how to write regression hypothesis

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

• Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
• Duis aute irure dolor in reprehenderit in voluptate
• Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

• At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)). 
• The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
• The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

• Prompt Library
• DS/AI Trends
• Stats Tools
• Interview Questions
• Generative AI
• Machine Learning
• Deep Learning

Linear regression hypothesis testing: Concepts, Examples

In relation to machine learning , linear regression is defined as a predictive modeling technique that allows us to build a model which can help predict continuous response variables as a function of a linear combination of explanatory or predictor variables. While training linear regression models, we need to rely on hypothesis testing in relation to determining the relationship between the response and predictor variables. In the case of the linear regression model, two types of hypothesis testing are done. They are T-tests and F-tests . In other words, there are two types of statistics that are used to assess whether linear regression models exist representing response and predictor variables. They are t-statistics and f-statistics. As data scientists , it is of utmost importance to determine if linear regression is the correct choice of model for our particular problem and this can be done by performing hypothesis testing related to linear regression response and predictor variables. Many times, it is found that these concepts are not very clear with a lot many data scientists. In this blog post, we will discuss linear regression and hypothesis testing related to t-statistics and f-statistics . We will also provide an example to help illustrate how these concepts work.

Table of Contents

What are linear regression models?

A linear regression model can be defined as the function approximation that represents a continuous response variable as a function of one or more predictor variables. While building a linear regression model, the goal is to identify a linear equation that best predicts or models the relationship between the response or dependent variable and one or more predictor or independent variables.

There are two different kinds of linear regression models. They are as follows:

• Simple or Univariate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and one predictor or independent variable. The form of the equation that represents a simple linear regression model is Y=mX+b, where m is the coefficients of the predictor variable and b is bias. When considering the linear regression line, m represents the slope and b represents the intercept.
• Multiple or Multi-variate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and more than one predictor or independent variable. The form of the equation that represents a multiple linear regression model is Y=b0+b1X1+ b2X2 + … + bnXn, where bi represents the coefficients of the ith predictor variable. In this type of linear regression model, each predictor variable has its own coefficient that is used to calculate the predicted value of the response variable.

While training linear regression models, the requirement is to determine the coefficients which can result in the best-fitted linear regression line. The learning algorithm used to find the most appropriate coefficients is known as least squares regression . In the least-squares regression method, the coefficients are calculated using the least-squares error function. The main objective of this method is to minimize or reduce the sum of squared residuals between actual and predicted response values. The sum of squared residuals is also called the residual sum of squares (RSS). The outcome of executing the least-squares regression method is coefficients that minimize the linear regression cost function .

The residual e of the ith observation is represented as the following where [latex]Y_i[/latex] is the ith observation and [latex]\hat{Y_i}[/latex] is the prediction for ith observation or the value of response variable for ith observation.

[latex]e_i = Y_i – \hat{Y_i}[/latex]

The residual sum of squares can be represented as the following:

[latex]RSS = e_1^2 + e_2^2 + e_3^2 + … + e_n^2[/latex]

The least-squares method represents the algorithm that minimizes the above term, RSS.

Once the coefficients are determined, can it be claimed that these coefficients are the most appropriate ones for linear regression? The answer is no. After all, the coefficients are only the estimates and thus, there will be standard errors associated with each of the coefficients.  Recall that the standard error is used to calculate the confidence interval in which the mean value of the population parameter would exist. In other words, it represents the error of estimating a population parameter based on the sample data. The value of the standard error is calculated as the standard deviation of the sample divided by the square root of the sample size. The formula below represents the standard error of a mean.

[latex]SE(\mu) = \frac{\sigma}{\sqrt(N)}[/latex]

Thus, without analyzing aspects such as the standard error associated with the coefficients, it cannot be claimed that the linear regression coefficients are the most suitable ones without performing hypothesis testing. This is where hypothesis testing is needed . Before we get into why we need hypothesis testing with the linear regression model, let’s briefly learn about what is hypothesis testing?

Train a Multiple Linear Regression Model using R

Before getting into understanding the hypothesis testing concepts in relation to the linear regression model, let’s train a multi-variate or multiple linear regression model and print the summary output of the model which will be referred to, in the next section. 

The data used for creating a multi-linear regression model is BostonHousing which can be loaded in RStudioby installing mlbench package. The code is shown below:

install.packages(“mlbench”) library(mlbench) data(“BostonHousing”)

Once the data is loaded, the code shown below can be used to create the linear regression model.

attach(BostonHousing) BostonHousing.lm <- lm(log(medv) ~ crim + chas + rad + lstat) summary(BostonHousing.lm)

Executing the above command will result in the creation of a linear regression model with the response variable as medv and predictor variables as crim, chas, rad, and lstat. The following represents the details related to the response and predictor variables:

• log(medv) : Log of the median value of owner-occupied homes in USD 1000’s
• crim : Per capita crime rate by town
• chas : Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
• rad : Index of accessibility to radial highways
• lstat : Percentage of the lower status of the population

The following will be the output of the summary command that prints the details relating to the model including hypothesis testing details for coefficients (t-statistics) and the model as a whole (f-statistics) 

Hypothesis tests & Linear Regression Models

Hypothesis tests are the statistical procedure that is used to test a claim or assumption about the underlying distribution of a population based on the sample data. Here are key steps of doing hypothesis tests with linear regression models:

• Hypothesis formulation for T-tests: In the case of linear regression, the claim is made that there exists a relationship between response and predictor variables, and the claim is represented using the non-zero value of coefficients of predictor variables in the linear equation or regression model. This is formulated as an alternate hypothesis. Thus, the null hypothesis is set that there is no relationship between response and the predictor variables . Hence, the coefficients related to each of the predictor variables is equal to zero (0). So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis for each test states that a1 = 0, a2 = 0, a3 = 0 etc. For all the predictor variables, individual hypothesis testing is done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. Thus, if there are, say, 5 features, there will be five hypothesis tests and each will have an associated null and alternate hypothesis.
• Hypothesis formulation for F-test : In addition, there is a hypothesis test done around the claim that there is a linear regression model representing the response variable and all the predictor variables. The null hypothesis is that the linear regression model does not exist . This essentially means that the value of all the coefficients is equal to zero. So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis states that a1 = a2 = a3 = 0.
• F-statistics for testing hypothesis for linear regression model : F-test is used to test the null hypothesis that a linear regression model does not exist, representing the relationship between the response variable y and the predictor variables x1, x2, x3, x4 and x5. The null hypothesis can also be represented as x1 = x2 = x3 = x4 = x5 = 0. F-statistics is calculated as a function of sum of squares residuals for restricted regression (representing linear regression model with only intercept or bias and all the values of coefficients as zero) and sum of squares residuals for unrestricted regression (representing linear regression model). In the above diagram, note the value of f-statistics as 15.66 against the degrees of freedom as 5 and 194. 
• Evaluate t-statistics against the critical value/region : After calculating the value of t-statistics for each coefficient, it is now time to make a decision about whether to accept or reject the null hypothesis. In order for this decision to be made, one needs to set a significance level, which is also known as the alpha level. The significance level of 0.05 is usually set for rejecting the null hypothesis or otherwise. If the value of t-statistics fall in the critical region, the null hypothesis is rejected. Or, if the p-value comes out to be less than 0.05, the null hypothesis is rejected.
• Evaluate f-statistics against the critical value/region : The value of F-statistics and the p-value is evaluated for testing the null hypothesis that the linear regression model representing response and predictor variables does not exist. If the value of f-statistics is more than the critical value at the level of significance as 0.05, the null hypothesis is rejected. This means that the linear model exists with at least one valid coefficients. 
• Draw conclusions : The final step of hypothesis testing is to draw a conclusion by interpreting the results in terms of the original claim or hypothesis. If the null hypothesis of one or more predictor variables is rejected, it represents the fact that the relationship between the response and the predictor variable is not statistically significant based on the evidence or the sample data we used for training the model. Similarly, if the f-statistics value lies in the critical region and the value of the p-value is less than the alpha value usually set as 0.05, one can say that there exists a linear regression model.

