WHAT IS SPEARMAN'S RHO?

This is an inferential test created by Charles Spearman (left). It is also known as "Spearman's Rank" and is sometimes represented by the Greek letter Rho (r).   It is used when:

  • You have a test of relationships (correlation) of two independent variables
  • The data is at least ordinal level*
 
(* it’s easy to turn interval/ratio level data into ordinal data: you just put the scores into rank order)
 
The Edexcel exam might ask you about the appropriateness of the Spearman's Rho test – when you would use it. You could also be asked to calculate the test. The equation and tables are provided at the front of the exam booklet.

If you are presented with an experiment comparing two or more different conditions, use another statistical test instead.

MAKING THE CALCULATIONS

Of course, most researchers use computers to do their statistical tests. There are lots of websites that will let you input your data from an experiment and carry out the Spearman's Rho test for you.
The Exam expects you to be able to work out the Spearman's test "by hand" and, although it's fiddly, it's not that difficult.

You are trying to work out the observed value of Rho.
 
The test uses ordinal level data, so you will probably need to turn your interval/ratio data into ordinal data. You do this by “ranking” the scores in each condition; the top scores gets rank 1, the next score gets rank 2 and so on; identical scores get the mean rank they share (so if there are 3 scores sharing 1st place, instead of getting ranks 1, 2 and 3, they all get rank 2).

  • You work out the ranks for each set of scores separately: this is different from the Mann-Whitney U-Test where you lump all the scores together and rank the whole lot

You will end up with two ranks for each person or entry: the rank in the first independent variable you are studying (say, how you rate yourself for aggression) and the rank in the second independent variable (perhaps, how your friend rates you for aggression).

You can put these in a table to see them clearly:
You can see that the first person rates himself as "2" for aggression and his friend rates him "3" - pretty low. For self-rated aggression, this person shares rank 6.5 with another person who rated themselves as "2"; these two were the two lowest self-ratings. For peer-rated aggression, this person is ranked as 6th because there's one other person rated lower by a friend.
The next step is to work out the differences between the ranks by deducting Rank 1 from Rank 2. This goes in a column labelled "d" (for "difference"):
Don't worry if you forget to put in the negative numbers or deduct Rank 2 from Rank 1. It all ends up the same.
The next step is to square all the differences - multiply each of the differences by itself:
Multiplying decimal fractions can be tricky so you might want a calculator for this. You'll notice the negative signs disappear when you square the values - that's why it didn't matter which way round you did the subtracting in the previous step.
You can now work out the sum of all the squared differences. There's a mathematical symbol for this which is used in Spearman's formula:
With this done, you can use the fairly simple formula for Spearman's Rho at the front of the exam booklet:
You could do this without a calculator, but it might take a while. Don't worry, because the Examiner should give you most of the marks if you write out the full equation even if you don't calculate the final answer. For example:
The number you're left with is Spearman's correlational coefficient:
  • If it's positive, then the data has a positive correlation; if it's negative, the data has a negative correlation
  • The closer it is to 0 (zero), the weaker the correlation
  • The closer it is to 1, the stronger the correlation
In the example above, 0.786 is a pretty strong positive correlation, but is it strong ENOUGH to be statistically significant?
You will compare the correlational coefficient (Rho, your observed value) to the critical value and you are looking for Rho to be higher than the critical value.
 
Before you can look up the critical value, you need to know two things:
  • Is your correlational hypothesis 1-tailed (directional) or 2-tailed (non-directional)? You will use a different column depending on the type of hypothesis.
  • What value of probability (p) are you considering? Normally, a classroom correlation would be considered at a value of p≤0.05

Once you know these two things, you can finish the calculation:

  1. Choose the column for your type of hypothesis and value of p
  2. Read down until you get to the row matching your score for n
  3. This is your critical value
If you look closely at these tables, you'll notice that the critical values for 2-tailed tests are a bit higher than the values for 1-tailed tests. This reflects the fact that, being a bit more vague, 2-tailed tests require stronger evidence of a correlation.

In the example above, the test will be 2-tailed (since it doesn't specify whether we are looking for a positive or a negative hypothesis) and the critical value is 0.679; our observed value is higher than that. In fact, 0.786 would be significant at p≤0.025 as well.
If your value of Rho is equal to or higher than the critical value, you can refute your null hypothesis (and cautiously accept your hypothesis).
 
If the value of Rho is less than the critical value, you must accept your null hypothesis and refute your correlational hypothesis.

A STATEMENT OF STATISTICAL SIGNIFICANCE

You can sum up your statistical test with a statement of statistical significance. This will include:
  1. The test used
  2. The observed (Rho) and critical values
  3. The direction of the hypothesis
  4. The chosen value of p
  5. The conclusion, in terms of the null hypothesis

For example:

The results were subjected to a Spearman's Rho test
The observed value (Rho) was 0.786, which is higher than the critical value of 0.378 for a 2-tailed test where p≤0.05
Therefore, the null hypothesis can be refuted

APPLYING SPEARMAN'S RHO IN PSYCHOLOGY
AO2

One of the main studies in this course that uses Spearman's Rho is Brendgen et al. (2005). Brendgen looks for a correlation between the aggression ratings of the children as given by the teacher and the aggression ratings as given by the children's friends.

Brendgen uses the correlational coefficients in an interesting way. If the correlation is strongly positive, this suggests these scores are valid because they agree with each other.

Brendgen found a “moderate” correlation between teacher and peer ratings for physical aggression (r=0.25) and for social aggression (r=0.33).

Brendgen also compared the strength of the correlation for the MZ twins with the correlation for the DZ twins. If the correlation for the DZ twins is weaker than for the MZ twins, it suggests teachers and friends are more uncertain about how similar the DZ twins are, with some people rating them as very alike but others rating them as different.

The MZ twins' correlations for physical aggression were twice as high as same-sex DZ twins' correlations.  For teacher-ratings, this was r=0.79, a strong correlation. This suggests the MZ twins were much more striking in their physical behaviour and there was little disagreement in rating them as aggressive or as non-aggressive. This makes it look more likely that there is a genetic component to physical aggression.

The MZ twins' correlations for social aggression were lower and more similar to the DZ twins' correlations.Since the correlation between teachers' and friends' ratings are weaker, this suggests more disagreement about how socially aggressive the children are. This adds to the impression that social aggression is less linked to genetics.

Schmolck et al. (2002) also point out a correlation, between the amount of damage to the patient's  temporal lobe and the amount of mistakes made in the semantic memory tests. However, Schmolck doesn't subject this correlation to Spearman's test.

WHAT ABOUT PEARSON'S CORRELATION?
WHAT'S THE DIFFERENCE?

Karl Pearson (left) is the handsome chap who created the horrible Chi-Squared test. He also created a different statistical test for correlation called the Pearson Correlation Coefficient.

Pearson's test and Spearman's test produce similar results: a coefficient between -1.0 (strongly negative) and +1.0 (strongly positive) with 0 meaning no correlation at all.

Pearson's test is designed to measure linear relationships where the two variables are closely locked together. For example, adding more rocks to a bag correlates with its weight: if you have 10 times as many rocks, the bag will be 10 times as heavy.

Spearman's test works for this too but is better for correlations where the relationship isn't linear. For example, if you work twice as hard on your revision you'll get more marks in the exam, but not twice as many marks. There's a correlation between time spent revising and overall mark, but it's not a linear one.
You don't need to know about Pearson's Correlation for the Psychology Exam, but you might come across it in some studies. There's a third correlational test called Kendall's Tau Coefficient, but there's quite enough Greek letters in this course as it is, so I won't bother you with any more.