# Calculating Slope and Intercept From Multiple Linear Regression

Consider this linear equation:

$$Y \sim \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4(X_2*X_3) + \epsilon$$

where

• $$Y$$ is what I'm trying to explain (it happens to be the evolved growth rate of a species growing and evolving in a community).
• $$B_0$$ is the intercept.
• $$X_1$$ is species richness, modeled as a continuous variable, taking values 1, 2, 3, or 4.
• $$X_2$$ and $$X_3$$ are the presence or absence of species A or species B. These are dummy variables coded either as 0 to indicate absence or 1 to indicate presence of that particular species.
• $$\beta_1$$, $$\beta_2$$, and $$\beta_3$$ are coefficients (or slopes) for $$X_1$$, $$X_2$$, and $$X_3$$.
• $$\beta_4$$ is the coefficient of the interaction term.

Notice that there in no interaction with $$X_1$$ and the other variables. This is because it doesn't make biological sense that this variable interacts with the other variables.

This is what the model looks like in R:

lm(Y ~ species_richness + A_present * B_present, data = .x)

When I run the model with my data, these are the coefficents (along with their P-values) predicted by the model:

B0 <- 0.1619 (P < 0.0001)
B1 <- -0.0112 (P = 0.0159)
B2 <- -0.0133 (P = 0.0924)
B3 <- -0.0224 (P = 0.0034)
B4 <- 0.0841 (P < 0.0001)


My questions are how do I calculate the slopes and intercepts of this model when a species is or is not absent?

Now if I want to calculate the slope and intercept of the model when species B, represented as $$X_3$$ is absent, it seems easy enough. Just plug 0 into the linear equation above and here is what you get:

$$Y ~ \sim \beta_0 + \beta_1X_1 + \beta_2X_2 + 0 + 0 + \epsilon$$

The way I interpret the equation now is that I want to get the slope of the line for a one unit change of species richness, then $$X_1$$, becomes a values of 1. And if I also consider that species A is present, the $$X_2$$ becomes one. Because $$X_1$$ and $$X_2$$ are both $$X$$ variables left in the equation, when you add them up, you get the slope of the line. So, $$slope = \beta_1X_1 + \beta_2X_2$$ The only coefficient left is $$B_0$$, so that's obviously the intercept. When I fit slope and this intercept on the data, it's clear that this is fitting the data where species B is absent. Even though it fits well, please let me know if my logic is correct.

I'm having much more trouble fitting the line when I want to consider that species B is present. In this case, I think I should subsitute a value of 1 for all times when $$X_3$$ is present in the equation. That looks like this:

$$Y \sim \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3*1 + \beta_4(X_2*1) + \epsilon$$

If this is true, then does $$slope = \beta_1X_1 + \beta_2X_2 + \beta_4(X_2*1)$$ and $$intercept = \beta_0 + \beta_3$$ ? When I plot this the incercept is definitely off and it's hard to tell if the slope is correct.

Thus, what are the slope and incercept of the model if we consider a one unit change in species richness, which is $$X_1$$, and we consider that species A, $$X_2$$, and species B, $$X_3$$. are both present.

Thank you and let me know if I can clarify anything further.

Before addressing the main problem. Notice that species richness isn't a continuous variable. It's a categorical variable with an ordinal structure, however let's ignore that.

As you said $$X_2,X_3,X_4$$ are indicator functions for the presence of species $$A$$, $$B$$ and $$A\cap B$$, thus their impact will $$0$$ or $$\beta_i$$, $$i\in\{2,3,4\}$$. Suppose $$X_2=1, X_3=0$$, notice that your regression will be

$$Y = (\beta_0 +\beta_2) + \beta_1 X_1$$ such that $$\beta_1$$ is the slope of the fitted line. Dummy variables only change the intercept of the line.

You are saying $$X_2=1$$ then you forget to actually replace it in the equation, also you are mixing parameters and variables, slope or angular coefficient is a term that refers to parameters. $$X\beta$$ isn't a coefficient, it's a linear predictor, the linear combination of parameters and covariates.

• Thank you for your reply :) On your first remark, I'm actually not sure how to think about richness. In the experiment it can take values 1, 2, 3, or 4. I put it into the lm() function as a numeric term. Not sure how that would change the outcome of the model.... For your second point. $X_4$ is not an indicator for the presence of species. It is the interaction coefficent when $X_2$ and $X_3$ (or species A and B) interact. There is no species C. Does that make my question clearer? EDIT: I realize I wrote this wrong in the question. I have now edited the question. My apologies. Mar 16 at 13:10
• @CAM_etal That's okay! The main idea remains unchanged, the interaction between dummies will also only affect the intercept of the estimated line. =) Mar 16 at 13:20
• @CAM_etal As for the ordinal thing, I think there's an ordinal factor data type in R, it will create dummies for $X_1$ with the appropriate encoding, as in factor(data, ordered = T) Mar 16 at 13:23
• So if I want to calculate the slope and intercept for the presence of species B, I have to consider that species A is absent (i.e., 0), reducing the equation to what you show? Mar 16 at 13:43
• I checked this as the correct answer. I've upvoted and althought the vote is recorded it cannot be seen because I need a reputation score of at least 15. Thank you very much for your help. Mar 16 at 14:00