ANLY 512 - Problem Set 2
Xiaohan Hou
9/12/2017
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 name your final output .html file as: YourName_ANLY512-Section-Year-Semester.html and upload 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.
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 6.89data <-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)3.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( x3, y3, main="Scatter Plot of x3 and y3 data", xlab="x3", ylab="y3")
plot( x4, y4, 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
attach(anscombe)## The following objects are masked from anscombe (pos = 3):
##
## x1, x2, x3, x4, y1, y2, y3, y4par(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(lm4, test="Chisq")Analysis of Variance Table
Response: x4 Df Sum Sq Mean Sq F value Pr(>F)
y4 1 73.338 73.338 18.003 0.002165 ** Residuals 9 36.662 4.074
— 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.
Learned that these four datasets are similar but have dramatically different graphs. Anscombe’s Quatet makes the dataset easier to be visualized and better understanding the data