Using Linear Models for t-tests and ANOVA, Clearly Explained!!!

Summary

In this StatQuest episode, the host delves into General Linear Models, focusing on the application of linear regression techniques to perform t-tests and ANOVA. The concept of a design matrix is introduced, allowing for the comparison of means and calculation of p-values to determine significant differences between groups, such as control and mutant mice in gene expression studies. The episode simplifies complex statistical methods, making them accessible for further exploration in future videos.

Takeaways

• 📊 **General Linear Models**: The video discusses the application of general linear models, specifically focusing on linear regression and its extension to T-tests and ANOVA.
• 🧬 **Gene Expression Study**: It uses a study comparing gene expression between control and mutant mice to illustrate statistical concepts.
• 📈 **Design Matrix Introduction**: Introduces the concept of a design matrix, a tool used to expand linear regression techniques to more complex tests.
• 🔍 **T-Test Application**: Explains how to apply linear regression techniques to perform a T-test to compare means between two groups.
• 📉 **Sum of Squared Residuals**: Describes the calculation of the sum of squared residuals around the mean and around the fitted line.
• 📝 **Fitting Lines to Data**: Demonstrates how to fit lines to control and mutant data separately and then combine them into a single equation.
• 🔢 **Calculating F and P Values**: Shows how to calculate F and P values for both linear regression and T-tests using the sums of squares.
• 📚 **ANOVA Test**: Discusses how to perform an ANOVA test to compare more than two categories, using the same principles as for T-tests.
• 🔄 **Design Matrix Variations**: Notes that there are different design matrices that can be used for T-tests and ANOVA, with one being more common than the other.
• 🔄 **Flexibility of Design Matrix**: Emphasizes the flexibility of the design matrix to allow for computer-based solutions to statistical problems without manual calculations.

Q & A

What is the main focus of the StatQuest video?

The main focus of the StatQuest video is to explain how to apply linear regression techniques to perform T-tests and ANOVA using a design matrix.

What is a design matrix?

A design matrix is a concept used in statistics, particularly in the context of general linear models, to represent the structure of the data and how different variables relate to each other.

What was the goal of the T-test discussed in the video?

The goal of the T-test discussed in the video was to compare gene expression between control mice and mutant mice to see if their means are significantly different.

What does 'mutant mice' refer to in the context of the video?

In the context of the video, 'mutant mice' refers to normal mice that have a specific gene that has been knocked out and is no longer functioning correctly.

How is the mean used in the context of a T-test?

In the context of a T-test, the mean is used as the least squares fit to the data, which is a horizontal line that intercepts the Y-axis at the mean value for each group.

What is the purpose of calculating the sum of squared residuals around the mean?

Calculating the sum of squared residuals around the mean helps to determine the variability of the data points relative to the overall mean, which is a step in both linear regression and T-tests.

How does the design matrix help in calculating F and the P-value?

The design matrix helps in calculating F and the P-value by providing a way to combine the data and parameters into a single equation, allowing for a unified approach to statistical analysis.

What is the difference between the design matrix used in the video and the standard design matrix for T-tests?

The design matrix used in the video is a simplified version created for the purpose of the tutorial, while the standard design matrix for T-tests is more commonly used in practice and may include more complex structures.

Why is it important to be able to calculate P-values for T-tests and ANOVA?

Being able to calculate P-values for T-tests and ANOVA is important because it allows researchers to determine if observed differences between groups are statistically significant and not due to chance.

What is the significance of the P-value in statistical tests?

The P-value in statistical tests indicates the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. A low P-value suggests that the results are significant and not due to chance.

How does the video demonstrate the application of linear regression to T-tests?

The video demonstrates the application of linear regression to T-tests by showing how the same techniques used to calculate P-values in linear regression can be adapted to perform T-tests by using a design matrix to fit lines to the data and calculate residuals.

Outlines

00:00

📊 Linear Regression and T-Tests with Design Matrices

This paragraph introduces the video's focus on General Linear Models, specifically linear regression and T-tests. It explains how these techniques can be applied using a design matrix, a tool that will be further explored in future videos. The host reviews the previous episode on linear regression, which involved measuring mouse weight and size to understand the predictability of size based on weight and the significance of their relationship. The goal is to extend these concepts to T-tests, which compare means to determine significant differences, using control and mutant mice as an example. Mutant mice have a gene that's been 'knocked out,' and the aim is to compare gene expression between control and mutant mice. The process involves calculating the overall mean, sum of squared residuals, fitting lines to the data, and combining these into a single equation for analysis.

05:00

🧬 Applying Design Matrices to T-Tests and Anova

The paragraph delves into the application of design matrices to perform T-tests and ANOVA. It describes how to create a design matrix that acts as a switch for different means, allowing for a unified approach to calculating F and P values. The process involves calculating the sum of squares of residuals around the fitted lines and using these values to determine the P value. The host contrasts the design matrix used in the video with a more standard one, indicating that both can achieve the same result but the latter is more commonly used. The video promises to cover these matrices in more detail in future episodes.

