Background

Source: Reddit Post - Help me choose approriate plot

I have data set containing measurements of wheat seed root lenght and weight of fresh leaves. I want to show correlation of root length and leaves mass with linear reggresion line on plot. Please help me choose plotting method

Raw data was furnished as a Microsoft Excel file (*.xlsx). This data was reformatted as shown in Table A1 in the Appendix below, then saved as a file named 000 Raw Data - Radicle Data.csv.

  dat1 <- read.csv("000 Raw Data - Radicle Data.csv")
  dat1$Treatment <- factor(dat1$Treatment, levels = c(0, 0.025, 0.05, 0.1, 0.2, 0.4), ordered = TRUE)
  names(dat1)[1] <- "treatment"

Edits to original data set

The biggest changes I made to the original data set was to delete the rows and columns that contained calculated averages, which were labeled as X. I also took the liberty of assuming that the control treatments corresponded to a numeric value of zero, which, in R parlance, would yield an ordered factor.

Do the replicates contain any inherent correlation structure?

A pairwise plot is useful for ascertaining whether or not there is any correlation structure between replicates. This information is important because it determines what type of statistical analyses we can perform with the data, as well as inherent limitations for reformatting the data set structure.

  color_array <- rainbow(length(unique(dat1$treatment)))

  pairs(
    dat1[-1],
    main = "Figure 1: Pairwise Plot",
    pch = 21,
    bg = color_array[as.integer(dat1$treatment)],
    oma = c(4, 4, 10, 18))
  
  legend(x = "right",
         legend = levels(dat1$treatment),
         pch = 21,
         pt.bg = color_array,
         title = "Treatment")

Figure 1 shows that there is no strong correlation structure within the data set. Therefore, we can reformat the data set to conform to a tidy data structure, which will simplify the process of summarizing the data.

Reformatting the Data in a Tidy Format

The following code sample illustrates how to convert the data set into a tidy format. Only a small subset of the full data is shown here. However, the full dataset is shown in the appendix below.

Table 1: Sample Tidy Data

This table is only a tiny subset of the full tidy data set. See Table A2 in the Appendix below for the full data set.

  library(tidyr, quietly = TRUE)
  library(kableExtra, quietly = TRUE)
  library(dplyr, quietly = TRUE)

  dat2 <- dat1 %>%
    gather(key = replicate, value = radicle_len, -treatment) %>% 
    arrange(treatment) %>% 
    mutate(replicate = as.factor(replicate))
  
  kableExtra::kable(head(dat2), align = "c") %>% 
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
treatment replicate radicle_len
0 r1 4.7
0 r1 5.1
0 r1 5.0
0 r1 6.2
0 r1 6.0
0 r1 4.2

Table 2: Tabular Summary

Although I find data plots to be the most useful items for gaining insights into my data, sometimes, a simple table of aggregrate data items can also be very useful. This table contains the mean, standard deviation, and coefficient of variation (CV) or relative standard deviation (RSD). The column designated n contains the number of valid values (i.e. not NA values) that were used to calculate the mean, standard deviation, and CV values.

  library(dplyr)
  library(kableExtra)

  dat2a <- dat2 %>% 
    group_by(treatment, replicate) %>% 
    summarize(
      n = sum(ifelse(is.na(radicle_len), 0, 1)),
      mean_radicle_len = round(mean(radicle_len, na.rm = TRUE), 2),
      sd_radicle_len = round(sd(radicle_len, na.rm = TRUE), 2),
      cv =  round(sd(radicle_len, na.rm = TRUE) / mean(radicle_len, na.rm = TRUE), 2)
    )

  kable(dat2a, align = "c") %>%
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
treatment replicate n mean_radicle_len sd_radicle_len cv
0 r1 25 6.81 1.81 0.27
0 r2 25 8.65 1.49 0.17
0 r3 25 9.76 1.22 0.12
0 r4 23 8.61 1.69 0.20
0.025 r1 25 8.80 1.51 0.17
0.025 r2 23 8.98 1.92 0.21
0.025 r3 23 10.20 2.21 0.22
0.025 r4 22 10.87 1.39 0.13
0.05 r1 25 8.15 1.80 0.22
0.05 r2 22 9.15 1.63 0.18
0.05 r3 23 8.93 1.59 0.18
0.05 r4 24 6.98 1.13 0.16
0.1 r1 22 8.16 2.29 0.28
0.1 r2 24 8.33 1.70 0.20
0.1 r3 24 8.75 1.89 0.22
0.1 r4 24 10.31 2.26 0.22
0.2 r1 21 9.72 2.02 0.21
0.2 r2 23 8.67 1.48 0.17
0.2 r3 25 8.94 1.49 0.17
0.2 r4 24 7.68 1.47 0.19
0.4 r1 25 7.12 1.07 0.15
0.4 r2 25 6.50 1.05 0.16
0.4 r3 22 7.36 1.06 0.14
0.4 r4 24 7.10 1.62 0.23

