Mathematics and Applied Statistics Lesson of the Day – Contrasts

A contrast is a linear combination of a set of variables such that the sum of the coefficients is equal to zero.  Notationally, consider a set of variables

\mu_1, \mu_2, ..., \mu_n.

Then the linear combination

c_1 \mu_1 + c_2 \mu_2 + ... + c_n \mu_n

is a contrast if

c_1 + c_2 + ... + c_n = 0.

There is a reason for why I chose to use \mu as the symbol for the variables in the above notation – in statistics, contrasts provide a very useful framework for comparing multiple population means in hypothesis testing.  In a later Statistics Lesson of the Day, I will illustrate some examples of contrasts, especially in the context of experimental design.

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Applied Statistics Lesson and Humour of the Day – Type I Error (False Positive) and Type 2 Error (False Negative)

In hypothesis testing,

  • a Type 1 error is the rejection of the null hypothesis when it is actually true
  • a Type 2 error is the acceptance of the null hypothesis when it is actually false.  (Some statisticians prefer to say “failure to reject” rather than “accept” the null hypothesis for Type 2 errors.)

A Type 1 error is also known as a false positive, and a Type 2 error is also known as a false negative.  This nomenclature comes from the conventional connotation of

  • the null hypothesis as the “negative” or the “boring” result
  • the alternative hypothesis as the “positive” or “exciting” result.

A great way to illustrate the meaning and the intuition of Type 1 errors and Type 2 errors is the following cartoon.

pregnant

Source of Image: Effect Size FAQs by Paul Ellis

In this case, the null hypothesis (or the “boring” result) is “You’re not pregnant”, and the alternative hypothesis (or the “exciting” result) is “You’re pregnant!”.

 

This is the most effective way to explain Type 1 error and Type 2 error that I have encountered!

Applied Statistics Lesson of the Day – The Independent 2-Sample t-Test with Unequal Variances (Welch’s t-Test)

A common problem in statistics is determining whether or not the means of 2 populations are equal.  The independent 2-sample t-test is a popular parametric method to answer this question.  (In an earlier Statistics Lesson of the Day, I discussed how data collected from a completely randomized design with 1 binary factor can be analyzed by an independent 2-sample t-test.  I also discussed its possible use in the discovery of argon.)  I have learned 2 versions of the independent 2-sample t-test, and they differ on the variances of the 2 samples.  The 2 possibilities are

  • equal variances
  • unequal variances

Most statistics textbooks that I have read elaborate at length about the independent 2-sample t-test with equal variances (also called Student’s t-test).  However, the assumption of equal variances needs to be checked using the chi-squared test before proceeding with the Student’s t-test, yet this check does not seem to be universally done in practice.  Furthermore, conducting one test based on the results of another can inflate the probability of committing a Type 1 error (Ruxton, 2006).

Some books give due attention to the independent 2-sample t-test with unequal variances (also called Welch’s t-test), but some barely mention its value, and others do not even mention it at all.  I find this to be puzzling, because the assumption of equal variances is often violated in practice, and Welch’s t-test provides an easy solution to this problem.  There is a seemingly intimidating but straightforward calculation to approximate the number of degrees of freedom for Welch’s t-test, and this calculation is automatically incorporated in most software, including R and SAS.  Finally, Welch’s t-test removes the need to check for equal variances, and it is almost as powerful as Student’s t-test when the variances are equal (Ruxton, 2006).

For all of these reasons, I recommend Welch’s t-test when using the parametric approach to compare the means of 2 populations.

Reference

Graeme D. Ruxton.  “The unequal variance t-test is an underused alternative to Student’s t-test and the Mann–Whitney U test“.  Behavioral Ecology (July/August 2006) 17 (4): 688-690 first published online May 17, 2006