Poplar Cutting Experiment - Exp V
Some tree species have the abilities to be propagated asexually. Some fruit trees like apples and pears can be grown from cuttings, as other some hardwood tree species such as maples, poplar and willow can be grown from cuttings. Poplar trees produce their rooting hormones, and can their ability to sprout roots from twigs can be tested by placing them in a water. In Sweden it is advised to establish poplar on old agricultural lands having fertile and well-drained soil.
source: bowhayestress.co.uk
The Experiment
An experiment was carried out to test the effect of fertilizer on the growth of three poplar clone cuttings separated into different blocks.
source: lignoplant.com
Questions
check if there’s an effect of the clones and fertilization treatment on the diameter and height of the seedlings
Investigate if the volume of 12 weeks seedlings is related to the initial cutting weight (poplar data).
Make a regression of height and dbh,then determine if dbh can be used to predict height if height data is missing (spruce data).
library(doBy)
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:doBy':
##
## order_by## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(lattice)
library(ggplot2)
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recodelibrary(data.table)##
## Attaching package: 'data.table'## The following objects are masked from 'package:dplyr':
##
## between, first, lastlibrary(TukeyC)
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutImporting data
# Importing data
pop2 <- read.table("https://raw.githubusercontent.com/xrander/Slu_experiment/master/Data/Lab%205/pop2.txt",
header = T,
sep = '\t',
dec = '.',
na.strings = 'NA',
strip.white = T)
poplar <- read.table("https://raw.githubusercontent.com/xrander/Slu_experiment/master/Data/Lab%205/poplar.txt",
header = T,
sep = '\t',
dec = '.',
na.strings = 'NA',
strip.white = T)
spruce2 <- read.table("https://raw.githubusercontent.com/xrander/Slu_experiment/master/Data/Lab%205/spruce.txt",
header = T,
sep = '\t',
dec = '.',
na.strings = 'NA',
strip.white = T)Data description
pop2 and poplar
block:1-5
cutw:cuttings width (g)
height:aboveground height (mm)
dia:root collar diameter (mm)
clone: A,B,C
fert: 1=fertilized, 3= control
Spruce
The data is just a short list of sampled saplings of Norway spruce with dbh (mm) and height (dm). The data is taken from a short interval of heights and the relation between height and diameter is still linear (compared to the range that you usually have in a stand). That is why, we are interested in testing whether we could use a linear model to fit a regression line of the relationship of height and dbh.
** Data exploration**
str(pop2)## 'data.frame': 189 obs. of 6 variables:
## $ block : int 1 1 1 1 2 2 2 2 2 2 ...
## $ cutw : num 2.4 0.7 6.5 1.1 2 4.9 0.8 1.3 8.8 2 ...
## $ height: int 71 67 211 69 116 123 68 79 166 91 ...
## $ dia : num 0.6 1.4 3.5 1 1.4 3.2 2.2 1.8 2.4 2.5 ...
## $ clone : chr "A" "A" "A" "A" ...
## $ fert : int 3 3 3 3 3 3 3 3 3 3 ...block , clone and fert columns
are factor data type but it is currently an integers and character data
type, thus, they have to be coerced to factor data type.
pop2$block <- as.factor(pop2$block)
pop2$fert <- as.factor(pop2$fert)
pop2$clone <- as.factor(pop2$clone)summary(pop2)## block cutw height dia clone fert
## 1:37 Min. : 0.10 Min. : 22.0 Min. : 0.600 A:51 1:98
## 2:39 1st Qu.: 1.10 1st Qu.:100.5 1st Qu.: 2.000 B:68 3:91
## 3:37 Median : 2.20 Median :262.5 Median : 3.200 C:70
## 4:37 Mean : 3.78 Mean :248.5 Mean : 3.046
## 5:39 3rd Qu.: 5.80 3rd Qu.:365.2 3rd Qu.: 3.600
## Max. :19.70 Max. :506.0 Max. :39.000
## NA's :1plot (pop2$dia, pop2$height,
xlab = 'diameter(mm)',
ylab = 'height (mm)',
main = 'Height vs Diameter',
col = pop2$fert)
After exploration, an exaggerated or wrong diameter value was found that is way far from the mean.Also, a missing value was found in the height variable when under the basic statistics summary. This is a common occurrence during data collection and recording.