Why hypothesis tests for linear regression models?

The reasons why we need to do hypothesis tests in case of a linear regression model are following:

• By creating the model, we are establishing a new truth (claims) about the relationship between response or dependent variable with one or more predictor or independent variables. In order to justify the truth, there are needed one or more tests. These tests can be termed as an act of testing the claim (or new truth) or in other words, hypothesis tests.
• One kind of test is required to test the relationship between response and each of the predictor variables (hence, T-tests)
• Another kind of test is required to test the linear regression model representation as a whole. This is called F-test.

While training linear regression models, hypothesis testing is done to determine whether the relationship between the response and each of the predictor variables is statistically significant or otherwise. The coefficients related to each of the predictor variables is determined. Then, individual hypothesis tests are done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. If at least one of the null hypotheses is rejected, it represents the fact that there exists no relationship between response and that particular predictor variable. T-statistics is used for performing the hypothesis testing because the standard deviation of the sampling distribution is unknown. The value of t-statistics is compared with the critical value from the t-distribution table in order to make a decision about whether to accept or reject the null hypothesis regarding the relationship between the response and predictor variables. If the value falls in the critical region, then the null hypothesis is rejected which means that there is no relationship between response and that predictor variable. In addition to T-tests, F-test is performed to test the null hypothesis that the linear regression model does not exist and that the value of all the coefficients is zero (0). Learn more about the linear regression and t-test in this blog – Linear regression t-test: formula, example .

Recent Posts

• Decision Tree Regression vs Linear Regression: Differences - August 13, 2024
• Parametric vs Non-Parametric Models: Differences, Examples - August 11, 2024
• How to know if Linear Regression Model is Appropriate? - August 11, 2024

Ajitesh Kumar

One response.

Very informative

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

• Search for:
• Excellence Awaits: IITs, NITs & IIITs Journey

ChatGPT Prompts (250+)

• Generate Design Ideas for App
• Expand Feature Set of App
• Create a User Journey Map for App
• Generate Visual Design Ideas for App
• Generate a List of Competitors for App
• Decision Tree Regression vs Linear Regression: Differences
• Parametric vs Non-Parametric Models: Differences, Examples
• How to know if Linear Regression Model is Appropriate?
• Lasso Regression in Machine Learning: Python Example
• Completion Model vs Chat Model: Python Examples

Data Science / AI Trends

• • Prepend any arxiv.org link with talk2 to load the paper into a responsive chat application
• • Custom LLM and AI Agents (RAG) On Structured + Unstructured Data - AI Brain For Your Organization
• • Guides, papers, lecture, notebooks and resources for prompt engineering
• • Common tricks to make LLMs efficient and stable
• • Machine learning in finance

Free Online Tools

• Create Scatter Plots Online for your Excel Data
• Histogram / Frequency Distribution Creation Tool
• Online Pie Chart Maker Tool
• Z-test vs T-test Decision Tool
• Independent samples t-test calculator

Recent Comments

I found it very helpful. However the differences are not too understandable for me

Very Nice Explaination. Thankyiu very much,

in your case E respresent Member or Oraganization which include on e or more peers?

Such a informative post. Keep it up

Thank you....for your support. you given a good solution for me.

Linear regression - Hypothesis testing

by Marco Taboga , PhD

This lecture discusses how to perform tests of hypotheses about the coefficients of a linear regression model estimated by ordinary least squares (OLS).

Table of contents

Normal vs non-normal model

The linear regression model, matrix notation, tests of hypothesis in the normal linear regression model, test of a restriction on a single coefficient (t test), test of a set of linear restrictions (f test), tests based on maximum likelihood procedures (wald, lagrange multiplier, likelihood ratio), tests of hypothesis when the ols estimator is asymptotically normal, test of a restriction on a single coefficient (z test), test of a set of linear restrictions (chi-square test), learn more about regression analysis.

The lecture is divided in two parts:

in the first part, we discuss hypothesis testing in the normal linear regression model , in which the OLS estimator of the coefficients has a normal distribution conditional on the matrix of regressors;

in the second part, we show how to carry out hypothesis tests in linear regression analyses where the hypothesis of normality holds only in large samples (i.e., the OLS estimator can be proved to be asymptotically normal).

We also denote:

We now explain how to derive tests about the coefficients of the normal linear regression model.

It can be proved (see the lecture about the normal linear regression model ) that the assumption of conditional normality implies that:

How the acceptance region is determined depends not only on the desired size of the test , but also on whether the test is:

one-tailed (only one of the two things, i.e., either smaller or larger, is possible).

For more details on how to determine the acceptance region, see the glossary entry on critical values .

The F test is one-tailed .

A critical value in the right tail of the F distribution is chosen so as to achieve the desired size of the test.

Then, the null hypothesis is rejected if the F statistics is larger than the critical value.

In this section we explain how to perform hypothesis tests about the coefficients of a linear regression model when the OLS estimator is asymptotically normal.

As we have shown in the lecture on the properties of the OLS estimator , in several cases (i.e., under different sets of assumptions) it can be proved that:

These two properties are used to derive the asymptotic distribution of the test statistics used in hypothesis testing.

The test can be either one-tailed or two-tailed . The same comments made for the t-test apply here.

Like the F test, also the Chi-square test is usually one-tailed .

The desired size of the test is achieved by appropriately choosing a critical value in the right tail of the Chi-square distribution.

The null is rejected if the Chi-square statistics is larger than the critical value.

Want to learn more about regression analysis? Here are some suggestions:

R squared of a linear regression ;

Gauss-Markov theorem ;

Generalized Least Squares ;

Multicollinearity ;

Dummy variables ;

Selection of linear regression models

Partitioned regression ;

Ridge regression .

How to cite

Please cite as:

Taboga, Marco (2021). "Linear regression - Hypothesis testing", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/linear-regression-hypothesis-testing.

Most of the learning materials found on this website are now available in a traditional textbook format.

• F distribution
• Beta distribution
• Conditional probability
• Central Limit Theorem
• Binomial distribution
• Mean square convergence
• Delta method
• Almost sure convergence
• Mathematical tools
• Fundamentals of probability
• Probability distributions
• Asymptotic theory
• Fundamentals of statistics
• About Statlect
• Cookies, privacy and terms of use
• Loss function
• Almost sure
• Type I error
• Precision matrix
• Integrable variable
• To enhance your privacy,
• we removed the social buttons,
• but don't forget to share .

• Skip to secondary menu
• Skip to main content
• Skip to primary sidebar

Statistics By Jim

Making statistics intuitive

Regression Tutorial with Analysis Examples

By Jim Frost 84 Comments

If you’re learning regression analysis, you might want to bookmark this tutorial!

When to Use Regression and the Signs of a High-Quality Analysis

Before we get to the regression tutorials, I’ll cover several overarching issues.

Why use regression at all? What are common problems that trip up analysts? And, how do you differentiate a high-quality regression analysis from a less rigorous study? Read these posts to find out:

• When Should I Use Regression Analysis? : Learn what regression can do for you and when you should use it.
• Five Regression Tips for a Better Analysis : These tips help ensure that you perform a top-quality regression analysis.

Tutorial: Choosing the Right Type of Regression Analysis

There are many different types of regression analysis. Choosing the right procedure depends on your data and the nature of the relationships, as these posts explain.

• Choosing the Correct Type of Regression Analysis : Reviews different regression methods by focusing on data types.
• How to Choose Between Linear and Nonlinear Regression : Determining which one to use by assessing the statistical output.
• The Difference between Linear and Nonlinear Models : Both kinds of models can fit curves, so what’s the difference?

