ANLY 512 - Problem Set 2
Anscombe’s quartet
Arti Gamara
2017-07-25
Objectives
The objectives of this problem set is to orient you to a number of activities in R. And to conduct a thoughtful exercise in appreciating the importance of data visualization. For each question create a code chunk or text response that completes/answers the activity or question requested. Finally, upon completion post your assignment on Rpubs and upload a link to it to the “Problem Set 2” assignmenet on Moodle.
Questions
- 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.
library(datasets)
data <- anscombe- 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!)
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. :12.500- Create scatter plots for each \(x, y\) pair of data.
attach(anscombe)
plot( x1, y1, main="Scatter Plot of x1 and y1 data", xlab="x1", ylab="y1")
plot( x2, y2, main="Scatter Plot of x2 and y2 data", xlab="x2", ylab="y2")
plot( x1, y1, main="Scatter Plot of x3 and y3 data", xlab="x3", ylab="y3")
plot( x1, y1, main="Scatter Plot of x4 and y4 data", xlab="x4", ylab="y4")
- 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="Scatter Plot of x1 and y1 data", xlab="x1", ylab="y1", pch=16)
plot( x2, y2, main="Scatter Plot of x2 and y2 data", xlab="x2", ylab="y2", pch=16)
plot( x3, y3, main="Scatter Plot of x3 and y3 data", xlab="x3", ylab="y3", pch=16)
plot( x4, y4, main="Scatter Plot of x4 and y4 data", xlab="x4", ylab="y4", pch=16)
- Now fit a linear model to each data set using the
lm()function.
lm1 <- lm( x1~y1, data)
summary(lm1)##
## Call:
## lm(formula = x1 ~ y1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6522 -1.5117 -0.2657 1.2341 3.8946
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.9975 2.4344 -0.410 0.69156
## y1 1.3328 0.3142 4.241 0.00217 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.019 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: 0.00217lm2 <- lm( x2~y2, data)
summary(lm2)##
## Call:
## lm(formula = x2 ~ y2, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8516 -1.4315 -0.3440 0.8467 4.2017
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.9948 2.4354 -0.408 0.69246
## y2 1.3325 0.3144 4.239 0.00218 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.02 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: 0.002179lm3 <- lm( x3~y3, data)
summary(lm3)##
## Call:
## lm(formula = x3 ~ y3, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.9869 -1.3733 -0.0266 1.3200 3.2133
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.0003 2.4362 -0.411 0.69097
## y3 1.3334 0.3145 4.239 0.00218 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.019 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: 0.002176lm4 <- lm( x4~y4, data)
summary(lm4)##
## Call:
## lm(formula = x4 ~ y4, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.7859 -1.4122 -0.1853 1.4551 3.3329
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.0036 2.4349 -0.412 0.68985
## y4 1.3337 0.3143 4.243 0.00216 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.018 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(lm1)
plot(lm2)
plot(lm3)
plot(lm4)
- Now compare the model fits for each model object.
anova(lm1, test="Chisq")Analysis of Variance Table
Response: x1 Df Sum Sq Mean Sq F value Pr(>F)
y1 1 73.32 73.320 17.99 0.00217 ** Residuals 9 36.68 4.076
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1
anova(lm2, test="Chisq")Analysis of Variance Table
Response: x2 Df Sum Sq Mean Sq F value Pr(>F)
y2 1 73.287 73.287 17.966 0.002179 ** Residuals 9 36.713 4.079
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1
anova(lm3, test="Chisq")Analysis of Variance Table
Response: x3 Df Sum Sq Mean Sq F value Pr(>F)
y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1
anova(lm3, test="Chisq")Analysis of Variance Table
Response: x3 Df Sum Sq Mean Sq F value Pr(>F)
y3 1 73.296 73.296 17.972 0.002176 ** Residuals 9 36.704 4.078
— Signif. codes: 0 ‘’ 0.001 ’’ 0.01 ’’ 0.05 ‘.’ 0.1 ‘’ 1
- In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization.