EPPS 6356 Assignment 3

Here is my Assignment 3 for EPPS 6356.

1.Run anscombe01.R (in classGitHub)

a. Compare the regression models.

Figure 1: Regression plot of X1 and Y1.

As seen in Figure 1, a regression plot is appropriate for data visualization and analysis. The plot points generally fit the regression line. However, because the plot points are very spaced out, the R^2 value may be very small, as the regression line does not fit plot points very well.

Figure 2: Regression plot of X2 and Y2.

As seen in Figure 2, a regression plot is most likely not the best tool for data analysis. This is because all of the plot points resemble a parabolic function. For this reason, a different or supplementary data visualization/analysis tools may be required.

Figure 3: Regression plot of X3 and Y3.

Figure 3 is similar to Figure 1, in that a regression plot is appropriate for data visualization and analysis. The plot points generally fit the regression line, except for one outlier point at x=13. It may be beneficial to take this outlier out, and therefore visualize how much better the regression line would fit the data.

Figure 4: Regression plot of X4 and Y4.

Like Figure 2, a regression plot is most likely not the best tool for data analysis for Figure 4. This is because all of the plot points except for one are all at x=8. For this reason, different or supplementary data visualization/analysis tools are required.

b. Compare different ways to create the plots (e.g. changing colors, line types, plot characters).

Figure 5: Aggregate regression plots from X1-X4 and Y1-Y4.

As seen in Figure 5, all four regression plots were generated as a composite chart by using the code to "plot for a loop". This composite/aggregate chart also was about to generate a title to describe the plots. This code was also able to add further design components such as regression line color, as well as plot point size, color, and shape.

2.Can you finetune the charts without using other packages (consult RGraphics by Murrell).

Figure 6: Finetuned aggregate regression plots for X1-X4 and Y1-Y4.

Above is Figure 6, which is the "finetuned" version of the regression plots for X1-X4 and Y1-Y4, using the RGraphics by Murrell. I changed the plot points to be smaller, so that the reader can more effectively read the charts and distinguish between the plot points. This is especially helpful for Plot 4, where many of the plot points are stacked on top of each other. Making the plot point sizes smaller also allows the reader to be able see the distancebetween plot points easier, as well. I also changed the color of the plot points and regression line to colors that are easier on the eye and less distracting.

3.How about with ggplot2? (use tidyverse package)

See the code below to see how the ggplot2 function (attached to the tidyverse package) can be used to efficiently generate the four regression plots for Anscombe's (1973) Quartlet. The generated chart is also seen below in Figure 7. This aggregate/composite plot was produced by using only two lines of code, instead of 13+ lines of code the traditional way without the ggplot2 function within the tidyverse package.

Figure 7: Aggregate regression plots from X1-X4 and Y1-Y4 generated using the ggplot2 function.

## Anscombe (1973) Quartlet

data(anscombe)  # Load Anscombe's data
summary(anscombe)

       x1             x2             x3             x4           y1        
 Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8   Min.   : 4.260  
 1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8   1st Qu.: 6.315  
 Median : 9.0   Median : 9.0   Median : 9.0   Median : 8   Median : 7.580  
 Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9   Mean   : 7.501  
 3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8   3rd Qu.: 8.570  
 Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19   Max.   :10.840  
       y2              y3              y4        
 Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
 1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
 Median :8.140   Median : 7.11   Median : 7.040  
 Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
 3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
 Max.   :9.260   Max.   :12.74   Max.   :12.500  

## Simple version
plot(anscombe$x1,anscombe$y1)

summary(anscombe)

       x1             x2             x3             x4           y1        
 Min.   : 4.0   Min.   : 4.0   Min.   : 4.0   Min.   : 8   Min.   : 4.260  
 1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 6.5   1st Qu.: 8   1st Qu.: 6.315  
 Median : 9.0   Median : 9.0   Median : 9.0   Median : 8   Median : 7.580  
 Mean   : 9.0   Mean   : 9.0   Mean   : 9.0   Mean   : 9   Mean   : 7.501  
 3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.:11.5   3rd Qu.: 8   3rd Qu.: 8.570  
 Max.   :14.0   Max.   :14.0   Max.   :14.0   Max.   :19   Max.   :10.840  
       y2              y3              y4        
 Min.   :3.100   Min.   : 5.39   Min.   : 5.250  
 1st Qu.:6.695   1st Qu.: 6.25   1st Qu.: 6.170  
 Median :8.140   Median : 7.11   Median : 7.040  
 Mean   :7.501   Mean   : 7.50   Mean   : 7.501  
 3rd Qu.:8.950   3rd Qu.: 7.98   3rd Qu.: 8.190  
 Max.   :9.260   Max.   :12.74   Max.   :12.500  

# Create four model objects
lm1 <- lm(y1 ~ x1, data=anscombe)
summary(lm1)

Call:
lm(formula = y1 ~ x1, data = anscombe)

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: 0.00217

lm2 <- lm(y2 ~ x2, data=anscombe)
summary(lm2)