Tutorial: Specifying the Regression Model

Model specification is an iterative process. The interpretation and assumption confirmation sections of this tutorial explain how to assess your model and how to change the model based on the statistical output and graphs.

• Model Specification: Choosing the Correct Regression Model : I review standard statistical approaches, difficulties you may face, and offer some real-world advice.
• Using Data Mining to Select Your Regression Model Can Create Problems : This approach to choosing a model can produce misleading results. Learn how to detect and avoid this problem.
• Guide to Stepwise Regression and Best Subsets Regression : Two common tools for identifying candidate variables during the investigative stages of model building.
• Overfitting Regression Models : Overly complicated models can produce misleading R-squared values, regression coefficients, and p-values. Learn how to detect and avoid this problem.
• Curve Fitting Using Linear and Nonlinear Regression : When your data don’t follow a straight line, the model must fit the curvature. This post covers various methods for fitting curves.
• Understanding Interaction Effects : When the effect of one variable depends on the value of another variable, you need to include an interaction effect in your model otherwise the results will be misleading.
• When Do You Need to Standardize the Variables? : In specific situations, standardizing the independent variables can uncover statistically significant results.
• Confounding Variables and Omitted Variable Bias : The variables that you leave out of the model can bias the variables that you include.
• Proxy Variables: The Good Twin of Confounding Variables : Find ways to incorporate valuable information in your models and avoid confounders.

Tutorial: Interpreting Regression Results

After choosing the type of regression and specifying the model, you need to interpret the results. The next set of posts explain how to interpret the results for various regression analysis statistics:

• Coefficients and p-values
• Constant (Y-intercept)
• Comparing regression slopes and constants with hypothesis tests
• How high does R-squared need be?
• Interpreting a model with a low R-squared
• Adjusted R-squared and Predicted R-squared
• Standard error of the regression (S) vs. R-squared
• Five Reasons Your R-squared can be Too High : A high R-squared can occasionally signify a problem with your model.
• F-test of overall significance
• Identifying the Most Important Independent Variables : After settling on a model, analysts frequently ask, “Which variable is most important?”

Tutorial: Using Regression to Make Predictions

Analysts often use regression analysis to make predictions. In this section of the regression tutorial, learn how to make predictions and assess their precision.

• Making Predictions with Regression Analysis : This guide uses BMI to predict body fat percentage.
• Predicted R-squared : This statistic evaluates how well a model predicts the dependent variable for new observations.
• Understand Prediction Precision to Avoid Costly Mistakes : Research shows that presentation affects the number of interpretation mistakes. Covers prediction intervals.
• Prediction intervals versus other intervals : Prediction intervals indicate the precision of the predictions. I compare prediction intervals to different types of intervals.

Tutorial: Checking Regression Assumptions and Fixing Problems

• The Seven Classical Assumptions of OLS Linear Regression
• Residual plots : Shows what the graphs should look like and why they might not!
• Heteroscedasticity : The residuals should have a constant scatter (homoscedasticity). Shows how to detect this problem and various methods of fixing it.
• Multicollinearity : Highly correlated independent variables can be problematic, but not always! Explains how to identify this problem and several ways of resolving it.

Examples of Different Types of Regression Analyses

The last part of the regression tutorial contains regression analysis examples. Some of the examples are included in previous tutorial sections. Most of these regression examples include the datasets so you can try it yourself! Also, try using Excel to perform regression analysis with a step-by-step example!

• Linear regression with a double-log transformation : Models the relationship between mammal mass and metabolic rate using a fitted line plot.
• Understanding Historians’ Rankings of U.S. Presidents using Regression Models : Models rankings of U.S. Presidents to various predictors.
• Modeling the relationship between BMI and Body Fat Percentage with linear regression .
• Curve fitting with linear and nonlinear regression .

If you’re learning regression and like the approach I use in my blog, check out my eBook!

Share this:

Reader Interactions

April 10, 2024 at 5:29 pm

Hello Jim, Finding your site super-helpful. Wondering if you can provide some examples to simply illustrate multiple regression output (from spss). I would like to illustrate the overall effects of the independent variables on the dependent without creating a histogram. Ideally something that shows the strength, direction and significance in a box plot, line graph, bubble chart or other smart graphic. So appreciate your guidance.

January 2, 2023 at 10:02 am

Hi Jim, I just bought all 3 of your books via your Website Store that took me to Amazon – $66.72 plus tax. I don’t see how to get the PDF versions though without an additional cost to buy them additionally. Can you help me with how to get access to the PDF’s? Also, I am reviewing these to see if I want to add them to the courses I teach in Data Analytics. Do you have academic pricing available for students? Both hardcopy and e-copy?

January 3, 2023 at 12:17 am

Hi Anthony,

Look for an email from me.

Edited to add: I just sent an email to the address you used for the comment, but it bounced back saying it was “Blocked.” Please provide a method for me to contact you. You can use the contact form on my website and provide a different email address. Thanks!

July 18, 2022 at 7:15 am

Dear Dr, how are you? Thank you very much for your contribution. I have one question, which might not be related to this post in My cross tabulation between two categorical variables, the cell of one of the variables have just 8 observations for the ” no” , and 33 observations for the “yes” of the of the second variable. Can I continue with is for the descriptive statistics or should collapse the categories to increase the sample size? Do I use the new variable with a fewer categories in my regression analysis? your help is much appreciated

May 5, 2021 at 1:46 pm

Thanks for teaching us about Stats intuitively.

Is your book Regression Analysis available in PDF format? I’m a student learning Stats and would like it only in PDF format (no Kindle)

May 5, 2021 at 1:49 pm

Yes, if you buy it through My Website Store , you’ll get it in PDF format.

February 22, 2021 at 4:08 pm

Thnak you for your valuable advice. Change is in the way I am putting data into the software. When I put average data, glm output shows ingestion has no significant effect on mortality. When I input data with replications, glm out shows significant effect of ingestion on mortality. My standard deviations are large but data shows homogscedacity and normal distribution.

Your comments wil really be helpful in this regard.

February 22, 2021 at 4:11 pm

If you have replications, I’d enter that data to maintain the separate data points and NOT use the average. That provides the model with more information!

February 22, 2021 at 6:39 am

I have a question about Generaližed linear model. I am getting different outputs of same response variable when I apply glm using 1) data with replications 2) using average data. Mortality is My response variable and no. Of particles ingested is My predictor variable, other two predictors are categorical.

Looking forward to your advice.

February 22, 2021 at 3:44 pm

I’m not sure what you’re changing in your analysis to get the different outputs?

Replications are good because they help the model estimate pure error.

Average data is okay too but just be aware of incorporating that into the interpretations.

September 7, 2020 at 1:02 pm

I know that we can use linear or nonlinear models to fit a line to a dataset with curvature. My question is that when we have too many independent variable, how could we understand if there is a curvature?

Do you think we should start with simple linear regression, then model polynomial, Reciprocal, log, and nonlinear regression and compare the result for all of them to find which model works the best?

Thanks a lot for your very good and easy to understand book.

August 6, 2020 at 8:49 pm

I wonder if you have any recommended articles as to how to interpret the actual p-values and confidence interval for multiple regressions? I am struggling to find examples/templates of reporting these results.

I truly appreciate your help.

August 6, 2020 at 10:29 pm

I’ve written a post about interpreting p-value for regression coefficients , which I think would be helpful.

For confidence intervals of regression coefficients, think about sample means and CIs for means as a starting point. You can use the mean as the sample estimate of the population mean. However, because we’re working with a sample, we know there is a margin of error around that estimate. The CI captures that margin of error. If you have a CI for a sample mean, you know that the true population parameter is likely to be in that range.