Box Plots

Set up the base plot

  library(ggplot2)

  p1 <- ggplot(dat2, aes(x = treatment, y = radicle_len)) +
    geom_boxplot(aes(fill = treatment)) +
    labs(x = "Treatment",
         y = "Radicle Length (cm)")

Figure 2a: Radicle Length vs Treatment

  p1 + labs(title = "Figure 2a: Radicle Length vs Treatment")
## Warning: Removed 32 rows containing non-finite values (stat_boxplot).

Figure 2b: Radicle Length vs Treatment by Replicate Group

  p1 + facet_wrap(. ~ replicate) +
    labs(title = "Figure 2b: Radicle Length vs Treatment by Replicate")
## Warning: Removed 32 rows containing non-finite values (stat_boxplot).

Histograms

Just like we did with the box plots, set up a base plot, which will be used as the basis for several plots.

  library(ggplot2)

  p2 <- ggplot(dat2, aes(x = radicle_len))

Figure 3a: Stacked Histogram by Treatment Condition

  p2 + geom_histogram(aes(fill = treatment))
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## Warning: Removed 32 rows containing non-finite values (stat_bin).

Figure 3b: Histogram by Treatment Condition

  p2 + geom_histogram(aes(fill = treatment)) +
    facet_grid(rows = vars(treatment))
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## Warning: Removed 32 rows containing non-finite values (stat_bin).

Figure 3c: Histogram by Treatment Condition Grouped by Replicate

  p2 + geom_histogram(aes(fill = treatment)) +
    facet_grid(rows = vars(treatment), cols = vars(replicate))
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## Warning: Removed 32 rows containing non-finite values (stat_bin).

Control Normalized Mean Radicle Values

Sometimes, a normalized value can be much more insightful than a raw value, especially in an experiment with a control. With R, there are many ways to accomplish this task. However, one of the easiest ways to create such a table is to use the tidyr package to reformat a summary table, then use the dplyr package to compute the normalized results.

Table 3a: Reformatted Summary Table

In this table, the mean radicle values calculated for each unique combination of treatment and replicate group were normalized by dividing them by the mean value for the control (i.e. the 0 value treatment group)

  library(tidyr)
  library(dplyr)
  
  dat3a <- dat2a %>%
    select(treatment, replicate, mean_radicle_len) %>%
    spread(key = treatment, value = mean_radicle_len)
  
  kable(dat3a, align = "c") %>%
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
replicate 0 0.025 0.05 0.1 0.2 0.4
r1 6.81 8.80 8.15 8.16 9.72 7.12
r2 8.65 8.98 9.15 8.33 8.67 6.50
r3 9.76 10.20 8.93 8.75 8.94 7.36
r4 8.61 10.87 6.98 10.31 7.68 7.10

Table 3b: Normalized Summary Table

  dat3b <- data.frame(replicate = dat3a[1],
                        L0 = dat3a[2] / dat3a[2],
                        L1 = dat3a[3] / dat3a[2],
                        L2 = dat3a[4] / dat3a[2],
                        L3 = dat3a[5] / dat3a[2],
                        L4 = dat3a[6] / dat3a[2],
                        L5 = dat3a[7] / dat3a[2]
  )

  kable(dat3b, align = "c") %>%
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
replicate X0 X0.025 X0.05 X0.1 X0.2 X0.4
r1 1 1.292217 1.1967695 1.1982379 1.4273128 1.0455213
r2 1 1.038150 1.0578035 0.9630058 1.0023121 0.7514451
r3 1 1.045082 0.9149590 0.8965164 0.9159836 0.7540984
r4 1 1.262485 0.8106852 1.1974448 0.8919861 0.8246225

