Logistic regression vs linear regression: When to use which approach

Data analysts often need to decide between logistic regression vs linear regression. Both linear regression and logistic regression are fundamental techniques, but there are numerous differences between them. Choosing the wrong one can waste time and produce misleading results.

The decision comes down to what you are trying to predict. Linear regression predicts continuous values, such as sales revenue or temperature. In contrast, logistic regression predicts probabilities and categories, such as whether an email is spam or if a customer will churn.

This post describes how to choose the right approach and implement it effectively.

What each method actually does

Linear regression: Predicts continuous outcomes

Linear regression finds the straight line that best fits your data points. It assumes your outcome variable can take any numerical value within a range. The algorithm finds coefficients that minimize the difference between predicted and actual values using ordinary least squares (OLS). The relationship follows the prediction equation y = β₀ + β₁x₁ + β₂x₂ + ... + βₙxₙ

Examples of when to use linear regression:

• Predicting house prices based on square footage, location, and amenities
• Forecasting sales revenue from advertising spend and seasonal factors
• Estimating insurance claims costs using driver age, vehicle type, and accident history
• Projecting energy consumption from weather patterns and historical usage

Logistic regression: Predicts probabilities and categories

Logistic regression uses “log-odds,” which is a way to express probabilities that make the math work cleanly. Odds compare the likelihood of something happening versus not happening. If there's a 75% chance of rain, the odds are 3:1 (75% vs 25%). Log-odds take the natural logarithm of those odds. For example, a 50% probability equals odds of 1:1, which becomes log-odds of 0. Probabilities above 50% create positive log-odds, while probabilities below 50% create negative log-odds. The beauty of log-odds is that they can range from negative infinity to positive infinity, giving logistic regression the mathematical space it needs to work with linear relationships before converting back to probabilities.

Logistic regression uses the sigmoid function to transform the continuous linear relationships to category probabilities between 0 and 1. Despite its name, it is a classification technique. The algorithm applies maximum likelihood estimation to find coefficients that best separate categories. The formula for logistic regression uses the logistic function: p = 1/(1 + e^-(β₀ + β₁x₁ + β₂x₂ + ... + βₙxₙ)).

Examples of when to use logistic regression:

• Determining fraud probability in financial transactions
• Predicting customer churn based on usage patterns and support interactions
• Diagnosing disease likelihood from patient symptoms and test results
• Classifying emails as spam or legitimate based on content and sender patterns

Key differences that affect your choice

Output type and interpretation

Linear regression produces specific numerical predictions. For example, if your model predicts a house will sell for $347,500, that's a point estimate you can use for pricing decisions.

Logistic regression produces probabilities. If your model gives a 0.73 probability that a transaction is fraudulent, you need to set a threshold (typically 0.5) to make binary decisions.

Mathematical foundations and assumptions

Linear regression assumes:

• A linear relationship between the variables
• Normally distributed residuals
• Constant variance across all prediction levels
• Independent observations

Logistic regression assumes:

• Linear relationship between variables and the log-odds
• Independent observations
• Large sample size for stable results
• No extreme outliers that distort coefficient estimates

Important clarification: The "linear relationship" in logistic regression can be confusing because we see the characteristic S-shaped sigmoid curve when plotting probabilities. However, logistic regression assumes linearity in the log-odds space, not in the probability space. Think of it as a two-step process: first, the model creates a straight-line relationship between the predictors and the log-odds (just like linear regression), then it transforms those log-odds through the sigmoid function to produce probabilities between 0 and 1. This transformation creates the curved probability relationship we observe.

Performance characteristics

Speed: Linear regression is computationally faster, especially with large datasets. Logistic regression requires iterative optimization, making it slower for complex models.

Interpretability: Both provide interpretable coefficients, but linear regression coefficients represent direct unit changes. Logistic regression coefficients represent changes in log-odds, requiring transformation for intuitive interpretation.

Scalability: Linear regression scales better with feature count and data volume. Logistic regression becomes computationally expensive with many features without regularization.

Real-world decision framework

The problem statement test

Look at your problem description:

• If you see words like "predict," "estimate," or "forecast" with numerical targets, then use linear regression.
• If you see "classify," "determine probability," or binary outcomes, then use logistic regression.

The outcome variable check

• Continuous outcomes (infinite possible values): temperature, price, duration, count data indicate to use linear regression.
• Categorical outcomes (limited discrete values): yes/no, approved/denied, high/medium/low risk indicate to use logistic regression.

The business context evaluation

• Regulatory requirements: In healthcare, finance, and legal contexts, you often need probability estimates with confidence intervals. Logistic regression provides these naturally.
• Decision-making needs: If stakeholders need specific numerical forecasts for budgeting or planning, linear regression gives direct estimates. If they need risk assessment or classification, logistic regression fits better.

Common mistakes and how to avoid them

1. Using linear regression for probabilities

Never use linear regression to predict probabilities directly. It can produce values outside the 0-1 range, making results meaningless.

2. Ignoring assumption violations

For linear regression: Check residual plots for patterns. If you see curved relationships or changing variance, consider transforming variables or using polynomial terms.