In the regression context, the coefficient is also a mean. It’s a mean effect or the mean change in the dependent variable given a one-unit change in the independent variable. However, because we’re working with a sample, we know there is a margin of error around that mean effect. Consequently, with a CI for a regression coefficient, we know that the true mean effect of that coefficient is likely to fall within that coefficient CI.

I hope that helps!

June 28, 2020 at 2:57 am

Hi Jim first of all, thanks for all your great work. I’m setting linear regression analysis, in which the standard coefficient is considered, but the problem is my dependent variable that is Energy usage intensity so it means the lower value is the better than a higher value. correct me if I’m wrong, I think SPSS evaluates high value as the best and lower one as the worst so in my case, it could lead to effect reversely on the result (standard coefficient beta). is it right? and what is your suggestion?

June 23, 2020 at 3:51 am

hello Jim i want help with an econometric model or equation that can be used if there is one independent variable (dam) and dependent variable(5 livelihood outcomes). here i am confused if i can use binary regression model considering the 5 outcomes as a indicators of the dependent variable which is livelihood outcomes or i have to consider the 5 livelihood outcomes as 5 dependent variables and use multivariate regression.please reply a.a.p thank you so much

June 28, 2020 at 12:31 am

It really depends on the nature of the variables. I don’t know what you’re assessing, but here are two possibilities.

You label the independent variable, which I’m assuming is continuous but I don’t know for sure, and the 5 indicator/binary outcomes. This is appropriate if you think the IV affects, or at least predicts, those five indicators. Use this aproach if the goal of your analysis is to use the IV to predict the probability of those binary outcomes. Use binary logistic regression. You’ll need to run five different models. In each model, one of the binary outcomes/indicators is your DV and you’d use the same IV for each model. This type of model allows you to use the value of the IV to predict the probability of the binary outcome.

However, if you instead want to use the binary indicators to predict the continuous variable, you’d need to use multiple regression. The continuous variable is your DV and the five indicators are your IVs. This type of model allows you to use the values of the five indicators to predict the mean value of the continuous variable.

Which approach you take is a mix of theory and want your study needs to learn.

June 3, 2020 at 5:17 pm

Is this normal that the signs in “Regression equation in uncoded units” are sometimes different from the signs in the “Coded coefficients table”? In my regression results, for some terms, while the sign of a coefficient is negative in “Coded coefficients table”, it is positive in the regression equation. I am a little confused here. I thought the signs should be the same.

Thanks, Behnaz

June 3, 2020 at 8:07 pm

There is nothing unusual about the coded and uncoded coefficients having different signs. Suppose a coded coefficient has a negative sign but the uncoded coefficient has a positive sign. Your software using one of several processes that translates the raw data (uncoded) into coded values that help the model estimate process. Sometimes that conversion process causes data values to switch signs.

October 17, 2019 at 5:31 am

Hello Jim, I am lookign to do a Rsquared line for a multiple regression series. i’m not so confident that the 3rd,4th,5th number in the correlations will help make a better line. i’m basically looking at data to predict stock prices (getting a better R2) so for example Enterprise Value/Sales to growth rate has a high R2 of like .48 but we know for sure that Free cash flow/revenue to percent down from 52 week high is like .299

i have no clue how to get this to work in a 3d chart or to make a formula and find the new r2. any help would be great.

i dont have excel..not a programmer..just have some google sheets experience.

October 17, 2019 at 3:42 pm

Hi Jonathan,

I’m not 100% sure what you mean by an R-squared line? Or, by the 3rd, 4th, 5th, number in the correlations? Are you fitting several models where each one has just one independent variable?

It sounds to me like you’ll to learn more about multiple regression. Fortunately, I’ve written an ebook about it that will take you from a novice to being able to perform multiple regression effectively. Learn about my intuitive guide to regression ebook .

It also sounds like you’ll need to obtain some statistical software! I’m not sure what statistics if any you can perform in Google Sheets.

July 18, 2019 at 9:44 am

Forgive me if these questions have obvious answers but I could not find the answers yet. Still reading and learning. Is 3 the minimum number of samples needed to calculate regression? Why? I’m guessing the equations used require at least 3 sets of X,Y data to calculate a regression but I do not see a good explanation of why. I’m am not wondering about how many sets make the strongest fit. And with only two sets we would get a straight line and no chance of curvature……

I am working on a stability analysis report. For some lots we only have two time points. Zero and Three months. The software will not calculate the regression. Obviously, it needs three time points…..but why? For example: standard error cannot be calculated with only two results and therefore the rest of the equations will not work…or maybe it is related to degrees of freedom? (in the meantime what I will do is run through the equations by hand. the problem is I’m so heavily relying on the software. in other words being lazy. at least I’m questioning though. i’ve been told not to worry about it and just submit the report with “regression requires three data points”.)

July 12, 2019 at 5:34 pm

Hello, Pls how can I construct a model on carbon pricing? Thanks in anticipation of a timely response.

July 15, 2019 at 11:12 am

The first step is to do a lot of research to see what others have done. That’ll get you started in the right direction. It’ll help you identify the data you’ll need to collect, variables to include in the model, and the type and form of model that is likely to fit your data. I’ve also written a post about choosing the correct regression model that you should read. That post describes the model fitting process and how to determine which model is the best.

Best of luck with your analysis!

June 25, 2019 at 11:47 am

Hi Jim: I recently read your book Regression Analysis and found it very helpful. It covered a lot of material but I continue to have some questions about basic workflow when conducting regression analysis in social science research. For someone who wants to create an explanatory multiple regression model(s) as part of an observational study in anthropology, what are the basic chronological steps one should follow to analyze the data (eg: choose model type based on type of data collected; create scatterplots between Y and X’s; calculate correlation coefficients; specify model . . .)? I am looking for the basic steps to follow in the order that they should be completed. Once a researcher has identified a research question and collected and stored data in a dataset, what should the step-by-step work flow look like for a regression / model building analysis? Having a basic chronology of steps will help me better organize (and use) the material in your book. Thanks!

June 25, 2019 at 10:03 pm

First, thanks so much for buying me ebook. I’m so happy to hear that it was helpful. You ask a great question. And, it my next book I tackle the actual process of performing statistical studies that use the scientific method. For now, I can point you towards a blog post that covers this topic: Five Steps for Conducting Studies with Statistical Analyses

And, because you’re talking about an observation study, I recommend my post about observational studies . It talks about how they’re helpful, what to watch out for, and some tips. Also be sure to read about confounding variables in regression analysis, which starts on page 158 in the book.

Additionally, starting on p. 150 in the ebook, I talk about how to determine which variables to include in the model.

Throughout all of those posts and the ebook, you’ll notice a common theme. That you need to do a lot of advance research to figure out what you need to measure and how to measure it. Also important to ensure that you don’t accidently not measure a variable and have omitted variable bias affect your results. That’s where all the literature research will be helpful.

Now, in terms of analyzing the data, it’s hard to come up with one general approach. Hopefully, the literature review will tell you what has worked and hasn’t worked for similar studies. For example, maybe you’ll see that similar studies use OLS but need to use a log transformation. It also strongly depends on the nature of your data. The type of dependent variable(s) plays a huge role in what type of model you should use. See page 315 for more about that. It’s really a mix of what type of data you have (particularly the DVs) and what has worked/not worked for similar studies.

Sometimes, even armed with all that advanced knowledge, you’ll go to fit the model with what seems to be the best choice, and it just doesn’t fit your data. Then, you need to go back to the drawing board and try something else. It’s definitely an iterative process. But, looking at what similar studies have done and understanding your data can give you a better chance of starting out with the right type of model. And, then use the tips starting on page 150 to see about the actual process of specifying the model, which is also an iterative process. You might well start out with the correct type of model, but have to go through several iterations to settle on the best form of it.

