Multiple Regression, Clearly Explained!!!

Summary

In this StatQuest episode, host Josh Starmer explains multiple regression, emphasizing it's not much different from simple linear regression. He reviews key concepts like fitting a plane to data, calculating R-squared, and adjusting for additional parameters. The episode also covers calculating p-values and F-values, comparing simple and multiple regression to determine the value of collecting more data, like tail length in mice. A companion video teaches how to perform multiple regression in R, detailing important aspects of the output.

Takeaways

• 📚 StatQuest is an educational series focused on statistics, hosted by Josh Starmer.
• 🔍 The episode discusses multiple regression, building on the concepts introduced in the linear regression episode.
• 📈 Simple regression involves fitting a line to data, while multiple regression involves fitting a plane or higher-dimensional object.
• 📊 R-squared is used to evaluate the fit of the model to the data, and its calculation remains the same for both simple and multiple regression.
• ⚖️ For multiple regression, R-squared is adjusted to account for additional parameters in the model.
• 🧮 Calculating the p-value involves comparing the sums of squares around the fit and the mean.
• 🔢 The number of parameters estimated (P fit) changes with the complexity of the regression model.
• 🆚 Multiple regression allows for comparison between models with different numbers of predictors to determine if additional data is beneficial.
• 📊 The F value is calculated similarly for both simple and multiple regression, but with different parameters.
• 💻 An additional StatQuest episode demonstrates how to perform multiple regression in R, detailing the interpretation of the output.

Q & A

What is the main topic of this StatQuest episode?

The main topic of this StatQuest episode is multiple regression, which is explained as an extension of linear regression.

Who is the presenter of StatQuest?

Josh Stommer is the presenter of StatQuest.

What is the relationship between simple linear regression and multiple regression?

Simple linear regression is fitting a line to data, while multiple regression involves fitting a plane or higher-dimensional object to data, which essentially means adding more variables to the model.

What is the purpose of R-squared in the context of regression?

R-squared is used to evaluate how well the regression model fits the data, and it is calculated in the same way for both simple and multiple regression.

How does the addition of more data affect the calculation of R-squared in multiple regression?

The R-squared value is adjusted to compensate for the additional parameters in the equation when more data is added to the model.

What is the role of the p-value in regression analysis?

The p-value is used to determine the statistical significance of the model, and it is calculated by comparing the sums of squares around the fit and the mean.

What does 'P fit' represent in the context of calculating the F-value for regression?

'P fit' represents the number of parameters that least-squares has to estimate in the regression equation.

Why is it necessary to compare simple and multiple regression?

Comparing simple and multiple regression helps determine if adding additional variables to the model is worthwhile, by assessing if the increase in R-squared and the decrease in p-value are significant.

What is the significance of a large difference in R-squared values between simple and multiple regression?

A large difference in R-squared values between simple and multiple regression indicates that including additional variables significantly improves the model's fit.

What does a small p-value suggest when comparing simple and multiple regression?

A small p-value when comparing simple and multiple regression suggests that the improvement in the model with additional variables is statistically significant.

Is there a follow-up StatQuest episode that demonstrates how to perform multiple regression in R?

Yes, there is a follow-up episode that shows how to perform multiple regression in R, detailing the interpretation of the output.

Outlines

00:00

📊 Introduction to Multiple Regression

Josh Stommer from StatQuest introduces multiple regression as an extension of linear regression, emphasizing that the concepts are not significantly different. He explains that simple regression involves fitting a line to data, whereas multiple regression involves fitting a plane or higher-dimensional object to data, which essentially means adding more variables to the model. The video script discusses how to evaluate the fit of the model using r-squared and p-value, and how these calculations remain consistent between simple and multiple regression. An important note is made about adjusting r-squared for additional parameters in multiple regression. The script also explains how to compare simple and multiple regression to determine if adding more data, such as tail length, improves the model significantly.

05:01

🔍 Conclusion and Practical Application

In the concluding paragraph, Josh Stommer highlights the completion of the multiple regression tutorial and directs viewers to another video that demonstrates how to perform multiple regression in R, focusing on interpreting the output. He encourages viewers to subscribe and ends the video with a motivational note to continue learning.

Keywords

💡StatQuest

StatQuest is an educational YouTube channel hosted by Josh Starmer, focused on explaining statistical concepts in an accessible manner. In the context of the video script, StatQuest serves as the platform through which the presenter shares his knowledge about multiple regression, making complex statistical concepts more understandable to a broad audience.

💡Multiple Regression

Multiple regression is a statistical method that allows for the analysis of the relationship between a dependent variable and two or more independent variables. In the video, multiple regression is the central topic, with the script explaining how it extends the concept of simple linear regression by adding more dimensions or variables to the model, such as mouse weight and tail length.

💡Linear Regression

Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables using a straight line. The script mentions linear regression as foundational knowledge for understanding multiple regression, emphasizing that simple regression is essentially fitting a line to data.

💡R-squared

R-squared is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. The script explains that calculating R-squared remains the same in both simple and multiple regression, and it's used to evaluate how well the model fits the data.

💡P-value

A p-value is a statistical measure used to determine the probability that a finding is due to chance. In the context of the script, the p-value is calculated for the R-squared to assess the significance of the model. A small p-value indicates that the model's fit is statistically significant and not due to random chance.

💡F-value

The F-value is a statistical measure used in the analysis of variance (ANOVA) to determine whether there are any significant differences between the means of different groups. In the script, the F-value is calculated to compare the fit of the simple regression model to the multiple regression model, helping to decide if adding more variables improves the model significantly.

💡Parameters

In statistics, parameters are the quantities that characterize a probability distribution or a statistical model. The script discusses how the number of parameters in a model affects the calculation of the F-value and the adjusted R-squared, with more parameters typically requiring more data to estimate accurately.

