How to Interpret Pr(>|z|) in Logistic Regression Output in R

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

Coefficients:
              Estimate Std. Error z value Pr(>|z|)  
(Intercept) -17.638452   9.165482  -1.924   0.0543 .
disp         -0.004153   0.006621  -0.627   0.5305  
drat          4.879396   2.268115   2.151   0.0315 * 

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

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

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

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

The following code shows how to fit a logistic regression model in R using the built-in mtcars dataset:

#fit logistic regression model
model #view model summary
summary(model)

Call:
glm(formula = am ~ disp + drat, family = binomial, data = mtcars)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.5773  -0.2273  -0.1155   0.5196   1.8957  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)  
(Intercept) -17.638452   9.165482  -1.924   0.0543 .
disp         -0.004153   0.006621  -0.627   0.5305  
drat          4.879396   2.268115   2.151   0.0315 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 43.230  on 31  degrees of freedom
Residual deviance: 21.268  on 29  degrees of freedom
AIC: 27.268

Number of Fisher Scoring iterations: 6

Here’s how to interpret the values in the Pr(>|z|) column:

• The p-value for the predictor variable “disp” is .5305. Since this value is not less than .05, it does not have a statistically significant relationship with the response variable in the model.
• The p-value for the predictor variable “drat” is .0315. Since this value is less than .05, it has a statistically significant relationship with the response variable in the model.

The significance codes under the coefficient table tell us that a single asterisk (*) next to the p-value of .0315 means the p-value is statistically significant at α = .05.

How is Pr(>|z|) Calculated?

Here’s how the value for Pr(>|z|) is actually calculated:

Step 1: Calculate the z value

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

• z value = Estimate / Std. Error

For example, here’s how to calculate the z value for the predictor variable “drat”:

#calculate z-value
4.879396 / 2.268115

[1] 2.151

Step 2: Calculate the p-value

Next, we calculate the two-tailed p-value. This represents the probability that the absolute value of the normal distribution is greater than 2.151 or less than -2.151.

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

• p-value = 2 * (1-pnorm(z value))

For example, here’s how to calculate the two-tailed p-value for a z-value of 2.151:

#calculate p-value
2*(1-pnorm(2.151))

[1] 0.0314762

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

Additional Resources

The following tutorials explain how to fit various regression models in R:

How to Perform Logistic Regression in R
How to Perform Simple Linear Regression in R
How to Perform Multiple Linear Regression in R