ANOVA

When Linear Models Don’t Fit Your Data, Now What?

When your dependent variable is not continuous, unbounded, and measured on an interval or ratio scale, linear models don’t fit. The data just will not meet the assumptions of linear models. But there’s good news, other models exist for many types of dependent variables.

Today I’m going to go into more detail about 6 common types of dependent variables that are either discrete, bounded, or measured on a nominal or ordinal scale and the tests that work for them instead. Some are all of these.

An Example of Specifying Within-Subjects Factors in Repeated Measures

by Karen Grace-Martin Leave a Comment

Some repeated measures designs make it quite challenging to specify within-subjects factors. Especially difficult is when the design contains two “levels” of repeat, but your interest is in testing just one.

Let’s look at a great example of what this looks like and how to deal with it in this great question from a reader :

The Design:

I want to do a GLM (repeated measures ANOVA) with the valence of some actions of my test-subjects (valence = desirability of actions) as a within-subject factor. My subjects have to rate a number of actions/behaviours in a pre-set list of 20 actions from ‘very likely to do’ to ‘will never do this’ on a scale from 1 to 7, and some of these actions are desirable (e.g. help a blind man crossing the street) and therefore have a positive valence (in psychology) and some others are non-desirable (e.g. play loud music at night) and therefore have negative valence in psychology.

My question is how I can use valence as a within-subjects factor in GLM. Is there a way to tell SPSS some actions have positive valence and others have negative valence ? I assume assigning labels to the actions will not do it, as SPSS does not make analyses based on labels …
Please help. Thank you.

Specifying Fixed and Random Factors in Mixed Models

by Karen Grace-Martin 10 Comments

One of the difficult decisions in mixed modeling is deciding which factors are fixed and which are random. And as difficult as it is, it’s also very important. Correctly specifying the fixed and random factors of the model is vital to obtain accurate analyses.

Now, you may be thinking of the fixed and random effects in the model, rather than the factors themselves, as fixed or random. If so, remember that each term in the model (factor, covariate, interaction or other multiplicative term) has an effect. We’ll come back to how the model measures the effects for fixed and random factors.

Sadly, the definitions in many texts don’t help much with decisions to specify factors as fixed or random. Textbook examples are often artificial and hard to apply to the real, messy data you’re working with.

Here’s the real kicker. The same factor can often be fixed or random, depending on the researcher’s objective.

That’s right, there isn’t always a right or a wrong decision here. It depends on the inferences you want to make about this factor. This article outlines a different way to think about fixed and random factors.

An Example

Consider an experiment that examines beetle damage on cucumbers. The researcher replicates the experiment at five farms and on four fields at each farm.

There are two varieties of cucumbers, and the researcher measures beetle damage on each of 50 plants at the end of the season. The researcher wants to measure differences in how much damage the two varieties sustain.

The experiment then has the following factors: VARIETY, FARM, and FIELD.

Fixed factors can be thought of in terms of differences.

The effect of a categorical fixed factor is measured by differences from the overall mean.

The effect of a continuous fixed predictor (usually called a covariate) is defined by its slope–how the mean of the dependent variable differs with differing values of the predictor.

The output for fixed predictors, then, gives estimates for mean-differences or slopes.

Conclusions regarding fixed factors are particular to the values of these factors. For example, if one variety of cucumber has significantly less damage than the other, you can make conclusions about these two varieties. This says nothing about cucumber varieties that were not tested.

Random factors, on the other hand, are defined by a distribution and not by differences.

The values of a random factor are assumed to be chosen from a population with a normal distribution with a certain variance.

The output for a random factor is an estimate of this variance and not a set of differences from a mean. Effects of random factors are measured in terms of variance, not mean differences.

For example, we may find that the variance among fields makes up a certain percentage of the overall variance in beetle damage. But we’re not measuring how much higher the beetle damage is in one field compared to another. We only care that they vary, not how much they differ.

Software specification of fixed and random factors

Before we get into specifying factors as fixed and random, please note:

When you’re specifying random effects in software, like intercepts and slopes, the random factor is itself the Subject variable in your software. This differs in different software procedures. Some specifically ask for code like “Subject=FARM.” Others just put FARM after a || or some other indicator of the subject for which random intercepts and slopes are measured.

Some software, like SAS and SPSS, allow you to specify the random factors two ways. One is to list the random effects (like intercept) along with the random factor listed as a subject.

/Random Intercept | Subject(Farm)

The other is to list the random factors themselves.

/Random Farm

Both of those give you the same output.

Other software, like R and Stata, only allow the former. And neither uses the term “Subject” anywhere. But that random factor, the subject, still needs to be specified after the | or || in the random part of the model.

Specifying fixed effects is pretty straightforward.

Situations that indicate a factor should be specified as fixed:

1. The factor is the primary treatment that you want to compare.

In our example, VARIETY is definitely fixed as the researcher wants to compare the mean beetle damage on the two varieties.

2. The factor is a secondary control variable, and you want to control for differences in the specific values of this factor.

Say the researcher chose these farms specifically for some feature they had, such as specific soil types or topographies that may affect beetle damage. If the researcher wants to compare the farms as representatives of those soil types, then FARM should be fixed.

3. The factor has only two values.

Even if everything else indicates that a factor should be random, if it has only two values, the variance cannot be calculated, and it should be fixed.

Situations that indicate random factors:

1. Your interest is in quantifying how much of the overall variance to attribute to this factor.

If you want to know how much of the variation in beetle damage you can attribute to the farm at which the damage took place, FARM would be random.

2. Your interest is not in knowing which specific means differ, but you want to account for the variation in this factor.

If the farms were chosen at random, FARM should be random.

