Introduction to Linear Trend Analysis: The Estimation Approach

This is an introduction to linear trend analysis from an estimation perspective, applied to estimating the linear trend among population means. The contents of this introduction is based on Maxwell, Delaney, and Kelley (2017) and Rosenthal, Rosnow, and Rubin (2000). I have taken the (invented) data from Haans (2018). The estimation perspective to statistical analysis is aimed at obtaining point and interval estimates of effect sizes. Here, I will use the frequentist perspective of obtaining a point estimate and a 95% Confidence Interval of the relevant effect size. For linear trend analysis, the relevant effect size is the slope coefficient of the linear trend, so, the purpose of the analysis is to estimate the value of the slope and the 95% confidence interval of the estimate. We will use contrast analysis to obtain the relevant data.

[Note: A pdf-file that differs only slightly from this blogpost can be found on my Researchgate page: here; I suggest Haans (2018) for an easy to follow introduction to contrast analysis, which should really help understanding what is being said below].

Planning for Precise Contrast Estimates: Introduction and Tutorial (Preprint)

I just finished a preprint of an introduction and tutorial to sample size planning for precision of contrast estimates. The tutorial focuses on single factor between and within subjects designs, and mixed factorial designs with one within and one between factor. The tutorial contains R-code for sample size planning in these designs.

The preprint is availabe on researchgate: Click (but I am just as happy to send it to you if you like; just let me know).

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)

Planning for a precise interaction contrast estimate

In my previous post (here),  I wrote about obtaining a confidence interval for the estimate of an interaction contrast. I demonstrated, for a simple two-way independent factorial design, how to obtain a confidence interval by making use of the information in an ANOVA source table and estimates of the marginal means and how a custom contrast estimate can be obtained with SPSS.

One of the results of the analysis in the previous post was that the 95% confidence interval for the interaction was very wide. The estimate was .77, 95% CI [0.04, 1.49]. Suppose that it is theoretically or practically important to know the value of the contrast to a more precise degree.  (I.e. some researchers will be content that the CI allows for a directional qualitative interpretation: there seems to exist a positive interaction effect, but others, more interested in the quantitative questions may not be so easily satisfied).  Let's see how we can plan the research to obtain a more precise estimate. In other words, let's plan for precision.

Of course, there are several ways in which the precision of the estimate can be increased. For instance, by using measurement procedures that are designed to obtain reliable data, we could change the experimental design, for example switching to a repeated measures (crossed) design, and/or increase the number of observations. An example of the latter would be to increase the number of participants and/or the number of observations per participant.  We will only consider the option of increasing the number of participants, and keep the independent factorial design, although in reality we would of course also strive for a measurement instrument that generally gives us highly reliable data. (By the way, it is possible to use my Precision application to investigate the effects of changing the experimental design on the expected precision of contrast estimates in studies with 1 fixed factor and 2 random factors).

The plan for the rest of this post is as follows. We will focus on getting a short confidence interval for our interaction estimate, and we will do that by considering the half-width of the interval, the Margin of Error (MOE). First we will try to find a sample size that gives us an expected MOE (in repeated replication of the experiment with new random samples) no more than a target MOE. Second, we will try to find a sample size that gives a MOE smaller than or equal to our target MOE in a specifiable percentage (say, 80% or 90%) of replication experiments. The latter approach is called planning with assurance.