Dealing with Missing Data
pop2[is.na(pop2$height), ]## block cutw height dia clone fert
## 109 4 0.5 NA 3.1 A 1## This shows we have one Na Value and it is in row 109
pop2[complete.cases(pop2), ] ## this shows all rows without missing values## block cutw height dia clone fert
## 1 1 2.4 71 0.6 A 3
## 2 1 0.7 67 1.4 A 3
## 3 1 6.5 211 3.5 A 3
## 4 1 1.1 69 1.0 A 3
## 5 2 2.0 116 1.4 A 3
## 6 2 4.9 123 3.2 A 3
## 7 2 0.8 68 2.2 A 3
## 8 2 1.3 79 1.8 A 3
## 9 2 8.8 166 2.4 A 3
## 10 2 2.0 91 2.5 A 3
## 11 2 2.5 32 1.9 A 3
## 12 3 2.5 127 2.1 A 3
## 13 3 1.3 79 1.5 A 3
## 14 3 1.0 81 1.6 A 3
## 15 3 6.3 186 2.9 A 3
## 16 3 12.1 234 2.7 A 3
## 17 4 1.6 94 1.7 A 3
## 18 4 0.5 43 0.6 A 3
## 19 4 8.5 249 3.2 A 3
## 20 4 1.1 86 1.6 A 3
## 21 5 1.7 92 1.9 A 3
## 22 5 7.2 241 3.7 A 3
## 23 5 2.7 154 2.4 A 3
## 24 5 1.4 92 1.9 A 3
## 25 1 1.4 36 1.4 B 3
## 26 1 2.6 186 2.2 B 3
## 27 1 2.2 197 2.0 B 3
## 28 1 14.7 349 3.5 B 3
## 29 1 0.7 66 1.3 B 3
## 30 1 1.9 233 2.2 B 3
## 31 1 4.6 251 2.8 B 3
## 32 1 0.8 26 1.0 B 3
## 33 2 0.9 32 0.8 B 3
## 34 2 2.1 89 2.4 B 3
## 35 2 5.2 246 3.5 B 3
## 36 2 11.3 238 3.0 B 3
## 37 2 5.7 245 2.8 B 3
## 38 3 7.0 247 3.0 B 3
## 39 3 2.8 161 2.1 B 3
## 40 3 2.0 117 1.7 B 3
## 41 3 4.3 163 3.2 B 3
## 42 3 0.7 51 1.3 B 3
## 43 4 0.5 81 1.5 B 3
## 44 4 1.9 144 2.1 B 3
## 45 4 19.7 324 2.8 B 3
## 46 4 3.1 164 2.0 B 3
## 47 4 6.8 242 3.0 B 3
## 48 4 1.5 101 1.6 B 3
## 49 4 14.3 323 3.6 B 3
## 50 5 11.8 294 3.9 B 3
## 51 5 4.1 144 2.8 B 3
## 52 5 3.8 173 2.3 B 3
## 53 5 2.3 61 2.0 B 3
## 54 5 2.4 58 1.3 B 3
## 55 5 1.4 99 2.0 B 3
## 56 1 0.6 79 0.9 C 3
## 57 1 0.9 24 0.9 C 3
## 58 1 4.5 208 3.2 C 3
## 59 1 0.8 93 1.2 C 3
## 60 1 0.6 93 0.9 C 3
## 61 2 0.7 116 1.1 C 3
## 62 2 4.1 261 2.7 C 3
## 63 2 7.4 282 3.5 C 3
## 64 2 1.1 62 0.9 C 3
## 65 2 0.6 48 0.9 C 3
## 66 2 10.3 281 3.6 C 3
## 67 2 14.7 284 3.8 C 3
## 68 2 1.8 79 1.9 C 3
## 69 3 2.4 42 2.4 C 3
## 70 3 2.2 77 2.1 C 3
## 71 3 1.2 114 1.7 C 3
## 72 3 0.8 81 1.3 C 3
## 73 3 0.7 27 0.9 C 3
## 74 3 5.6 22 2.1 C 3
## 75 3 8.5 73 4.0 C 3
## 76 3 10.1 276 3.9 C 3
## 77 4 8.6 247 3.4 C 3
## 78 4 0.9 93 1.8 C 3
## 79 4 0.9 76 1.2 C 3
## 80 4 7.1 221 3.3 C 3
## 81 4 2.1 49 1.4 C 3
## 82 4 0.7 82 1.4 C 3
## 83 4 9.2 241 3.2 C 3
## 84 5 1.1 48 1.3 C 3
## 85 5 2.0 74 1.5 C 3
## 86 5 4.7 51 3.2 C 3
## 87 5 2.9 31 1.3 C 3
## 88 5 8.4 221 3.6 C 3
## 89 5 6.1 227 2.8 C 3
## 90 5 0.5 79 1.0 C 3
## 91 5 0.8 128 1.6 C 3
## 92 1 4.6 