The Analysis Factor
Statistical Consulting, Resources, and Statistics Workshops for Researchers
Learning statistics is difficult enough; throw in some especially confusing terminology and it can feel impossible! There are many ways that statistical language can be confusing.
Some terms mean one thing in the English language, but have another (usually more specific) meaning in statistics. [Read more…] about Member Training: Confusing Statistical Terms
Analysis of Covariance (ANCOVA) is a type of linear model that combines the best abilities of linear regression with the best of Analysis of Variance.
It allows you to test differences in group means and interactions, just like ANOVA, while covarying out the effect of a continuous covariate.
Through examples and graphs, we’ll talk about what it really means to covary out the effect of a continuous variable and how to interpret results.
Primary to the discussion will be when ANCOVA is and is not appropriate and how correlations and interactions between the covariate and the independent variables affect interpretation.
Note: This training is an exclusive benefit to members of the Statistically Speaking Membership Program and part of the Stat’s Amore Trainings Series. Each Stat’s Amore Training is approximately 90 minutes long.
Karen Grace-Martin helps statistics practitioners gain an intuitive understanding of how statistics is applied to real data in research studies.
She has guided and trained researchers through their statistical analysis for over 15 years as a statistical consultant at Cornell University and through The Analysis Factor. She has master’s degrees in both applied statistics and social psychology and is an expert in SPSS and SAS.
Not a Member Yet?
It’s never too early to set yourself up for successful analysis with support and training from expert statisticians.
Just head over and sign up for Statistically Speaking.
You'll get access to this training webinar and 85+ other stats trainings — plus the expert guidance you need to progress with live Q&A sessions and an ask-a-mentor forum.
There are two oft-cited assumptions for Analysis of Covariance (ANCOVA), which is used to assess the effect of a categorical independent variable on a numerical dependent variable while controlling for a numerical covariate:
1. The independent variable and the covariate are independent of each other.
2. There is no interaction between independent variable and the covariate.
In a previous post, I showed a detailed example for an observational study where the first assumption is irrelevant, but I have gotten a number of questions about the second.
So what does it mean, and what should you do, if you find an interaction between the categorical IV and the continuous covariate? [Read more…] about ANCOVA Assumptions: When Slopes are Unequal
Linear, Logistic, Tobit, Cox, Poisson, Zero Inflated… The list of regression models goes on and on before you even get to things like ANCOVA or Linear Mixed Models.
In this webinar, we will explore types of regression models, how they differ, how they’re the same, and most importantly, when to use each one.
Note: This training is an exclusive benefit to members of the Statistically Speaking Membership Program and part of the Stat’s Amore Trainings Series. Each Stat’s Amore Training is approximately 90 minutes long.
Karen Grace-Martin helps statistics practitioners gain an intuitive understanding of how statistics is applied to real data in research studies.
She has guided and trained researchers through their statistical analysis for over 15 years as a statistical consultant at Cornell University and through The Analysis Factor. She has master’s degrees in both applied statistics and social psychology and is an expert in SPSS and SAS.
Not a Member Yet?
It’s never too early to set yourself up for successful analysis with support and training from expert statisticians.
Just head over and sign up for Statistically Speaking.
You'll get access to this training webinar and 85+ other stats trainings — plus the expert guidance you need to progress with live Q&A sessions and an ask-a-mentor forum.
Every once in a while, I work with a client who is stuck between a particular statistical rock and hard place.
It happens when they’re trying to run an analysis of covariance (ANCOVA) model because they have a categorical independent variable and a continuous covariate.
The problem arises when a coauthor, committee member, or reviewer insists that ANCOVA is inappropriate in this situation because one of the following ANCOVA assumptions are not met:
1. The independent variable and the covariate are independent of each other.
2. There is no interaction between independent variable and the covariate.
If you look them up in any design of experiments textbook, which is usually where you’ll find information about ANOVA and ANCOVA, you will indeed find these assumptions. So the critic has nice references.
However, this is a case where it’s important to stop and think about whether the assumptions apply to your situation, and how dealing with the assumption will affect the analysis and the conclusions you can draw.
If you’re like most researchers, your statistical training focused on Regression or ANOVA, but not both. It all depends on whether your field focuses more on experimental data (Biology, Psychology) or observed data (Sociology, Economics). Maybe one class covered a bit of the other, but most people are comfortable in one, but not the other.
This, in my opinion, is a shame. (Okay, I was going to say tragedy, but let’s be real. Tsunami that kills thousands=tragedy. Different scale here).
First of all, the distinction between ANOVA and linear regression is arbitrary. They’re really the same model with different outfits on.
Second, regardless of which one you normally use, you’re going to occasionally have to use the other kind of predictor variables–categorical or continuous. And we can come up with nice names for these models–a regression with dummy variables or an Analysis of Covariance.
But real understanding of the relationships among variables comes only when you dispense of the names and can focus on analyzing and interpreting the model using the kinds of variables you have.
There are other examples, but today I’m going to focus on an ANOVA model with a continuous covariate.
A common model is one in which one predictor is categorical (we’ll use 4 categories) and the other is continuous. Here is an example of a scatterplot of just such a model:
There are four groups, each of which received a different training. The continuous moderator is Age, and the outcome is OverallPost, which is the post-training test score to see how well they learned the material in each training program.
As you can see, the effect of the training program is moderated by age. Another way to say that is there is a significant interaction between Age and Training Group. The effect of the training is depending on the trainee’s age.
One way to interpret this significant interaction is to compare the slopes of the four lines, which is easily done with any regression coefficient table. (Okay, not always easily done, but easily found in…)
But this doesn’t make very much sense when Age is really a moderator–a predictor we want to control for, and see how it affects the relationship between the independent (IV) and dependent variables (DV), but not really the IV we’re interested in.
A better way to do it in this situation is to compare the means among groups at a low value of Age, say 20, and again at a high value of Age, say 50. You can get p-values, adjusted for multiple comparisons, using either SAS or SPSS GLM.
SAS Proc GLM uses the LSMeans statement and SPSS GLM uses EMMeans. They do the same thing–calculate the mean of Y for each group, at a specific value of the covariate.
If you use the menus in SPSS, you can only get those EMMeans at the Covariate’s mean, which in this example is about 25, where the vertical black line is. This isn’t very useful for our purposes. But we can change the value of the covariate at which to compare the means using syntax.
So it would tell us that at a young age of say 20, the three treatment groups (green, tan, and purple lines) all have means higher than the control (blue). Young people learned more in all three treatment groups.
But at an older age, say 50, the means of the purple and tan groups were not significantly different from the control group’s (blue), and the green (EIQ group) did worse!
In SPSS GLM, the syntax would be:
UNIANOVA OverallPost BY group WITH NEWAGE
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/EMMEANS=TABLES(group) WITH(NEWAGE=MEAN) COMPARE ADJ(SIDAK)
/EMMEANS=TABLES(group) WITH(NEWAGE=45) COMPARE ADJ(SIDAK)
/EMMEANS=TABLES(group) WITH(NEWAGE=20) COMPARE ADJ(SIDAK)
/PRINT=PARAMETER
/CRITERIA=ALPHA(.05)
/DESIGN=NEWAGE group NEWAGE*group.
And if you want to learn a lot more about SPSS GLM, in detail, it was because of questions like this that I created the Running Regressions and ANOVAs in SPSS GLM Workshop. The entire third week is devoted to ANOVA models, with and without covariates.
