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

  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.
summary(anscombe)
##        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
data <- anscombe

str(data)
## 'data.frame':    11 obs. of  8 variables:
##  $ x1: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x2: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x3: num  10 8 13 9 11 14 6 4 12 7 ...
##  $ x4: num  8 8 8 8 8 8 8 19 8 8 ...
##  $ y1: num  8.04 6.95 7.58 8.81 8.33 ...
##  $ y2: num  9.14 8.14 8.74 8.77 9.26 8.1 6.13 3.1 9.13 7.26 ...
##  $ y3: num  7.46 6.77 12.74 7.11 7.81 ...
##  $ y4: num  6.58 5.76 7.71 8.84 8.47 7.04 5.25 12.5 5.56 7.91 ...
  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(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
fBasics::ghMean(data)
##    x1 x2 x3 x4 y1 y2 y3 y4
## 1   0  0  0  0  0  0  0  0
## 2   0  0  0  0  0  0  0  0
## 3   0  0  0  0  0  0  0  0
## 4   0  0  0  0  0  0  0  0
## 5   0  0  0  0  0  0  0  0
## 6   0  0  0  0  0  0  0  0
## 7   0  0  0  0  0  0  0  0
## 8   0  0  0  0  0  0  0  0
## 9   0  0  0  0  0  0  0  0
## 10  0  0  0  0  0  0  0  0
## 11  0  0  0  0  0  0  0  0
  1. Create scatter plots for each \(x, y\) pair of data.
library(ggplot2)

plot(data$x1,data$y1, main = "x1 & y1 Scatter Plot")

plot(data$x2,data$y2, main = "x2 & y2 Scatter Plot")

plot(data$x3,data$y3, main = "x3 & y3 Scatter Plot")

plot(data$x4,data$y4, main = "x4 & y4 Scatter Plot")

  1. Now change the symbols on the scatter plots to solid circles and plot them together as a 4 panel graphic
# P1 <- ggplot(data,aes(data$x1,data$y1)) + 
#                title("x1 & y1 Scatter Plot") +
#                geom_point(shape = 19)
#     p1         


# install.packages("gridExtra")
# 
# gridExtra::grid.arrange(P1,P2,P3,P4,
#             LABELS = c("1","2","3","4"),
#             ncol = 2, nrow = 2,
#             heights = c(1,2,3,4))
# 
# 
# 
# gridExtra::grid.arrange(P1,P2,P3,P4, nrow = 2)
# 
# gridExtra::grid.arrange(
#                         grobs = gl,
#                         width = c(2,1,1)
#                         layout_matrix = rbind()
#                         
# P1 <- ggplot(data, aes(data$x1,data$y1))+geom_point(shape=19)+title("x1 & y1 Scatter Plot")

par(mfrow = c(2,2))
P1 <- plot(data$x1,data$y1, main = "x1 & y1 Scatter Plot", pch = 19)
P2 <- plot(data$x2,data$y2, main = "x2 & y2 Scatter Plot", pch = 19)
P3 <- plot(data$x3,data$y3, main = "x3 & y3 Scatter Plot", pch = 19)
P4 <- plot(data$x4,data$y4, main = "x4 & y4 Scatter Plot", pch = 19)

  1. Now fit a linear model to each data set using the lm() function.
L1 <- lm(data$x1~data$y1)
L2 <- lm(data$x2~data$y2)
L3 <- lm(data$x3~data$y3)
L4 <- lm(data$x4~data$y4)

L1
## 
## Call:
## lm(formula = data$x1 ~ data$y1)
## 
## Coefficients:
## (Intercept)      data$y1  
##     -0.9975       1.3328
summary(L1)
## 
## Call:
## lm(formula = data$x1 ~ data$y1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6522 -1.5117 -0.2657  1.2341  3.8946 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9975     2.4344  -0.410  0.69156   
## data$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
L2
## 
## Call:
## lm(formula = data$x2 ~ data$y2)
## 
## Coefficients:
## (Intercept)      data$y2  
##     -0.9948       1.3325
summary(L2)
## 
## Call:
## lm(formula = data$x2 ~ data$y2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8516 -1.4315 -0.3440  0.8467  4.2017 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -0.9948     2.4354  -0.408  0.69246   
## data$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
L3
## 
## Call:
## lm(formula = data$x3 ~ data$y3)
## 
## Coefficients:
## (Intercept)      data$y3  
##      -1.000        1.333
summary(L3)
## 
## Call:
## lm(formula = data$x3 ~ data$y3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9869 -1.3733 -0.0266  1.3200  3.2133 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0003     2.4362  -0.411  0.69097   
## data$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
L4
## 
## Call:
## lm(formula = data$x4 ~ data$y4)
## 
## Coefficients:
## (Intercept)      data$y4  
##      -1.004        1.334
summary(L4)
## 
## Call:
## lm(formula = data$x4 ~ data$y4)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7859 -1.4122 -0.1853  1.4551  3.3329 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -1.0036     2.4349  -0.412  0.68985   
## data$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(L1)

