## 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

## Applied Statistics Lesson of the Day – The Matched Pairs Experimental Design

The matched pairs design is a special type of the randomized blocked design in experimental design.  It has only 2 treatment levels (i.e. there is 1 factor, and this factor is binary), and a blocking variable divides the $n$ experimental units into $n/2$ pairs.  Within each pair (i.e. each block), the experimental units are randomly assigned to the 2 treatment groups (e.g. by a coin flip).  The experimental units are divided into pairs such that homogeneity is maximized within each pair.

For example, a lab safety officer wants to compare the durability of nitrile and latex gloves for chemical experiments.  She wants to conduct an experiment with 30 nitrile gloves and 30 latex gloves to test her hypothesis.  She does her best to draw a random sample of 30 students in her university for her experiment, and they all perform the same organic synthesis using the same procedures to see which type of gloves lasts longer.

She could use a completely randomized design so that a random sample of 30 hands get the 30 nitrile gloves, and the other 30 hands get the 30 latex gloves.  However, since lab habits are unique to each person, this poses a confounding variable – durability can be affected by both the material and a student’s lab habits, and the lab safety officer only wants to study the effect of the material.  Thus, a randomized block design should be used instead so that each student acts as a blocking variable – 1 hand gets a nitrile glove, and 1 hand gets a latex glove.  Once the gloves have been given to the student, the type of glove is randomly assigned to each hand; some may get the nitrile glove on their left hand, and some may get it on their right hand.  Since this design involves one binary factor and blocks that divide the experimental units into pairs, this is a matched pairs design.

## Applied Statistics Lesson of the Day – Blocking and the Randomized Complete Blocked Design (RCBD)

A completely randomized design works well for a homogeneous population – one that does not have major differences between any sub-populations.  However, what if a population is heterogeneous?

Consider an example that commonly occurs in medical studies.  An experiment seeks to determine the effectiveness of a drug on curing a disease, and 100 patients are recruited for this double-blinded study – 50 are men, and 50 are women.  An abundance of biological knowledge tells us that men and women have significantly physiologies, and this is a heterogeneous population with respect to gender.  If a completely randomized design is used for this study, gender could be a confounding variable; this is especially true if the experimental group has a much higher proportion of one gender, and the control group has a much higher proportion of the other gender.  (For instance, purely due to the randomness, 45 males may be assigned to the experimental group, and 45 females may be assigned to the control group.)  If a statistically significant difference in the patients’ survival from the disease is observed between such a pair of experimental and control groups, this effect could be attributed to the drug or to gender, and that would ruin the goal of determining the cause-and-effect relationship between the drug and survival from the disease.

To overcome this heterogeneity and control for the effect of gender, a randomized blocked design could be used.  Blocking is the division of the experimental units into homogeneous sub-populations before assigning treatments to them.  A randomized blocked design for our above example would divide the males and females into 2 separate sub-populations, and then each of these 2 groups is split into the experimental and control group.  Thus, the experiment actually has 4 groups:

1. 25 men take the drug (experimental)
2. 25 men take a placebo (control)
3. 25 women take the drug (experimental)
4. 25 women take a placebo (control)

Essentially, the population is divided into blocks of homogeneous sub-populations, and a completely randomized design is applied to each block.  This minimizes the effect of gender on the response and increases the precision of the estimate of the effect of the drug.

## Applied Statistics Lesson of the Day – The Completely Randomized Design with 1 Factor

The simplest experimental design is the completely randomized design with 1 factor.  In this design, each experimental unit is randomly assigned to a factor level.  This design is most useful for a homogeneous population (one that does not have major differences between any sub-populations).  It is appealing because of its simplicity and flexibility – it can be used for a factor with any number of levels, and different treatments can have different sample sizes.  After controlling for confounding variables and choosing the appropriate range and number of levels of the factor, the different treatments are applied to the different groups, and data on the resulting responses are collected.  The means of the response variable in the different groups are compared; if there are significant differences, then there is evidence to suggest that the factor and the response have a causal relationship.  The single-factor analysis of variance (ANOVA) model is most commonly used to analyze the data in such an experiment, but it does assume that the data in each group have a normal distribution, and that all groups have equal variance.  The Kruskal-Wallis test is a non-parametric alternative to ANOVA in analyzing data from single-factor completely randomized experiments.

If the factor has 2 levels, you may think that an independent 2-sample t-test with equal variance can also be used to analyze the data.  This is true, but the square of the t-test statistic in this case is just the F-test statistic in a single-factor ANOVA with 2 groups.  Thus, the results of these 2 tests are the same.  ANOVA generalizes the independent 2-sample t-test with equal variance to more than 2 groups.

Some textbooks state that “random assignment” means random assignment of experimental units to treatments, whereas other textbooks state that it means random assignment of treatments to experimental units.  I don’t think that there is any difference between these 2 definitions, but I welcome your thoughts in the comments.