441 4.0 A 1
## 93 1 3.6 316 3.6 A 1
## 94 1 2.1 353 3.3 A 1
## 95 1 0.3 284 3.2 A 1
## 96 2 0.1 366 3.4 A 1
## 97 2 0.8 421 4.1 A 1
## 98 2 7.3 407 39.0 A 1
## 99 2 0.6 182 1.6 A 1
## 100 2 14.2 414 3.6 A 1
## 101 2 3.2 305 3.4 A 1
## 102 3 0.9 306 3.0 A 1
## 103 3 7.7 418 4.5 A 1
## 104 3 0.6 373 3.7 A 1
## 105 3 0.9 188 2.3 A 1
## 106 3 2.1 328 3.1 A 1
## 107 4 1.3 371 3.6 A 1
## 108 4 1.4 377 3.8 A 1
## 110 4 2.2 416 3.4 A 1
## 111 4 6.0 406 3.8 A 1
## 112 5 5.3 261 3.0 A 1
## 113 5 0.4 68 2.0 A 1
## 114 5 4.1 218 2.3 A 1
## 115 5 1.6 334 3.2 A 1
## 116 5 5.1 371 3.8 A 1
## 117 5 3.1 274 3.1 A 1
## 118 5 1.1 262 2.8 A 1
## 119 1 2.7 394 3.4 B 1
## 120 1 12.2 431 4.3 B 1
## 121 1 0.8 386 3.3 B 1
## 122 1 1.3 349 3.1 B 1
## 123 1 7.6 361 4.0 B 1
## 124 1 0.9 343 3.2 B 1
## 125 1 0.7 306 2.7 B 1
## 126 1 6.1 424 3.5 B 1
## 127 2 1.9 406 3.4 B 1
## 128 2 2.0 434 3.4 B 1
## 129 2 1.2 356 3.3 B 1
## 130 2 0.8 404 3.3 B 1
## 131 2 6.8 378 3.8 B 1
## 132 2 0.5 391 3.5 B 1
## 133 2 7.2 364 4.2 B 1
## 134 3 1.9 357 3.8 B 1
## 135 3 2.4 313 3.4 B 1
## 136 3 1.3 236 2.8 B 1
## 137 3 3.2 366 3.8 B 1
## 138 3 7.7 346 3.8 B 1
## 139 3 1.3 348 3.3 B 1
## 140 3 6.3 347 3.9 B 1
## 141 4 6.5 408 4.2 B 1
## 142 4 0.6 371 3.5 B 1
## 143 4 5.6 414 4.1 B 1
## 144 4 2.6 413 3.4 B 1
## 145 4 1.1 459 3.4 B 1
## 146 4 2.5 339 3.6 B 1
## 147 4 4.0 423 3.7 B 1
## 148 5 5.2 342 3.5 B 1
## 149 5 4.7 254 2.6 B 1
## 150 5 1.2 293 2.8 B 1
## 151 5 1.2 323 2.7 B 1
## 152 5 1.0 397 3.2 B 1
## 153 5 6.9 373 3.6 B 1
## 154 5 0.9 334 3.0 B 1
## 155 5 1.6 186 2.0 B 1
## 156 1 1.6 391 3.4 C 1
## 157 1 1.0 382 3.7 C 1
## 158 1 0.5 336 3.3 C 1
## 159 1 5.1 411 4.0 C 1
## 160 1 6.6 316 4.0 C 1
## 161 1 12.8 401 4.5 C 1
## 162 1 1.3 366 3.7 C 1
## 163 1 4.6 353 2.4 C 1
## 164 2 5.8 506 4.2 C 1
## 165 2 10.2 499 4.9 C 1
## 166 2 1.5 213 2.9 C 1
## 167 2 0.7 414 3.4 C 1
## 168 2 6.8 431 4.1 C 1
## 169 2 3.9 365 4.1 C 1
## 170 3 12.1 340 4.2 C 1
## 171 3 0.7 341 3.4 C 1
## 172 3 6.3 307 4.1 C 1
## 173 3 3.7 349 4.0 C 1
## 174 3 1.1 259 3.2 C 1
## 175 3 2.1 302 3.6 C 1
## 176 3 1.0 267 3.4 C 1
## 177 4 1.7 393 3.7 C 1
## 178 4 0.8 394 3.6 C 1
## 179 4 6.2 403 4.4 C 1
## 180 4 10.2 437 4.5 C 1
## 181 4 3.5 454 4.4 C 1
## 182 4 4.8 307 3.6 C 1
## 183 4 1.2 418 3.6 C 1
## 184 5 1.3 263 3.3 C 1
## 185 5 5.1 323 3.8 C 1
## 186 5 11.6 394 4.4 C 1
## 187 5 6.8 336 4.1 C 1
## 188 5 1.3 393 3.1 C 1
## 189 5 6.0 339 3.8 C 1pop2[complete.cases(pop2$height), ]## Another way to right the code above, but focusing more on the height's column.