Let`s talk Spearman rank correlation
Melanie Tietje
February 2019
History
Charles Edward Spearman (1863 – 1945) was an English psychologist.
- studied “new psychology”—one that used the scientific method instead of metaphysical speculation
- elected member of the Royal Society 1924: “[For his] pioneer work in the application of mathematical methods to the analysis of the human mind, and his original studies of correlation in this sphere.”
- feud with Karl Pearson, who also taught at the University College London (RLY?)
Spearman rank correlation test - recap
- the non-parametric version for Pearson correlation
- rank correlation = statistical dependence between the rankings of two variables
- assesses how well the relationship between two variables can be described using a monotonic function
- the math: \( \rho = 1 - \frac{6\sum d ^2_i}{n(n^2-1)} \)
- \( d_i \) = the difference between two ranks of each observation
- \( n \) = number of observations
P-value?
“The p-value is the probability of observing data at least as favorable to the alternative hypothesis as our current data set, if the null hypothesis is true” (Open Intro Statistics)
- Probability of observing this amount of correlation given there actually is no correlation (=observing by chance)
- A small p-value (usually < 0.05) corresponds to sufficient evidence to reject \( H_0 \) in favor of \( H_A \)
- z-statistic –> t-distribution –> p-value
Some test data
library(magrittr)
x <- runif(100, min = 0, max = 1) # our x variable
start <- 10 # minimum degree variation in y
end <- 95 # maximum degree variation in y
# create data
aval <- list()
for(step in start:end){
scramble <- sample(c(1:100), step*0.01*length(x), replace = FALSE) # define which numbers to alter
temp <- data.frame(x=x, y=x)
temp$y[scramble] %<>% runif # actual randomization
aval[[step]] <-list(visible = FALSE,
name = paste0('v = ', step),
x=x,
y=temp$y,
rho=cor.test(x,temp$y,method="s")$estimate[[1]],
p=cor.test(x,temp$y,method="s")$p.[[1]]
)
}
- creating a dataset with 2 variables, in which y is a copy of x with increasing degrees of randomization
Some test data
load("spearman.RData")
head(dat, 20)
x y random
1 0.63860736 0.63860736 10
2 0.64373766 0.64373766 10
3 0.09359379 0.09359379 10
4 0.42091535 0.53961715 10
5 0.03532037 0.03532037 10
6 0.31653066 0.31653066 10
7 0.89893546 0.89893546 10
8 0.23923984 0.23923984 10
9 0.69392201 0.69392201 10
10 0.89247378 0.89247378 10
11 0.68513072 0.68513072 10
12 0.10910229 0.10910229 10
13 0.82045396 0.82045396 10
14 0.70001250 0.70001250 10
15 0.71046496 0.71046496 10
16 0.45163949 0.45163949 10
17 0.14559774 0.14559774 10
18 0.11367611 0.11859527 10
19 0.89437855 0.89437855 10
20 0.19451756 0.19451756 10
Some test data
Some test data - adding sample size
library(dplyr)
samplesize <- c()
rho <- c()
p.value <- c()
randomization <- c()
for(j in 1:length(samples_per_group)){ # run for each samplesize
sub <- dat %>% # take subset from each randomization group
group_by(random) %>%
slice(sample(n(), min(samples_per_group[j], n()))) %>%
ungroup()
rho.sub <- c()
p.sub <- c()
for(i in start:end){ # calculate correlation for each randomization
rho <- c(rho, cor.test(sub$x[sub$random==i],sub$y[sub$random==i],
method="s")$estimate[[1]])
p.value <- c(p.value, cor.test(sub$x[sub$random==i],sub$y[sub$random==i],
method="s")$p.[[1]])
randomization <- c(randomization, i)
samplesize <- c(samplesize, samples_per_group[j])
}
}
all <- data.frame(rho=rho, p.value=p.value, randomization=randomization, samplesize=samplesize)
- DF with rho and p-values for each degree of randomization (=correlation) for differing sample sizes
Some test data
head(all, 20)
rho p.value randomization samplesize
1 0.9130879 0 10 95
2 0.9218785 0 11 95
3 0.8516797 0 12 95
4 0.8853443 0 13 95
5 0.9409854 0 14 95
6 0.9175392 0 15 95
7 0.8464306 0 16 95
8 0.8506019 0 17 95
9 0.8666433 0 18 95
10 0.8618841 0 19 95
11 0.7955067 0 20 95
12 0.6642497 0 21 95
13 0.8080627 0 22 95
14 0.7210386 0 23 95
15 0.8103863 0 24 95
16 0.7507279 0 25 95
17 0.7162514 0 26 95
18 0.8047312 0 27 95
19 0.7423992 0 28 95
20 0.7844765 0 29 95
Randomization effect on rho
- The smaller sample size gets, the less predictable is the connection between randomization + rho
Rho and p-values
Threshold for p-values + sample size
summary(lm(data=res, log(smallest_significant_rho)~samplesize))
Call:
lm(formula = log(smallest_significant_rho) ~ samplesize, data = res)
Residuals:
Min 1Q Median 3Q Max
-0.11339 -0.08851 -0.01100 0.05852 0.23767
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.5553022 0.0559552 -9.924 3.05e-08 ***
samplesize -0.0115485 0.0009555 -12.086 1.86e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1052 on 16 degrees of freedom
Multiple R-squared: 0.9013, Adjusted R-squared: 0.8951
F-statistic: 146.1 on 1 and 16 DF, p-value: 1.855e-09
Back to the sqrt(question)
- When I got a low correlation, how do I need to change the data to lose / get significance?
- How do I get a highly significant “non”-correlation?
- \[ \rho = 1 - \frac{6\sum d ^2_i}{n(n^2-1)} \]
–> Looks like every sort of correlation gets strong if sample size is large enough
–> low significant correlations always require large sample sizes