Variance, Covariance, and Correlation

1. Two factors (x,y)

x=c(1,2,3,4)              #(n=4)
y=c(10,15,24,36)

1.1. Variance of x | y

sum((x-mean(x))^2)/(4-1)    # calculation of variance of x 

## [1] 1.666667

var(x)                      # R code

## [1] 1.666667

sum((y-mean(y))^2)/(4-1)    # calculation of variance of y 

## [1] 130.25

var(y)                      # R code

## [1] 130.25

1.2. Covariance of x versus y

(sum((x-mean(x))*(y-mean(y)))) / (4-1)  # calculation of covariance of x and y 

## [1] 14.5

cov(x,y)

## [1] 14.5

var(x,y)                                # R code

## [1] 14.5

1.3. Correlation of x versus y

cov(x,y) / (sd(x)*sd(y))  # calculation of covariance of x and y 

## [1] 0.9841352

cor(x,y)                                # R code

## [1] 0.9841352

2. Three factors

V1=c(2,3,4,5)               # Three vectors: 3 variables; 4 subjects (n=4)
V2=c(45,35,24,36)
V3=c(2.1,3.5,6.2,4.3)

matrix=cbind(V1,V2,V3)      #3x4 matrix
matrix

##      V1 V2  V3
## [1,]  2 45 2.1
## [2,]  3 35 3.5
## [3,]  4 24 6.2
## [4,]  5 36 4.3

cov(matrix)                 ## # R code for Covariance matrix

##           V1         V2         V3
## V1  1.666667  -6.333333   1.550000
## V2 -6.333333  74.000000 -14.300000
## V3  1.550000 -14.300000   2.929167

## Covariance matrix manual
Cv11=sum((V1-mean(V1))^2)/(4-1)                  ;Cv11

## [1] 1.666667

Cv12=(sum((V1-mean(V1))*(V2-mean(V2)))) / (4-1) ; Cv12

## [1] -6.333333

Cv22=sum((V2-mean(V2))^2)/(4-1)                 ; Cv22

## [1] 74

Cv23=(sum((V2-mean(V2))*(V3-mean(V3)))) / (4-1) ; Cv23

## [1] -14.3

3. Validation parameters

3.1. Standard error of measurement (SEM)

SEM=sqrt(sd/n)

3.2. Within subject coefficient of variation (CV)

wCV=SEM/m*100 in which m=sample size

3.3. Intraclass correlation coefficient (ICC)

Description

The Intraclass Correlation Coefficient (ICC) is a measure of the reliability of measurements or ratings. = Coefficient of reliability

ICC = var.t/(var.t+var.e)

in which var.t = individual variance (variation explained by the model)

———var.e = residual variance

x1=c(2.3,3.2,5.6,4.5,7.6)
x2=c(2.5,3.5,4.6,5.8,6.6)

k=2
n=length(x1)
meani=(x1+x2)/k                         #[1] 2.40 3.35 5.10 5.15 7.10
overall.mean <- mean(meani)             #[1] 4.62  = mean(c(x1,x2))
#Between sum square (bss) and between mean square (bsm)
bssi=k*(meani-overall.mean)^2           #[1]  9.8568  3.2258  0.4608  0.5618 12.3008 
bss=sum(bssi)                           #between sum square (bss)
bms=bss/(n-1)                               #between mean square (bsm)

#Within sum square
wssi=(x1-x2)^2/2
wss=sum(wssi)                           #within sum square (wss)
wms=wss/(k*n-n)                         #within mean square (wms)

# ICC calculation
var.t=(bms-wms)/k
var.e=wms
ICC=var.t/(var.t+var.e)
ICC

## [1] 0.8905993

#Using ICC package
value=c(x1,x2)
ID=rep(1:5,2)
dat=cbind(ID,value)
dat=as.data.frame(dat)
dat$ID=factor(dat$ID)

library(ICC)
ICCbare(ID,value,dat)

## [1] 0.8905993