Poisson Regression Models and its extensions (Zero-Inflated Poisson, Negative Binomial Regression, etc.) are used to model counts and rates. A few examples of count variables include:
– Number of words an eighteen month old can say
– Number of aggressive incidents performed by patients in an impatient rehab center
Most count variables follow one of these distributions in the Poisson family. Poisson regression models allow researchers to examine the relationship between predictors and count outcome variables.
Using these regression models gives much more accurate parameter (more…)
Adding interaction terms to a regression model has real benefits. It greatly expands your understanding of the relationships among the variables in the model. And you can test more specific hypotheses. But interpreting interactions in regression takes understanding of what each coefficient is telling you.
The example from Interpreting Regression Coefficients was a model of the height of a shrub (Height) based on the amount of bacteria in the soil (Bacteria) and whether the shrub is located in partial or full sun (Sun). Height is measured in cm, Bacteria is measured in thousand per ml of soil, and Sun = 0 if the plant is in partial sun, and Sun = 1 if the plant is in full sun.
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…)
One of the most common causes of multicollinearity is when predictor variables are multiplied to create an interaction term or a quadratic or higher order terms (X squared, X cubed, etc.).
Why does this happen? When all the X values are positive, higher values produce high products and lower values produce low products. So the product variable is highly correlated with the component variable. I will do a very simple example to clarify. (Actually, if they are all on a negative scale, the same thing would happen, but the correlation would be negative).
In a small sample, say you have the following values of a predictor variable X, sorted in ascending order:
2, 4, 4, 5, 6, 7, 7, 8, 8, 8
It is clear to you that the relationship between X and Y is not linear, but curved, so you add a quadratic term, X squared (X2), to the model. The values of X squared are:
4, 16, 16, 25, 49, 49, 64, 64, 64
The correlation between X and X2 is .987–almost perfect.
Plot of X vs. X squared
To remedy this, you simply center X at its mean. The mean of X is 5.9. So to center X, I simply create a new variable XCen=X-5.9.
The correlation between XCen and XCen2 is -.54–still not 0, but much more managable. Definitely low enough to not cause severe multicollinearity. This works because the low end of the scale now has large absolute values, so its square becomes large.
The scatterplot between XCen and XCen2 is:
Plot of Centered X vs. Centered X squared
If the values of X had been less skewed, this would be a perfectly balanced parabola, and the correlation would be 0.
Tonight is my free teletraining on Multicollinearity, where we will talk more about it. Register to join me tonight or to get the recording after the call.
I was recently asked about whether centering (subtracting the mean) a predictor variable in a regression model has the same effect as standardizing (converting it to a Z score). My response:
They are similar but not the same.
In centering, you are changing the values but not the scale. So a predictor that is centered at the mean has new values–the entire scale has shifted so that the mean now has a value of 0, but one unit is still one unit. The intercept will change, but the regression coefficient for that variable will not. Since the regression coefficient is interpreted as the effect on the mean of Y for each one unit difference in X, it doesn’t change when X is centered.
And incidentally, despite the name, you don’t have to center at the mean. It is often convenient, but there can be advantages of choosing a more meaningful value that is also toward the center of the scale.
But a Z-score also changes the scale. A one-unit difference now means a one-standard deviation difference. You will interpret the coefficient differently. This is usually done so you can compare coefficients for predictors that were measured on different scales. I can’t think of an advantage for doing this for an interaction.
Statistical models, such as general linear models (linear regression, ANOVA, MANOVA), linear mixed models, and generalized linear models (logistic, Poisson, regression, etc.) all have the same general form.
On the left side of the equation is one or more response variables, Y. On the right hand side is one or more predictor variables, X, and their coefficients, B. The variables on the right hand side can have many forms and are called by many names.
There are subtle distinctions in the meanings of these names. Unfortunately, though, there are two practices that make them more confusing than they need to be.
First, they are often used interchangeably. So someone may use “predictor variable” and “independent variable” interchangably and another person may not. So the listener may be reading into the subtle distinctions that the speaker may not be implying.
Second, the same terms are used differently in different fields or research situations. So if you are an epidemiologist who does research on mostly observed variables, you probably have been trained with slightly different meanings to some of these terms than if you’re a psychologist who does experimental research.
