# How to compute the standard error of the estimate for a model (in R)

## Task

One measure of the goodness of fit of a model is the standard error of its estimates. If the actual values are $y_i$ and the estimates are $\hat y_i$, the definition of this quantity is as follows, for $n$ data points.

\[\sigma_{\text{est}} = \sqrt{ \frac{ \sum (y_i-\hat y_i)^2 }{ n } }\]If we’ve fit a linear model, how do we compute the standard error of its estimates?

## Solution

Let’s assume that you already fit the linear model, as shown in the code below. This one uses a small amount of fake data, but it’s just an example. See also how to fit a linear model to two columns of data.

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x <- c(34, 9, 78, 60, 22, 45, 83, 59, 25)
y <- c(126, 347, 298, 309, 450, 187, 266, 385, 400)
model <- lm(y ~ x)

The standard error for each estimate is shown as part of the model summary, reported by R’s
built-in `summary`

function. See the column entitled “Std. Error” in the output below.

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summary(model)

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Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-193.776 -4.334 15.459 71.143 118.116
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 354.082 76.733 4.614 0.00244 **
x -1.009 1.473 -0.685 0.51536
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 107.1 on 7 degrees of freedom
Multiple R-squared: 0.06283, Adjusted R-squared: -0.07106
F-statistic: 0.4693 on 1 and 7 DF, p-value: 0.5154

If we need to extract just the model coefficients table, or even just the “Std. Error” column of it, we can use code like the following.

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coef(summary(model))
coef(summary(model))[,2]

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Estimate Std. Error t value Pr(>|t|)
(Intercept) 354.082248 76.732772 4.6144853 0.002441995
x -1.009013 1.472939 -0.6850334 0.515358250
(Intercept) x
76.732772 1.472939

The standard error of the estimate for the intercept is is 76.733 and the standard error of the estimate for the slope is 1.473.

Content last modified on 24 July 2023.

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Contributed by:

- Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)
- Nathan Carter (ncarter@bentley.edu)