Understanding GLM Models and Analysis of Deviance Tables: A Tale of Two P-Values

Understanding GLM Models and Analysis of Deviance Tables

Generalized Linear Model (GLM) is a statistical model that extends traditional linear regression by allowing the dependent variable to take on non-continuous values. In this article, we’ll delve into the world of GLMs, specifically focusing on Gamma-GLM models and their analysis using the stats package in R.

Introduction to Gamma-GLM Models

Gamma-GLM is a type of generalized linear model that assumes the response variable follows a gamma distribution. This distribution is often used when modeling count data or response variables with a non-negative lower bound. The log-link function is commonly used with the gamma distribution, which results in an exponential family of distributions.

Understanding Analysis of Deviance Tables

The analysis of deviance table is a key component of any ANOVA or GLM analysis. It provides information on the contribution of each term to the overall model fit and helps identify the significance of each term in the model.

In this article, we’ll explore how the p-value calculated from the deviance table can differ from the estimated p-value using the pchisq() function in R.

Examining the Analysis of Deviance Table

Let’s examine the deviance table generated by the model.test() function in R:

Analysis of Deviance Table

Model: Gamma, link: log

Response: cont

Terms added sequentially (first to last)


     Df Deviance Resid. Df Resid. Dev  Pr(&gt;Chi)    
NULL                    23    11.5897              
trt   7   11.106        16     0.4839 &lt; 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

From this table, we can see that the deviance for each term is reported along with its residual degrees of freedom and residual deviance. The p-value corresponding to the overall model fit is also reported as Pr(>Chi). In this case, the p-value is less than 2.2e-16, indicating strong evidence against the null hypothesis.

Calculating P-Values Using pchisq()

However, when we calculate the p-value using the pchisq() function in R, we get a different result:

p <- 1 - pchisq(11.1057, 7, lower.tail = FALSE)
[1] 0.1340744

This calculated p-value is greater than 0.13, which seems to contradict the reported p-value in the deviance table.

Why the Difference?

So, what’s going on here? What transformation or calculation method is used when calculating the p-value from the deviance table?

The Deviance Table and P-Value Calculation

The deviance table calculates the contribution of each term to the overall model fit by comparing the deviance of the full model with that of a reduced model (with only one degree of freedom). This results in an F-statistic, which is then converted into an p-value using the F-distribution.

## Convert F-statistic to p-value
p_value <- pf(anova(model.test)$Fstatistic[2], anova(model.test)$Df[2])

However, this calculated p-value does not directly correspond to the p-value reported in the deviance table. Instead, it’s the p-value for the overall model fit, excluding the intercept term.

## Calculate p-value for overall model fit (excluding intercept)
p_value <- 1 - pchisq(anova(model.test)$Deviance[2] / 
                    summary(model.test)$dispersion, 
                   anova(model.test)$Df[2], 
                   lower.tail = FALSE)

In this code snippet, we divide the deviance by the dispersion to get the scale factor for the F-statistic. This scale factor is then used to convert the F-statistic into a p-value.

Conclusion

The analysis of deviance table and calculated p-values can sometimes be confusing due to the different methods used to calculate these values. By understanding how the p-values are generated in R, we can better interpret the results and make more informed decisions about our models.

In the next section, we’ll explore some common applications of GLM models and provide code examples for each scenario.

Applications of GLM Models

GLM models have a wide range of applications across various fields. In this section, we’ll examine two common scenarios:

Scenario 1: Modeling Count Data with Gamma-GLM

In this example, we’ll use the stats package in R to fit a Gamma-GLM model to count data.

## Load necessary libraries
library(ggplot2)
library(statmod)

## Load sample dataset
data("mtcars")

## Fit GLM model
model <- glm(mpg ~ wt + cyl, data = mtcars, family = poisson())

## Print summary of model
summary(model)

This code snippet fits a Poisson-GLM model to the mpg column in the mtcars dataset.

Scenario 2: Modeling Binary Response with Gamma-GLM

In this example, we’ll use the stats package in R to fit a Gamma-GLM model to binary response data.

## Load necessary libraries
library(ggplot2)
library(statmod)

## Load sample dataset
data("mtcars")

## Fit GLM model
model <- glm(vs ~ wt + cyl, data = mtcars, family = binomial())

## Print summary of model
summary(model)

This code snippet fits a binary-GLM model to the vs column in the mtcars dataset.

In the next section, we’ll explore some common issues and limitations with GLM models.

Last modified on 2024-04-09