## block cutw height dia clone fert
## 1 1 2.4 71 0.6 A 3
## 2 1 0.7 67 1.4 A 3
## 3 1 6.5 211 3.5 A 3
## 4 1 1.1 69 1.0 A 3
## 5 2 2.0 116 1.4 A 3
## 6 2 4.9 123 3.2 A 3
## 7 2 0.8 68 2.2 A 3
## 8 2 1.3 79 1.8 A 3
## 9 2 8.8 166 2.4 A 3
## 10 2 2.0 91 2.5 A 3
## 11 2 2.5 32 1.9 A 3
## 12 3 2.5 127 2.1 A 3
## 13 3 1.3 79 1.5 A 3
## 14 3 1.0 81 1.6 A 3
## 15 3 6.3 186 2.9 A 3
## 16 3 12.1 234 2.7 A 3
## 17 4 1.6 94 1.7 A 3
## 18 4 0.5 43 0.6 A 3
## 19 4 8.5 249 3.2 A 3
## 20 4 1.1 86 1.6 A 3
## 21 5 1.7 92 1.9 A 3
## 22 5 7.2 241 3.7 A 3
## 23 5 2.7 154 2.4 A 3
## 24 5 1.4 92 1.9 A 3
## 25 1 1.4 36 1.4 B 3
## 26 1 2.6 186 2.2 B 3
## 27 1 2.2 197 2.0 B 3
## 28 1 14.7 349 3.5 B 3
## 29 1 0.7 66 1.3 B 3
## 30 1 1.9 233 2.2 B 3
## 31 1 4.6 251 2.8 B 3
## 32 1 0.8 26 1.0 B 3
## 33 2 0.9 32 0.8 B 3
## 34 2 2.1 89 2.4 B 3
## 35 2 5.2 246 3.5 B 3
## 36 2 11.3 238 3.0 B 3
## 37 2 5.7 245 2.8 B 3
## 38 3 7.0 247 3.0 B 3
## 39 3 2.8 161 2.1 B 3
## 40 3 2.0 117 1.7 B 3
## 41 3 4.3 163 3.2 B 3
## 42 3 0.7 51 1.3 B 3
## 43 4 0.5 81 1.5 B 3
## 44 4 1.9 144 2.1 B 3
## 45 4 19.7 324 2.8 B 3
## 46 4 3.1 164 2.0 B 3
## 47 4 6.8 242 3.0 B 3
## 48 4 1.5 101 1.6 B 3
## 49 4 14.3 323 3.6 B 3
## 50 5 11.8 294 3.9 B 3
## 51 5 4.1 144 2.8 B 3
## 52 5 3.8 173 2.3 B 3
## 53 5 2.3 61 2.0 B 3
## 54 5 2.4 58 1.3 B 3
## 55 5 1.4 99 2.0 B 3
## 56 1 0.6 79 0.9 C 3
## 57 1 0.9 24 0.9 C 3
## 58 1 4.5 208 3.2 C 3
## 59 1 0.8 93 1.2 C 3
## 60 1 0.6 93 0.9 C 3
## 61 2 0.7 116 1.1 C 3
## 62 2 4.1 261 2.7 C 3
## 63 2 7.4 282 3.5 C 3
## 64 2 1.1 62 0.9 C 3
## 65 2 0.6 48 0.9 C 3
## 66 2 10.3 281 3.6 C 3
## 67 2 14.7 284 3.8 C 3
## 68 2 1.8 79 1.9 C 3
## 69 3 2.4 42 2.4 C 3
## 70 3 2.2 77 2.1 C 3
## 71 3 1.2 114 1.7 C 3
## 72 3 0.8 81 1.3 C 3
## 73 3 0.7 27 0.9 C 3
## 74 3 5.6 22 2.1 C 3
## 75 3 8.5 73 4.0 C 3
## 76 3 10.1 276 3.9 C 3
## 77 4 8.6 247 3.4 C 3
## 78 4 0.9 93 1.8 C 3
## 79 4 0.9 76 1.2 C 3
## 80 4 7.1 221 3.3 C 3
## 81 4 2.1 49 1.4 C 3
## 82 4 0.7 82 1.4 C 3
## 83 4 9.2 241 3.2 C 3
## 84 5 1.1 48 1.3 C 3
## 85 5 2.0 74 1.5 C 3
## 86 5 4.7 51 3.2 C 3
## 87 5 2.9 31 1.3 C 3
## 88 5 8.4 221 3.6 C 3
## 89 5 6.1 227 2.8 C 3
## 90 5 0.5 79 1.0 C 3
## 91 5 0.8 128 1.6 C 3
## 92 1 4.6 441 4.0 A 1
## 93 1 3.6 316 3.6 A 1
## 94 1 2.1 353 3.3 A 1
## 95 1 0.3 284 3.2 A 1
## 96 2 0.1 366 3.4 A 1
## 97 2 0.8 421 4.1 A 1
## 98 