• See what other studies have done
• Understand your own data.
• Use information from step 1 and 2 to settle on a good type of model to start with and what variables to include in it.
• Try to obtain a good fit using that type of model. This step is an iterative process of fitting models, assessing the fit and significance, and possibly making adjustments.
• If you can obtain a good fit in step 4, you’re done after settling on the best form.
• If you cannot obtain a good fit in step 4, do more research to find another type of model you can try and go back to step 3.

Best of luck with your study!

June 5, 2019 at 11:50 am

Thank you! Much appreciated!!

June 5, 2019 at 12:24 pm

You’re very welcome, Svend! Because you’re study uses regression, you might consider buying my ebook about regression . I cover a lot more in it than I do on the blog.

June 4, 2019 at 6:17 am

Hi Jim! Did you notice my question from 28. May…?? Svend

June 4, 2019 at 11:06 am

Hi Svend, Sorry about the delay in replying. Sometimes life gets busy! I will reply to your previous comment right now.

June 2, 2019 at 5:12 am

Thank you so much for such timely responses! They helped clarify a lot of things for me 🙂

May 28, 2019 at 3:53 am

Thank you for a very informative blog! I have a question regarding “overfitting” of a multivariable regression analysis that I have performed; 368 patients (ACL-reconstructed + concomitant cartilage lesions) with 5-year FU after ACL-reconstruction. The dependent variable was continuous (PROM). I have included 14 independent variables (sex/age/time from surgery etc, all of which formerely shown to be clinically important for the outcome) including two different types of surgery for the concomitant cartilage injury. No surgery to the concomitant lesions was used as reference (n=203), debridement (n=70), and Microfracture (n=95). My main objective was to investigate the effect on PROMs of those 2 treatments. My initial understanding was that it was OK to include that many independent variables as long as there were 368 patients included/PROMs at FU. But I have had comments that as long as the number of patients for some of the independent variables, ex. (debridement and microfracture) is lower than the model as a whole, the number of independent variables should be based on the variable with least observations…? I guess my question is: does the lowest number of observations for an independent variable dictate the size of the model/how many predictors you can use..? -And also the power..? Thanks!

June 4, 2019 at 11:23 am

I’m not sure if you’ve read my post about overfitting . If you haven’t, you should read it. It’ll answer some of your questions.

For your specific case, in general, yes, I think you have enough observations. In my blog post, I’m talking mainly about continuous variables. However, if I’m understanding correctly, you’re referring to a categorical variable for reference/debridement? If so, the rules are a bit different but I still think you’re good.

Regression and ANOVA are really the same analysis. So, you can thinking of your analysis as an ANOVA where you’re comparing groups in your data. And, it’s true that groups with smaller numbers will produce less precise estimates than groups with larger numbers. And, you generally require more observations for categorical variables than you do for continuous variables. However, it appears like your smallest group has an n=70 and that’s a very good sample size. In ANOVA, having more than 15-20 observations per group is usually good from a assumptions point of view (might not be produce sufficient statistical power depending on the effect size). So, you’re way over that. If some of your groups had very few observation, you might have need to worry about the estimates for that variable–but that’s not the case.

And, given your number of observations (368) and number of model terms requiring estimates overall (14), I don’t see any obvious reason to worry about overfitting on that basis either. Just be sure that you’re counting interaction terms and polynomials in the number model terms. Additionally, a categorical variable can use more degrees of freedom than a single continuous variable.

In short, I don’t see any reason for concern about overfitting given what you have written. Power depends on the effect size, which I don’t know. However, based on the number of observations/terms in model, I again don’t see an obvious problem.

I hope this helps! Best of luck with your analysis!

May 26, 2019 at 5:25 am

Also, another query. I want to run a multiple regression but my demographics and one of my IVs weren’t significant in the initial correlation I ran. What variables should I put in my regression test now? Should I skip all those that weren’t significant? Or just the demographics? I have read that if you have literature backing up the relationship, you can run a regression analysis regardless of how it appeared in your preliminary analysis. How true it that? What would be the best approach in this case? would mean a lot if you help me out on this one

May 27, 2019 at 10:25 pm

Hi again Aisha,

Two different answers for you. One, be wary of the correlation results. The problem is, again, the potential for confounding variables. Correlation doesn’t factor in other variables. Confounding variables can mess up the correlation results just like it can bias a regression model as I explained in my other comment. You have reason to believe that some of your demographic variables won’t be significant until you add your main IVs. So, you should try that to see what happens. Read the post about confounding variables and keep that in mind as you work through this!

And, yes, if you have strong theory or evidence from other studies for including IVs in the model, it’s ok to include them in your model even if it’s not significant. Just explain that in the write up.

For more about that, and model building in general, read my post about specifying the correct model !

May 25, 2019 at 8:13 am

Hi! I can’t believe I didn’t find this blog earlier, would have saved me a lot of trouble for my research 😀 Anyway, I have a question. Is it possible for your demographic variables to become significant predictors in the final model of a Hierarchical regression? I cant seem to understand why it is the case with mine when they came out to be non significant in the first model (even in the correlation test when tested earlier) but became significant when I put them with the rest of my (main) IVs. Are there practical reasons for that or is it poor statistical skills? :-/

May 27, 2019 at 10:19 pm

Thanks for writing with a fantastic question. It really touches on a number of different issues.

Statistics is a funny field. There’s the field of statistics, but then many scientists/researchers in different fields use statistics within their own fields. And, I’ve observed in different fields that there are different terminology and practices for statistical procedures. Often I’ll hear a term for a statistical procedure and at first I won’t know what it is. But, then the person will describe it to me and I’ll know it by another name.

At one point, hierarchical regression was like this for me. I’ve never used it myself, but it appears to be common in social sciences research. The idea is you add variables to model in several groups, such as the demographic variables in one group, and then some other variables in the next group. There’s usually a logic behind the grouping. The idea is to see how much the model improves with the addition of each group.

I have some issues with this practice, and I think your case illustrates them. The idea behind this method is that each model in the process isn’t as good as the subsequent model, but it’s still a valid comparison. Unfortunately, if you look at a model knowing that you’re leaving out significant predictors, there’s a chance that the model with fewer IVs is biased. This problem occurs more frequently with observational studies, which I believe are more common in the social sciences. It’s the problem of confounding variables. And, what you describe is consistent with there being confounding variables that are not in the model with demographic variables until you add the main IVs. For more details, read my post about how confounding variables that are not in the model can bias your results .

Chances are that some of your main IVs are correlated with one more demographic variables and the DV. That condition will bias coefficients in your demographic IV model because that model excludes the confounding variables.

So, that’s the likely practical reason for what you’re observing. Not poor statistical skills! And, I’m not a fan of hierarchical regression for that reason. Perhaps there’s value to it that I’m not understanding. I’ve never used it in practice. But there doesn’t seem to be much to gain by assessing that first (in your case) demographic IV model when it appears to be excluding confounding variables and is, consequently, biased!

However, I know that methodology is common in some fields, so it’s probably best to roll with it! 🙂 But, that’s what I think is happening.

May 19, 2019 at 6:30 am

Hello Jim I need your help please. I have this eq: Can you perform a multiple regression with two independent variables but one of them constant ? for example I have this data

Angle (Theta) Length ratio (%) Force (kn) 0 1 52.1 0.174444444 1 52.9 0.261666667 1 53.3 0.348888889 1 55.5 0.436111111 1 58.1

May 20, 2019 at 2:42 pm

Hi Ibrahim,

Thanks for writing with the good question!

The heart of regression analysis is determining how changes in an independent variable correlates with changes in the dependent variable. However, if an independent variable does not change (i.e., it is constant), there is no way for the analysis to determine how changes in it correlate to changes in the DV. It’s just not possible. So, to answer your question, you can’t perform regression with a constant variable.