For logistic regression: Verify adequate sample sizes using the "10 events per variable" rule. You need at least 10 occurrences of your outcome (like 10 customers who bought) for each predictor variable in your model. So if you're predicting purchases using 4 variables (age, income, time on site, previous purchases), you need at least 40 customers who actually bought. Some statisticians recommend 15-20 events per variable for more reliable results.

Also, check for outliers that might skew results, as extreme values can disproportionately influence the model's decision boundaries. Extreme values, often called high-leverage points, can pull the decision boundary towards them, disproportionately influencing the model's coefficients and potentially misrepresenting the true relationship for the rest of the data.

3. Misinterpreting coefficients

Linear regression coefficients show direct impact, such as "Each additional square foot increases house price by $150."

Logistic regression coefficients require transformation to make business sense. The raw coefficient represents a change in log-odds, which is not intuitive. To interpret it, you raise the mathematical constant e (approximately 2.718) to the power of your coefficient to get an odds ratio. This is called “exponentiation.” For example, a coefficient of 0.5 means e^0.5 ≈ 1.65, which indicates the odds are 1.65 times higher (or 65% higher) when that variable increases by one unit.

In other words, each additional year of age multiplies the odds of buying by 1.65. This means the odds increase by 65%, but the effect on probability is not a simple 65% increase because it depends on the baseline probability. In Example 1 below, if someone starts with a 10% probability of buying, increasing their odds by 65% might raise their probability to around 16%. In Example 2, if they start with a 50% probability, the same odds increase might raise it to around 62%. Coefficients above 0 increase odds, coefficients below 0 decrease odds, and a coefficient of exactly 0 means no effect on the outcome.

Example 1: A 10% probability equals odds of 0.1 / 0.9 = 0.111. Multiplying by 1.65 gives new odds of 0.183. Converting back to probability gives 0.183 / (1 + 0.183) = 15.5%, which is "around 16%."

Example 2: A 50% probability equals odds of 0.5 / 0.5 = 1.0. Multiplying by 1.65 gives new odds of 1.65. Converting back to probability gives 1.65 / (1 + 1.65) = 62.3%, which is "around 62%."

Advanced considerations

When simple approaches aren't enough

Non-linear relationships: Both techniques assume linear relationships. For curved patterns, consider polynomial terms, splines, or tree-based methods.

Multiple categories: Standard logistic regression handles binary outcomes. For multiple categories, use multinomial logistic regression or one-vs-rest approaches.

Mixed data types: When you have both continuous predictions and categorical classifications, consider ensemble methods or separate models for different aspects.

Model evaluation strategies

Linear regression: Use R-squared, mean absolute error (MAE), and root-mean-square error (RMSE). Check residual plots for assumption violations.

Logistic regression: Use accuracy, precision, recall, and AUC-ROC. Examine confusion matrices to understand where the model makes classification errors.

Understanding ROC curves and AUC scores: ROC stands for "Receiver Operating Characteristic," a graph that evaluates how well your logistic regression model performs across different probability thresholds. Since logistic regression produces probabilities (like a 0.73 chance of customer conversion), you must choose a cutoff point to make binary decisions. The cutoff is typically 0.5, but this might not be optimal for your business needs.

The ROC curve plots the True Positive Rate (how often you correctly identify conversions) against the False Positive Rate (how often you incorrectly predict conversions that don't happen) as you vary this threshold from 0 to 1. A perfect model would create a curve that hugs the top-left corner, catching all positive cases with no false alarms. A useless model follows the diagonal line, performing no better than random guessing.

The AUC (Area Under the Curve) score summarizes the entire ROC curve into a single number: 0.5 indicates random performance, 1.0 represents perfect prediction, and scores above 0.7 generally indicate good model performance. Business teams use ROC analysis to select optimal thresholds based on the relative costs of missing true positives versus accepting false positives.

How Quadratic enhances both approaches

Traditional linear regression Excel workflows require manual formula construction, assumption checking, and result interpretation. Standard logistic regression in Excel implementations struggle with iterative optimization algorithms and probability calculations. In contrast, Quadratic's AI-powered environment eliminates many of the manual actions common in implementation.

Exploring your data: The embedded AI can analyze your data and determine what questions to ask about it. The AI can find hidden patterns not obvious to human eyes.

Method selection: Describe your problem in natural language, and Quadratic suggests whether linear regression vs logistic regression fits better based on your data structure and goals. It can then write the Python to do the correct analysis.

Assumption checking: Ask the AI to test assumptions and flag violations, with suggestions for addressing problems like multicollinearity or heteroscedasticity.

Real-time interpretation: As you work, you can ask the embedded AI to give you plain-language explanations of coefficients, confidence intervals, and prediction qualities.

Integrated workflows: Connect data preparation, model building, and results presentation in one environment, with AI assistance at each step.

When you are ready to do a regression analysis

Start with these questions:

1. What type of output does your business problem require?
2. What assumptions does your data satisfy?
3. How will stakeholders use the predictions?
4. What performance trade-offs matter most?

Most analysts develop intuition through practice, but these frameworks provide reliable guidance when facing new problem types. The difference between logistic and linear regression comes down to solving different types of problems. The choice depends on your outcome variable type, assumption compliance, and business application requirements.