# Intercept

### Interpreting Regression Coefficients

December 20th, 2021 by

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:

### What is Degrees of Freedom?

May 27th, 2020 by

No, degrees of freedom is not “having one foot out the door”!

Definitions are rarely very good at explaining the meaning of something. At least not in statistics. Degrees of freedom: “the number of independent values or quantities which can be assigned to a statistical distribution”.

This is no exception.

### Member Training: Reporting Structural Equation Modeling Results

October 1st, 2019 by The last, and sometimes hardest, step for running any statistical model is writing up results.

As with most other steps, this one is a bit more complicated for structural equation models than it is for simpler models like linear regression.

Any good statistical report includes enough information that someone else could replicate your results with your data.

### Member Training: Segmented Regression

April 3rd, 2017 by

Linear regression with a continuous predictor is set up to measure the constant relationship between that predictor and a continuous outcome. This relationship is measured in the expected change in the outcome for each one-unit change in the predictor.

One big assumption in this kind of model, though, is that this rate of change is the same for every value of the predictor. It’s an assumption we need to question, though, because it’s not a good approach for a lot of relationships.

Segmented regression allows you to generate different slopes and/or intercepts for different segments of values of the continuous predictor. This can provide you with a wealth of information that a non-segmented regression cannot.

In this webinar, we will cover (more…)

### Should You Always Center a Predictor on the Mean?

December 2nd, 2011 by

Centering predictor variables is one of those simple but extremely useful practices that is easily overlooked.

It’s almost too simple.

Centering simply means subtracting a constant from every value of a variable.  What it does is redefine the 0 point for that predictor to be whatever value you subtracted.  It shifts the scale over, but retains the units.

The effect is that the slope between that predictor and the response variable doesn’t (more…)

### Confusing Statistical Terms #2: Alpha and Beta

December 11th, 2009 by

Oh so many years ago I had my first insight into just how ridiculously confusing all the statistical terminology can be for novices.

I was TAing a two-semester applied statistics class for graduate students in biology.  It started with basic hypothesis testing and went on through to multiple regression.

It was a cross-listed class, meaning there were a handful of courageous (or masochistic) undergrads in the class, and they were having trouble keeping (more…)