Questions

  1. 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 anscombe data that is part of the library(datasets) in R. And assign that data to a new object called data.
library(datasets)
data <- anscombe
  1. 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 of data
library(fBasics)
## Loading required package: timeDate
## Loading required package: timeSeries
basicStats(data)
##                    x1        x2        x3        x4        y1        y2
## nobs        11.000000 11.000000 11.000000 11.000000 11.000000 11.000000
## NAs          0.000000  0.000000  0.000000  0.000000  0.000000  0.000000
## Minimum      4.000000  4.000000  4.000000  8.000000  4.260000  3.100000
## Maximum     14.000000 14.000000 14.000000 19.000000 10.840000  9.260000
## 1. Quartile  6.500000  6.500000  6.500000  8.000000  6.315000  6.695000
## 3. Quartile 11.500000 11.500000 11.500000  8.000000  8.570000  8.950000
## Mean         9.000000  9.000000  9.000000  9.000000  7.500909  7.500909
## Median       9.000000  9.000000  9.000000  8.000000  7.580000  8.140000
## Sum         99.000000 99.000000 99.000000 99.000000 82.510000 82.510000
## SE Mean      1.000000  1.000000  1.000000  1.000000  0.612541  0.612568
## LCL Mean     6.771861  6.771861  6.771861  6.771861  6.136083  6.136024
## UCL Mean    11.228139 11.228139 11.228139 11.228139  8.865735  8.865795
## Variance    11.000000 11.000000 11.000000 11.000000  4.127269  4.127629
## Stdev        3.316625  3.316625  3.316625  3.316625  2.031568  2.031657
## Skewness     0.000000  0.000000  0.000000  2.466911 -0.048374 -0.978693
## Kurtosis    -1.528926 -1.528926 -1.528926  4.520661 -1.199123 -0.514319
##                    y3        y4
## nobs        11.000000 11.000000
## NAs          0.000000  0.000000
## Minimum      5.390000  5.250000
## Maximum     12.740000 12.500000
## 1. Quartile  6.250000  6.170000
## 3. Quartile  7.980000  8.190000
## Mean         7.500000  7.500909
## Median       7.110000  7.040000
## Sum         82.500000 82.510000
## SE Mean      0.612196  0.612242
## LCL Mean     6.135943  6.136748
## UCL Mean     8.864057  8.865070
## Variance     4.122620  4.123249
## Stdev        2.030424  2.030579
## Skewness     1.380120  1.120774
## Kurtosis     1.240044  0.628751
# Co-relation between columns
cor(data, use = "complete.obs",method = "kendall")
##             x1          x2          x3         x4         y1          y2
## x1  1.00000000  1.00000000  1.00000000 -0.4264014  0.6363636  0.56363636
## x2  1.00000000  1.00000000  1.00000000 -0.4264014  0.6363636  0.56363636
## x3  1.00000000  1.00000000  1.00000000 -0.4264014  0.6363636  0.56363636
## x4 -0.42640143 -0.42640143 -0.42640143  1.0000000 -0.4264014 -0.42640143
## y1  0.63636364  0.63636364  0.63636364 -0.4264014  1.0000000  0.56363636
## y2  0.56363636  0.56363636  0.56363636 -0.4264014  0.5636364  1.00000000
## y3  0.96363636  0.96363636  0.96363636 -0.4264014  0.6000000  0.60000000
## y4 -0.09090909 -0.09090909 -0.09090909  0.4264014 -0.1636364 -0.01818182
##             y3          y4
## x1  0.96363636 -0.09090909
## x2  0.96363636 -0.09090909
## x3  0.96363636 -0.09090909
## x4 -0.42640143  0.42640143
## y1  0.60000000 -0.16363636
## y2  0.60000000 -0.01818182
## y3  1.00000000 -0.05454545
## y4 -0.05454545  1.00000000
  1. Create scatter plots for each \(x, y\) pair of data.
attach(data)
plot(x1, y1, main="Scatter Plot - (x1, y1)", xlab="x1", ylab="y1")

plot(x2, y2, main="Scatter Plot - (x2, y2)", xlab="x2", ylab="y2")

plot(x3, y3, main="Scatter Plot - (x3, y3)", xlab="x3", ylab="y3")

plot(x4, y4, main="Scatter Plot - (x4, y4)", xlab="x4", ylab="y4")

  1. 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 - (x1, y1)", xlab="x1", ylab="y1", pch=19)
plot(x2, y2, main="Scatter Plot - (x2, y2)", xlab="x2", ylab="y2", pch=19)
plot(x3, y3, main="Scatter Plot - (x3, y3)", xlab="x3", ylab="y3", pch=19)
plot(x4, y4, main="Scatter Plot - (x4, y4)", xlab="x4", ylab="y4", pch=19)

  1. Now fit a linear model to each data set using the lm() function.
xy1 <- lm(x1 ~ y1, data)
summary(xy1)
## 
## 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: 0.00217
xy2 <- lm(x2 ~ y2, data)
summary(xy2)
## 
## 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: 0.002179
xy3 <- lm(x3 ~ y3, data)
summary(xy3)
## 
## 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: 0.002176
xy4 <- lm(x4 ~ y4, data)
summary(xy4)
## 
## 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
  1. 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(xy1)

par(mfrow=c(2,2))
plot(xy2)

par(mfrow=c(2,2))
plot(xy3)

par(mfrow=c(2,2))
plot(xy4)

  1. Now compare the model fits for each model object.
anova(xy1, 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(xy2, 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(xy3, 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(xy4, 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
  1. In text, summarize the lesson of Anscombe’s Quartet and what it says about the value of data visualization. The motive of anscombe was to show that data that look similar might represent different scenarios when plotted using graphs. In current world, where the cost of storage is low there will be plenty of statistically similar data that might not be consider of a value for business but having it visualized in turn can assist with a value. To conclude, data with visualization can speak more than data without visualization.