ProblemSet2
Kirti
4/22/2019
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- Anscombes quartet is a set of 4 \(x,y\) data sets that were published by Francis Anscombe in a 1973 paper Graphs in statistical analysis. For this first question load the
anscombedata that is part of thelibrary(datasets)inR. And assign that data to a new object calleddata.
#install.packages("datasets", repos = "http://cran.us.r-project.org")
library(datasets)
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 6.89View(anscombe)
data <- anscombe
View(data)- Summarise the data by calculating the mean, variance, for each column and the correlation between each pair (eg. x1 and y1, x2 and y2, etc) (Hint: use the
fBasics()package!)
#install.packages("fBasics")
library(fBasics)## Loading required package: timeDate## Loading required package: timeSeriesView(data)
summary(data)## x1 x2 x3 x4
## Min. : 4.0 Min. : 4.0 Min. : 4.0 Min. : 8
## 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 8
## Median : 9.0 Median : 9.0 Median : 9.0 Median : 8
## Mean : 9.0 Mean : 9.0 Mean : 9.0 Mean : 9
## 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.: 8
## Max. :14.0 Max. :14.0 Max. :14.0 Max. :19
## y1 y2 y3 y4
## Min. : 4.260 Min. :3.100 Min. : 5.39 Min. : 5.250
## 1st Qu.: 6.315 1st Qu.:6.695 1st Qu.: 6.25 1st Qu.: 6.170
## Median : 7.580 Median :8.140 Median : 7.11 Median : 7.040
## Mean : 7.501 Mean :7.501 Mean : 7.50 Mean : 7.501
## 3rd Qu.: 8.570 3rd Qu.:8.950 3rd Qu.: 7.98 3rd Qu.: 8.190
## Max. :10.840 Max. :9.260 Max. :12.74 Max. :12.500x1<-data[,1]
x2<-data[,2]
x3<-data[,3]
x4<-data[,4]
y1<-data[,5]
y2<-data[,6]
y3<-data[,7]
y4<-data[,8]
#mean(x1)
#mean(x2)mean(x3)mean(x4)mean(y1)mean(y2)mean(y3)mean(y4)
#var(x1)var(x2)var(x3)var(x4)var(y1)var(y2)var(y3)var(y4)
colMeans(data)## x1 x2 x3 x4 y1 y2 y3 y4
## 9.000000 9.000000 9.000000 9.000000 7.500909 7.500909 7.500000 7.500909colVars(data)## x1 x2 x3 x4 y1 y2 y3
## 11.000000 11.000000 11.000000 11.000000 4.127269 4.127629 4.122620
## y4
## 4.123249C1 <- cor(data$x1, data$y1)
C1## [1] 0.8164205C2 <- cor(data$x2, data$y2)
C3 <- cor(data$x3, data$y3)
C4 <- cor(data$x4, data$y4)- Create scatter plots for each \(x, y\) pair of data.
plot(x1, y1, main = "Correlation1",
xlab = "X1", ylab = "Y1",
pch = 11, frame = FALSE)
plot(x2, y2, main = "Correlation2",
xlab = "X2", ylab = "Y2",
pch = 11, frame = FALSE)
plot(x3, y3, main = "Correlation3",
xlab = "X3", ylab = "Y3",
pch = 12, frame = FALSE)
plot(x4, y4, main = "Correlation4",
xlab = "X4", ylab = "Y4",
pch = 13, frame = FALSE)
- Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
par(mfrow=c(2,2))
plot(x1, y1, main = "Correlation1",
xlab = "X1", ylab = "Y1",
pch = 19, frame = FALSE)
plot(x2, y2, main = "Correlation2",
xlab = "X2", ylab = "Y2",
pch = 19, frame = FALSE)
plot(x3, y3, main = "Correlation3",
xlab = "X3", ylab = "Y3",
pch = 19, frame = FALSE)
plot(x4, y4, main = "Correlation4",
xlab = "X4", ylab = "Y4",
pch = 19, frame = FALSE)
- Now fit a linear model to each data set using the
lm()function.
Model1 <- lm(y1~x1, data = data)
summary(Model1)##
## Call:
## lm(formula = y1 ~ x1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.92127 -0.45577 -0.04136 0.70941 1.83882
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.0001 1.1247 2.667 0.02573 *
## x1 0.5001 0.1179 4.241 0.00217 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.237 on 9 degrees of freedom
## Multiple R-squared: 0.6665, Adjusted R-squared: 0.6295
## F-statistic: 17.99 on 1 and 9 DF, p-value: 0.00217Model2 <- lm(y2~x2, data = data)
summary(Model2)##
## Call:
## lm(formula = y2 ~ x2, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9009 -0.7609 0.1291 0.9491 1.2691
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.001 1.125 2.667 0.02576 *
## x2 0.500 0.118 4.239 0.00218 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## 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: 0.002179Model3 <- lm(y3~x3, data = data)
summary(Model3)##
## Call:
## lm(formula = y3 ~ x3, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.1586 -0.6146 -0.2303 0.1540 3.2411
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.0025 1.1245 2.670 0.02562 *
## x3 0.4997 0.1179 4.239 0.00218 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 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: 0.002176Model4 <- lm(y4~x4, data = data)
summary(Model4)##
## Call:
## lm(formula = y4 ~ x4, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.751 -0.831 0.000 0.809 1.839
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.0017 1.1239 2.671 0.02559 *
## x4 0.4999 0.1178 4.243 0.00216 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.236 on 9 degrees of freedom
## Multiple R-squared: 0.6667, Adjusted R-squared: 0.6297
## F-statistic: 18 on 1 and 9 DF, p-value: 0.002165- Now combine the last two tasks. Create a four panel scatter plot matrix that has both the data points and the regression lines. (hint: the model objects will carry over chunks!)
par(mfrow=c(2,2))
plot(x1, y1, main = "Correlation1",
xlab = "X1", ylab = "Y1",
pch = 19, frame = FALSE, abline(Model1))
plot(x2, y2, main = "Correlation2",
xlab = "X2", ylab = "Y2",
pch = 19, frame = FALSE, abline(Model2))
plot(x3, y3, main = "Correlation3",
xlab = "X3", ylab = "Y3",
pch = 19, frame = FALSE, abline(Model3))
plot(x4, y4, main = "Correlation4",
xlab = "X4", ylab = "Y4",
pch = 19, frame = FALSE, abline(Model4))
- Now compare the model fits for each model object.
anova(Model1)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
anova(Model2)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
anova(Model3)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
anova(Model4)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