercept) 0010.ר x4.new NA NA NA NA reg.new2-lm (y2 poly (x2,2) ) summary (reg.new2) Signif. codes: 0 0.001 0.01 0.05 0.1 1 Residual standard error: 1.236 on 9 degrees of freedom x3.new-c (10,8,9, 11,14, 6, 4, 12, 7, 5) y 3.new-c ( 7.46, 6.77, 7 . 11,7.81,8.84,6.08, 5.39, 8 .15, 6.42, 5 . 73) reg.new3-lm (y3.new x3.new) summary (reg.new3) x4.new-c (rep (8,10) ) y4.new-c (6.58,5.76,7.71,8.84,8.47,7.04, 5.25,5.56, 7.91, 6.89) reg.new4=lm ( y4 . new~x4.new) summary (reg.new4)

Here, after removing the outlier we can see that there exists a singularity in x values so that no regression can be fitted and hence no multiple R-squared values can be found but as the standard deviation of x becomes zero.