Karen Grace-Martin

Specifying Fixed and Random Factors in Mixed Models

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.

That’s right, there isn’t always a right or a wrong decision here. It depends on the inferences you want to make about this factor. This article outlines a different way to think about fixed and random factors.

An Example

Consider an experiment that examines beetle damage on cucumbers. The researcher replicates the experiment at five farms and on four fields at each farm.

There are two varieties of cucumbers, and the researcher measures beetle damage on each of 50 plants at the end of the season. The researcher wants to measure differences in how much damage the two varieties sustain.

The experiment then has the following factors: VARIETY, FARM, and FIELD.

Fixed factors can be thought of in terms of differences.

The effect of a categorical fixed factor is measured by differences from the overall mean.

The effect of a continuous fixed predictor (usually called a covariate) is defined by its slope–how the mean of the dependent variable differs with differing values of the predictor.

The output for fixed predictors, then, gives estimates for mean-differences or slopes.

Conclusions regarding fixed factors are particular to the values of these factors. For example, if one variety of cucumber has significantly less damage than the other, you can make conclusions about these two varieties. This says nothing about cucumber varieties that were not tested.

Random factors, on the other hand, are defined by a distribution and not by differences.

The values of a random factor are assumed to be chosen from a population with a normal distribution with a certain variance.

The output for a random factor is an estimate of this variance and not a set of differences from a mean. Effects of random factors are measured in terms of variance, not mean differences.

For example, we may find that the variance among fields makes up a certain percentage of the overall variance in beetle damage. But we’re not measuring how much higher the beetle damage is in one field compared to another. We only care that they vary, not how much they differ.

Software specification of fixed and random factors

Before we get into specifying factors as fixed and random, please note:

When you’re specifying random effects in software, like intercepts and slopes, the random factor is itself the Subject variable in your software. This differs in different software procedures. Some specifically ask for code like “Subject=FARM.” Others just put FARM after a || or some other indicator of the subject for which random intercepts and slopes are measured.

Some software, like SAS and SPSS, allow you to specify the random factors two ways. One is to list the random effects (like intercept) along with the random factor listed as a subject.

/Random Intercept | Subject(Farm)

The other is to list the random factors themselves.

/Random Farm

Both of those give you the same output.

Other software, like R and Stata, only allow the former. And neither uses the term “Subject” anywhere. But that random factor, the subject, still needs to be specified after the | or || in the random part of the model.

Specifying fixed effects is pretty straightforward.

Situations that indicate a factor should be specified as fixed:

1. The factor is the primary treatment that you want to compare.

In our example, VARIETY is definitely fixed as the researcher wants to compare the mean beetle damage on the two varieties.

2. The factor is a secondary control variable, and you want to control for differences in the specific values of this factor.

Say the researcher chose these farms specifically for some feature they had, such as specific soil types or topographies that may affect beetle damage. If the researcher wants to compare the farms as representatives of those soil types, then FARM should be fixed.

3. The factor has only two values.

Even if everything else indicates that a factor should be random, if it has only two values, the variance cannot be calculated, and it should be fixed.

Situations that indicate random factors:

1. Your interest is in quantifying how much of the overall variance to attribute to this factor.

If you want to know how much of the variation in beetle damage you can attribute to the farm at which the damage took place, FARM would be random.

2. Your interest is not in knowing which specific means differ, but you want to account for the variation in this factor.

If the farms were chosen at random, FARM should be random.

This choice of the specific farms involved in the study is key. If you can rerun the study using different specific farms–different values of the Farm factor–and still be able to draw the same conclusions, then Farm should be random. However, if you want to compare or control for these particular farms, then Farm is a fixed factor.

3. You would like to generalize the conclusions about this factor to the whole population.

There is nothing about comparing these specific fields that is of interest to the researcher. Rather, the researcher wants to generalize the results of this experiment to all fields, so FIELD is random.

4. Any interaction with a random factor is also random.

How the factors of a model are specified can have great influence on the results of the analysis and on the conclusions drawn.

Interpreting Regression Coefficients

by Karen Grace-Martin 30 Comments

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:

Y_i = B₀ + B₁*X_1i + B₂*X_2i + e_i.

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:

Interpreting the Intercept
Removing the Intercept when X is Continuous or Categorical
Interpreting Interactions in Regression
How Changing the Scale of X affects Interpreting its Regression Coefficient
Interpreting Coefficients with a Centered Predictor

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Member Training: Matrix Algebra for Data Analysts: A Primer

by Karen Grace-Martin 4 Comments

If you’ve been doing data analysis for very long, you’ve certainly come across terms, concepts, and processes of matrix algebra. Not just matrices, but:

Matrix addition and multiplication
Traces and determinants
Eigenvalues and Eigenvectors
Inverting and transposing
Positive and negative definite

Member Training: Elements of Experimental Design

by Karen Grace-Martin

Whether or not you run experiments, there are elements of experimental design that affect how you need to analyze many types of studies.

The most fundamental of these are replication, randomization, and blocking. These key design elements come up in studies under all sorts of names: trials, replicates, multi-level nesting, repeated measures. Any data set that requires mixed or multilevel models has some of these design elements. [Read more…] about Member Training: Elements of Experimental Design

When to Use Logistic Regression for Percentages and Counts

by Karen Grace-Martin 2 Comments

One important yet difficult skill in statistics is choosing a type model for different data situations. One key consideration is the dependent variable.

For linear models, the dependent variable doesn’t have to be normally distributed, but it does have to be continuous, unbounded, and measured on an interval or ratio scale.

Percentages don’t fit these criteria. Yes, they’re continuous and ratio scale. The issue is the [Read more…] about When to Use Logistic Regression for Percentages and Counts

Why ANOVA is Really a Linear Regression, Despite the Difference in Notation

by Karen Grace-Martin 3 Comments

When I was in graduate school, stat professors would say “ANOVA is just a special case of linear regression.” But they never explained why.

And I couldn’t figure it out.

The model notation is different.

The output looks different.

The vocabulary is different.

The focus of what we’re testing is completely different. How can they be the same model?