What do the Pr(>|t|) values in Regression Model Output in R mean?

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Whenever you perform linear regression in R, the output of your regression model will be displayed in the following format:

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  10.0035     5.9091   1.693   0.1513  
x1            1.4758     0.5029   2.935   0.0325 *
x2           -0.7834     0.8014  -0.978   0.3732 

The Pr(>|t|) column represents the p-value associated with the value in the t value column.

If the p-value is less than a certain significance level (e.g. α = .05) then the predictor variable is said to have a statistically significant relationship with the response variable in the model.

The following example shows how to interpret values in the Pr(>|t|) column for a given regression model.

Example: How to Interpret Pr(>|t|) Values

Suppose we would like to fit a using predictor variables x1 and x2 and a single response variable y.

The following code shows how to create a data frame and fit a regression model to the data:

#create data frame
df <- data.frame(x1=c(1, 3, 3, 4, 4, 5, 6, 6),
                 x2=c(7, 7, 5, 6, 5, 4, 5, 6),
                 y=c(8, 8, 9, 9, 13, 14, 17, 14))

#fit multiple linear regression model
model <- lm(y ~ x1 + x2, data=df)

#view model summary
summary(model)

Call:
lm(formula = y ~ x1 + x2, data = df)

Residuals:
      1       2       3       4       5       6       7       8 
 2.0046 -0.9470 -1.5138 -2.2062  1.0104 -0.2488  2.0588 -0.1578 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  10.0035     5.9091   1.693   0.1513  
x1            1.4758     0.5029   2.935   0.0325 *
x2           -0.7834     0.8014  -0.978   0.3732  
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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.867 on 5 degrees of freedom
Multiple R-squared:  0.7876,	Adjusted R-squared:  0.7026 
F-statistic: 9.268 on 2 and 5 DF,  p-value: 0.0208

Here’s how to interpret the values in the Pr(>|t|) column:

• The p-value for the predictor variable x1 is .0325. Since this value is less than .05, it has a statistically significant relationship with the response variable in the model.
• The p-value for the predictor variable x2 is .3732. Since this value is not less than .05, it does not have a statistically significant relationship with the response variable in the model.

The under the coefficient table tell us that a single asterik (*) next to the p-value of .0325 means the p-value is statistically significant at α = .05.

How is Pr(>|t|) Actually Calculated?

Here’s how the value for Pr(>|t|) is actually calculated:

Step 1: Calculate the t value

First, we calculate the t value using the following formula:

• t value = Estimate / Std. Error

#calculate t-value
1.4758 / .5029

[1] 2.934579

Step 2: Calculate the p-value

Next, we calculate the p-value. This represents the probability that the absolute value of the t-distribution is greater than 2.935.

We can use the following formula in R to calculate this value:

• p-value = 2 * pt(abs(t value), residual df, lower.tail = FALSE)

For example, here’s how to calculate the p-value for a t-value of 2.935 with 5 residual degrees of freedom:

#calculate p-value
2 * pt(abs(2.935), 5, lower.tail = FALSE)

[1] 0.0324441

Notice that this p-value matches the p-value in the regression output from above.

Note: The value for the residual degrees of freedom can be found near the bottom of the regression output. In our example, it turned out to be 5:

Residual standard error: 1.867 on 5 degrees of freedom