Table 3c: Normalized Summary Table in Tidy Format

  library(tidyr)

  dat3c <- dat3b %>% 
    gather(key = "treatment", value = mean_radicle_len, -replicate)
  
  dat3c$treatment <- as.numeric(substring(as.character(dat3c$treatment), 2))
  
  dat3c$mean_radicle_len <- round(dat3c$mean_radicle_len, 2)
  
  kable(dat3c, align = "c") %>%
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
replicate treatment mean_radicle_len
r1 0.000 1.00
r2 0.000 1.00
r3 0.000 1.00
r4 0.000 1.00
r1 0.025 1.29
r2 0.025 1.04
r3 0.025 1.05
r4 0.025 1.26
r1 0.050 1.20
r2 0.050 1.06
r3 0.050 0.91
r4 0.050 0.81
r1 0.100 1.20
r2 0.100 0.96
r3 0.100 0.90
r4 0.100 1.20
r1 0.200 1.43
r2 0.200 1.00
r3 0.200 0.92
r4 0.200 0.89
r1 0.400 1.05
r2 0.400 0.75
r3 0.400 0.75
r4 0.400 0.82

Figure 4: Normalized Mean Radicle Length vs Treatment Level

  ggplot(dat3c, aes(x = treatment, y = mean_radicle_len, color = replicate)) +
    geom_point(size = 4) +
    geom_line() +
    labs(
      title = "Figure 4: Normalized Mean Radicle Length vs Treatment Level",
      subtitle = "All values normalized to control (Treatment Level 0)",
      x = "Treatment Level (Unkown Units)",
      y = "Normalized Mean Radicile Length (unitless)",
      color = "Replicate"
    )

Conclusions

Although there are some anomolies in Figure 4, namely, replicate 1 having a large relative positive offset for all values and treatment levels 2 and 3 for replicate 4 appearing to be outliers, there is a clear negative trend for mean normalized radicle length and treatment level.

Appendix

Table A1: Raw Data - Original Data Set

  library(kableExtra, quietly = TRUE)