Call:
lm(formula = y2 ~ x2, data = anscombe)

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: 0.002179

lm3 <- lm(y3 ~ x3, data=anscombe)
summary(lm3)

Call:
lm(formula = y3 ~ x3, data = anscombe)

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: 0.002176

lm4 <- lm(y4 ~ x4, data=anscombe)
summary(lm4)

Call:
lm(formula = y4 ~ x4, data = anscombe)

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

plot.new

function () 
{
    for (fun in getHook("before.plot.new")) {
        if (is.character(fun)) 
            fun <- get(fun)
        try(fun())
    }
    .External2(C_plot_new)
    grDevices:::recordPalette()
    for (fun in getHook("plot.new")) {
        if (is.character(fun)) 
            fun <- get(fun)
        try(fun())
    }
    invisible()
}
<bytecode: 0x7fcc40a494e0>
<environment: namespace:graphics>

plot(y1 ~ x1, data=anscombe) + abline(lm(y1 ~ x1, data=anscombe))

integer(0)

plot(y2 ~ x2, data=anscombe) + abline(lm(y2 ~ x2, data=anscombe))

integer(0)

plot(y3 ~ x3, data=anscombe) + abline(lm(y3 ~ x3, data=anscombe))

integer(0)

plot(y4 ~ x4, data=anscombe) + abline(lm(y4 ~ x4, data=anscombe))

integer(0)

## Fancy version (per help file)

ff <- y ~ x
mods <- setNames(as.list(1:4), paste0("lm", 1:4))

# Plot using for loop
for(i in 1:4) {
  ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
  ## or   ff[[2]] <- as.name(paste0("y", i))
  ##      ff[[3]] <- as.name(paste0("x", i))
  mods[[i]] <- lmi <- lm(ff, data = anscombe)
  print(anova(lmi))
}

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
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
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
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

sapply(mods, coef)  # Note the use of this function

                  lm1      lm2       lm3       lm4
(Intercept) 3.0000909 3.000909 3.0024545 3.0017273
x1          0.5000909 0.500000 0.4997273 0.4999091

lapply(mods, function(fm) coef(summary(fm)))

$lm1
             Estimate Std. Error  t value    Pr(>|t|)
(Intercept) 3.0000909  1.1247468 2.667348 0.025734051
x1          0.5000909  0.1179055 4.241455 0.002169629

$lm2
            Estimate Std. Error  t value    Pr(>|t|)
(Intercept) 3.000909  1.1253024 2.666758 0.025758941
x2          0.500000  0.1179637 4.238590 0.002178816

$lm3
             Estimate Std. Error  t value    Pr(>|t|)
(Intercept) 3.0024545  1.1244812 2.670080 0.025619109
x3          0.4997273  0.1178777 4.239372 0.002176305

$lm4
             Estimate Std. Error  t value    Pr(>|t|)
(Intercept) 3.0017273  1.1239211 2.670763 0.025590425
x4          0.4999091  0.1178189 4.243028 0.002164602

# Preparing for the plots
op <- par(mfrow = c(2, 2), mar = 0.1+c(4,4,1,1), oma =  c(0, 0, 2, 0))

# Plot charts using for loop
for(i in 1:4) {
  ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
  plot(ff, data = anscombe, col = "darkorchid4", pch = 23, bg = "mediumpurple", 
       cex = 0.75, xlim = c(3, 19), ylim = c(3, 13))
  abline(mods[[i]], col = "steelblue")
}
mtext("Anscombe's 4 Regression Data Sets", outer = TRUE, cex = 1.5)

par(op)


## 3.How about with ggplot2? (use tidyverse package)
library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✓ ggplot2 3.3.6     ✓ purrr   0.3.4
✓ tibble  3.1.6     ✓ dplyr   1.0.8
✓ tidyr   1.2.0     ✓ stringr 1.4.0
✓ readr   2.1.1     ✓ forcats 0.5.1
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
x dplyr::filter() masks stats::filter()
x dplyr::lag()    masks stats::lag()

library(dplyr)
library(broom)

tidy_anscombe <- anscombe %>%
  pivot_longer(cols = everything(),
               names_to = c(".value", "set"),
               names_pattern = "(.)(.)")

tidy_anscombe

# A tibble: 44 × 3
   set       x     y
   <chr> <dbl> <dbl>
 1 1        10  8.04
 2 2        10  9.14
 3 3        10  7.46
 4 4         8  6.58
 5 1         8  6.95
 6 2         8  8.14
 7 3         8  6.77
 8 4         8  5.76
 9 1        13  7.58
10 2        13  8.74
# … with 34 more rows

ggplot(tidy_anscombe,
       aes(x = x,
           y = y)) +
  geom_point() + 
  facet_wrap(~set) +
  geom_smooth(method = "lm", se = FALSE)

`geom_smooth()` using formula 'y ~ x'

#> `geom_smooth()` using formula 'y ~ x'