### Introduction

When working with Generalized Linear Models (GLM) in R, you may encounter the warning message:

Warning: glm.fit: algorithm did not converge

This warning typically arises when the model-fitting process fails to find optimal parameters after the maximum number of iterations. The underlying causes vary but often relate to the characteristics of the data or the initial parameter estimates.

In this blog post, we will explore why this warning occurs and how to resolve it.

### What Does “Algorithm Did Not Converge” Mean?

In R, the `glm()`

function fits models using an iterative algorithm (usually Iteratively Reweighted Least Squares, IRLS) to find parameter estimates. If the algorithm cannot make sufficient progress after a specified number of iterations, it triggers the warning that the algorithm did not converge.

Common reasons for this warning include:

**Poorly scaled data**: Large differences in the scale of predictor variables can slow convergence.**Perfect separation**: When the outcome variable can be perfectly predicted by a combination of predictors, the algorithm cannot properly estimate coefficients.**Multicollinearity**: Highly correlated predictors make it difficult for the model to differentiate their individual effects.**Outliers or extreme values**: These can distort the fitting process.

### Example to Reproduce the Warning ⚠️

# Create data frame x = c(0.2, 0.4, 0.5, 0.7, 0.85, 1, 1.05, 1.2, 1.4, 1.6, 1.65, 1.8, 2, 2.2, 2.4) y = c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1) df <- data.frame(x, y) # Attempt to fit a logistic regression model model <- glm(y ~ x, data = df, family = "binomial")

**Output:**

Warning messages: 1: glm.fit: algorithm did not converge 2: glm.fit: fitted probabilities numerically 0 or 1 occurred

### How to Fix the Warning

#### Check for Perfect Separation

Perfect separation is a frequent culprit when fitting GLMs. In logistic regression, perfect separation occurs when one or more predictors can perfectly predict the outcome. You can use the `brglm2`

package to fit bias-reduced GLMs, which can handle separation more gracefully:

# Install and load the brglm2 package install.packages("brglm2") library(brglm2) # Refit the model using bias-reduced GLM model_brglm <- glm(y ~ x, family = binomial, method = "brglmFit") # Model summary summary(model_brglm)

**Output:**

Call: glm(formula = y ~ x, family = binomial, method = "brglmFit") Deviance Residuals: Min 1Q Median 3Q Max -1.09730 -0.27741 0.05491 0.27082 1.14057 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -5.765 3.252 -1.773 0.0763 . x 5.573 3.026 1.842 0.0655 . --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 20.1928 on 14 degrees of freedom Residual deviance: 4.6835 on 13 degrees of freedom AIC: 8.6835 Type of estimator: AS_mixed (mixed bias-reducing adjusted score equations) Number of Fisher Scoring iterations: 14

#### Standardize Predictors

If your data contains predictors with large differences in scale, standardizing these variables can improve the convergence of the algorithm. Standardization transforms predictors to have a mean of 0 and a standard deviation of 1.

x_standardized <- scale(x) # Fit the GLM again with standardized predictor model_standardized <- glm(y ~ x_standardized, family = binomial) summary(model_standardized)

**Output:**

Call: glm(formula = y ~ x_standardized, family = binomial) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 187.9 78477.3 0.002 0.998 x_standardized 530.9 220465.0 0.002 0.998 (Dispersion parameter for binomial family taken to be 1) Null deviance: 2.0190e+01 on 14 degrees of freedom Residual deviance: 1.0981e-08 on 13 degrees of freedom AIC: 4 Number of Fisher Scoring iterations: 25

#### Increase Iterations

Sometimes, the default number of iterations is insufficient for the algorithm to converge. You can adjust the `maxit`

parameter to increase the maximum number of iterations allowed during model fitting:

# Increase maximum number of iterations model_more_iter <- glm(y ~ x, data = df, family = binomial, control = list(maxit = 100)) # Model summary summary(model_more_iter)

**Output:**

Call: glm(formula = y ~ x, family = binomial, data = df, control = list(maxit = 100)) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -931.0 1500645.6 -0.001 1 x 908.3 1463664.5 0.001 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 2.0190e+01 on 14 degrees of freedom Residual deviance: 5.4945e-10 on 13 degrees of freedom AIC: 4 Number of Fisher Scoring iterations: 28

This solution provides more iterations for the algorithm, helping it converge in difficult cases.

#### Use Penalized Regression

Penalized regression methods like Ridge (L2 regularization) or Lasso (L1 regularization) can stabilize coefficient estimates and address convergence issues. These techniques work by adding a penalty to large coefficients, effectively reducing their influence on the model.

You can implement penalized regression using the `glmnet`

package:

# Create predictor matrix with an intercept term # install.packages("glmnet") # install if needed library(glmnet) x_matrix <- model.matrix(~ df$x) # Create response vector y_vector <- as.numeric(df$y) # Fit penalized logistic regression with Ridge penalty (alpha = 0) model_ridge <- cv.glmnet(x_matrix, y_vector, alpha = 0, family = "binomial") # Display model coefficients coef(model_ridge)

**Output:**

3 x 1 sparse Matrix of class "dgCMatrix" s1 (Intercept) -2.230315 (Intercept) . df$x 2.246259

### Explanation:

: Creates a design matrix with an intercept column and the`model.matrix(~ df$x)`

`x`

variable, ensuring at least two columns.: This fits a penalized logistic regression model using cross-validation to find the optimal penalty parameter (lambda).`cv.glmnet()`

**Ridge Regression** adds an L2 penalty, which helps prevent extremely large coefficients and improves convergence. If multicollinearity or perfect separation is an issue, this regularization can be particularly effective.

For Lasso Regression, which adds an L1 penalty and can set some coefficients to zero, simply change the `alpha`

parameter to 1:

# Install the glmnet package if not already installed install.packages("glmnet") # Load the glmnet package library(glmnet) # Create predictor matrix with an intercept term x_matrix <- model.matrix(~ df$x) # Create response vector y_vector <- as.numeric(df$y) # Fit Lasso penalized regression with L1 penalty (alpha = 1) model_lasso <- cv.glmnet(x_matrix, y_vector, alpha = 1, family = "binomial") # Display model coefficients coef(model_lasso)

**Output:**

3 x 1 sparse Matrix of class "dgCMatrix" s1 (Intercept) -16.80925 (Intercept) . df$x 16.39463

### Conclusion

The warning “glm.fit: algorithm did not converge” is common when fitting GLMs in R, especially with challenging datasets. By addressing issues like perfect separation, scaling predictors, increasing iterations, or using regularization, you can often resolve the issue and obtain a well-fitting model.

For further reading on R-related errors, go to the articles:

- How to Solve R Error in sort.int(x, na.last = na.last, decreasing = decreasing, …) : ‘x’ must be atomic
- How to Solve R Error: Arguments imply differing number of rows
- How to Solve R Error: number of levels of each grouping factor must be < number of observations

Go to the online courses page on R to learn more about coding in R for data science and machine learning.

Have fun and happy researching!

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.