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.
As it has been said a picture is worth a thousand words and so it is with graphics too. A well constructed graph can summarize information collected from tens to hundreds or even thousands of data points. But not every graph has the same power to convey complex information clearly. (more…)
Every time you analyze data, you start with a research question and end with communicating an answer. But in between those start and end points are twelve other steps. I call this the Data Analysis Pathway. It’s a framework I put together years ago, inspired by a client who kept getting stuck in Weed #1. But I’ve honed it over the years of assisting thousands of researchers with their analysis.
(more…)
Have you ever compared the list of model assumptions for linear regression across two sources? Whether they’re textbooks, lecture notes, or web pages, chances are the assumptions don’t quite line up.
Why? Sometimes the authors use different terminology. So it just looks different.
And sometimes they’re including not only model assumptions, but inference assumptions and data issues. All are important, but understanding the role of each can help you understand what applies in your situation.
Model Assumptions
The actual model assumptions are about the specification and performance of the model for estimating the parameters well.
1. The errors are independent of each other
2. The errors are normally distributed
3. The errors have a mean of 0 at all values of X
4. The errors have constant variance
5. All X are fixed and are measured without error
6. The model is linear in the parameters
7. The predictors and response are specified correctly
8. There is a single source of unmeasured random variance
Not all of these are always explicitly stated. And you can’t check them all. How do you know you’ve included all the “correct” predictors?
But don’t skip the step of checking what you can. And for those you can’t, take the time to think about how likely they are in your study. Report that you’re making those assumptions.
Assumptions about Inference
Sometimes the assumption is not really about the model, but about the types of conclusions or interpretations you can make about the results.
These assumptions allow the model to be useful in answering specific research questions based on the research design. They’re not about how well the model estimates parameters.
Is this important? Heck, yes. Studies are designed to answer specific research questions. They can only do that if these inferential assumptions hold.
But if they don’t, it doesn’t mean the model estimates are wrong, biased, or inefficient. It simply means you have to be careful about the conclusions you draw from your results. Sometimes this is a huge problem.
But these assumptions don’t apply if they’re for designs you’re not using or inferences you’re not trying to make. This is a situation when reading a statistics book that is written for a different field of application can really be confusing. They focus on the types of designs and inferences that are common in that field.
It’s hard to list out these assumptions because they depend on the types of designs that are possible given ethics and logistics and the types of research questions. But here are a few examples:
1. ANCOVA assumes the covariate and the IV are uncorrelated and do not interact. (Important only in experiments trying to make causal inferences).
2. The predictors in a regression model are endogenous. (Important for conclusions about the relationship between Xs and Y where Xs are observed variables).
3. The sample is representative of the population of interest. (This one is always important!)
Data Issues that are Often Mistaken for Assumptions
And sometimes the list of assumptions includes data issues. Data issues are a little different.
They’re important. They affect how you interpret the results. And they impact how well the model performs.
But they’re still different. When a model assumption fails, you can sometimes solve it by using a different type of model. Data issues generally stay around.
That’s a big difference in practice.
Here are a few examples of common data issues:
1. Small Samples
2. Outliers
3. Multicollinearity
4. Missing Data
5. Truncation and Censoring
6. Excess Zeros
So check for these data issues, deal with them if the solution doesn’t create more problems than you solved, and be careful with the inferences you draw when you can’t.
Go to the next article or see the full series on Easy-to-Confuse Statistical Concepts
When you’re model building, a key decision is which interaction terms to include. And which interactions to remove.
As a general rule, the default in regression is to leave them out. Add interactions only with a solid reason. It would seem like data fishing to simply add in all possible interactions.
And yet, that’s a common practice in most ANOVA models: put in all possible interactions and only take them out if there’s a solid reason. Even many software procedures default to creating interactions among categorical predictors.
(more…)
There are important ‘rules’ of statistical analysis. Like
- Always run descriptive statistics and graphs before running tests
- Use the simplest test that answers the research question and meets assumptions
- Always check assumptions.
But there are others you may have learned in statistics classes that don’t serve you or your analysis well once you’re working with real data.
When you are taking statistics classes, there is a lot going on. You’re learning concepts, vocabulary, and some really crazy notation. And probably a software package on top of that.
In other words, you’re learning a lot of hard stuff all at once.
Good statistics professors and textbook authors know that learning comes in stages. Trying to teach the nuances of good applied statistical analysis to students who are struggling to understand basic concepts results in no learning at all.
And yet students need to practice what they’re learning so it sticks. So they teach you simple rules of application. Those simple rules work just fine for students in a stats class working on sparkling clean textbook data.
But they are over-simplified for you, the data analyst, working with real, messy data.
Here are three rules of data analysis practice that you may have learned in classes that you need to unlearn. They are not always wrong. They simply don’t allow for the nuance involved in real statistical analysis.
The Rules of Statistical Analysis to Unlearn:
1. To check statistical assumptions, run a test. Decide whether the assumption is met by the significance of that test.
Every statistical test and model has assumptions. They’re very important. And they’re not always easy to verify.
For many assumptions, there are tests whose sole job is to test whether the assumption of another test is being met. Examples include the Levene’s test for constant variance and Kolmogorov-Smirnov test, often used for normality. These tests are tools to help you decide if your model assumptions are being met.
But they’re not definitive.
When you’re checking assumptions, there are a lot of contextual issues you need to consider: the sample size, the robustness of the test you’re running, the consequences of not meeting assumptions, and more.
What to do instead:
Use these test results as one of many pieces of information that you’ll use together to decide whether an assumption is violated.
2. Delete outliers that are 3 or more standard deviations from the mean.
This is an egregious one. Really. It’s bad.
Yes, it makes the data look pretty. Yes, there are some situations in which it’s appropriate to delete outliers (like when you have evidence that it’s an error). And yes, outliers can wreak havoc on your parameter estimates.
But don’t make it a habit. Don’t follow a rule blindly.
Deleting outliers because they’re outliers (or using techniques like Winsorizing) is a great way to introduce bias into your results or to miss the most interesting part of your data set.
What to do instead:
When you find an outlier, investigate it. Try to figure out if it’s an error. See if you can figure out where it came from.
3. Check Normality of Dependent Variables before running a linear model
In a t-test, yes, there is an assumption that Y, the dependent variable, is normally distributed within each group. In other words, given the group as defined by X, Y follows a normal distribution.
ANOVA has a similar assumption: given the group as defined by X, Y follows a normal distribution.
In linear regression (and ANCOVA), where we have continuous variables, this same assumption holds. But it’s a little more nuanced since X is not necessarily categorical. At any specific value of X, Y has a normal distribution. (And yes, this is equivalent to saying the errors have a normal distribution).
But here’s the thing: the distribution of Y as a whole doesn’t have to be normal.
In fact, if X has a big effect, the distribution of Y, across all values of X, will often be skewed or bimodal or just a big old mess. This happens even if the distribution of Y, at each value of X, is perfectly normal.
What to do instead:
Because normality depends on which Xs are in a model, check assumptions after you’ve chosen predictors.
Conclusion:
The best rule in statistical analysis: always stop and think about your particular data analysis situation.
If you don’t understand or don’t have the experience to evaluate your situation, discuss it with someone who does. Investigate it. This is how you’ll learn.