  kableExtra::kable(dat1, align = "c") %>% 
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
treatment r1 r2 r3 r4
0 4.7 11.1 12.2 NA
0 5.1 7.9 9.3 7.3
0 5.0 7.4 10.2 8.3
0 6.2 6.9 7.5 10.1
0 6.0 8.4 7.2 8.7
0 4.2 9.2 9.1 9.0
0 5.7 8.5 10.3 7.5
0 4.6 5.5 11.2 4.0
0 3.5 6.2 9.0 NA
0 4.7 6.6 9.2 10.3
0 7.2 9.5 10.3 8.4
0 10.0 9.1 10.4 9.1
0 8.5 8.5 9.5 7.3
0 6.5 7.1 9.7 9.2
0 7.2 9.2 9.5 12.2
0 8.6 10.1 8.3 10.4
0 9.0 11.0 10.2 10.5
0 7.2 8.1 8.2 8.2
0 7.1 9.2 9.1 9.1
0 8.4 8.4 10.3 9.1
0 6.6 9.2 11.8 10.1
0 9.1 11.1 9.5 7.4
0 9.2 8.6 11.3 8.2
0 7.4 9.2 10.1 6.4
0 8.5 10.3 10.6 7.3
0.4 7.1 6.2 NA 8.3
0.4 7.5 6.6 NA 8.1
0.4 7.0 7.1 6.7 8.2
0.4 7.2 7.5 7.0 7.9
0.4 8.3 4.5 7.3 8.1
0.4 7.4 7.0 8.5 7.5
0.4 6.0 6.2 5.1 8.1
0.4 7.2 6.2 8.1 7.6
0.4 9.3 5.3 7.2 6.5
0.4 6.5 6.2 9.1 8.1
0.4 7.2 4.5 7.5 8.3
0.4 9.3 5.1 7.1 0.4
0.4 6.5 5.5 7.3 6.1
0.4 7.2 7.3 9.2 6.5
0.4 5.5 5.6 8.4 7.2
0.4 6.1 6.1 5.6 7.5
0.4 7.3 8.0 7.3 5.4
0.4 7.4 8.2 7.6 7.1
0.4 9.2 7.6 8.5 7.0
0.4 7.5 6.4 6.2 NA
0.4 6.5 6.1 8.1 8.1
0.4 5.5 7.2 6.2 7.3
0.4 7.4 7.4 6.5 6.6
0.4 6.1 8.1 7.5 7.1
0.4 5.8 6.5 NA 7.5
0.2 11.2 NA 10.1 7.1
0.2 11.9 NA 10.5 7.3
0.2 11.0 6.1 10.6 7.2
0.2 11.2 8.1 11.2 7.1
0.2 9.2 8.5 7.0 9.2
0.2 10.2 7.3 9.1 7.4
0.2 12.3 8.3 7.7 8.5
0.2 11.9 8.4 12.1 9.2
0.2 9.1 8.5 9.2 7.1
0.2 NA 9.2 7.1 7.2
0.2 11.2 9.5 8.2 6.3
0.2 7.3 8.2 9.1 7.4
0.2 NA 9.1 8.5 7.1
0.2 10.7 10.7 9.1 10.1
0.2 7.5 8.3 8.2 8.2
0.2 9.3 7.9 8.3 6.5
0.2 8.1 8.2 9.2 9.1
0.2 7.1 10.0 8.2 6.5
0.2 10.8 7.3 9.3 7.1
0.2 10.6 12.2 9.5 10.5
0.2 9.3 8.3 10.1 8.1
0.2 10.2 7.6 8.3 3.5
0.2 4.0 12.2 9.1 NA
0.2 NA 7.1 9.2 9.5
0.2 NA 8.5 4.7 7.1
0.1 10.1 NA 9.5 NA
0.1 NA 9.1 9.2 8.3
0.1 NA 8.2 9.3 9.2
0.1 9.1 7.6 10.3 9.7
0.1 8.2 7.2 11.3 10.3
0.1 NA 9.2 10.1 13.4
0.1 8.2 6.1 11.1 12.2
0.1 12.5 11.1 10.8 15.0
0.1 9.2 8.2 7.8 15.0
0.1 10.2 5.1 10.0 7.3
0.1 6.1 11.1 NA 8.5
0.1 9.5 9.2 6.2 9.8
0.1 8.2 10.1 6.5 10.0
0.1 8.1 8.5 8.7 12.7
0.1 8.0 7.3 10.0 8.5
0.1 9.2 8.2 7.5 10.1
0.1 9.8 7.4 10.1 9.5
0.1 7.4 7.7 11.2 10.0
0.1 1.0 7.3 8.2 8.5
0.1 8.7 6.3 7.4 7.3
0.1 6.5 8.2 6.5 11.1
0.1 9.1 6.6 3.7 8.5
0.1 6.3 8.1 8.1 13.5
0.1 4.8 10.0 9.2 11.0
0.1 9.3 12.2 7.3 8.1
0.05 9.1 11.1 7.1 8.1
0.05 9.3 NA NA 7.3
0.05 4.7 10.0 9.3 6.6
0.05 9.2 11.1 8.5 6.1
0.05 9.8 9.9 8.7 5.5
0.05 5.5 12.1 9.0 5.4
0.05 9.1 7.5 11.1 8.3
0.05 8.5 8.4 10.4 7.5
0.05 8.7 9.1 10.7 8.2
0.05 7.2 8.3 11.1 6.5
0.05 8.5 NA 9.0 7.1
0.05 8.1 7.3 5.5 7.5
0.05 9.5 9.1 9.5 7.0
0.05 8.1 12.3 7.0 7.7
0.05 7.5 6.2 10.1 6.2
0.05 8.5 8.1 9.5 5.5
0.05 7.5 9.5 10.1 7.2
0.05 8.1 7.5 8.4 6.2
0.05 8.8 9.3 9.3 5.5
0.05 2.7 9.3 11.2 7.2
0.05 12.1 8.1 7.1 6.2
0.05 8.5 11.1 6.8 9.1
0.05 7.5 8.5 9.1 9.5
0.05 8.1 7.4 6.8 6.1
0.05 9.2 NA NA NA
0.025 9.2 NA NA 12.2
0.025 10.2 11.3 10.5 NA
0.025 11.0 9.2 6.5 10.5
0.025 11.0 10.5 10.5 11.1
0.025 7.2 8.3 9.1 11.7
0.025 8.1 11.3 NA 12.2
0.025 9.5 NA 8.2 12.4
0.025 10.1 10.2 9.3 11.7
0.025 8.2 8.5 8.2 9.2
0.025 9.9 9.1 7.4 11.2
0.025 7.0 8.3 11.3 9.7
0.025 9.0 9.2 8.5 10.7
0.025 10.2 8.4 11.4 9.1
0.025 10.5 11.6 9.2 11.1
0.025 8.2 7.8 15.0 10.6
0.025 8.4 6.2 12.2 13.6
0.025 9.1 8.3 11.4 11.1
0.025 6.5 11.4 10.0 12.3
0.025 9.1 12.2 11.2 11.7
0.025 9.2 6.3 14.0 7.4
0.025 9.7 10.3 7.4 10.1
0.025 8.7 9.2 10.3 9.5
0.025 7.4 7.4 9.7 10.1
0.025 4.5 6.1 14.2 NA
0.025 8.2 5.4 9.1 NA