2 7.3 407 39.0 A 1
## 99 2 0.6 182 1.6 A 1
## 100 2 14.2 414 3.6 A 1
## 101 2 3.2 305 3.4 A 1
## 102 3 0.9 306 3.0 A 1
## 103 3 7.7 418 4.5 A 1
## 104 3 0.6 373 3.7 A 1
## 105 3 0.9 188 2.3 A 1
## 106 3 2.1 328 3.1 A 1
## 107 4 1.3 371 3.6 A 1
## 108 4 1.4 377 3.8 A 1
## 110 4 2.2 416 3.4 A 1
## 111 4 6.0 406 3.8 A 1
## 112 5 5.3 261 3.0 A 1
## 113 5 0.4 68 2.0 A 1
## 114 5 4.1 218 2.3 A 1
## 115 5 1.6 334 3.2 A 1
## 116 5 5.1 371 3.8 A 1
## 117 5 3.1 274 3.1 A 1
## 118 5 1.1 262 2.8 A 1
## 119 1 2.7 394 3.4 B 1
## 120 1 12.2 431 4.3 B 1
## 121 1 0.8 386 3.3 B 1
## 122 1 1.3 349 3.1 B 1
## 123 1 7.6 361 4.0 B 1
## 124 1 0.9 343 3.2 B 1
## 125 1 0.7 306 2.7 B 1
## 126 1 6.1 424 3.5 B 1
## 127 2 1.9 406 3.4 B 1
## 128 2 2.0 434 3.4 B 1
## 129 2 1.2 356 3.3 B 1
## 130 2 0.8 404 3.3 B 1
## 131 2 6.8 378 3.8 B 1
## 132 2 0.5 391 3.5 B 1
## 133 2 7.2 364 4.2 B 1
## 134 3 1.9 357 3.8 B 1
## 135 3 2.4 313 3.4 B 1
## 136 3 1.3 236 2.8 B 1
## 137 3 3.2 366 3.8 B 1
## 138 3 7.7 346 3.8 B 1
## 139 3 1.3 348 3.3 B 1
## 140 3 6.3 347 3.9 B 1
## 141 4 6.5 408 4.2 B 1
## 142 4 0.6 371 3.5 B 1
## 143 4 5.6 414 4.1 B 1
## 144 4 2.6 413 3.4 B 1
## 145 4 1.1 459 3.4 B 1
## 146 4 2.5 339 3.6 B 1
## 147 4 4.0 423 3.7 B 1
## 148 5 5.2 342 3.5 B 1
## 149 5 4.7 254 2.6 B 1
## 150 5 1.2 293 2.8 B 1
## 151 5 1.2 323 2.7 B 1
## 152 5 1.0 397 3.2 B 1
## 153 5 6.9 373 3.6 B 1
## 154 5 0.9 334 3.0 B 1
## 155 5 1.6 186 2.0 B 1
## 156 1 1.6 391 3.4 C 1
## 157 1 1.0 382 3.7 C 1
## 158 1 0.5 336 3.3 C 1
## 159 1 5.1 411 4.0 C 1
## 160 1 6.6 316 4.0 C 1
## 161 1 12.8 401 4.5 C 1
## 162 1 1.3 366 3.7 C 1
## 163 1 4.6 353 2.4 C 1
## 164 2 5.8 506 4.2 C 1
## 165 2 10.2 499 4.9 C 1
## 166 2 1.5 213 2.9 C 1
## 167 2 0.7 414 3.4 C 1
## 168 2 6.8 431 4.1 C 1
## 169 2 3.9 365 4.1 C 1
## 170 3 12.1 340 4.2 C 1
## 171 3 0.7 341 3.4 C 1
## 172 3 6.3 307 4.1 C 1
## 173 3 3.7 349 4.0 C 1
## 174 3 1.1 259 3.2 C 1
## 175 3 2.1 302 3.6 C 1
## 176 3 1.0 267 3.4 C 1
## 177 4 1.7 393 3.7 C 1
## 178 4 0.8 394 3.6 C 1
## 179 4 6.2 403 4.4 C 1
## 180 4 10.2 437 4.5 C 1
## 181 4 3.5 454 4.4 C 1
## 182 4 4.8 307 3.6 C 1
## 183 4 1.2 418 3.6 C 1
## 184 5 1.3 263 3.3 C 1
## 185 5 5.1 323 3.8 C 1
## 186 5 11.6 394 4.4 C 1
## 187 5 6.8 336 4.1 C 1
## 188 5 1.3 393 3.1 C 1
## 189 5 6.0 339 3.8 C 1The value for the missing data was found on the sheets where data collected were written. We can simply replace the missing data using the chunk below.
