*19*

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. Î± = .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 multiple linear regression model 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 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 #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 --- Signif. codes: 0 â€˜***â€™ 0.001 â€˜**â€™ 0.01 â€˜*â€™ 0.05 â€˜.â€™ 0.1 â€˜ â€™ 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

Hereâ€™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 significance codes 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 Î± = .05.

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

Hereâ€™s how the value for Pr(>|t|) is actually calculated:

**Step 1: Calculate the t value**

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

**t value**= Estimate / Std. Error

For example, hereâ€™s how to calculate the t value for the predictor variable x1:

#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, hereâ€™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
**

**Additional Resources**

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Plot Multiple Linear Regression Results in R