Table A2: Raw Data - Full Tidy Data Set

  library(kableExtra, quietly = TRUE)

  kableExtra::kable(dat2, align = "c") %>% 
    kable_styling(bootstrap_options = c("striped", "hover", "condensed")) %>% 
    kable_styling(full_width = FALSE)
treatment replicate radicle_len
0 r1 4.7
0 r1 5.1
0 r1 5.0
0 r1 6.2
0 r1 6.0
0 r1 4.2
0 r1 5.7
0 r1 4.6
0 r1 3.5
0 r1 4.7
0 r1 7.2
0 r1 10.0
0 r1 8.5
0 r1 6.5
0 r1 7.2
0 r1 8.6
0 r1 9.0
0 r1 7.2
0 r1 7.1
0 r1 8.4
0 r1 6.6
0 r1 9.1
0 r1 9.2
0 r1 7.4
0 r1 8.5
0 r2 11.1
0 r2 7.9
0 r2 7.4
0 r2 6.9
0 r2 8.4
0 r2 9.2
0 r2 8.5
0 r2 5.5
0 r2 6.2
0 r2 6.6
0 r2 9.5
0 r2 9.1
0 r2 8.5
0 r2 7.1
0 r2 9.2
0 r2 10.1
0 r2 11.0
0 r2 8.1
0 r2 9.2
0 r2 8.4
0 r2 9.2
0 r2 11.1
0 r2 8.6
0 r2 9.2
0 r2 10.3
0 r3 12.2
0 r3 9.3
0 r3 10.2
0 r3 7.5
0 r3 7.2
0 r3 9.1
0 r3 10.3
0 r3 11.2
0 r3 9.0
0 r3 9.2
0 r3 10.3
0 r3 10.4
0 r3 9.5
0 r3 9.7
0 r3 9.5
0 r3 8.3
0 r3 10.2
0 r3 8.2
0 r3 9.1
0 r3 10.3
0 r3 11.8
0 r3 9.5
0 r3 11.3
0 r3 10.1
0 r3 10.6
0 r4 NA
0 r4 7.3
0 r4 8.3
0 r4 10.1
0 r4 8.7
0 r4 9.0
0 r4 7.5
0 r4 4.0
0 r4 NA
0 r4 10.3
0 r4 8.4
0 r4 9.1
0 r4 7.3
0 r4 9.2
0 r4 12.2
0 r4 10.4
0 r4 10.5
0 r4 8.2
0 r4 9.1
0 r4 9.1
0 r4 10.1
0 r4 7.4
0 r4 8.2
0 r4 6.4
0 r4 7.3
0.025 r1 9.2
0.025 r1 10.2
0.025 r1 11.0
0.025 r1 11.0
0.025 r1 7.2
0.025 r1 8.1
0.025 r1 9.5
0.025 r1 10.1
0.025 r1 8.2
0.025 r1 9.9
0.025 r1 7.0
0.025 r1 9.0
0.025 r1 10.2
0.025 r1 10.5
0.025 r1 8.2
0.025 r1 8.4
0.025 r1 9.1
0.025 r1 6.5
0.025 r1 9.1
0.025 r1 9.2
0.025 r1 9.7
0.025 r1 8.7
0.025 r1 7.4
0.025 r1 4.5
0.025 r1 8.2
0.025 r2 NA
0.025 r2 11.3
0.025 r2 9.2
0.025 r2 10.5
0.025 r2 8.3
0.025 r2 11.3
0.025 r2 NA
0.025 r2 10.2
0.025 r2 8.5
0.025 r2 9.1
0.025 r2 8.3
0.025 r2 9.2
0.025 r2 8.4
0.025 r2 11.6
0.025 r2 7.8
0.025 r2 6.2
0.025 r2 8.3
0.025 r2 11.4
