Intro to Multivariate Stats

Difference between Transformations and Standardizations

• Transformations are applied to each element in a matrix
• Standardization adjust elements in a matrix by a row or column statistic

Create some data

rawdata <- matrix(c(1,1,1,3,3,1,
                    2,2,4,6,6,0,
                    10,10,20,30,30,0,
                    3,3,2,1,1,0,
                    0,0,0,20,0,0), ncol = 6, byrow = TRUE)
colnames(rawdata) <- paste("species",toupper(letters[1:6]), sep = "_")
rawdata

     species_A species_B species_C species_D species_E species_F
[1,]         1         1         1         3         3         1
[2,]         2         2         4         6         6         0
[3,]        10        10        20        30        30         0
[4,]         3         3         2         1         1         0
[5,]         0         0         0        20         0         0

Calculating row and column statistics

Rows

# Row sums
rowSums(rawdata)

[1]  10  20 100  10  20

apply(rawdata, 1, sum)

[1]  10  20 100  10  20

# Max values
apply(rawdata, 1, max)

[1]  3  6 30  3 20

Columns

# Sums
apply(rawdata, 2, sum)

species_A species_B species_C species_D species_E species_F 
       16        16        27        60        40         1 

colSums(rawdata)

species_A species_B species_C species_D species_E species_F 
       16        16        27        60        40         1 

# Max
apply(rawdata, 2, max)

species_A species_B species_C species_D species_E species_F 
       10        10        20        30        30         1 

Monotonic transformations

Log transformations

• Useful for when you have a wide spread in data values

• Ir is important that you add 1 to values to account for zeros log10(x+1)

logdata <- apply(rawdata , c(1,2), function(x) log10(x + 1))

library(tidyverse)
hemlock <- read_csv("https://raw.githubusercontent.com/chrischizinski/SNR_R_Group/master/data/hemlock_cover.csv")
hemlock$logTsuga<- log10(hemlock$Tsuga.canadensis +1) # log transform 
glimpse(hemlock)

Observations: 98
Variables: 3
$ Site             <int> 1001, 1002, 1003, 1004, 1005, 1006, 1011, 1102, 1108, 1110, 1112, 1201, 1203, 1206, 1305, 1306, 1307, 1...
$ Tsuga.canadensis <dbl> 0.1, 34.0, 42.0, 21.0, 37.0, 18.0, 9.0, 0.0, 0.0, 0.0, 43.0, 8.0, 0.0, 4.0, 45.0, 25.0, 0.0, 0.0, 0.0, ...
$ logTsuga         <dbl> 0.04139269, 1.54406804, 1.63346846, 1.34242268, 1.57978360, 1.27875360, 1.00000000, 0.00000000, 0.00000...

ggplot(data = hemlock) + 
  geom_histogram(aes(Tsuga.canadensis), binwidth = 5, colour = "black", fill = "dodgerblue") + 
  coord_cartesian(ylim = c(0, 30), expand = FALSE) +
  theme_bw()

ggplot(data = hemlock) + 
  geom_histogram(aes(logTsuga), bins = 30, colour = "black", fill = "red") + 
  coord_cartesian(ylim = c(0, 30), expand = FALSE) +
  theme_bw()

Power tranformations

• Square root transformation is most often used for Poisson type date (count data)
• Greater the power, the greater the compression of the data
• Flexible for a wide range of data
• Applied when the data is > 0

Write power function

pwr_trans <- function(x, trans){
  x <- ifelse(x>0,x^(1/trans),0)
  return(x)
}
pwr_trans(25,2)

[1] 5

pwr_trans(0,2)

[1] 0

Display the effect of the power function

newdata <- data.frame(x = 0:100, 
                      cubic = pwr_trans(x=0:100, trans = 3),
                      power10 = pwr_trans(x=0:100, trans = 10))
head(newdata)

ggplot(data = newdata) +
  geom_line(aes(x = x, y = cubic), size = 1, colour = "blue") +
  geom_line(aes(x = x, y = power10), size = 1, colour = "red") +
  labs(y = "Value") +
  coord_cartesian(xlim = c(0,100.5), ylim = c(0,5), expand = F)+
  theme_classic()

NA

Presence absence transformation

• Transforms quantitative data to non-quanitative (binary)
• Applicable to species data
• Most useful when there is not a lot of quantitative data available (i.e., LOTS of zeros)
• Severe transformation (i.e., loose lots of information)

library(vegan)

Loading required package: permute
Loading required package: lattice
This is vegan 2.4-3

decostand(rawdata, method = "pa")

     species_A species_B species_C species_D species_E species_F
[1,]         1         1         1         1         1         1
[2,]         1         1         1         1         1         0
[3,]         1         1         1         1         1         0
[4,]         1         1         1         1         1         0
[5,]         0         0         0         1         0         0
attr(,"decostand")
[1] "pa"

Arcsine transformation

Please NOTE: The arcsine is asinine: the analysis of proportions in ecology

• Transformations on proportion data (0-1)
• Useful when you have a positive skew in data
  • Spreads the end of the scale while compressing the middle

Standardizations

Sums

• Can be applied to any range of x
• Output will range 0 - 1
• Converts values to a relative value (equalizes the area under the curve)
• Useful when you have large difference in total abundance

Rows

ttl_species <- apply(rawdata, 1, sum)
rowprop_data <- rawdata / ttl_species
  
rowprop_data

     species_A species_B species_C species_D species_E species_F
[1,]       0.1       0.1       0.1       0.3       0.3       0.1
[2,]       0.1       0.1       0.2       0.3       0.3       0.0
[3,]       0.1       0.1       0.2       0.3       0.3       0.0
[4,]       0.3       0.3       0.2       0.1       0.1       0.0
[5,]       0.0       0.0       0.0       1.0       0.0       0.0

decostand(rawdata, margin = 1, method = "total")

     species_A species_B species_C species_D species_E species_F
[1,]       0.1       0.1       0.1       0.3       0.3       0.1
[2,]       0.1       0.1       0.2       0.3       0.3       0.0
[3,]       0.1       0.1       0.2       0.3       0.3       0.0
[4,]       0.3       0.3       0.2       0.1       0.1       0.0
[5,]       0.0       0.0       0.0       1.0       0.0       0.0
attr(,"decostand")
[1] "total"

Columns

colprop_data <- rawdata %*% diag(1/apply(rawdata,2,sum))