Even worse, statistical software packages use different names for similar concepts, even among their own procedures. This quest for accuracy often renders confusion. (It’s hard enough without switching the words!).
Here are some common terms that all refer to a variable in a model that is proposed to affect or predict another variable.
I’ll give you the different definitions and implications, but it’s very likely that I’m missing some. If you see a term that means something different than you understand it, please add it to the comments. And please tell us which field you primarily work in.
Predictor Variable, Predictor
This is the most generic of the terms. There are no implications for being manipulated, observed, categorical, or numerical. It does not imply causality.
A predictor variable is simply used for explaining or predicting the value of the response variable. Used predominantly in regression.
Independent Variable
I’ve seen Independent Variable (IV) used different ways.
1. It implies causality: the independent variable affects the dependent variable. This usage is predominant in ANOVA models where the Independent Variable is manipulated by the experimenter. If it is manipulated, it’s generally categorical and subjects are randomly assigned to conditions.
2. It does not imply causality, but it is a key predictor variable for answering the research question. In other words, it is in the model because the researcher is interested in understanding its relationship with the dependent variable. In other words, it’s not a control variable.
3. It does not imply causality or the importance of the variable to the research question. But it is uncorrelated (independent) of all other predictors.
Honestly, I only recently saw someone define the term Independent Variable this way. Predictor Variables cannot be independent variables if they are at all correlated. It surprised me, but it’s good to know that some people mean this when they use the term.
Explanatory Variable
A predictor variable in a model where the main point is not to predict the response variable, but to explain a relationship between X and Y.
Control Variable
A predictor variable that could be related to or affecting the dependent variable, but not really of interest to the research question.
Covariate
Generally a continuous predictor variable. Used in both ANCOVA (analysis of covariance) and regression. Some people use this to refer to all predictor variables in regression, but it really means continuous predictors. Adding a covariate to ANOVA (analysis of variance) turns it into ANCOVA (analysis of covariance).
Sometimes covariate implies that the variable is a control variable (as opposed to an independent variable), but not always.
And sometimes people use covariate to mean control variable, either numerical or categorical.
These terms are used differently in different fields. In experimental design, it’s used to mean a variable whose effect cannot be distinguished from the effect of an independent variable.
In observational fields, it’s used to mean one of two situations. The first is a variable that is so correlated with an independent variable that it’s difficult to separate out their effects on the response variable. The second is a variable that causes the independent variable’s effect on the response.
The distinction in those interpretations are slight but important.
Exposure Variable
This is a term for independent variable in some fields, particularly epidemiology. It’s the key predictor variable.
Risk Factor
Another epidemiology term for a predictor variable. Unlike the term “Factor” listed below, it does not imply a categorical variable.
Factor
A categorical predictor variable. It may or may not indicate a cause/effect relationship with the response variable (this depends on the study design, not the analysis).
Independent variables in ANOVA are almost always called factors. In regression, they are often referred to as indicator variables, categorical predictors, or dummy variables. They are all the same thing in this context.
Also, please note that Factor has completely other meanings in statistics, so it too got its own Confusing Statistical Terms article.
Feature
Used in Machine Learning and Predictive models, this is simply a predictor variable.
Grouping Variable
Same as a factor.
Fixed factor
A categorical predictor variable in which the specific values of the categories are intentional and important, often chosen by the experimenter. Examples include experimental treatments or demographic categories, such as sex and race.
A categorical predictor variable in which the specific values of the categories were randomly assigned. Generally used in mixed modeling. Examples include subjects or random blocks.
This term is generally used in experimental design, but I’ve also seen it in randomized controlled trials.
A blocking variable is a variable that indicates an experimental block: a cluster or experimental unit that restricts complete randomization and that often results in similar response values among members of the block.
Blocking variables can be either fixed or random factors. They are never continuous.
Dummy variable
A categorical variable that has been dummy coded. Dummy coding (also called indicator coding) is usually used in regression models, but not ANOVA. A dummy variable can have only two values: 0 and 1. When a categorical variable has more than two values, it is recoded into multiple dummy variables.
Indicator variable
Same as dummy variable.
The Take Away Message
Whenever you’re using technical terms in a report, an article, or a conversation, it’s always a good idea to define your terms. This is especially important in statistics, which is used in many, many fields, each of whom adds their own subtleties to the terminology.
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