Spearman's Correlation

Spearman's Correlation

Table of content

  • When to use the Spearman Correlation test
  • What does the Speaman Correlation measure?
  • What are the assumptions of the Spearman
  • Spearman Correlation formula
  • What is the definition of the Spearman Correlation test?
  • Example: how to calculate Spearman Correlation?
When to use the Spearman Correlation test

When to use the Spearman Correlation test

What does the Speaman Correlation measure?

  • The Spearman correlation (denoted as p (rho) or rs) measures the strength and direction of association between two ranked variables.
  • It is most commonly used to measure the degree and direction of a linear relation between two variables that are of the ordinal type.

What are the assumptions of the Spearman Correlation test?

  • two variables that are either ordinal, interval or ratio (note: if the data are interval or ratio, normally you would use a Pearson correlation test)
  • the relationship between the variables is monotonic, or (curvi)linear
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Spearman Correlation formula

Spearman Correlation formula

What is the definition of the Spearman Correlation test?

\[\rho = r_s = 1 - \frac{6\sum{d^2_i}}{n(n^2-1)}\]

  • p : Spearman correlation
  • rs : Spearman correlation
  • di : rg(Xi) - rg(Yi): difference between the two ranks of each observation (for example, one can have the second best score on variable X, but the ninth on variable Y.)
  • n : number of scores

Note: if there are tied ranks, the following method can be used to calculate the Spearman Correlation:

\[\rho = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt {\sum_i (x_i - \bar{x})^2 \sum_i (y_i - \bar{y})^2}}\]

  • x : value of x
  • x̄ : mean of x
  • y : value of y
  • ȳ : mean of y
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Example: how to calculate Spearman Correlation?

Example: how to calculate Spearman Correlation?

Sample question

The scores for nine students in history and algebra are as follows:

  • History: 35, 23, 47, 17, 10, 43, 9, 6, 28
  • Algebra: 30, 33, 45, 23, 8, 49, 12, 4, 31

Compute the Spearman rank correlation.

Answer

  1. Step 1: rank each student
Student #History ScoreHistory RankAlgebra ScoreAlgebra Rank
1353305
2235333
3471452
4176236
510788
6432491
798127
86949
9284314
  1. step 2: calculate difference between the ranks (d) and square d
Student #History ScoreHistory RankAlgebra ScoreAlgebra Rankdd2
135330524
223533324
347145211
417623600
51078811
643249111
79812711
8694900
928431400
  1. step 3: sum (add up) all the d2 scores
  • Σ d2 = 4 + 4 + 1 + 0 + 1 + 1 + 1 + 0 + 0 = 12
  1. step 4: insert the values in the formula. These ranks are not tied, so use the first formula:

\[\rho = 1 - \frac{6\sum{d^2_i}}{n(n^2-1)}\]

= 1 – (6*12)/(9(81-1))
= 1 – 72/720
= 1-0.1
= 0.9

> The Spearman Rank Correlation for this set of data is 0.9.

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SPOTLIGHT

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Introduction to Statistics: in short

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How to triumph over the theory of statistics (without understanding everything)?

Stats of students

  • The first years that you follow statistics, it is often a case of taking knowledge for granted and simply trying to pass the courses. Don't worry if you don't understand everything right away: in later years it will fall into place, and you will see the importance of the theory you had to know before.
  • The book you need to study may be difficult to understand at first. Be patient: later in your studies, the effort you put in now will pay off.
  • Be a Gestalt Scientist! In other words, recognize that the whole of statistics is greater than the sum of its parts. It is very easy to get hung up on nit-picking details and fail to see the forest because of the trees
  • Tip: Precise use of language is important in research. Try to reproduce the theory verbatim (i.e. learn by heart) where possible. With that, you don't have to understand it yet, you show that you've been working on it, you can't go wrong by using the wrong word and you practice for later reporting of research.
  • Tip: Keep study material, handouts, sheets, and other publications from your teacher for future reference.

How to score points with formulas of statistics (without learning them all)?

