Regression Analysis

What Regression Analysis Helps You Do

Regression analysis offers several advantages, especially for beginners who want to make sense of data:

• It helps you forecast what might happen in the future based on past information. For example, businesses can predict sales based on marketing spend.
• It shows whether two factors are strongly connected, weakly connected, or not connected at all.
• Regression uncovers hidden trends. For example, seasonal shifts in customer behaviour or patterns in patient vitals.
• Whether you are a researcher, healthcare provider, or business owner, regression gives you solid evidence to make smart, confident decisions.

Key Terms You Must Know First

Before running a regression test, it is important to understand a few basic terms:

Types Of Regression Analysis

Each type of regression analysis helps you understand different kinds of relationships in your data.

Simple Linear Regression

Simple linear regression is the easiest form of regression. It uses one independent variable (predictor) to explain or predict a dependent variable.

Example: Hours studied → Exam score

If you want to know whether studying more leads to higher marks, simple linear regression can show that relationship and predict expected scores.

Use it when:

• You want to test or predict the effect of one factor.
• The relationship looks like a straight line.

Multiple Linear Regression

Multiple linear regression uses two or more predictors to explain the outcome. This gives a more realistic and accurate picture, especially when real-life situations involve many factors.

Example: Exam score → hours studied + sleep hours + attendance

Use it when:

• Many independent variables affect your dependent variable.
• You want to control for other factors.
• You want better prediction accuracy.

Logistic Regression

Logistic regression is used when your outcome is categorical, not numerical.
Instead of predicting a number, it predicts probabilities.

Examples:

• Will a patient be readmitted? (yes/no)
• Will a customer click the ad? (click/no click)
• Will a loan get approved? (approved/rejected)

Use it when:

• Your dependent variable has categories (binary or multi-class).
• You need classification instead of prediction.

Polynomial Regression

Polynomial regression is used when the relationship between variables is curved, not straight.

If the effect increases at first, slows down later, or changes direction, a straight line won’t fit well, but a curve will.

Use cases:

• Growth patterns (children’s height, plant growth)
• Sales trends over long periods
• Complex scientific or medical relationships
• When data clearly shows a bend or curve

Other Variants 

These are advanced forms of regression, often used in research, machine learning, and data science:

✔ Ridge Regression

Handles multicollinearity by adding a penalty to large coefficients.

✔ Lasso Regression

Can shrink some coefficients to zero, helping with variable selection.

✔ Elastic Net

Combines Ridge + Lasso strengths.

✔ Stepwise Regression

Automatically adds or removes predictors to find the best model.

✔ Multivariate Regression

Used when there are multiple dependent variables instead of just one.

Assumptions Of Regression Analysis

To get accurate and trustworthy results, regression analysis relies on a few key assumptions. These assumptions make sure your results are valid.

Linearity

The relationship between the independent and dependent variable should be a straight line. If the relationship is curved, simple linear regression will not work well.

Independence of Errors

The errors (residuals) should be independent of each other. This means one error should not influence another.

Why it matters: If errors are related, your predictions may be biased (example: time-series data with trends).

Homoscedasticity

This means the spread of residuals should be consistent across all values of the independent variable.

In simple terms:

• The variance of errors should stay the same.
• If errors get bigger at higher values, your model becomes unreliable.

Normality of Residuals

Residuals should follow a normal distribution.
This helps your regression coefficients and p-values remain accurate.

How to check:

• Histogram
• Q-Q plot
• Shapiro-Wilk test

No Multicollinearity

Multicollinearity happens when two predictors are highly correlated with each other.
This makes it hard to know which variable is actually influencing the outcome.

Why it matters:

• It inflates standard errors
• It makes coefficients unstable
• It weakens model reliability

How to detect: VIF (Variance Inflation Factor)

How To Perform Regression Analysis

Running a regression analysis becomes much easier when you break it down into clear steps. 

Step 1: Define Your Research Question

Start by asking what you want to find out. For example:

• Does marketing spend affect sales?
• Do hours of sleep influence productivity?
• Which factors predict patient recovery time?

Step 2: Choose Your Variables

You need two types of variables:

• Dependent Variable (Outcome): The variable you want to predict or explain.
• Independent Variables (Predictors): Factors that influence the dependent variable.

Example: If your question is “Does exercise affect weight loss?”

• Dependent variable: weight loss
• Independent variable: hours of exercise per week

Step 3: Collect and Clean Your Data

Good data leads to good results. Make sure your dataset is:

• Complete (no major missing values)
• Clean (correct formats, no duplicates)
• Accurate (no outliers unless justified)
• Suitable for regression (numeric values for predictors and outcomes)

Step 4: Check Assumptions

Before running regression, ensure that your data meets key assumptions:

• Linearity
• Independence of errors
• Homoscedasticity
• Normal distribution of residuals
• No multicollinearity

Step 5: Run the Regression (SPSS, R, Python, Excel)

You can run regression using many tools:

Step 6: Interpret the Results

Interpretation helps you understand what your numbers actually mean. Key elements to interpret:

• Coefficients: Tell you how much the dependent variable changes when the predictor changes.
• P-values: Show whether the relationship is statistically significant.
• R-squared: Explains how much of the outcome is predicted by your model.
• Standard error & confidence intervals: Show how stable and reliable your estimates are.
• F-statistic: Shows whether your overall model is significant.

Step 7: Validate the Model

Model validation checks whether your regression works well on new data.

How To Interpret Regression Output

Once you run a regression, you will see a table full of numbers with coefficients, p-values, R², and more. Below is a breakdown of each key output.

Coefficients (β values)

Coefficients show how much the dependent variable changes when one independent variable increases by one unit, while keeping all other variables constant.

How to interpret a coefficient

• Positive coefficient: the dependent variable increases
• Negative coefficient: the dependent variable decreases
• Zero or very small coefficient: little or no relationship

Example: If β = 2.5 for hours studied, it means: 

For every additional hour studied, the exam score increases by 2.5 points (on average).

P-values

P-values show whether a predictor has a statistically significant effect on the outcome.

How to interpret p-values

• p < 0.05 → statistically significant
• p ≥ 0.05 → not statistically significant

This means:

• If p < 0.05, the predictor meaningfully contributes to the model.
• If p ≥ 0.05, the predictor likely has little or no effect.

Example: If “sleep hours” has p = 0.002, it significantly affects the outcome. If “coffee intake” has p = 0.45, it does not significantly affect the outcome.

R-squared & Adjusted R-squared

These values tell you how well your model explains the variation in your dependent variable.

R-squared (R²)

Shows the percentage of variance explained by your predictors.

Example: R² = 0.70 → your model explains 70% of the variation.

Adjusted R-squared

More reliable for multiple regression. It adjusts for the number of variables and penalises unnecessary predictors. Use it when:

• You have more than one independent variable
• You want a realistic measure of model performance

Standard Error

Standard error shows how accurately the coefficient is estimated.

If the standard error is large compared to the coefficient, you may need:

• More data
• Fewer predictors
• Better model specification

Confidence Intervals

Confidence intervals (often 95%) show the range where the true coefficient value is likely to fall.

How to interpret

If the CI does not include zero, the variable is usually significant. If the CI includes zero, the effect may be weak or questionable.

Example: Coefficient for exercise = 1.2

CI = [0.5, 1.8] → does not include zero → significant effect.

F-statistic

The F-statistic tells you whether your entire model is statistically significant.

Frequently Asked Questions