One of the difficult decisions in mixed modeling is deciding which factors are fixed and which are random. And as difficult as it is, it’s also very important. Correctly specifying the fixed and random factors of the model is vital to obtain accurate analyses.
Now, you may be thinking of the fixed and random effects in the model, rather than the factors themselves, as fixed or random. If so, remember that each term in the model (factor, covariate, interaction or other multiplicative term) has an effect. We’ll come back to how the model measures the effects for fixed and random factors.
Sadly, the definitions in many texts don’t help much with decisions to specify factors as fixed or random. Textbook examples are often artificial and hard to apply to the real, messy data you’re working with.
Here’s the real kicker. The same factor can often be fixed or random, depending on the researcher’s objective. (more…)
In this nearly 6-hour tutorial you will learn menu-based R libraries so you can use R without having to fuss with R code. These libraries don’t cover everything R can do, but they do quite a bit and can set you up to make running R much easier.
(more…)
Updated 12/20/2021
Despite its popularity, interpreting regression coefficients of any but the simplest models is sometimes, well….difficult.
So let’s interpret the coefficients in a model with two predictors: a continuous and a categorical variable. The example here is a linear regression model. But this works the same way for interpreting coefficients from any regression model without interactions.
A linear regression model with two predictor variables results in the following equation:
Yi = B0 + B1*X1i + B2*X2i + ei.
The variables in the model are:
- Y, the response variable;
- X1, the first predictor variable;
- X2, the second predictor variable; and
- e, the residual error, which is an unmeasured variable.
The parameters in the model are:
- B0, the Y-intercept;
- B1, the first regression coefficient; and
- B2, the second regression coefficient.
One example would be a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2).
Height is measured in cm. Bacteria is measured in thousand per ml of soil. And type of sun = 0 if the plant is in partial sun and type of sun = 1 if the plant is in full sun.
Let’s say it turned out that the regression equation was estimated as follows:
Y = 42 + 2.3*X1 + 11*X2
Interpreting the Intercept
B0, the Y-intercept, can be interpreted as the value you would predict for Y if both X1 = 0 and X2 = 0.
We would expect an average height of 42 cm for shrubs in partial sun with no bacteria in the soil. However, this is only a meaningful interpretation if it is reasonable that both X1 and X2 can be 0, and if the data set actually included values for X1 and X2 that were near 0.
If neither of these conditions are true, then B0 really has no meaningful interpretation. It just anchors the regression line in the right place. In our case, it is easy to see that X2 sometimes is 0, but if X1, our bacteria level, never comes close to 0, then our intercept has no real interpretation.
Interpreting Coefficients of Continuous Predictor Variables
Since X1 is a continuous variable, B1 represents the difference in the predicted value of Y for each one-unit difference in X1, if X2 remains constant.
This means that if X1 differed by one unit (and X2 did not differ) Y will differ by B1 units, on average.
In our example, shrubs with a 5000/ml bacteria count would, on average, be 2.3 cm taller than those with a 4000/ml bacteria count. They likewise would be about 2.3 cm taller than those with 3000/ml bacteria, as long as they were in the same type of sun.
(Don’t forget that since the measurement unit for bacteria count is 1000 per ml of soil, 1000 bacteria represent one unit of X1).
Interpreting Coefficients of Categorical Predictor Variables
Similarly, B2 is interpreted as the difference in the predicted value in Y for each one-unit difference in X2 if X1 remains constant. However, since X2 is a categorical variable coded as 0 or 1, a one unit difference represents switching from one category to the other.
B2 is then the average difference in Y between the category for which X2 = 0 (the reference group) and the category for which X2 = 1 (the comparison group).
So compared to shrubs that were in partial sun, we would expect shrubs in full sun to be 11 cm taller, on average, at the same level of soil bacteria.
Interpreting Coefficients when Predictor Variables are Correlated
Don’t forget that each coefficient is influenced by the other variables in a regression model. Because predictor variables are nearly always associated, two or more variables may explain some of the same variation in Y.
Therefore, each coefficient does not measure the total effect on Y of its corresponding variable. It would if it were the only predictor variable in the model. Or if the predictors were independent of each other.
Rather, each coefficient represents the additional effect of adding that variable to the model, if the effects of all other variables in the model are already accounted for.
This means that adding or removing variables from the model will change the coefficients. This is not a problem, as long as you understand why and interpret accordingly.
Interpreting Other Specific Coefficients
I’ve given you the basics here. But interpretation gets a bit trickier for more complicated models, for example, when the model contains quadratic or interaction terms. There are also ways to rescale predictor variables to make interpretation easier.
So here is some more reading about interpreting specific types of coefficients for different types of models:
Odds ratios have a unique part to play in describing the effects of logistic regression models. But that doesn’t mean they’re easy to communicate to an audience who is likely to misinterpret them. So writing up your odds ratios has to be done with care. (more…)
Updated 11/22/2021
Probability and odds measure the same thing: the likelihood or propensity or possibility of a specific outcome.
People use the terms odds and probability interchangeably in casual usage, but that is unfortunate. It just creates confusion because they are not equivalent.
How Odds and Probability Differ
They measure the same thing on different scales. Imagine how confusing it would be if people used degrees Celsius and degrees Fahrenheit interchangeably. “It’s going to be 35 degrees today” could really make you dress the wrong way.
In measuring the likelihood of any outcome, we need to know (more…)
by Kim Love and Karen Grace-Martin
Statistics terminology is confusing.
Sometimes different terms are used to mean the same thing, often in different fields of application. Sometimes the same term is used to mean different things. And sometimes very similar terms are used to describe related but distinct statistical concepts.
(more…)
