Diagnosing residual plots in linear regression models

tavish
tavish 25 Jun, 2020
5 min read

My first analytics project involved predicting business from each sales agent and coming up with a targeted intervention for each agent.

I built my first linear regression model after devoting a good amount of time on data cleaning and variable preparation. Now was the time to access the predictive power of the model. I got a MAPE of 5%, Gini coefficient of 82% and a high R-square. Gini and MAPE are metrics to gauge the predictive power of linear regression model. Such Gini coefficient and MAPE for an insurance industry sales prediction are considered to be way better than average. To validate the overall prediction we found the aggregate business in an out of time sample. I was shocked to see that the total expected business was not even 80% of the actual business. With such high lift and concordant ratio, I failed to understand what was going wrong. I decided to read more on statistical details of the model. With a better understanding of the model, I started analyzing the model on different dimensions. After a close examination of residual plots, I found that one of the predictor variables had a square relationship with the output variable.

3d doctor

Since then, I validate all the assumptions of the model even before reading the predictive power of the model. This article will take you through all the assumptions in a linear regression and how to validate assumptions and diagnose relationship using residual plots.

[stextbox id=”section”]Assumptions of Linear Regression Model :[/stextbox]

There are number of assumptions of a linear regression model. In modeling, we normally check for five of the assumptions. These are as follows :

1. Relationship between the outcomes and the predictors is linear.
2. Error term  has mean almost equal to zero for each value of outcome.
3. Error term  has constant variance.
4. Errors are uncorrelated.
5. Errors are normally distributed or we have an adequate sample size to rely on large sample theory.

The point to be noted here is that none of these assumptions can be validated by R-square chart, F-statistics or any other model accuracy plots. On the other hand, if any of the assumptions are violated, chances are high that accuracy plot can give misleading results.

 

[stextbox id=”section”]How to use residual for diagnostics :[/stextbox]

Residual analysis is usually done graphically. Following are the two category of graphs we normally look at:

1. Quantile plots : This type of is to assess whether the distribution of the residual is normal or not. The graph is between the actual distribution of residual quantiles and a perfectly normal distribution residuals. If the graph is perfectly overlaying on the diagonal, the residual is normally distributed. Following is an illustrative graph of approximate normally distributed residual.Quantile_plot

Let’s try to visualize a quantile plot of a biased residual distribution.

skewed

In the graph above, we see the assumption of the residual normal distribution being clearly violated.

2. Scatter plots:  This type of graph is used to assess model assumptions, such as constant variance and linearity, and to identify potential outliers. Following is a scatter plot of perfect residual distribution

Scatter_plot

Let’s try to visualize a scatter plot of residual distribution which has unequal variance.

Scatter_plot_2

In the graph above, we see the assumption of the residual normal distribution being clearly violated.

[stextbox id=”section”]Example :[/stextbox]

For simplicity, I have taken an example of single variable regression model to analyze residual curves. Similar kind of approach is followed for multi-variable as well.

Say, the actual relation of the predictor and the output variable is as follows:

Actual eq

Ignorant of the type of relationship, we start the analysis with the following equation.

Example2

Can we diagnose this misfit using residual curves?

After making a comprehensive model, we check all the diagnostic curves. Following is the Q-Q plot for the residual of the final linear equation.

Example3

Q-Q plot looks slightly deviated from the baseline, but on both the sides of the baseline. This indicated residuals are distributed approximately in a normal fashion.

Following is the scatter plot of the residual :

Example4

Clearly, we see the mean of residual not restricting its value at zero. We also see a parabolic trend of the residual mean. This indicates the predictor variable is also present in squared form. Now, let’s modify the initial equation to  the following equation :

Example5

Following is the new scatter plot for the residual of the new equation :

Example6

We now clearly see a random distribution and a approximate zero residual mean.

[stextbox id=”section”]End Notes:[/stextbox]

Every linear regression model should be validated on all the residual plots . Such regression plots directionaly guides us to the right form of equations to start with. You might also be interested in the previous article on regression ( https://www.analyticsvidhya.com/blog/2013/10/trick-enhance-power-regression-model-2/ )

Do you think this provides a solution to any problem you face? Are there any other techniques you use to detect the right form of relationship between predictor and output variables ? Do let us know your thoughts in the comments below.

If you like what you just read & want to continue your analytics learning, subscribe to our emails or like our facebook page.

Business Analyticslinear regressionPredictive modelingregressionResidualresidual plotsSASvalidation
tavish
tavish 25 Jun, 2020

Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naïve Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Proven Ways for improving the “Accuracyâ€_x009d_ of a Machine Learning Model
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

Frequently Asked Questions

Lorem ipsum dolor sit amet, consectetur adipiscing elit,

Responses From Readers

Clear

Tuhin Chattopadhyay
Tuhin Chattopadhyay 02 Dec, 2013

Good one Kunal, though I prefer to use designated tests to verify each of the assumptions.

123

Tavish Srivastava
Tavish Srivastava 02 Dec, 2013

Tuhin, I agree doing designated test to verify each assumption is a more sturdy approach. However, using the approach mentioned in the article will tell you two facts. First, whether assumption is validated or not. Second, if assumptions are not validated, what will be the best starting point for the next iteration. For instance, in the example mentioned in the article, residual plot gave us a better starting equation. Please share your thoughts of how will designated tests help on the second point. Thanks, Tavish

123
Kunal Jain
Kunal Jain 02 Dec, 2013

Tuhin, While the designated tests would provide you the output, they do not provide as much details about why things have failed. Also, even if you use designated tests, it makes sense to understand what has failed. Thanks, Kunal

123
Tuhin Chattopadhyay
Tuhin Chattopadhyay 02 Dec, 2013

Sorry, it's by Tavish. Liked it Tavish.

123

Deepak Rai
Deepak Rai 13 Dec, 2013

Hi, How to we check whether Error Terms are independent to each other i.e. they are not correlated with one another (Assumption of Independence of the Error Terms) and Co-linearity of the independent variables? Regards, Deepak Rai

123

Tavish
Tavish 09 Jan, 2015

Deepak, This will be visible from your residual scatter plot with independent variables. You will see that your error will show you autocorrelation i.e. you will that a positive error tend to follow a positive error and vice versa. Hope this helps. Tavish

123

Write for us

Write, captivate, and earn accolades and rewards for your work
  • Reach a Global Audience
  • Get Expert Feedback
  • Build Your Brand & Audience
  • Cash In on Your Knowledge
  • Join a Thriving Community
  • Level Up Your Data Science Game
imag
Rahul Shah img

Rahul Shah

27
Sion Chakrabarti img

Sion Chakrabarti

16
CHIRAG GOYAL img

CHIRAG GOYAL

87
Barney Darlington img

Barney Darlington

5
Suvojit Hore img

Suvojit Hore

9
Arnab Mondal img

Arnab Mondal

15
Prateek Majumder img

Prateek Majumder

68