💡Sums of Squares

Sums of squares are calculations used in regression analysis to partition the total variability in the data into components that can be attributed to the model and the error. The script mentions the sums of squares around the fit and the mean as inputs for calculating the F-value and R-squared in both simple and multiple regression.

💡Genetics Department

The genetics department at the University of North Carolina at Chapel Hill is mentioned as the sponsor of StatQuest. This department likely has an interest in statistical methods like multiple regression for analyzing genetic data, emphasizing the practical application of the concepts discussed in the video.

💡Higher-Dimensional Object

In the context of the script, a higher-dimensional object refers to the extension of a regression model beyond two dimensions. While it sounds complex, it simply means adding more variables to the model, such as incorporating tail length and food intake in addition to mouse weight when predicting body length.

💡Model Fit

Model fit refers to how well a statistical model represents the data it is designed to explain. The script discusses evaluating the fit of both simple and multiple regression models using R-squared and comparing them using the F-value to determine if adding more variables improves the model's predictive power.

Highlights

StatQuest is brought to you by the genetics department at the University of North Carolina at Chapel Hill.

Today's topic is multiple regression, which will be clearly explained.

This StatQuest builds on the one for linear regression.

Simple regression is just fitting a line to data, while multiple regression fits a plane or higher-dimensional object.

Multiple regression involves adding additional data to the model, such as mouse weight and tail length.

Calculating R squared is the same for both simple and multiple regression.

R squared calculation compensates for additional parameters in multiple regression.

Calculating the F value and p-value for multiple regression is similar to simple regression.

P fit is adjusted based on the number of parameters in the multiple regression equation.

Comparing simple and multiple regression can indicate the value of additional data collection.

The F value is calculated using sums of squares around the simple regression.

A significant difference in R squared values and a small p value suggest that additional data is beneficial.

StatQuest also provides a tutorial on how to perform multiple regression in R.

The tutorial covers important details and aspects of interpreting R's output.

Subscribe to StatQuest for more educational content.

Quest on!

Transcripts

StatQuest, StatQuest, StatQuest, StatQuest! Yeah!

StatQuest!

Hello, I'm Josh Stommer and welcome to Stat Quest. StatQuest is brought to you by the friendly folks in the genetics department at

the University of North Carolina at Chapel Hill.

Today we're gonna be talking about multiple regression, and it's gonna be clearly explained.

This StatQuest builds on the one for linear regression.

So, if you haven't already seen that one yet, check it out. Alright, now let's get to it!

People who don't understand linear regression tend to make a big deal out of the "differences" between simple and multiple regression.

It's not a big deal and a StatQuest on simple linear regression

already covered most of the concepts we're going to cover here. You might recall from the StatQuest on the linear regression

that simple regression is just fitting a line to data.

We're interested in the r-squared and the p-value to evaluate how well that line fits the data. In

that same stat quest. I also showed you how to fit a plane to data.

Well, that's what multiple regression is. You fit a plane or some higher-dimensional object to your data. A

term like higher-dimensional object sounds really fancy and complicated,

but it's not. All it means is that we're adding additional data to the model. In

the previous example all that meant was that instead of just modeling body length by mouse weight,

we modeled body length using mouse weight and tail length. If we added additional

factors like the amount of food eaten or the amount of time spent running on a wheel, well those would be considered

additional dimensions, but they're really just additional pieces of data that we can add to our fancy equation.

So, from the StatQuest on linear regression you may remember the first thing we did was calculate R squared.

Well, the good news is calculating r squared is the exact same for both simple regression and

multiple regression. There's absolutely no difference.

Here's the equation for R squared, and we plug in the values for the sums of squares around the fit and

then we plug in the sums of squares around the mean value for the body length.

Regardless of how much additional data we add to our fancy equation, if we're using it to predict body lengths,

then we use the sums of squares around the body length.

One caveat is for multiple regression you adjust r-squared to compensate for the additional parameters in the equation.

We covered this in the StatQuest for linear regression, so it's no big deal.

Now we want to calculate a p-value for our r-squared.

Calculating F And the p-value is pretty much the same. You plug in the sums of squares around the fit and

then you plug in the sums of squares around the mean.

For simple regression P fit equals 2 because we have two parameters in the equation that least-squares has to estimate. And

for this specific example

the multiple regression version of P fit equals 3 because least-squares had to estimate three different parameters. If we added

additional data to the model for example the amount of time a mouse spins running on a wheel

then we have to change P fit to equal the number of parameters in our new equation. And

for both simple regression and multiple regression, P

mean equals 1 because we only have to estimate the mean value of the body length.

So far we have compared this simple regression to the mean and this multiple regression to the mean,

but we can compare them to each other. And this is where multiple regression really starts to shine.

This will tell us if it's worth the time and trouble to collect the tail length data

because we will compare a fit without it, the simple regression,

to a fit with it, the multiple regression.

Calculating the F value is the exact same as before only this time we replace the mean stuff

with the simple regression stuff.

So instead of plugging in the sums of squares around the mean, we plug in the sums of squares around the simple regression. Ane

instead of plugging in P Mean we plug in P Simple, which equals the number of parameters in the simple regression.

That's 2. And

then we plug in the sums of squares for the multiple regression and we

plug in the number of parameters in our multiple regression equation.

BAM! If

the difference in r squared values between the simple and multiple regression is big and the p value is small

then adjusting tail length to the model is

worth the trouble

Hooray, we've made it to the end of another exciting StatQuest! Now for this StatQuest I've made another one

that shows you how to do multiple regression in R. It

shows all the little details and sort of what's important and what's not important about the output that R gives you.

So, check that one out and don't forget to subscribe!

Ok, until next time, Quest on!