Effect size statistics are extremely important for interpreting statistical results. The emphasis on reporting them has been a great development over the past decade. [Read more…] about Odds Ratio: Standardized or Unstandardized Effect Size?

## Why report estimated marginal means?

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*

## Overfitting in Regression Models

The practice of choosing predictors for a regression model, called model building, is an area of real craft.

There are many possible strategies and approaches and they all work well in some situations. Every one of them requires making a lot of decisions along the way. As you make decisions, one danger to look out for is overfitting—creating a model that is too complex for the the data. [Read more…] about Overfitting in Regression Models

## Six Easy Ways to Complicate Your Analysis

It’s easy to make things complex without meaning to. Especially in statistical analysis.

Sometimes that complexity is unavoidable. You have ethical and practical constraints on your study design and variable measurement. Or the data just don’t behave as you expected. Or the only research question of interest is one that demands many variables.

But sometimes it isn’t. Seemingly innocuous decisions lead to complicated analyses. These decisions occur early in the design, research questions, or variable choice.

[Read more…] about Six Easy Ways to Complicate Your Analysis

## Why Generalized Linear Models Have No Error Term

Even if you’ve never heard the term Generalized Linear Model, you may have run one. It’s a term for a family of models that includes logistic and Poisson regression, among others.

It’s a small leap to generalized linear models, if you already understand linear models. Many, many concepts are the same in both types of models.

But one thing that’s perplexing to many is why generalized linear models have no error term, like linear models do. [Read more…] about Why Generalized Linear Models Have No Error Term

## What is a Chi-Square Test?

Just about everyone who does any data analysis has used a chi-square test. Probably because there are quite a few of them, and they’re all useful.

But it gets confusing because very often you’ll just hear them called “Chi-Square test” without their full, formal name. And without that context, it’s hard to tell exactly what hypothesis that test is testing. [Read more…] about What is a Chi-Square Test?