Contrast Analysis with R: Tutorial for obtaining contrast estimates in a 2-way factorial design

In this post, I want to show how contrast estimates can be obtained with R. In particular, I want to show how we can replicate, with R, a contrast analysis of an interaction contrast in a 2 x 4 between subjects design.

Our example is from Haans (2018; see also this post). It considers the effect of students' seating distance from the teacher and the educational performance of the students: the closer to the teacher the student is seated, the higher the performance. A "theory "explaining the effect is that the effect is mainly caused by the teacher having decreased levels of eye contact with the students sitting farther to the back in the lecture hall.

To test that theory, a experiment was conducted with N = 72 participants attending a lecture. The lecture was given to two independent groups of 36 participants. The first group attended the lecture while the teacher was wearing dark sunglasses, the second group attented the lecture while the teacher was not wearing sunglasses. All participants were randomly assigned to 1 of 4 possible rows, with row 1 being closest to the teacher and row 4 the furthest from the teacher The dependent variable was the score on a 10-item questionnaire about the contents of the lecture. So, we have a 2 by 4 factorial design, with n = 9 participants in each combination of the factor levels. 

Here we focus on obtaining an interaction contrast: we will estimate the extent to which the difference between the mean retention score of the participants on the first row and those on the other rows differs between the conditions with and without sunglasses. 

The interaction contrast with SPSS

Planning for precise contrast estimates in between subjects designs

Here I would like to explain the procedure for sample size planning for one-way and two-way (factorial) between subjects designs. We will consider examples based on and described in Haans (2018).

The first example: one-way design

The first example considers the effect of seating location  of students on their educational performance. Seating location is defined as distance from the teacher and operationalized in terms of the row the student is seated in, with first row being the closest to the teacher and the fourth row being the furthest away. 20 Students are randomly assigned to one of the four possible rows, so N = 20, n = 5. The dependent variable is the course grade of the student. (Note: the data and study are hypothetical).

As Haans (2018) explains, one psychological theory explaining the effect of seating position on educational performance is based on social influence. This theory posits that due to the social influence of the teacher, the students that are seated closest to the teacher find themselves in a state of undivided attention. This undivided attention causes their educational performance to be better than the students who are seated further away.

In operational terms, then, we may expect that first row students will have a better average grade than students seated on the other rows. So, the quantitative research question we are interested in is:

"How much do the average grades differ between students seated first row and the students seated on other rows?"

We can estimate this quantity with a Helmert Contrast, where we assign a contrast weight of 1 to mean of the first row grades and weights -1/3 to the means of the grades in the other rows.

Haans (2018) gives us the following results. The contrast estimate equals 2.00 , 95% CI [0.27, 3.73]. In order to interpret this more easily, we divide this estimate by the square root Mean Square Error, to obtain the standardized estimate and standardized confidence interval (not to be confused with the confidence interval of the standardized estimate, but that's a different story. The result is: 1.26, 95% CI [0.17, 2.36].

To answer the research question, the estimated difference equals 1.26 standard deviations, which according to rule-of-thumbs frequently used in psychology is a large difference. The CI shows the enormous amount of uncertainty of this estimate: population values between 0.17 (small) and 2.36 (very large) are also consistent with the observed data and our statistical assumptions. So, it seems safe to conclude that it looks like there is a positive effect of seating position, but the wide range of the CI makes it clear that the data do not tell us enough about the size of the effect, the precision is simply too low.

The precision is f = 1.09, which according to my rules-of-thumb is very imprecise (I consider f = 0.65, to be barely tolerable).

So, let's plan for a replication study with a reasonably precise estimate of  f = 0.40, with 80% assurance. (Note: for some advice on setting target Moe: Planning with assurance, with assurance. ) I've used the app: https://gmulder.shinyapps.io/PlanningFactorialContrasts/ with the default values for a single factor between subjects design with 4 conditions.  According to the app, we need n = 36 participants per condition (making a total of  N = 144).

(For more detailed information considering sample size planning for contrast analysis see: https://the-small-s-scientist.blogspot.com/2019/04/sample-size-planning-for-contrast-estimates.html and for some guidelines for setting target MoE: https://the-small-s-scientist.blogspot.com/2019/03/planning-with-assurance-with-assurance.html)