# par(mfrow = c(2,2))
plot(L2)

# par(mfrow = c(2,2))
plot(L3)

# par(mfrow = c(2,2))
plot(L4)

  1. Now compare the model fits for each model object.
str(L1)

List of 12 $ coefficients : Named num [1:2] -0.998 1.333 ..- attr(, “names”)= chr [1:2] “(Intercept)” “data$y1” $ residuals : Named num [1:11] 0.281 -0.266 3.895 -1.745 0.895 … ..- attr(, “names”)= chr [1:11] “1” “2” “3” “4” … $ effects : Named num [1:11] -29.85 8.563 3.832 -1.865 0.797 … ..- attr(, “names”)= chr [1:11] “(Intercept)” “data$y1” “” “” … $ rank : int 2 $ fitted.values: Named num [1:11] 9.72 8.27 9.11 10.74 10.11 … ..- attr(, “names”)= chr [1:11] “1” “2” “3” “4” … $ assign : int [1:2] 0 1 $ qr :List of 5 ..$ qr : num [1:11, 1:2] -3.317 0.302 0.302 0.302 0.302 … .. ..- attr(, “dimnames”)=List of 2 .. .. ..$ : chr [1:11] “1” “2” “3” “4” … .. .. ..$ : chr [1:2] “(Intercept)” “data\(y1" .. ..- attr(*, "assign")= int [1:2] 0 1 ..\) qraux: num [1:2] 1.3 1.11 ..$ pivot: int [1:2] 1 2 ..$ tol : num 1e-07 ..$ rank : int 2 ..- attr(, “class”)= chr “qr” $ df.residual : int 9 $ xlevels : Named list() $ call : language lm(formula = data\(x1 ~ data\)y1) $ terms :Classes ‘terms’, ‘formula’ language data\(x1 ~ data\)y1 .. ..- attr(,”variables“)= language list(data\(x1, data\)y1) .. ..- attr(, “factors”)= int [1:2, 1] 0 1 .. .. ..- attr(,”dimnames“)=List of 2 .. .. .. ..$ : chr [1:2]”data\(x1" "data\)y1" .. .. .. ..$ : chr “data\(y1" .. ..- attr(*, "term.labels")= chr "data\)y1” .. ..- attr(, “order”)= int 1 .. ..- attr(, “intercept”)= int 1 .. ..- attr(, “response”)= int 1 .. ..- attr(, “.Environment”)=<environment: R_GlobalEnv> .. ..- attr(, “predvars”)= language list(data\(x1, data\)y1) .. ..- attr(, “dataClasses”)= Named chr [1:2] “numeric” “numeric” .. .. ..- attr(, “names”)= chr [1:2] “data\(x1" "data\)y1” $ model :‘data.frame’: 11 obs. of 2 variables: ..$ data\(x1: num [1:11] 10 8 13 9 11 14 6 4 12 7 ... ..\) data\(y1: num [1:11] 8.04 6.95 7.58 8.81 8.33 ... ..- attr(*, "terms")=Classes 'terms', 'formula' language data\)x1 ~ data\(y1 .. .. ..- attr(*, "variables")= language list(data\)x1, data\(y1) .. .. ..- attr(*, "factors")= int [1:2, 1] 0 1 .. .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. .. ..\) : chr [1:2] “data\(x1" "data\)y1” .. .. .. .. ..$ : chr “data\(y1" .. .. ..- attr(*, "term.labels")= chr "data\)y1” .. .. ..- attr(, “order”)= int 1 .. .. ..- attr(, “intercept”)= int 1 .. .. ..- attr(, “response”)= int 1 .. .. ..- attr(, “.Environment”)=<environment: R_GlobalEnv> .. .. ..- attr(, “predvars”)= language list(data\(x1, data\)y1) .. .. ..- attr(, “dataClasses”)= Named chr [1:2] “numeric” “numeric” .. .. .. ..- attr(, “names”)= chr [1:2] “data\(x1" "data\)y1” - attr(, “class”)= chr “lm”

anova(L1)

Analysis of Variance Table

Response: data\(x1 Df Sum Sq Mean Sq F value Pr(>F) data\)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(L2)

Analysis of Variance Table

Response: data\(x2 Df Sum Sq Mean Sq F value Pr(>F) data\)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(L3)

Analysis of Variance Table

Response: data\(x3 Df Sum Sq Mean Sq F value Pr(>F) data\)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(L4)

Analysis of Variance Table

Response: data\(x4 Df Sum Sq Mean Sq F value Pr(>F) data\)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.

Anscombe’s Quartet essentially has four sets of data, by juxtaposing them, we can compare and contrast each set individually. When I used the summary function to examine individual data, I noticed that X1,X2,X3 have similar characteristics. My initial assumption is that these datasets are identical. However, further along the process, I cam to realize that they are rather different. P1 appears to be randomly scattered, P2 appears to be a curve trend which can like be fitted with a regression. P3 has one outlier, while the rest appear to be follow a linear relationship, and P4 is rather interesting, majority of the datapoints appear to be stacked together, with one outlier. I do find that visualization is a good way of examine data, as sometime, the data does not tell the whole story. It is fairly quick to examine data in a graphical way as well.