If your graduate statistical training was anything like mine, you learned ANOVA in one class and Linear Regression in another. My professors would often say things like “ANOVA is just a special case of Regression,” but give vague answers when pressed.
It was not until I started consulting that I realized how closely related ANOVA and regression are. They’re not only related, they’re the same thing. Not a quarter and a nickel–different sides of the same coin.
So here is a very simple example that shows why. When someone showed me this, a light bulb went on, even though I already knew both ANOVA and multiple linear (more…)
A Linear Regression Model with an interaction between two predictors (X1 and X2) has the form:
Y = B0 + B1X1 + B2X2 + B3X1*X2.
It doesn’t really matter if X1 and X2 are categorical or continuous, but let’s assume they are continuous for simplicity.
One important concept is that B1 and B2 are not main effects, the way they would be if (more…)
Back when I was doing psychology research, I knew ANOVA pretty well. I’d taken a number of courses on it and could run it backward and forward. I kept hearing about ANCOVA, but in every ANOVA class that was the last topic on the syllabus, and we always ran out of time.
The other thing that drove me crazy was those stats professors kept saying “ANOVA is just a special case of Regression.” I could not for the life of me figure out why or how.
It was only when I switched over to statistics that I finally took a regression class and figured out what ANOVA was all about. And only when I started consulting, and seeing hundreds of different ANOVA and regression models, that I finally made the connection.
But if you don’t have the driving curiosity about ANOVA and regression, why should you, as a researcher in Psychology, Education, or Agriculture, who is trained in ANOVA, want to learn regression? There are 3 main reasons.
1. There a many, many continuous independent variables and covariates that need to be included in models. Without the tools to analyze them as continuous, you are left forcing them into ANOVA using an arbitrary technique like median splits. At best, you’re losing power. At worst, you’re not publishing your article because you’re missing real effects.
2. Having a solid understanding of the General Linear Model in its various forms equips you to really understand your variables and their relationships. It allows you to try a model different ways–not for data fishing, but for discovering the true nature of the relationships. Having the capacity to add an interaction term or a squared term allows you to listen to your data and makes you a better researcher.
3. The multiple linear regression model is the basis for many other statistical techniques–logistic regression, multilevel and mixed models, Poisson regression, Survival Analysis, and so on. Each of these is a step (or small leap) beyond multiple regression. If you’re still struggling with what it means to center variables or interpret interactions, learning one of these other techniques becomes arduous, if not painful.
Having guided thousands of researchers through their statistical analysis over the past 10 years, I am convinced that having a strong, intuitive understanding of the general linear model in its variety of forms is the key to being an effective and confident statistical analyst. You are then free to learn and explore other methodologies as needed.
There are many dependent variables that no matter how many transformations you try, you cannot get to be normally distributed. The most common culprits are count variables–the variable that measures the count or rate of some event in a sample. Some examples I’ve seen from a variety of disciplines are:
Number of eggs in a clutch that hatch
Number of domestic violence incidents in a month
Number of times juveniles needed to be restrained during tenure at a correctional facility
Number of infected plants per transect
A common quality of these variables is that 0 is the mode–the most common value. 1 is the next most common, 2 the next, and so on. In variables with low expected counts (number of cars in a household, number of degrees earned), (more…)
The beauty of the Univariate GLM procedure in SPSS is that it is so flexible. You can use it to analyze regressions, ANOVAs, ANCOVAs with all sorts of interactions, dummy coding, etc.
The down side of this flexibility is it is often confusing what to put where and what it all means.
So here’s a quick breakdown.
The dependent variable I hope is pretty straightforward. Put in your continuous dependent variable.
Fixed Factors are categorical independent variables. It does not matter if the variable is (more…)
Level is a term in statistics that is confusing because it has multiple meanings in different contexts (much like alpha and beta).
There are three different uses of the term Level in statistics that mean completely different things. What makes this especially confusing is that all three of them can be used in the exact same research context.
I’ll show you an example of that at the end.
So when you’re talking to someone who is learning statistics or who happens to be thinking of that term in a different context, this gets especially confusing.
Levels of Measurement
The most widespread of these is levels of measurement. Stanley Stevens came up with this taxonomy of assigning numerals to variables in the 1940s. You probably learned about them in your Intro Stats course: the nominal, ordinal, interval, and ratio levels.
Levels of measurement is really a measurement concept, not a statistical one. It refers to how much and the type of information a variable contains. Does it indicate an unordered category, a quantity with a zero point, etc?
It is important in statistics because it has a big impact on which statistics are appropriate for any given variable. For example, you would not do the same test of association between two nominal variables as you would between two interval variables.
Levels of a Factor
A related concept is a Factor. Although Factor itself has multiple meanings in statistics, here we are talking about a categorical predictor variable.
The typical use of Factor as a categorical predictor variable comes from experimental design. In experimental design, the predictor variables (also often called Independent Variables) are generally categorical and nominal. They represent different experimental conditions, like treatment and control traditions.
Each of these categorical conditions is called a level.
Here are a few examples:
In an agricultural study, a fertilizer treatment variable has three levels: Organic fertilizer (composted manure); High concentration of chemical fertilizer; low concentration of chemical fertilizer.
In a medical study, a drug treatment has four levels: Placebo; low dosage; medium dosage; high dosage.
In a linguistics study, a word frequency variable has two levels: high frequency words; low frequency words.
Although this use of level is very widespread, I try to avoid it personally. Instead I use the word “value” or “category” both of which are accurate, but without other meanings.
Level in Multilevel Models
A completely different use of the term is in the context of multilevel models. Multilevel models is a term for some mixed models. (The terms multilevel models and mixed models are often used interchangably, though mixed model is a bit more flexible).
Multilevel models have two or more sources of random variation. A two level model has two sources of random variation and can have predictors at each level.
A common example is a model from a design where the response variable of interest is measured on students. It’s hard though, to sample students directly or to randomly assign them to treatments, since there is a natural clustering of students within schools.
So the resource-efficient way to do this research is to sample students within schools.
Predictors can be measured at the student level (eg. gender, SES, age) or the school level (enrollment, % who go on to college). The dependent variable has variation from student to student (level 1) and from school to school (level 2).
We always count these levels from the bottom up. So if we have students clustered within classroom and classroom clustered within school and school clustered within district, we have:
- Level 1: Students
- Level 2: Classroom
- Level 3: School
- Level 4: District
So this use of the term level describes the design of the study, not the measurement of the variables or the categories of the factors.
Putting them together
So this is the truly unfortunate part. There are situations where all three definitions of level are relevant within the same statistical analysis context.
I find this unfortunate because I think using the same word to mean completely different things just confuses people. But here it is:
Picture that study in which students are clustered within school (a two-level design). Each school is assigned to use one of three math curricula (the independent variable, which happens to be categorical).
So, the variable “math curriculum” is a factor with 3 levels (ie, three categories). Because those three categories of “math curriculum” are unordered, “math curriculum” has a nominal level of measurement. And since “math curriculum” is assigned to each school, it is considered a level 2 variable in the two-level model.
See the rest of the Confusing Statistical Terms series.
