# Interpreting a p-value of a spearman correlation

In the following plot, I am trying to find the correlation between two different measures:

I'm confused about the result of the p-value which is below the cut-off p-value of 0.05. Looking at the plot, I was expecting there will be no correlation between the two measures.

My question is it really true there is a significant correlation between the two measures? and should I say that we reject the null hypothesis and accept the alternative instead?

• What is the blue line along the top of the plot? How many observations do you have as I suspect there is much over-printing here which makes it difficult to see what is going on? – mdewey Aug 4 '19 at 16:45
• Agree with @mdewey. Perhaps the few points at bottom center along with a slight concentration of points at upper right are enough to give a significantly positive Spearman correlation. Also many overplotted points at upper left might be slightly lower than points at upper right. Maybe start with temporary deletion of points below .5 on y axis, trying for a scale with less overplotting. // I'm wondering if there any useful association. – BruceET Aug 4 '19 at 17:13
• If you consider Spearman's correlation appropriate to summarize the association between the variables and rejecting its null hypothesis is of relevance, then everything seems okay. – Michael M Aug 4 '19 at 18:05
• @mdewey The blue line is a regression line (I accidentally added it to the ggscatter plot in R). The number of observations is 7290 so there is no overplotting. – Adam Amin Aug 5 '19 at 10:57

Comment continued, with graphs using R.

set.seed(2019)
y = rbeta(1000, 10, .4)
x = runif(1000)
par(mfrow=c(1,2))
plot(x,y,pch=20,ylim=0:1)
plot(x,y^20,pch=20, ylim=0:1)
par(mfrow=c(1,1))


Data for both plots have the same Spearman correlation.

cor(x,y, meth="sp")
[1] 0.01978538
cor(x,y^20, meth="sp")
[1] 0.01978538

• Can I say in this case there is a significant correlation between measure x and measure y? – Adam Amin Aug 5 '19 at 11:03
• I couldn't edit my last comment so I have to add a new comment. If there is a significant correlation, is this because of some data points, but not all points, that show a correlation between measure x and y? – Adam Amin Aug 5 '19 at 11:29