0.025 r2 12.2
0.025 r2 6.3
0.025 r2 10.3
0.025 r2 9.2
0.025 r2 7.4
0.025 r2 6.1
0.025 r2 5.4
0.025 r3 NA
0.025 r3 10.5
0.025 r3 6.5
0.025 r3 10.5
0.025 r3 9.1
0.025 r3 NA
0.025 r3 8.2
0.025 r3 9.3
0.025 r3 8.2
0.025 r3 7.4
0.025 r3 11.3
0.025 r3 8.5
0.025 r3 11.4
0.025 r3 9.2
0.025 r3 15.0
0.025 r3 12.2
0.025 r3 11.4
0.025 r3 10.0
0.025 r3 11.2
0.025 r3 14.0
0.025 r3 7.4
0.025 r3 10.3
0.025 r3 9.7
0.025 r3 14.2
0.025 r3 9.1
0.025 r4 12.2
0.025 r4 NA
0.025 r4 10.5
0.025 r4 11.1
0.025 r4 11.7
0.025 r4 12.2
0.025 r4 12.4
0.025 r4 11.7
0.025 r4 9.2
0.025 r4 11.2
0.025 r4 9.7
0.025 r4 10.7
0.025 r4 9.1
0.025 r4 11.1
0.025 r4 10.6
0.025 r4 13.6
0.025 r4 11.1
0.025 r4 12.3
0.025 r4 11.7
0.025 r4 7.4
0.025 r4 10.1
0.025 r4 9.5
0.025 r4 10.1
0.025 r4 NA
0.025 r4 NA
0.05 r1 9.1
0.05 r1 9.3
0.05 r1 4.7
0.05 r1 9.2
0.05 r1 9.8
0.05 r1 5.5
0.05 r1 9.1
0.05 r1 8.5
0.05 r1 8.7
0.05 r1 7.2
0.05 r1 8.5
0.05 r1 8.1
0.05 r1 9.5
0.05 r1 8.1
0.05 r1 7.5
0.05 r1 8.5
0.05 r1 7.5
0.05 r1 8.1
0.05 r1 8.8
0.05 r1 2.7
0.05 r1 12.1
0.05 r1 8.5
0.05 r1 7.5
0.05 r1 8.1
0.05 r1 9.2
0.05 r2 11.1
0.05 r2 NA
0.05 r2 10.0
0.05 r2 11.1
0.05 r2 9.9
0.05 r2 12.1
0.05 r2 7.5
0.05 r2 8.4
0.05 r2 9.1
0.05 r2 8.3
0.05 r2 NA
0.05 r2 7.3
0.05 r2 9.1
0.05 r2 12.3
0.05 r2 6.2
0.05 r2 8.1
0.05 r2 9.5
0.05 r2 7.5
0.05 r2 9.3
0.05 r2 9.3
0.05 r2 8.1
0.05 r2 11.1
0.05 r2 8.5
0.05 r2 7.4
0.05 r2 NA
0.05 r3 7.1
0.05 r3 NA
0.05 r3 9.3
0.05 r3 8.5
0.05 r3 8.7
0.05 r3 9.0
0.05 r3 11.1
0.05 r3 10.4
0.05 r3 10.7
0.05 r3 11.1
0.05 r3 9.0
0.05 r3 5.5
0.05 r3 9.5
0.05 r3 7.0
0.05 r3 10.1
0.05 r3 9.5
0.05 r3 10.1
0.05 r3 8.4
0.05 r3 9.3
0.05 r3 11.2
0.05 r3 7.1
0.05 r3 6.8
0.05 r3 9.1
0.05 r3 6.8
0.05 r3 NA
0.05 r4 8.1
0.05 r4 7.3
0.05 r4 6.6
0.05 r4 6.1
0.05 r4 5.5
0.05 r4 5.4
0.05 r4 8.3
0.05 r4 7.5
