Implementing Log-linear Regression in R

In this article, we discuss widely used Generalized Linear Models (GLMs) in the industry, focusing on:
Log-linear regression
Interpreting log-transformations
Binary logistic regression
We also review the underlying distributions and applicable link functions. To illustrate concepts in R, we use the sample datasets:
cola.csv
penalty.csv

Introduction to Generalized Linear Models
A Generalized Linear Model (GLM) represents the dependent variable as a function of independent variables. The traditional form of GLM is simple linear regression, which assumes that the dependent variable is normally distributed.
However, in real-world scenarios, this assumption is often violated. For example, if the dependent variable is the number of coffee cups sold (always positive and often with a fat-tailed distribution) and the independent variable is temperature, simple linear regression may produce nonsensical results — e.g., predicting negative sales at low temperatures.
GLMs overcome these limitations by applying a link function to the dependent variable, allowing the model to accommodate distributions other than normal (e.g., Poisson, Bernoulli).

Linear Regression in R
Simple linear regression models a linear relationship:
Yi=α+βXiY_i = \alpha + \beta X_iYi​=α+βXi​
The coefficients are computed using Ordinary Least Squares (OLS).
Example: Cola Sales vs Temperature

Read data

data = read.csv("Cola.csv", header = TRUE)
head(data)

Scatter plot

plot(data, main = "Scatter Plot")

Install hydroGOF for RMSE

install.packages("hydroGOF")
library(hydroGOF)

Fit linear model

model = lm(Cola ~ Temperature, data)

Overlay best-fit line

abline(model)

Calculate RMSE

PredCola = predict(model, data)
RMSE = rmse(PredCola, data$Cola)

The simple linear regression fit is poor, with an RMSE of 241.49. The model predicts negative sales for low temperatures — highlighting the limitations of linear regression when the dependent variable has a non-normal distribution.

Log-linear Regression
Log-linear regression is useful when the dependent variable grows exponentially with the independent variable. For instance, expected salary vs. education or compound interest over time.
The model form is:
Y=a⋅bXY = a \cdot b^XY=a⋅bX
Taking the logarithm of both sides:
log⁡(Y)=log⁡(a)+(log⁡b)⋅X\log(Y) = \log(a) + (\log b) \cdot Xlog(Y)=log(a)+(logb)⋅X
Now the relationship is linear and can be estimated using OLS. Here, log⁡(a)\log(a)log(a) is the intercept, and log⁡(b)\log(b)log(b) represents the growth rate.
Implementing Log-linear Regression in R

Transform dependent variable

data$LCola = log(data$Cola)

Scatter plot

plot(LCola ~ Temperature, data = data, main = "Scatter Plot")

Fit linear model on transformed data

model1 = lm(LCola ~ Temperature, data)
abline(model1)

Calculate RMSE

PredCola1 = predict(model1, data)
RMSE = rmse(PredCola1, data$LCola)

The RMSE drops dramatically to 0.24, and predictions are now meaningful (no negative sales).

Interpreting Log Transformations
Log transformations help model non-linear relationships using linear techniques.
Log-linear: Log-transform dependent variable; independent variable remains unchanged.
Linear-log: Log-transform independent variable; dependent variable remains unchanged.
Log-log: Log-transform both dependent and independent variables.
Model TypeEquationInterpretation
Log-linear
log(Y) = α + βX
Y changes by a constant percentage per unit change in X
Linear-log
Y = α + β log(X)
Y changes by β units per percentage change in X
Log-log
log(Y) = α + β log(X)
Y changes by β% per 1% change in X

Binary Logistic Regression
Binary logistic regression is used when the dependent variable is categorical (0 or 1). The conditional distribution is Bernoulli.
Example: Predicting success in a football penalty based on practice hours.

Read data

data1 = read.csv("Penalty.csv", header = TRUE)
head(data1)

Scatter plot

plot(data1, main = "Scatter Plot")

Fitting a Logistic Regression Model
fit = glm(Outcome ~ Practice, family = binomial(link = "logit"), data = data1)

Plot predicted probabilities

plot(data1, main = "Scatter Plot")
curve(predict(fit, data.frame(Practice = x), type = "resp"), add = TRUE)
points(data1$Practice, fitted(fit), pch = 20)

The probability of success increases with practice hours. A positive β1\beta_1β1​ indicates that probability of success increases as X increases.

Conclusion
GLMs generalize linear regression to handle non-normal dependent variables and non-linear relationships.
Log-linear regression is useful for exponential relationships, log-normal distributions, and Poisson counts.
Binary logistic regression models probabilities for 0/1 outcomes using the logistic function.
R implementations are straightforward, and data transformations help resolve issues like negative predictions or non-linear growth.
Along with theory, we provided commented R code to implement these models on sample datasets.
We hope this article helps you understand and implement GLMs in real-world scenarios.

If you want, I can create a companion RMarkdown notebook for this article that includes:
All code chunks runnable in R
Plots for Cola sales and penalty data
Step-by-step interpretation of results
This makes it ready for sharing or teaching purposes.
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