Anscombe’s Quartet
Dario DeBlasis
2022-03-23
> fourdata <- read.table("C:/Users/Dario/Downloads/fourData.csv", header=TRUE,
+ stringsAsFactors=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)> summary(fourdata) x123 y1 y2 y3 x4
Min. : 4.0 Min. : 4.260 Min. :3.100 Min. : 5.39 Min. : 8
1st Qu.: 6.5 1st Qu.: 6.315 1st Qu.:6.695 1st Qu.: 6.25 1st Qu.: 8
Median : 9.0 Median : 7.580 Median :8.140 Median : 7.11 Median : 8
Mean : 9.0 Mean : 7.501 Mean :7.501 Mean : 7.50 Mean : 9
3rd Qu.:11.5 3rd Qu.: 8.570 3rd Qu.:8.950 3rd Qu.: 7.98 3rd Qu.: 8
Max. :14.0 Max. :10.840 Max. :9.260 Max. :12.74 Max. :19
y4
Min. : 5.250
1st Qu.: 6.170
Median : 7.040
Mean : 7.501
3rd Qu.: 8.190
Max. :12.500 Linear Regression
> RegModel.1 <- lm(y1~x4+x123, data=fourdata)
> summary(RegModel.1)
Call:
lm(formula = y1 ~ x4 + x123, data = fourdata)
Residuals:
Min 1Q Median 3Q Max
-1.87818 -0.18500 -0.01036 0.54964 1.88818
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.33291 2.21774 1.954 0.0865 .
x4 -0.09873 0.14012 -0.705 0.5011
x123 0.45073 0.14012 3.217 0.0123 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.273 on 8 degrees of freedom
Multiple R-squared: 0.686, Adjusted R-squared: 0.6075
F-statistic: 8.74 on 2 and 8 DF, p-value: 0.009718x123 is significant given by the * for an alpha of 0.05. The adjusted R-squared value is 0.607 which is a strong positve correlation between the variables. x123 is a good estimator of y1.
> RegModel.2 <- lm(y2~x4+x123, data=fourdata)
> summary(RegModel.2)
Call:
lm(formula = y2 ~ x4 + x123, data = fourdata)
Residuals:
Min 1Q Median 3Q Max
-1.5213 -0.5060 0.2522 0.7590 1.0156
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.4225 1.7937 3.581 0.00718 **
x4 -0.2535 0.1133 -2.236 0.05573 .
x123 0.3733 0.1133 3.294 0.01096 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.029 on 8 degrees of freedom
Multiple R-squared: 0.7946, Adjusted R-squared: 0.7433
F-statistic: 15.48 on 2 and 8 DF, p-value: 0.001779x123 is significant given by the * for an alpha of 0.05. The adjusted R-squared value is 0.743 which is a strong positve correlation between the variables. x123 is a good estimator of y2.
> RegModel.3 <- lm(y3~x4+x123, data=fourdata)
> summary(RegModel.3)
Call:
lm(formula = y3 ~ x4 + x123, data = fourdata)
Residuals:
Min 1Q Median 3Q Max
-1.2364 -0.6016 -0.1525 0.1159 3.1893
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.30291 2.26650 1.016 0.3393
x4 0.05182 0.14320 0.362 0.7268
x123 0.52564 0.14320 3.671 0.0063 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.301 on 8 degrees of freedom
Multiple R-squared: 0.6717, Adjusted R-squared: 0.5896
F-statistic: 8.184 on 2 and 8 DF, p-value: 0.01162x123 is significant given by the ** for an alpha of 0.01. The adjusted R-squared value is 0.589 which is a strong positve correlation between the variables. x123 is a good estimator of y3.
> RegModel.4 <- lm(y4~x4+x123, data=fourdata)
> summary(RegModel.4)
Call:
lm(formula = y4 ~ x4 + x123, data = fourdata)
Residuals:
Min 1Q Median 3Q Max
-1.6333 -0.7925 0.0000 0.7705 1.8775
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.96345 2.24292 0.875 0.40687
x4 0.53836 0.14171 3.799 0.00524 **
x123 0.07691 0.14171 0.543 0.60212
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.287 on 8 degrees of freedom
Multiple R-squared: 0.6785, Adjusted R-squared: 0.5982
F-statistic: 8.443 on 2 and 8 DF, p-value: 0.01068x4 is significant given by the ** for an alpha of 0.01. The adjusted R-squared value is 0.598 which is a strong positve correlation between the variables. x4 is a good estimator of y4.
Scatterplots
> scatterplot(y1~x123, regLine=FALSE, smooth=FALSE, boxplots=FALSE,
+ data=fourdata)
Linear regression is suitable for this relationship
> scatterplot(y2~x123, regLine=FALSE, smooth=FALSE, boxplots=FALSE,
+ data=fourdata)
Linear regression is not suitable for this relationship as the data seems to be exponentially distributed.
> scatterplot(y3~x123, regLine=FALSE, smooth=FALSE, boxplots=FALSE,
+ data=fourdata)
Linear regression could be suited but there is a major outlier in line 3.
> scatterplot(y4~x4, regLine=FALSE, smooth=FALSE, boxplots=FALSE,
+ data=fourdata)
Linear regression is not suitable and the line of best fit would run straight along the x axis if it were not for the outlier in line 8.
Boxplots
> Boxplot( ~ y1, data=fourdata, id=list(method="y"))
y1 is normally distributed. Even lengths on the whiskers and mean line split in the middle
> Boxplot( ~ y2, data=fourdata, id=list(method="y"))
[1] "8"y2 is not normally distributed. The bottom whisker is longer than the top whisker and there is an outlier at line 8.
> Boxplot( ~ y3, data=fourdata, id=list(method="y"))
[1] "3"y3 is normally distributed. The box has even width and the whiskers are evenly spread from eachother. However there is a large outlier at line 3.
> Boxplot( ~ y4, data=fourdata, id=list(method="y"))
[1] "8"y4 is normally distributed. The box has even width and whiskers are venely spread from eachother. However, there is a large outlier at line 8.
Conclusion
My conclusion from the Anscombe’s Quartet dataset is that further analysis must be conducted to fully undersatand how these variables are correlated. After looking at the linear regression it seemed that x123 was a good estimator of y1, y2, y3 and linear regression would be appropriate for all. However, further analysis was completed and by looking at the scatterplots and boxplots we examined many outliers and different relationships among the variables which wouldn’t make linear regression suitable. Therefore, it is important to not rely soley on linear regression and R-squared values and conduct further analysis and visualize the data to confirm your hypothesis.