10:01

🔍 Summarizing the Process and Looking Forward

This final paragraph summarizes the process covered in the video, from calculating the sum of squares around the mean to fitting lines and using design matrices for T-tests and ANOVA. It highlights the importance of understanding how to calculate F and P values using these methods. The host also encourages viewers to subscribe for more content and invites suggestions for future videos, promising more exciting Stat Quests to come.

Keywords

💡Stat Quest

Stat Quest is a series of educational videos that focus on statistics, particularly as it relates to genetics. In the transcript, Stat Quest is the show's title, and it's brought to the audience by the genetics department at the University of North Carolina at Chapel Hill. The series aims to demystify statistical concepts and make them accessible.

💡General linear models

General linear models (GLMs) are a class of statistical models that estimate the relationships between a dependent variable and one or more independent variables using regression analysis. In the video, GLMs are the overarching theme, with the host discussing how to apply linear regression techniques to t-tests and ANOVA.

💡Design Matrix

A design matrix is a matrix of values used in regression analysis to represent the effects of different variables on the dependent variable. In the script, the design matrix is introduced as a tool to expand upon in future videos and is used to combine data from different groups into a single equation for analysis.

💡Linear regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The video script reviews linear regression from the previous episode, where mouse weight was used to predict mouse size, and R-squared and P-value were discussed.

💡T-test

A t-test is a type of statistical hypothesis test that determines if there is a significant difference between the means of two groups. In the script, the host explains how to use linear regression techniques to perform a t-test comparing gene expression between control and mutant mice.

💡ANOVA

ANOVA stands for Analysis of Variance and is used to compare the means of three or more groups. The script describes how to apply the concepts of linear regression to perform an ANOVA to see if there are significant differences between gene expression in control, mutant, and heterozygote mice.

💡Residuals

Residuals are the differences between the observed values and the values predicted by a model. The script discusses calculating the sum of squared residuals around the mean and around the fitted lines, which is crucial for determining the fit of the model to the data.

💡P-value

The P-value is a statistical measure used to determine the probability that the observed data could have occurred by chance. The script explains how P-values are calculated in the context of both linear regression and t-tests, indicating whether the relationship between variables is statistically significant.

💡F-test

The F-test is used to compare two or more groups to determine if their variances are significantly different from each other. In the script, the F-test is mentioned in the context of ANOVA, where it is used to calculate the P-value for the test.

💡Mean

The mean, or average, is a measure of central tendency in statistics. In the video script, the mean is used to describe the central value of the data for control and mutant mice, and it plays a crucial role in fitting lines to the data in linear regression and t-tests.

💡Sum of Squares

Sum of Squares (SS) is a statistical method used to measure the total variability in a dataset. The script describes calculating the sum of squares of residuals around the mean and around the fitted lines, which is part of the process for performing ANOVA and t-tests.

Highlights

Introduction to part two of the General linear models series by Stat Quest

Explanation of how linear regression techniques can be used for t-tests and ANOVA