I hope this helps!

February 27, 2019 at 6:13 pm

Thank you very much for this awesome site!

February 27, 2019 at 11:01 am

Hello sir, i need to know about regression and anova could you help me please.

February 27, 2019 at 11:46 am

You’re in the right spot! Read through my blog posts and you’ll learn about these topics. Additionally, within a couple of weeks, I’ll be releasing an ebook that’s all about learning regression!

February 20, 2019 at 12:09 pm

Very nice tutorial. I’m reading them all! Are there any articles explaining how the regression model gets trained? Something about gradient descent?

February 11, 2019 at 11:55 am

Thanks alot for your precious time sir

February 11, 2019 at 11:58 am

You’re very welcome! 🙂

February 10, 2019 at 5:05 am

Hey sir,hope you will be fine.It is really wonderful platform to learn regression. Sir i have some problem as I’m using cross sectional data and dependent variable is continuous.Its basically MICS data and I’m using OLS but the problem is that there are some missing observation in some variables.So the sample size is not equal across all the variables.So its make sense in OLS?

February 11, 2019 at 11:40 am

In the normal course of events, yes, when an observation has a missing value in one of the variables, OLS will exclude the entire observation when it fits the model. If observations with missing values are a small portion of your dataset, it’s probably not a problem. You do have to be aware of whether certain types of respondents are more likely to have missing values because that can skew your results. You want the missing values to occur randomly through the observations rather than systematically occurring more frequently in particular types of observations. But, again, if the vast majority of your observations don’t have missing values, OLS can still be a good choice.

Assuming that OLS make sense for your data, one difficulty with missing values is that there really is no alternative analysis that you can use to handle them. If OLS is appropriate for your data, you’re pretty much stuck with it even if you have problematic missing values. However, there are methods of estimating the missing values so you can use those observations. This process is particularly helpful if the missing values don’t occur randomly (as I describe above). I don’t know which software you are using, but SPSS has a particularly good method for imputing missing values. If you think missing values are a problem for your dataset, you should investigate ways to estimate those missing values, and then use OLS.

January 20, 2019 at 10:33 am

Hi Jim, I was quite excited to see you post this, but then there was no following article, only related subjects.

Binary logistic regression

By Jim Frost

Binary logistic regression models the relationship between a set of predictors and a binary response variable. A binary response has only two possible values, such as win and lose. Use a binary regression model to understand how changes in the predictor values are associated with changes in the probability of an event occurring.

Is the lesson on binary logistic regression to follow, or what am I missing?

Thank you for your time.

Antonio Padua

January 20, 2019 at 1:20 pm

Hi Antonio,

That’s a glossary term. On my blog, glossary terms have a special link. If you hover the pointer over the link, you’ll see a tooltip that displays the glossary term. Or, if you click the link, you go to the glossary term itself. You can also find all the glossary terms by clicking Glossary in the menu across the top of the screen. It seems like you probably clicked the link to get to the glossary term for binary logistic regression.

I’ve had several requests for articles about this topic. So, I’m putting it on my to-do list! Although, it probably won’t be for a number of months. In the mean time, you can read my post where I show an example of binary logistic regression .

Thanks for writing!

November 2, 2018 at 1:24 pm

Thanks so much, your blog is really helpful! I was wondering whether you have some suggestions on published articles that use OLS (nothing fancy, just very plain OLS) and that could be used in class for learning interpreting regression outputs. I’d love to use “real” work and make students see that what they learn is relevant in academia. I mostly find work that is too complicated for someone just starting to learn regression techniques, so any advice would be appreciated!

Thanks, Hanna

October 25, 2018 at 7:52 pm

Hi Jim. Did you write on Instrumental variable and 2 SLS method? I am interested in them. Thanks so all excellent things you did on this site.

October 25, 2018 at 10:29 pm

I haven’t yet, but those might be good topics for the future!

October 23, 2018 at 2:33 pm

Jim. Thank you so much. Especially for such a prompt response! The slopes are coming from IT segment stock valuations over 150 years. The slopes are derived from valuation troughs and peaks. So it is a graph like you’d see for the S&P. Sorry I was not clear on this.

October 23, 2018 at 12:14 pm

Jim, could you recommend a model based on the following:

1. I can see a strong visual correlation between the left side trough and peak and the right side. When the left has a steep vector so does the left, for example.

2. This does not need to be the case, the left could provide a much steeper slope compared to right or a much more narrow slope.

3. The parallels intrigue me and I would like to measure if the left slope can be explained by the right to any degree.

4. I am measuring the rise and fall of industry valuations over time. (it is the rise and fall in these valuations over time that create these ~ parallel slopes.

5. My data set since 1886 only provides 6 events, but they are consistent as described.

6. I attempted correlate rising slope against declining.

October 23, 2018 at 2:04 pm

I’m having time figuring out what you’re describing. I’m not sure what slopes you’re referring and I don’t know what you mean by the left versus right slopes?

If you only have 6 data points, you’ll only be able to fit an extremely simple model. You’ll usually need at least 10 data points (absolute minimum but probably more) to even include one independent variable.

If you have two slopes for something and you want to see if one slope explains the other, you could try using linear regression. Use one slope as an independent variable and another as a dependent variable. Slopes would be a continuous variable and so that might work. The underlying data for each slope would have to be independent from data used for other slopes. And, you’ll have to worry about time order effects such as autocorrelation.

October 2, 2018 at 1:37 am

Thank you Jim.

October 2, 2018 at 1:31 am

Hi Jim, I have a doubt regarding which regression analysis is to be conducted. The data set consists of categorical independent variables (ordinal) and one dependent variable which is of continuous type. Moreover, most of the data pertaining to an independent variable is concentrated towards first category (70%). My objective is to capture the factors influencing the dependent variable and its significance. In that case should I consider the ind. variables to be continuous or as categorical? Thanks in advance.

October 2, 2018 at 2:26 am

I think I already answered your question on this. Although, it looks like you’re now saying that you have an ordinal independent variable rather than a categorical variable. Ordinal data can be difficult. I’d still try using linear regression to fit the data.

You have two options that you can try.

1) You can include the ordinal data as continuous data. Doing this assumes that going from 1 to 2 is the same scale change as going from 2 to 3 and so on. Just like with actual continuous data. Although, you can add polynomials and transformations to improve the fit.

2) However, that doesn’t always work. Sometimes ordinal data don’t behave like continuous data. For example, the 2nd place finisher in a race doesn’t necessarily take twice as long as the 1st place finisher. And the difference between 3rd and 2nd isn’t the same as between 1st and 2nd. Etc. In that case, you can include it as a categorical variable. Using this approach, you estimate the mean differences between the different ordinal levels and you don’t have to assume they’ll be the same.

There’s an important caveat about including them as categorical variables. When you include categorical variables, you’re actually using indicator variables. A 5 point Likert scale (ordinal) actually includes 4 indicator variables. If you have many Likert variables, you’re actually including 4 variables for each one. That can be problematic. If you add enough of these variables, it can lead to overfitting . Depending on your software, you might not even see these indicator variables because they code and include them behind the scenes. It’s something to be aware of. If you have many such variables, it’s preferable to include them as continuous variables if possible.

You’ll have to think about whether your data seems more like continuous or categorical data. And, try both methods if you’re not sure. Check the residuals to make sure the model provides a good fit.

Ordinal data can be tricky because they’re not really continuous data nor categorical data–a bit of both! So, you’ll have to experiment and assess how well the different approaches work.

Good luck with your analysis!

October 1, 2018 at 2:32 am

Hello Jim, I have a set of data consisting of dependent variable which is of continuous type and independent variables which are of categorical type. The interesting thing which I found is that majority (more than 70%)of the independent variables belong to the category 1. The category values range from scale 1 to 5. I would like to know the appropriate sampling technique to be used. Is it appropriate to use linear regression or should I use other alternatives? Or any preprocessing of data is required? Please help me with the above.