pop2[is.na(pop2$height), 3] <- 331Now we run the summary again to see if there’s a missing data
pop2[is.na(pop2), ]## [1] block cutw height dia clone fert
## <0 rows> (or 0-length row.names)pop2$fert_name<- ifelse(pop2$fert == 1, 'fertilized', 'control')The result returns no values, this implies that we’ve fixed the missing data.
Dealing with Outliers
Outliers are observations that is distant from other observations. This observations have values that are significantly different from other observation values. With outliers we need to careful as they are not errors sometimes.
Identifying the outlier
summary(pop2)## block cutw height dia clone fert
## 1:37 Min. : 0.10 Min. : 22.0 Min. : 0.600 A:51 1:98
## 2:39 1st Qu.: 1.10 1st Qu.:101.0 1st Qu.: 2.000 B:68 3:91
## 3:37 Median : 2.20 Median :263.0 Median : 3.200 C:70
## 4:37 Mean : 3.78 Mean :248.9 Mean : 3.046
## 5:39 3rd Qu.: 5.80 3rd Qu.:365.0 3rd Qu.: 3.600
## Max. :19.70 Max. :506.0 Max. :39.000
## fert_name
## Length:189
## Class :character
## Mode :character
##
##
## We check the mean, quantiles, min, and max value to get an idea of the range of the data within a normal distribution. The min and max are the main values of interest. There is a common rule that says that a data point is an outlier if it is more than 1.5 IQR(Interquartile Range) above the third quartile or below the first quartile. This implies that low outliers are below the ‘1st quarter - (1.5 multipied by IQR)’ and the high outliers are above ‘3rd quarter + (1.5 * IQR)’. Remember IQR is the difference between the 1st and 3rd quarter. We can also draw boxplot or histogram to see the distribution and identify where the outlier is.
Using a histogram
diahist<- ggplot(pop2, aes(dia)) +
geom_histogram(binwidth = 0.45,
show.legend = F)+
labs(title = 'Histogram of diameter',
xlab = 'diameter')
ggplotly(diahist)Using a boxplot
bwplot(dia~block, data = pop2,
xlab = 'block',
ylab = 'diameter (cm)',
main = 'Diameter distribution across Blocks') #bwplot is the lattice package boxplot function
another approach is to look for the data above or below the outlier common rule that uses IQR.
pop2[(pop2$dia> 5),]## block cutw height dia clone fert fert_name
## 98 2 7.3 407 39 A 1 fertilizedrow 98 is having the data with an outlier with a value of 39. On checking the original data written from the field, the result was 3.9
## changing data
pop2$dia <- ifelse(pop2$dia > 38, 3.9, pop2$dia)Experimental Design To check the experimental design we use the ftable funtion
ftable(pop2$block, pop2$fert, pop2$clone)## A B C
##
## 1 1 4 8 8
## 3 4 8 5
## 2 1 6 7 6
## 3 7 5 8
## 3 1 5 7 7
## 3 5 5 8
## 4 1 5 7 7
## 3 4 7 7
## 5 1 7 8 6
## 3 4 6 8Height performance
bwplot(height~clone|fert_name,
data = pop2,
ylab = 'Height(mm)',
xlab = 'Clone')
Diameter performance
diapor <- ggplot(pop2, aes(clone, dia, col = fert_name))+
geom_boxplot()+
labs(x = 'clone',
y = 'diameter')+
facet_wrap(~fert_name)+
labs(title = 'Diameter performance')
ggplotly(diapor)Analysis of Variance for Height
Test of difference with post-hoc to see the effect of the clones and fertilizer on height
Clone
pophd <- lm(height ~ block + clone, data = pop2)anova
anova (pophd)## Analysis of Variance Table
##
## Response: height
## Df Sum Sq Mean Sq F value Pr(>F)
## block 4 104269 26067 1.4851 0.2085
## clone 2 58674 29337 1.6715 0.1908
## Residuals 182 3194450 17552Result of anova shows that the clone variants is having no effect on the height growth of the seedlings propagated through cuttings. Thus, there is no need for a post-hoc test, as the clone effects are more or less the same.