This choice of the specific farms involved in the study is key. If you can rerun the study using different specific farms–different values of the Farm factor–and still be able to draw the same conclusions, then Farm should be random. However, if you want to compare or control for these particular farms, then Farm is a fixed factor.

3. You would like to generalize the conclusions about this factor to the whole population.

There is nothing about comparing these specific fields that is of interest to the researcher. Rather, the researcher wants to generalize the results of this experiment to all fields, so FIELD is random.

4. Any interaction with a random factor is also random.

How the factors of a model are specified can have great influence on the results of the analysis and on the conclusions drawn.

Member Training: Classic Experimental Designs

by TAF Support 1 Comment

Have you ever wondered why there are so many different types of experimental designs, and how a researcher would go about choosing among them to best address their research questions? [Read more…] about Member Training: Classic Experimental Designs

Member Training: ANOVA Post-hoc Tests: Practical Considerations

by TAF Support Leave a Comment

Post-hoc tests, pairwise or other linear contrasts, are typical in an analysis of variance (ANOVA) setting to understand which group means differ. They incorporate p-value adjustments to avoid concluding that group means differ when they actually do not. There are several adjustments that can be considered for conducting multiple post-hoc tests, including single-step and stepwise adjustments. [Read more…] about Member Training: ANOVA Post-hoc Tests: Practical Considerations

Why report estimated marginal means?

by Karen Grace-Martin 55 Comments

Updated 8/18/2021

I recently was asked whether to report means from descriptive statistics or from the Estimated Marginal Means with SPSS GLM.

The short answer: Report the Estimated Marginal Means (almost always).

To understand why and the rare case it doesn’t matter, let’s dig in a bit with a longer answer.

First, a marginal mean is the mean response for each category of a factor, adjusted for any other variables in the model (more on this later).

Just about any time you include a factor in a linear model, you’ll want to report the mean for each group. The F test of the model in the ANOVA table will give you a p-value for the null hypothesis that those means are equal. And that’s important.

But you need to see the means and their standard errors to interpret the results. The difference in those means is what measures the effect of the factor. While that difference can also appear in the regression coefficients, looking at the means themselves give you a context and makes interpretation more straightforward. This is especially true if you have interactions in the model.

Some basic info about marginal means

In SPSS menus, they are in the Options button and in SPSS’s syntax they’re EMMEANS.
These are called LSMeans in SAS, margins in Stata, and emmeans in R’s emmeans package.
Although I’m talking about them in the context of linear models, all the software has them in other types of models, including linear mixed models, generalized linear models, and generalized linear mixed models.
They are also called predicted means, and model-based means. There are probably a few other names for them, because that’s what happens in statistics.

When marginal means are the same as observed means

Let’s consider a few different models. In all of these, our factor of interest, X, is a categorical predictor for which we’re calculating Estimated Marginal Means. We’ll call it the Independent Variable (IV).

Model 1: No other predictors

If you have just a single factor in the model (a one-way anova), marginal means and observed means will be the same.

Observed means are what you would get if you simply calculated the mean of Y for each group of X.

Model 2: Other categorical predictors, and all are balanced

Likewise, if you have other factors in the model, if all those factors are balanced, the estimated marginal means will be the same as the observed means you got from descriptive statistics.

Model 3: Other categorical predictors, unbalanced

Now things change. The marginal mean for our IV is different from the observed mean. It’s the mean for each group of the IV, averaged across the groups for the other factor.

When you’re observing the category an individual is in, you will pretty much never get balanced data. Even when you’re doing random assignment, balanced groups can be hard to achieve.

In this situation, the observed means will be different than the marginal means. So report the marginal means. They better reflect the main effect of your IV—the effect of that IV, averaged across the groups of the other factor.

Model 4: A continuous covariate

When you have a covariate in the model the estimated marginal means will be adjusted for the covariate. Again, they’ll differ from observed means.

It works a little bit differently than it does with a factor. For a covariate, the estimated marginal mean is the mean of Y for each group of the IV at one specific value of the covariate.

By default in most software, this one specific value is the mean of the covariate. Therefore, you interpret the estimated marginal means of your IV as the mean of each group at the mean of the covariate.

This, of course, is the reason for including the covariate in the model–you want to see if your factor still has an effect, beyond the effect of the covariate. You are interested in the adjusted effects in both the overall F-test and in the means.

If you just use observed means and there was any association between the covariate and your IV, some of that mean difference would be driven by the covariate.

For example, say your IV is the type of math curriculum taught to first graders. There are two types. And say your covariate is child’s age, which is related to the outcome: math score.

It turns out that curriculum A has slightly older kids and a higher mean math score than curriculum B. Observed means for each curriculum will not account for the fact that the kids who received that curriculum were a little older. Marginal means will give you the mean math score for each group at the same age. In essence, it sets Age at a constant value before calculating the mean for each curriculum. This gives you a fairer comparison between the two curricula.

But there is another advantage here. Although the default value of the covariate is its mean, you can change this default. This is especially helpful for interpreting interactions, where you can see the means for each group of the IV at both high and low values of the covariate.

In SPSS, you can change this default using syntax, but not through the menus.

For example, in this syntax, the EMMEANS statement reports the marginal means of Y at each level of the categorical variable X at the mean of the Covariate V.

UNIANOVA Y BY X WITH V
/INTERCEPT=INCLUDE
/EMMEANS=TABLES(X) WITH(V=MEAN)
/DESIGN=X V.

If instead, you wanted to evaluate the effect of X at a specific value of V, say 50, you can just change the EMMEANS statement to:

/EMMEANS=TABLES(X) WITH(V=50)

Another good reason to use syntax.

Article updated 8/17/2021