0.05 r4 8.2
0.05 r4 6.5
0.05 r4 7.1
0.05 r4 7.5
0.05 r4 7.0
0.05 r4 7.7
0.05 r4 6.2
0.05 r4 5.5
0.05 r4 7.2
0.05 r4 6.2
0.05 r4 5.5
0.05 r4 7.2
0.05 r4 6.2
0.05 r4 9.1
0.05 r4 9.5
0.05 r4 6.1
0.05 r4 NA
0.1 r1 10.1
0.1 r1 NA
0.1 r1 NA
0.1 r1 9.1
0.1 r1 8.2
0.1 r1 NA
0.1 r1 8.2
0.1 r1 12.5
0.1 r1 9.2
0.1 r1 10.2
0.1 r1 6.1
0.1 r1 9.5
0.1 r1 8.2
0.1 r1 8.1
0.1 r1 8.0
0.1 r1 9.2
0.1 r1 9.8
0.1 r1 7.4
0.1 r1 1.0
0.1 r1 8.7
0.1 r1 6.5
0.1 r1 9.1
0.1 r1 6.3
0.1 r1 4.8
0.1 r1 9.3
0.1 r2 NA
0.1 r2 9.1
0.1 r2 8.2
0.1 r2 7.6
0.1 r2 7.2
0.1 r2 9.2
0.1 r2 6.1
0.1 r2 11.1
0.1 r2 8.2
0.1 r2 5.1
0.1 r2 11.1
0.1 r2 9.2
0.1 r2 10.1
0.1 r2 8.5
0.1 r2 7.3
0.1 r2 8.2
0.1 r2 7.4
0.1 r2 7.7
0.1 r2 7.3
0.1 r2 6.3
0.1 r2 8.2
0.1 r2 6.6
0.1 r2 8.1
0.1 r2 10.0
0.1 r2 12.2
0.1 r3 9.5
0.1 r3 9.2
0.1 r3 9.3
0.1 r3 10.3
0.1 r3 11.3
0.1 r3 10.1
0.1 r3 11.1
0.1 r3 10.8
0.1 r3 7.8
0.1 r3 10.0
0.1 r3 NA
0.1 r3 6.2
0.1 r3 6.5
0.1 r3 8.7
0.1 r3 10.0
0.1 r3 7.5
0.1 r3 10.1
0.1 r3 11.2
0.1 r3 8.2
0.1 r3 7.4
0.1 r3 6.5
0.1 r3 3.7
0.1 r3 8.1
0.1 r3 9.2
0.1 r3 7.3
0.1 r4 NA
0.1 r4 8.3
0.1 r4 9.2
0.1 r4 9.7
0.1 r4 10.3
0.1 r4 13.4
0.1 r4 12.2
0.1 r4 15.0
0.1 r4 15.0
0.1 r4 7.3
0.1 r4 8.5
0.1 r4 9.8
0.1 r4 10.0
0.1 r4 12.7
0.1 r4 8.5
0.1 r4 10.1
0.1 r4 9.5
0.1 r4 10.0
0.1 r4 8.5
0.1 r4 7.3
0.1 r4 11.1
0.1 r4 8.5
0.1 r4 13.5
0.1 r4 11.0
0.1 r4 8.1
0.2 r1 11.2
0.2 r1 11.9
0.2 r1 11.0
0.2 r1 11.2
0.2 r1 9.2
0.2 r1 10.2
0.2 r1 12.3
0.2 r1 11.9
0.2 r1 9.1
0.2 r1 NA
0.2 r1 11.2
0.2 r1 7.3
0.2 r1 NA
0.2 r1 10.7
0.2 r1 7.5
0.2 r1 9.3
0.2 r1 8.1
0.2 r1 7.1
0.2 r1 10.8
0.2 r1 10.6
0.2 r1 9.3
0.2 r1 10.2
0.2 r1 4.0
0.2 r1 NA
0.2 r1 NA
0.2 r2 NA
0.2 r2 NA
0.2 r2 6.1
0.2 r2 8.1
0.2 r2 8.5
0.2 r2 7.3
0.2 r2 8.3
0.2 r2 8.4
0.2 r2 8.5
0.2 r2 9.2
0.2 r2 9.5
0.2 r2 8.2
0.2 r2 9.1
0.2 r2 10.7
0.2 r2 8.3