Fertilizer
pophdf <- lm(height~fert_name+block, data = pop2)anova
anova(pophdf)## Analysis of Variance Table
##
## Response: height
## Df Sum Sq Mean Sq F value Pr(>F)
## fert_name 1 2191675 2191675 383.7363 < 2.2e-16 ***
## block 4 120530 30132 5.2758 0.0004814 ***
## Residuals 183 1045188 5711
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Result of anova shows that there’s a significant effect of fertilizer on the height growth of poplar cuttings. However, we don’t have to perform a post-hoc test because there’s just one degree of freedom for fert. If the test is to be conducted anyways the control is lower.
summary(TukeyC(pophdf, where = fert_name))## Goups of means at sig.level = 0.05
## Means G1 G2
## fertilized 353.29 a
## control 136.81 b
##
## Matrix of the difference of means above diagonal and
## respective p-values of the Tukey test below diagonal values
## fertilized control
## fertilized 0 216.486
## control 0 0.000Clone and Fertilizer Interaction
pop_cf <- lm(height ~clone*fert+block, data = pop2)anova
anova(pop_cf)## Analysis of Variance Table
##
## Response: height
## Df Sum Sq Mean Sq F value Pr(>F)
## clone 2 62644 31322 5.7041 0.0039671 **
## fert 1 2177491 2177491 396.5475 < 2.2e-16 ***
## block 4 119244 29811 5.4289 0.0003784 ***
## clone:fert 2 15103 7552 1.3752 0.2554346
## Residuals 179 982911 5491
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Result of anova shows that fertilizer has a large of effect on the height of poplar cuttings seedlings, as clone is having little effect on the height performance of the cuttings seedlings.
post-hoc
summary(TukeyC(pop_cf))## Goups of means at sig.level = 0.05
## Means G1 G2
## B 263.38 a
## C 245.11 a b
## A 222.11 b
##
## Matrix of the difference of means above diagonal and
## respective p-values of the Tukey test below diagonal values
## B C A
## B 0.000 18.276 41.277
## C 0.318 0.000 23.001
## A 0.008 0.213 0.000The post-hoc shows the clear difference on the clone performance, with clone B producing having greater effect when fertilizer is added.
Analysis of Variance for Diameter
Test of difference with post-hoc to see the effect of the clones and fertilizer on diameter
Clone
popd <- lm(dia ~ block + clone, data = pop2)anova
anova (popd)## Analysis of Variance Table
##
## Response: dia
## Df Sum Sq Mean Sq F value Pr(>F)
## block 4 2.411 0.60272 0.5706 0.6843
## clone 2 1.679 0.83928 0.7945 0.4534
## Residuals 182 192.263 1.05639Result of anova shows that the clone variants is having no effect on the diameter development of the seedlings propagated through cuttings. Thus, there is no need for a post-hoc test, as the clone effects are more or less the same.
Fertilizer
popdf <- lm(dia~fert_name+block, data = pop2)anova
anova(popdf)## Analysis of Variance Table
##
## Response: dia
## Df Sum Sq Mean Sq F value Pr(>F)
## fert_name 1 85.695 85.695 146.3392 <2e-16 ***
## block 4 3.494 0.873 1.4914 0.2066
## Residuals 183 107.164 0.586
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Result of anova shows that there’s a significant effect of fertilizer on the development of diameter of poplar cuttings. However, we don’t have to perform a post-hoc test because there’s just one degree of freedom for fert. If the test is to be conducted anyways the control is lower.
summary(TukeyC(popdf, where = fert_name))## Goups of means at sig.level = 0.05
## Means G1 G2
## fertilized 3.51 a
## control 2.16 b
##
## Matrix of the difference of means above diagonal and
## respective p-values of the Tukey test below diagonal values
## fertilized control
## fertilized 0 1.357
## control 0 0.000Clone and Fertilizer Interaction
pop_dcf <- lm(dia ~fert_name*clone+block, data = pop2)pop_dcf2 <- lm(dia ~clone+fert_name+block, data = pop2)anova
anova(pop_dcf)## Analysis of Variance Table
##
## Response: dia
## Df Sum Sq Mean Sq F value Pr(>F)
## fert_name 1 85.695 85.695 150.8916 < 2e-16 ***
## clone 2 2.189 1.094 1.9268 0.14862
## block 4 3.627 0.907 1.5966 0.17716
## fert_name:clone 2 3.183 1.591 2.8021 0.06335 .
## Residuals 179 101.659 0.568
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Result of anova shows that fertilizer has a large of effect on the diameter development of the poplar cuttings seedlings, as other factors or interaction of factors have no effect.