Thanks in advance Raju.

October 1, 2018 at 9:40 pm

I’d try linear regression first. You can include that categorical variable as the independent variable with no problem. As always, be sure to check the residual plots. You can also use one-way ANOVA, which would be the more usual choice for this type of analysis. But, linear regression and ANOVA are really the same analysis “under the hood.” So, you can go either way.

September 23, 2018 at 4:28 am

Hello Jim I’d like to Know what your suggestions are with regards to choice of regression for predicting: dependent variable is count data but it does not follow a poisson distribution independent variables include categorical and continuous data I’d appreciate your thoughts on it …. thanks!

September 24, 2018 at 11:08 pm

Hi Sarkhani,

Having count data that don’t follow the Poisson happens fairly often. The top alternatives that I’m aware of are negative binomial regression and zero inflated models. I talk about those options a bit in my post about choosing the correct type of regression analysis . The count data section is near the end. I hope this information points you in the right direction!

August 29, 2018 at 9:38 am

Hi jim i’m really happy to find your blog

August 11, 2018 at 1:42 pm

Independent variables range from 0 to 1 and corresponding dependent variables range from 1 to 5 . If we apply regression analysis to above and predict the value of y for any value of x that also ranges from 0 to 1, whether the value of y will always lie in the range 1 to 5?

August 11, 2018 at 4:18 pm

In my experience, the predicted values will fall outside the range of the actual dependent variable. Assuming that you are referring to actual limits at 1 and 5, the regression analysis does not “understand” that those are hard limits. The extent that the predicted values fall outside these limits depends on the amount of error in the model.

August 8, 2018 at 4:18 am

Very Good Explanation about regression ….Thank you sir for such a wonderful post….

March 29, 2018 at 11:43 am

Hi Jim, I would like to see you writing something about Cross Validation (Training and test).

February 20, 2018 at 8:30 am

thank you Jim this is helpful

February 21, 2018 at 4:08 pm

You’re very welcome, Lisa! I’m glad you found it to be helpful!

January 21, 2018 at 10:39 am

Hello Jim I’d like to Know what your suggestions are with regards to choice of regression for predicting: the likelihood of participants falling into One of two categories (low Fear group codes 1 and high Fear 2 … when looking at scores from several variables ( e.g. external Other locus of control, external social locus of control , internal locus of control and social phobia and sleep quality ) It was suggested that I break the question up to smaller components … I’d appreciate your thoughts on it …. thanks!

January 22, 2018 at 2:30 pm

Because you have a binary response (dependent variable), you’ll need to use binary logistic regression. I don’t know what types of predictors you have. If they’re continuous, you can just use them in the model and see how it works.

If they’re ordinal data, such as a Likert scale, you can still try using them as predictors in the model. However, ordinal data are less likely to satisfy all the assumptions. Check the residual plots. If including the ordinal data in the model doesn’t work, you can recode them as indicator variables (1s and 0s only based on whether an observation meets a criteria or not. For example, if you have a scale of -2, -1. 0, 1, 2 you could recode it so observations with a positive score get a 1 while all other scores get a 0.

Those are some ideas to try. Of course, what works best for your case depends on the subject area and types of data that you have.

January 21, 2018 at 5:04 am

I am using Step-wise regression to select significant variables in the model for prediction.how to interpret BIC in variable selection?

regards, Zishan

January 22, 2018 at 5:36 pm

Hi, when comparing candidate models, you look for models with a lower BIC. A lower BIC indicates that a model is more likely to be the true model. BIC identifies the model that is more likely to have generated the observed data.

January 18, 2018 at 2:44 pm

yes.the language of the topic is very easy , i would appreciate you sir ,if you let me know that ,If rank

correlation is r =0.8,sum of “D”square=33.how we will calculate /find no. observations (n).

January 18, 2018 at 3:00 pm

I’m not sure what you mean by “D” square, but I believe you’ll need more information for that.

January 6, 2018 at 11:08 pm

Hi, Jim! I’m really happy to find your blog. It’s really helping, especially that you use basic English so non-native speaker can understand it better than reading most textbooks. Thanks!

January 7, 2018 at 12:49 am

Hi Dina, you’re welcome! And, thanks so much for your kind words–you made my day!

December 21, 2017 at 12:30 am

Can you write on Logistic regression please!

December 21, 2017 at 12:45 am

Hi! You bet! I plan to write about it in the near future!

December 16, 2017 at 2:33 am

great work by great man,, it is easily accessible source to access the scholars,, sir i am going to analyse data plz send me guidlines for selection of best simple linear/ multiple linear regression model, thanks

December 17, 2017 at 12:21 am

Hi, thank you so much for your kind words. I really appreciate it! I’ve written a blog post that I think is exactly what you need. It’ll help you choose the best regression model .

December 8, 2017 at 8:47 am

such a splendid compilation, Thanks Jim

December 8, 2017 at 11:09 am

December 3, 2017 at 10:00 pm

would you also throw some ideas on Instrumental variable and 2 SLS method please?

December 3, 2017 at 10:40 pm

Those are great ideas! I’ll write about them in future posts.

Comments and Questions Cancel reply

Save 10% on All AnalystPrep 2024 Study Packages with Coupon Code BLOG10 .

• Payment Plans
• Product List
• Partnerships

• Try Free Trial
• Study Packages
• Levels I, II & III Lifetime Package
• Video Lessons
• Study Notes
• Practice Questions
• Levels II & III Lifetime Package
• About the Exam
• About your Instructor
• Part I Study Packages
• Part I & Part II Lifetime Package
• Part II Study Packages
• Exams P & FM Lifetime Package
• Quantitative Questions
• Verbal Questions
• Data Insight Questions
• Live Tutoring
• About your Instructors
• EA Practice Questions
• Data Sufficiency Questions
• Integrated Reasoning Questions

Hypothesis Testing in Regression Analysis

Hypothesis testing is used to confirm if the estimated regression coefficients bear any statistical significance.  Either the confidence interval approach or the t-test approach can be used in hypothesis testing. In this section, we will explore the t-test approach.

The t-test Approach

The following are the steps followed in the performance of the t-test:

• Set the significance level for the test.
• Formulate the null and the alternative hypotheses.

$$t=\frac{\widehat{b_1}-b_1}{s_{\widehat{b_1}}}$$

\(b_1\) = True slope coefficient.

\(\widehat{b_1}\) = Point estimate for \(b_1\)

\(b_1 s_{\widehat{b_1\ }}\) = Standard error of the regression coefficient.

• Compare the absolute value of the t-statistic to the critical t-value (t_c). Reject the null hypothesis if the absolute value of the t-statistic is greater than the critical t-value i.e., \(t\ >\ +\ t_{critical}\ or\ t\ <\ –t_{\text{critical}}\).

Example: Hypothesis Testing of the Significance of Regression Coefficients

An analyst generates the following output from the regression analysis of inflation on unemployment:

$$\small{\begin{array}{llll}\hline{}& \textbf{Regression Statistics} &{}&{}\\ \hline{}& \text{Multiple R} & 0.8766 &{} \\ {}& \text{R Square} & 0.7684 &{} \\ {}& \text{Adjusted R Square} & 0.7394 & {}\\ {}& \text{Standard Error} & 0.0063 &{}\\ {}& \text{Observations} & 10 &{}\\ \hline {}& & & \\ \hline{} & \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t-Stat}\\ \hline \text{Intercept} & 0.0710 & 0.0094 & 7.5160 \\\text{Forecast (Slope)} & -0.9041 & 0.1755 & -5.1516\\ \hline\end{array}}$$

At the 5% significant level, test the null hypothesis that the slope coefficient is significantly different from one, that is,

$$ H_{0}: b_{1} = 1\ vs. \ H_{a}: b_{1}≠1 $$

The calculated t-statistic, \(\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}\) is equal to:

$$\begin{align*}\text{t}& = \frac{-0.9041-1}{0.1755}\\& = -10.85\end{align*}$$

The critical two-tail t-values from the table with \(n-2=8\) degrees of freedom are:

$$\text{t}_{c}=±2.306$$

Notice that \(|t|>t_{c}\) i.e., (\(10.85>2.306\))

Therefore, we reject the null hypothesis and conclude that the estimated slope coefficient is statistically different from one.

