Determining the direction of a significant Spearman's Rho Correlation

Question

I have conducted Spearman's Rho tests with two ordinal variables (one with 4 possible answers and the other with 6). I have obtained a statistically significant correlation between the two. My question is, how can I graphically (or some other way) determine which answer of each correlate together - as a scatterplot would not work with my data (since it is not scale).

Answer 1

Let's create some toy data in R:

> set.seed(1)
> nn <- 60
> xx <- floor(4*runif(nn))+1
> yy <- pmin(6,pmax(1,xx+rnorm(nn,0,3)))
> 
> Q1 <- ordered(letters[xx],levels=letters[1:4])
> Q2 <- ordered(LETTERS[yy],levels=LETTERS[1:6])

Here is a Spearman's rho correlation test:

> cor.test(as.numeric(Q1),as.numeric(Q2),method="spearman")

        Spearman's rank correlation rho

data:  as.numeric(Q1) and as.numeric(Q2)
S = 24468, p-value = 0.01264
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.3201582 

Warnmeldung:
In cor.test.default(as.numeric(Q1), as.numeric(Q2), method = "spearman") :
  Kann exakten p-Wert bei Bindungen nicht berechnen

Now, one way of understanding this would be a simple incidence table:

> table(Q1,Q2)
   Q2
Q1   A  B  C  D  E  F
  a 10  1  1  0  1  0
  b  6  2  1  1  2  5
  c  4  3  3  4  1  1
  d  3  2  2  1  4  2

Alternatively, you can certainly plot your data. Simply transform them to the underlying integers and add a little jittering:

> delta <- 0.2
> plot(as.numeric(Q1)+runif(nn,-delta,delta),as.numeric(Q2)+runif(nn,-delta,delta),pch=19,xlab="Q1",ylab="Q2",xaxt="n",yaxt="n")
> axis(1,1:4,levels(Q1))
> axis(2,1:6,levels(Q2),las=2)

Alternatively, create a sunflowerplot:

sunflowerplot(Q1,Q2,xaxt="n",yaxt="n")
axis(1,1:4,levels(Q1))
axis(2,1:6,levels(Q2),las=2)

(To be honest, I'm not too keen on the default sunflowerplot, but you can customize the colors and the widths of the segments rather well, see ?sunflowerplot.)

Answer 2

Sometimes people overlook the convenient way in which a Chi-square procedure can answer questions like this. Specifically, each of your 24 cells' Chi-square residuals -- or better yet, standardized residuals -- will tell to what extent the count of observations in that cell is disproportionate, given the entire bivariate distribution. In SPSS:

cross Q1 by Q2 / stat chi /cells count sresid.

Standardized residuals beyond +/- 2 are typically the ones that signal notably disproportionate counts (although this is only a rule of thumb). You'll probably be most interested in values >2 since you want to know which pairs of answers tend to co-occur.