0.2 r2 7.9
0.2 r2 8.2
0.2 r2 10.0
0.2 r2 7.3
0.2 r2 12.2
0.2 r2 8.3
0.2 r2 7.6
0.2 r2 12.2
0.2 r2 7.1
0.2 r2 8.5
0.2 r3 10.1
0.2 r3 10.5
0.2 r3 10.6
0.2 r3 11.2
0.2 r3 7.0
0.2 r3 9.1
0.2 r3 7.7
0.2 r3 12.1
0.2 r3 9.2
0.2 r3 7.1
0.2 r3 8.2
0.2 r3 9.1
0.2 r3 8.5
0.2 r3 9.1
0.2 r3 8.2
0.2 r3 8.3
0.2 r3 9.2
0.2 r3 8.2
0.2 r3 9.3
0.2 r3 9.5
0.2 r3 10.1
0.2 r3 8.3
0.2 r3 9.1
0.2 r3 9.2
0.2 r3 4.7
0.2 r4 7.1
0.2 r4 7.3
0.2 r4 7.2
0.2 r4 7.1
0.2 r4 9.2
0.2 r4 7.4
0.2 r4 8.5
0.2 r4 9.2
0.2 r4 7.1
0.2 r4 7.2
0.2 r4 6.3
0.2 r4 7.4
0.2 r4 7.1
0.2 r4 10.1
0.2 r4 8.2
0.2 r4 6.5
0.2 r4 9.1
0.2 r4 6.5
0.2 r4 7.1
0.2 r4 10.5
0.2 r4 8.1
0.2 r4 3.5
0.2 r4 NA
0.2 r4 9.5
0.2 r4 7.1
0.4 r1 7.1
0.4 r1 7.5
0.4 r1 7.0
0.4 r1 7.2
0.4 r1 8.3
0.4 r1 7.4
0.4 r1 6.0
0.4 r1 7.2
0.4 r1 9.3
0.4 r1 6.5
0.4 r1 7.2
0.4 r1 9.3
0.4 r1 6.5
0.4 r1 7.2
0.4 r1 5.5
0.4 r1 6.1
0.4 r1 7.3
0.4 r1 7.4
0.4 r1 9.2
0.4 r1 7.5
0.4 r1 6.5
0.4 r1 5.5
0.4 r1 7.4
0.4 r1 6.1
0.4 r1 5.8
0.4 r2 6.2
0.4 r2 6.6
0.4 r2 7.1
0.4 r2 7.5
0.4 r2 4.5
0.4 r2 7.0
0.4 r2 6.2
0.4 r2 6.2
0.4 r2 5.3
0.4 r2 6.2
0.4 r2 4.5
0.4 r2 5.1
0.4 r2 5.5
0.4 r2 7.3
0.4 r2 5.6
0.4 r2 6.1
0.4 r2 8.0
0.4 r2 8.2
0.4 r2 7.6
0.4 r2 6.4
0.4 r2 6.1
0.4 r2 7.2
0.4 r2 7.4
0.4 r2 8.1
0.4 r2 6.5
0.4 r3 NA
0.4 r3 NA
0.4 r3 6.7
0.4 r3 7.0
0.4 r3 7.3
0.4 r3 8.5
0.4 r3 5.1
0.4 r3 8.1
0.4 r3 7.2
0.4 r3 9.1
0.4 r3 7.5
0.4 r3 7.1
0.4 r3 7.3
0.4 r3 9.2
0.4 r3 8.4
0.4 r3 5.6
0.4 r3 7.3
0.4 r3 7.6
0.4 r3 8.5
0.4 r3 6.2
0.4 r3 8.1
0.4 r3 6.2
0.4 r3 6.5
0.4 r3 7.5
0.4 r3 NA
0.4 r4 8.3
0.4 r4 8.1
0.4 r4 8.2
0.4 r4 7.9
0.4 r4 8.1
0.4 r4 7.5
0.4 r4 8.1
0.4 r4 7.6
0.4 r4 6.5
0.4 r4 8.1
0.4 r4 8.3
0.4 r4 0.4
0.4 r4 6.1
0.4 r4 6.5
0.4 r4 7.2
0.4 r4 7.5
0.4 r4 5.4
0.4 r4 7.1
0.4 r4 7.0
0.4 r4 NA
0.4 r4 8.1
0.4 r4 7.3
0.4 r4 6.6
0.4 r4 7.1
0.4 r4 7.5