  • The direct relationship between data and results consists of mathematical formulas. These follow their own logic, are written in their own language, and can therefore be complex to comprehend.
  • If you don't understand the math behind statistics, you don't understand statistics. This does not have to be a problem, because statistics is an applied science from which you can also get excellent results without understanding. None of your teachers will understand all the statistical formulas.
  • Please note: you will probably have to know and understand a number of formulas, so that you can demonstrate that you know the principle of how statistics work. Which formulas you need to know differs from subject to subject and lecturer to lecturer, but in general these are relatively simple formulas that occur frequently, and your lecturer will likely tell you (often several times) that you should know this formula.
  • Tip: if you want to recognize statistical symbols, you can use: Recognizing commonly used statistical symbols
  • Tip: have fun with LaTeX! LaTeX code gives us a simple way to write out mathematical formulas and make them look professional. Play with LaTeX. With that, you can include used formulas in your own papers and you learn to understand how a formula is built up – which greatly benefits your understanding and remembering that formula. See also (in Dutch): How to create formulas like a pro on JoHo WorldSupporter?
  • Tip: Are you interested in a career in sciences or programming? Then take your formulas seriously and go through them again after your course.

How to practice your statistics (with minimal effort)?

How to select your data?

  • Your teacher will regularly use a dataset for lessons during the first years of your studying. It is instructive (and can be a lot of fun) to set up your own research for once with real data that is also used by other researchers.
  • Tip: scientific articles often indicate which datasets have been used for the research. There is a good chance that those datasets are valid. Sometimes there are also studies that determine which datasets are more valid for the topic you want to study than others. Make use of datasets other researchers point out.
  • Tip: Do you want an interesting research result? You can use the same method and question, but use an alternative dataset, and/or alternative variables, and/or alternative location, and/or alternative time span. This allows you to validate or falsify the results of earlier research.
  • Tip: for datasets you can look at Discovering datasets for statistical research

How to operationalize clearly and smartly?

  • For the operationalization, it is usually sufficient to indicate the following three things:
    • What is the concept you want to study?
    • Which variable does that concept represent?
    • Which indicators do you select for those variables?
  • It is smart to argue that a variable is valid, or why you choose that indicator.
  • For example, if you want to know whether someone is currently a father or mother (concept), you can search the variables for how many children the respondent has (variable) and then select on the indicators greater than 0, or is not 0 (indicators). Where possible, use the terms 'concept', 'variable', 'indicator' and 'valid' in your communication. For example, as follows: “The variable [variable name] is a valid measure of the concept [concept name] (if applicable: source). The value [description of the value] is an indicator of [what you want to measure].” (ie.: The variable "Number of children" is a valid measure of the concept of parenthood. A value greater than 0 is an indicator of whether someone is currently a father or mother.)

How to run analyses and draw your conclusions?

  • The choice of your analyses depends, among other things, on what your research goal is, which methods are often used in the existing literature, and practical issues and limitations.
  • The more you learn, the more independently you can choose research methods that suit your research goal. In the beginning, follow the lecturer – at the end of your studies you will have a toolbox with which you can vary in your research yourself.
  • Try to link up as much as possible with research methods that are used in the existing literature, because otherwise you could be comparing apples with oranges. Deviating can sometimes lead to interesting results, but discuss this with your teacher first.
  • For as long as you need, keep a step-by-step plan at hand on how you can best run your analysis and achieve results. For every analysis you run, there is a step-by-step explanation of how to perform it; if you do not find it in your study literature, it can often be found quickly on the internet.
  • Tip: Practice a lot with statistics, so that you can show results quickly. You cannot learn statistics by just reading about it.
  • Tip: The measurement level of the variables you use (ratio, interval, ordinal, nominal) largely determines the research method you can use. Show your audience that you recognize this.
  • Tip: conclusions from statistical analyses will never be certain, but at the most likely. There is usually a standard formulation for each research method with which you can express the conclusions from that analysis and at the same time indicate that it is not certain. Use that standard wording when communicating about results from your analysis.
  • Tip: see explanation for various analyses: Introduction to statistics
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