Note that we used the confidence interval approach and arrived at the same conclusion.

Question Neeth Shinu, CFA, is forecasting price elasticity of supply for a certain product. Shinu uses the quantity of the product supplied for the past 5months as the dependent variable and the price per unit of the product as the independent variable. The regression results are shown below. $$\small{\begin{array}{lccccc}\hline \textbf{Regression Statistics} & & & & & \\ \hline \text{Multiple R} & 0.9971 & {}& {}&{}\\ \text{R Square} & 0.9941 & & & \\ \text{Adjusted R Square} & 0.9922 & & & & \\ \text{Standard Error} & 3.6515 & & & \\ \text{Observations} & 5 & & & \\ \hline {}& \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t Stat} & \textbf{P-value}\\ \hline\text{Intercept} & -159 & 10.520 & (15.114) & 0.001\\ \text{Slope} & 0.26 & 0.012 & 22.517 & 0.000\\ \hline\end{array}}$$ Which of the following most likely reports the correct value of the t-statistic for the slope and most accurately evaluates its statistical significance with 95% confidence?     A. \(t=21.67\); slope is significantly different from zero.     B. \(t= 3.18\); slope is significantly different from zero.     C. \(t=22.57\); slope is not significantly different from zero. Solution The correct answer is A . The t-statistic is calculated using the formula: $$\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}$$ Where: \(b_{1}\) = True slope coefficient \(\widehat{b_{1}}\) = Point estimator for \(b_{1}\) \(\widehat{S_{b_{1}}}\) = Standard error of the regression coefficient $$\begin{align*}\text{t}&=\frac{0.26-0}{0.012}\\&=21.67\end{align*}$$ The critical two-tail t-values from the t-table with \(n-2 = 3\) degrees of freedom are: $$t_{c}=±3.18$$ Notice that \(|t|>t_{c}\) (i.e \(21.67>3.18\)). Therefore, the null hypothesis can be rejected. Further, we can conclude that the estimated slope coefficient is statistically different from zero.

Offered by AnalystPrep

Analysis of Variance (ANOVA)

Predicted value of a dependent variable, simple random vs. stratified random sa ....

Simple random and stratified random sampling are both sampling techniques used by analysts... Read More

Chi-square and F-Distributions

Chi-square Distribution A chi-square distribution is an asymmetrical family of distributions. A chi-square... Read More

Tests of Independence Using Contingenc ...

With categorical or discrete data, correlation is not suitable for assessing relationships between... Read More

Calculating Probabilities Given the Di ...

We can calculate and interpret probabilities of random variables that assume either the... Read More

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base

Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Prevent plagiarism. Run a free check.

Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

• Academic style
• Vague sentences
• Style consistency

See an example

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

McCombes, S. (2023, November 20). How to Write a Strong Hypothesis | Steps & Examples. Scribbr. Retrieved August 12, 2024, from https://www.scribbr.com/methodology/hypothesis/

Is this article helpful?

Shona McCombes

Other students also liked, construct validity | definition, types, & examples, what is a conceptual framework | tips & examples, operationalization | a guide with examples, pros & cons, "i thought ai proofreading was useless but..".

I've been using Scribbr for years now and I know it's a service that won't disappoint. It does a good job spotting mistakes”

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

Writing hypothesis for linear multiple regression models

I struggle writing hypothesis because I get very much confused by reference groups in the context of regression models.

For my example I'm using the mtcars dataset. The predictors are wt (weight), cyl (number of cylinders), and gear (number of gears), and the outcome variable is mpg (miles per gallon).

Say all your friends think you should buy a 6 cylinder car, but before you make up your mind you want to know how 6 cylinder cars perform miles-per-gallon-wise compared to 4 cylinder cars because you think there might be a difference.

Would this be a fair null hypothesis (since 4 cylinder cars is the reference group)?: There is no difference between 6 cylinder car miles-per-gallon performance and 4 cylinder car miles-per-gallon performance.

Would this be a fair model interpretation ?: 6 cylinder vehicles travel fewer miles per gallon (p=0.010, β -4.00, CI -6.95 - -1.04) as compared to 4 cylinder vehicles when adjusting for all other predictors, thus rejecting the null hypothesis.

Sorry for troubling, and thanks in advance for any feedback!

• multiple-regression
• linear-model
• interpretation

Yes, you already got the right answer to both of your questions.

• Your null hypothesis in completely fair. You did it the right way. When you have a factor variable as predictor, you omit one of the levels as a reference category (the default is usually the first one, but you also can change that). Then all your other levels’ coefficients are tested for a significant difference compared to the omitted category. Just like you did.

If you would like to compare 6-cylinder cars with 8-cylinder car, then you would have to change the reference category. In your hypothesis you just could had added at the end (or as a footnote): "when adjusting for weight and gear", but it is fine the way you did it.

• Your model interpretation is correct : It is perfect the way you did it. You could even had said: "the best estimate is that 6 cylinder vehicles travel 4 miles per gallon less than 4 cylinder vehicles (p-value: 0.010; CI: -6.95, -1.04), when adjusting for weight and gear, thus rejecting the null hypothesis".

Let's assume that your hypothesis was related to gears, and you were comparing 4-gear vehicles with 3-gear vehicles. Then your result would be β: 0.65; p-value: 0.67; CI: -2.5, 3.8. You would say that: "There is no statistically significant difference between three and four gear cars in fuel consumption, when adjusting for weight and engine power, thus failing to reject the null hypothesis".

Your Answer

Sign up or log in, post as a guest.

Required, but never shown

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy .

Not the answer you're looking for? Browse other questions tagged r regression multiple-regression linear-model interpretation or ask your own question .

• Featured on Meta
• Bringing clarity to status tag usage
• Introducing an accessibility dashboard and some upcoming changes to display...
• We've made changes to our Terms of Service & Privacy Policy - July 2024

Hot Network Questions

• HTTPs compression, CSRF and mobile apps
• GNOME Shell on Wayland was skipped because of an unmet condition
• Shift right by half a trit
• efficiently reverse EISH to HSIE
• Possible bug in DateList, DateObject etc returning negative years in 14.1
• What does it mean to echo multiple strings into a UNIX pipe?
• Proving an inequality involving a definite integral.
• Zsh: enable extended_glob inline in a filename generation glob
• Is Marisa Tomei in the film the Toxic Avenger?
• Strategies for handling Maternity leave the last two weeks of the semester
• Using elastic-net only for feature selection
• Dummit Foote, Group Theory, Section 1.6, Problem 23
• Variant of global failure of GCH
• Claims of "badness" without a moral framework?
• Looking for a story about a colonized solar system that has asteroid belts but no habitable planet
• Tiling a rectangle with one-sided P-pentominoes
• Overlap two Chess Games
• Every time I see her, she said you've not been doing me again
• Is an infinite composition of bijections always a bijection?
• Why do instructions for various goods sold in EU nowadays lack pages in English?
• What was I thinking when I drew this diagram?
• How to deal with a boss that is using non-related arguments in a dissent?
• To "Add Connector" or not to "Add Connector", that is a question
• How can one design earplugs so that they provide protection from loud noises, such as explosions or gunfire, while still allowing user to hear voices?