anscombe
Anscombe’s Quartet Of ‘Identical’ Simple Linear Regressions
A pre-loaded example dataset in R
Main page: https://www.rdocumentation.org/packages/datasets/versions/3.6.2/topics/anscombe
anscombe## x1 x2 x3 x4 y1 y2 y3 y4
## 1 10 10 10 8 8.04 9.14 7.46 6.58
## 2 8 8 8 8 6.95 8.14 6.77 5.76
## 3 13 13 13 8 7.58 8.74 12.74 7.71
## 4 9 9 9 8 8.81 8.77 7.11 8.84
## 5 11 11 11 8 8.33 9.26 7.81 8.47
## 6 14 14 14 8 9.96 8.10 8.84 7.04
## 7 6 6 6 8 7.24 6.13 6.08 5.25
## 8 4 4 4 19 4.26 3.10 5.39 12.50
## 9 12 12 12 8 10.84 9.13 8.15 5.56
## 10 7 7 7 8 4.82 7.26 6.42 7.91
## 11 5 5 5 8 5.68 4.74 5.73 6.89require(stats); require(graphics)
summary(anscombe)## x1 x2 x3 x4 y1
## Min. : 4.0 Min. : 4.0 Min. : 4.0 Min. : 8 Min. : 4.260
## 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 8 1st Qu.: 6.315
## Median : 9.0 Median : 9.0 Median : 9.0 Median : 8 Median : 7.580
## Mean : 9.0 Mean : 9.0 Mean : 9.0 Mean : 9 Mean : 7.501
## 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.: 8 3rd Qu.: 8.570
## Max. :14.0 Max. :14.0 Max. :14.0 Max. :19 Max. :10.840
## y2 y3 y4
## Min. :3.100 Min. : 5.39 Min. : 5.250
## 1st Qu.:6.695 1st Qu.: 6.25 1st Qu.: 6.170
## Median :8.140 Median : 7.11 Median : 7.040
## Mean :7.501 Mean : 7.50 Mean : 7.501
## 3rd Qu.:8.950 3rd Qu.: 7.98 3rd Qu.: 8.190
## Max. :9.260 Max. :12.74 Max. :12.500##-- now some "magic" to do the 4 regressions in a loop:
ff <- y ~ x
mods <- setNames(as.list(1:4), paste0("lm", 1:4))
for(i in 1:4) {
ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
## or ff[[2]] <- as.name(paste0("y", i))
## ff[[3]] <- as.name(paste0("x", i))
mods[[i]] <- lmi <- lm(ff, data = anscombe)
print(anova(lmi))
}## Analysis of Variance Table
##
## Response: y1
## Df Sum Sq Mean Sq F value Pr(>F)
## x1 1 27.510 27.5100 17.99 0.00217 **
## Residuals 9 13.763 1.5292
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Analysis of Variance Table
##
## Response: y2
## Df Sum Sq Mean Sq F value Pr(>F)
## x2 1 27.500 27.5000 17.966 0.002179 **
## Residuals 9 13.776 1.5307
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Analysis of Variance Table
##
## Response: y3
## Df Sum Sq Mean Sq F value Pr(>F)
## x3 1 27.470 27.4700 17.972 0.002176 **
## Residuals 9 13.756 1.5285
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Analysis of Variance Table
##
## Response: y4
## Df Sum Sq Mean Sq F value Pr(>F)
## x4 1 27.490 27.4900 18.003 0.002165 **
## Residuals 9 13.742 1.5269
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## See how close they are (numerically!)
sapply(mods, coef)## lm1 lm2 lm3 lm4
## (Intercept) 3.0000909 3.000909 3.0024545 3.0017273
## x1 0.5000909 0.500000 0.4997273 0.4999091lapply(mods, function(fm) coef(summary(fm)))## $lm1
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.0000909 1.1247468 2.667348 0.025734051
## x1 0.5000909 0.1179055 4.241455 0.002169629
##
## $lm2
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.000909 1.1253024 2.666758 0.025758941
## x2 0.500000 0.1179637 4.238590 0.002178816
##
## $lm3
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.0024545 1.1244812 2.670080 0.025619109
## x3 0.4997273 0.1178777 4.239372 0.002176305
##
## $lm4
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.0017273 1.1239211 2.670763 0.025590425
## x4 0.4999091 0.1178189 4.243028 0.002164602## Now, do what you should have done in the first place: PLOTS
op <- par(mfrow = c(2, 2), mar = 0.1+c(4,4,1,1), oma = c(0, 0, 2, 0))
for(i in 1:4) {
ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
plot(ff, data = anscombe, col = "red", pch = 21, bg = "orange", cex = 1.2,
xlim = c(3, 19), ylim = c(3, 13))
abline(mods[[i]], col = "blue")
}
mtext("Anscombe's 4 Regression data sets", outer = TRUE, cex = 1.5)
par(op)