Estimate Linear Model with Fixed Intercept in R (2 Examples)

 

In this post you’ll learn how to set a fixed intercept when estimating a linear regression model in the R programming language.

The post is structured as follows:

It’s time to dive into the programming part.

 

Creation of Example Data

The data below will be used as basement for this R tutorial:

set.seed(653897)                                   # Create example data
x <- rnorm(1000, 3)
y <- rnorm(1000, 2) + x

Our example data consists of two randomly distributed numeric vectors that are correlated with each other.

Let’s estimate a linear regression model without specifying the intercept manually (i.e. the default specification of the lm function):

mod_default <- lm(y ~ x)                           # Estimate linear model
summary(mod_default)                               # Summary statistics
# Call:
# lm(formula = y ~ x)
# 
# Residuals:
#     Min      1Q  Median      3Q     Max 
# -3.3152 -0.6598  0.0209  0.6563  3.4294 
# 
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept)  2.05729    0.09966   20.64   <2e-16 ***
# x            0.98086    0.03156   31.08   <2e-16 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Residual standard error: 0.9891 on 998 degrees of freedom
# Multiple R-squared:  0.4919,	Adjusted R-squared:  0.4914 
# F-statistic: 966.1 on 1 and 998 DF,  p-value: < 2.2e-16

The previous output of the RStudio console shows the descriptive summary statistics of our linear regression model. As you can see, we have estimated an intercept of 2.05729 and a regression coefficient for x of 0.98086.

Let’s estimate another model with fixed intercept…

 

Example 1: Estimate Linear Model with Fixed Intercept Using I() Function

Example 1 illustrates how to estimate a generalized linear model with known intercept.

For this, we first have to specify our fixed intercept:

intercept <- 3                                     # Define fixed intercept

Next, we can estimate our linear model using the I() function as shown below:

mod_intercept_1 <- lm(I(y - intercept) ~ 0 + x)    # Model with fixed intercept

Finally, we can apply the summary function to return our descriptive statistics:

summary(mod_intercept_1)                           # Summary statistics
# Call:
# lm(formula = I(y - intercept) ~ 0 + x)
# 
# Residuals:
#     Min      1Q  Median      3Q     Max 
# -3.0314 -0.7734 -0.0577  0.6222  3.1767 
# 
# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)    
# x  0.69743    0.01033   67.49   <2e-16 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Residual standard error: 1.032 on 999 degrees of freedom
# Multiple R-squared:  0.8201,	Adjusted R-squared:   0.82 
# F-statistic:  4555 on 1 and 999 DF,  p-value: < 2.2e-16

As you can see, the previously estimated model did not return an intercept value, since we have manually specified this intercept in the forefront.

You can also see that the x estimate has changed to 0.69743.

 

Important notes on models with fixed intercept:

The summary output of models with fixed intercept has to be interpreted carefully. Metrics such as the R-squared, the t-value, and the F-statistic are much larger than in the model without fixed intercept.

Furthermore, it is often not advisable to specify a fixed intercept from a theoretical & methodological viewpoint. You may find a detailed discussion on this topic in this thread on Cross Validated.

 

Example 2: Estimate Linear Model with Fixed Intercept Using offset() & rep() Functions

This example shows a second alternative to the syntax of the previous example.

In this example we’ll use the offset and rep functions to estimate our linear model with known intercept:

mod_intercept_2 <- lm(y ~ x + 0 +                  # Model with fixed intercept
                        offset(rep(intercept, 1000)))

The following summary statistics are exactly the same as in Example 1, even though we have used a different R syntax:

summary(mod_intercept_2)                           # Summary statistics
# Call:
# lm(formula = y ~ x + 0 + offset(rep(intercept, 1000)))
# 
# Residuals:
#     Min      1Q  Median      3Q     Max 
# -3.0314 -0.7734 -0.0577  0.6222  3.1767 
# 
# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)    
# x  0.69743    0.01033   67.49   <2e-16 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# Residual standard error: 1.032 on 999 degrees of freedom
# Multiple R-squared:  0.9613,	Adjusted R-squared:  0.9612 
# F-statistic: 2.479e+04 on 1 and 999 DF,  p-value: < 2.2e-16

 

Video, Further Resources & Summary

In case you need further explanations on the content of this article, you may have a look at the following video on my YouTube channel. I’m illustrating the R codes of this article in the video:

 

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In addition, you could read the other tutorials of my website.

 

In summary: At this point you should know how to define a known constant in a linear regression model in R programming. Let me know in the comments section, if you have further questions or comments on regression models, constants, or any other related topics.

 

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