anova(pop_dcf2)## Analysis of Variance Table
##
## Response: dia
## Df Sum Sq Mean Sq F value Pr(>F)
## clone 2 1.553 0.777 1.3406 0.2643
## fert_name 1 86.331 86.331 149.0427 <2e-16 ***
## block 4 3.627 0.907 1.5655 0.1854
## Residuals 181 104.842 0.579
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1post-hoc
summary(TukeyC(pop_dcf, where = fert_name))## Goups of means at sig.level = 0.05
## Means G1 G2
## fertilized 3.50 a
## control 2.15 b
##
## Matrix of the difference of means above diagonal and
## respective p-values of the Tukey test below diagonal values
## fertilized control
## fertilized 0 1.349
## control 0 0.000The post-hoc shows the clear difference on the clone performance, with clone B producing having greater effect when fertilizer is added.
summary(TukeyC(pop_dcf2, where = clone))## Goups of means at sig.level = 0.05
## Means G1
## C 2.94 a
## B 2.86 a
## A 2.66 a
##
## Matrix of the difference of means above diagonal and
## respective p-values of the Tukey test below diagonal values
## C B A
## C 0.000 0.084 0.278
## B 0.795 0.000 0.195
## A 0.118 0.353 0.000No difference in the effect of the clones.
Poplar
Linear Model for Poplar
plot(poplar$cutw, poplar$vol)
# Linear model
lmpop <-lm(vol~cutw, data = poplar)
anova (lmpop)## Analysis of Variance Table
##
## Response: vol
## Df Sum Sq Mean Sq F value Pr(>F)
## cutw 1 33029 33029 233.48 < 2.2e-16 ***
## Residuals 279 39470 141
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1checking the distribution of the data.
hist(lmpop$residuals)
Checking for homoscedactisity, i.e, the assumption for similar variance for a group being compared.
plot(lmpop$fitted.values, lmpop$residuals,
xlab = 'Fitted Values',
ylab = 'Residuals')
abline(c(0,0), col = 2)
Making a qqplot (quantile-quantile plot) to check for normal distribution
qqnorm(lmpop$residuals)
qqline(lmpop$residuals, col = 'red')
Now we can use the predicted values of the linear model as a function to estimate a value when we have the cutting weight available. Let’s use the function which we have To get the value for the function, we extract the intercept and slope.
##intercept
intcpt <- coef(lmpop)[1]
## slope
slp <- coef(lmpop)[2]Creating a simulated value for cutting weights
simul.cutw <- c(0.1, 2, 4,5,6, 9, 10, 14, 16)
### Applying the model
simul.vol <- intcpt + (slp * simul.cutw)
###
simul <- data.frame(simul.cutw, simul.vol)Plotting the data the relationship
plot(simul$simul.cutw, simul$simul.vol,
pch = 16,
col = 'red',
xlab = 'cutting width',
ylab = 'volume',
main = 'Cutting width vs volume relationship in Poplar')
lines(simul$simul.cutw, simul$simul.vol,
lwd = 1.1,
col = 'blue')
points(poplar$cutw, poplar$vol,
pch = 2,
cex = 0.6,
col = 'black')
Spruce Stand Linear Model
Data exploration
summary(spruce2)## height dbh
## Min. :13.00 Min. : 4.00
## 1st Qu.:20.00 1st Qu.:13.00
## Median :25.00 Median :20.00
## Mean :25.56 Mean :19.89
## 3rd Qu.:32.00 3rd Qu.:27.00
## Max. :41.00 Max. :39.00## quick visualization
plot(spruce2$height, spruce2$dbh)
Fitting linear model
lmspruce <- lm(spruce2$height~spruce2$dbh)
anova(lmspruce)## Analysis of Variance Table
##
## Response: spruce2$height
## Df Sum Sq Mean Sq F value Pr(>F)
## spruce2$dbh 1 3541.3 3541.3 1341.4 < 2.2e-16 ***
## Residuals 69 182.2 2.6
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Checking the distribution of the residuals
hist(lmspruce$residuals)
Checking for homoscedactisity,i.e the assumption for similar variance for a group being compared.
plot(lmspruce$fitted.values, lmspruce$residuals,
xlab = 'fitted values',
ylab = 'residuals')
abline(c(0,0), col = 'red')
QQplot
qqnorm(lmspruce$residuals)
qqline(lmspruce$residuals, col = 'red')
Testing the model
int_spr <-coef(lmspruce)[1]
slp_spr <- coef(lmspruce)[2]
### Generating random numbers
sprce_dbh = sample(5:50, replace = TRUE)
sprce_height = int_spr + (slp_spr * sprce_dbh)
sprce <- data.frame(sprce_height, sprce_dbh)plot(sprce$sprce_height, sprce_dbh,
col = 'red')
lines(sprce$sprce_height, sprce_dbh,
col = 'black')
points(spruce2$height, spruce2$dbh)