Introduction of the design Matrix concept

Quick review of linear regression from the previous episode

Importance of R-squared and P-value in understanding relationships and chance

Application of linear regression concepts to a T Test

Comparison of gene expression between control and mutant mice

Goal of a T Test to compare means and check for significant differences

Step-by-step guide on how to perform a T Test using linear regression techniques

Explanation of calculating the sum of squared residuals around the mean

Fitting a line to control and mutant data using least squares fit

Combining control and mutant data lines into a single equation

Introduction of the design Matrix as a tool for computer-based calculations

Calculation of F and P-value using the design Matrix

Review of the process from original data to calculating P-value

Introduction to ANOVA and the comparison of five categories

Calculation of sum of squares around the mean for ANOVA

Explanation of the design Matrix for ANOVA with five parameters

Final calculation of F and P-value for ANOVA

Discussion on the different design matrices used for T tests and ANOVA

Conclusion and invitation to subscribe for more Stat Quest episodes

Transcripts

stat Quest stat Quest stat

Quest

yeah hello and welcome to stat Quest

stat Quest is brought to you by the

friendly folks in the genetics

department at the University of North

Carolina at Chapel Hill today we're

doing part two of our series on General

linear

models last time we talked about how to

do linear regression

this time we're going to talk about how

to use those exact same techniques to do

tea tests and a Nova we'll do this using

something called a design Matrix which

is a cool concept that will expand upon

in future stat quests on General linear

models let's start with a super quick

review of linear

regression last time we measured mouse

weight and mouse size and we wanted to

learn two things from it we wanted to

learn how useful mouse weight was for

predicting Mouse size R squar told us

this and we wanted to know if the

relationship was due to Chance the P

value told us this now let's see if we

can apply those Concepts to a T Test in

this specific example we're going to be

comparing gene expression between

control mice and mutant mice mutant mice

are just normal mice that have a

specific Gene that's been knocked out

and is no longer functioning

correctly the goal of a T Test is to

compare means and see if they are

significantly different from each other

if the same method can calculate P

values for a linear regression and a t

test then we can easily calculate P

values for more complicated

situations so now I'm going to walk you

through the steps for using the

techniques from linear regression to do

a t test on the left side of the screen

our remind you how each step applies to

linear regression on the right side of

the screen I'll show you how those steps

apply to T tests step one ignore the

xais and find the overall

mean to emphasize that we want to focus

on the Y AIS I've removed the labels on

the X

AIS here are the overall means for the

linear regression and the T

Test the next step is to calculate the

sum of squared residuals around the mean

this is SS mean these are the residuals

the distance from the data points to the

lines in this case the lines are the

overall

means bam calculating the sum of squared

residuals around the mean was

easy step three fit a line to the data

Note this is when we start caring about

the X AIS again on the left side we have

the least squares fit to the data

however how do we do a least squares fit

to a T Test let's start by just fitting

a line to the Control Data we start by

finding a least squares fit to the

Control Data it turns out that the mean

is the least squares fit the mean

intercepts the Y AIS at

2.2 this is the equation for horizontal

line that intercepts the Y AIS at

2.2 thus this is the line that we fit to

the Control

Data now let's fit a line to the mutant

data the least squares fit is the mean

of the mutant data the mean intercepts

the y axis at

3.6 this is the equation for a

horizontal line that intercepts the Y

AIS at

3.6 thus this is the line that we fit to

the mutant data data we have fit two

lines to the data originally when we did

the regression we fit a single line to

the

data however there is a way to combine

these two lines into a single

equation this will make the steps for

computing F the exact same for the

regression and the test which in turn

means a computer can do it

automatically this is key because we

don't want to do this by hand hand ever

this is going to look a little weird but

just bear with me keep in mind that the

goal is to have a flexible way for a

computer to solve this and every other

least squares based problem without

having to create a whole new method each

time this is the equation which combines

both lines for this

point we have 1 time the mean of the

Control Data 0er times the mean of the

mutant

data plus the

residual yes this is strange especially

multiplying the mutant mean by zero but

bear with me if we multiplied things out

the equation for this point would be y =

2.2 plus the

residual and that sort of makes sense

but just bear with me this is the

equation for the next point the only

difference is the residual this one is

smaller this is the equation for the

next

Point again the only difference is the

residual this is the equation for the

next point and again the only difference

is the

residual this is the equation for the

first point in the mutant data

set now we are multiplying the control

mean by

zero and multiplying the mutant mean by

one

these are the equations for the

remaining

points now let's focus on the ones and

zeros they function like on and off

switches for the two means a one turns

the mean on and a zero turns the mean

off when we isolate the ones and zeros

they form a matrix called a design

Matrix the design Matrix can be combined

with an abstract version of the equation

to represent a fit to the data column

one turns the control mean on or

off column two turns the mutant mean on

or off in practice the role of each

column is assumed and the equation is

written out like this y equals the mean

of the Control Data plus the mean of the

mutant

data now that we have the fit for the

control and mutant data down to a single

equation plus design Matrix we can move

on to calculating F and the P

value so step four calculate the sum of

squares of the residuals around the

fitted

lines with the linear regression that

means the sum of these squared

residuals the sum of squares around the

fit for the T Test is the sum of these

squared

residuals to viw what we've done so far

we've calculated the sum of squared

residuals around the mean and then we

calculated the sum of squared residuals

around the fitted line now we can just

plug these things in to our equation for

f f will lead to a P value for the

linear regression p mean refers to the

number of parameters in the equation for

the mean Mouse size that's one parameter

in the T Test p mean refers to the

number of parameters in the equation for

the mean of the gene expression that's

also just one

parameter for the linear regression P

fit refers to the number of parameters

in the equation for the fitted line in

this case that's two the parameters are

the intercept and the

slope for the T Test P fit refers to the

number of parameters in the line that we

fit to the T Test data in this case P

fit equals 2 because we had to estimate

two parameters one for the mean of the

control data and one for the mean of the

mutant

data now we can calculate a P value for

the T Test

bam let's review what we've done so far

here's the original data gene expression

for control mice and mutant

mice the first thing we did is we

calculated the sum of squares of the

resist residuals around the overall mean

then we calculated the sum of squares of

the residuals around the fit in order to

do this with a single equation we had to

create a design

Matrix once we've calculated the sums of

squares all we have to do is plug the

values into the equation for f and then

we'll get our P

value now let's do an

anova an NOA tests if all five

categories are the same

here we have control and mutant mice

just like before but we also have

control and mutant mice on a funky diet

and we also have heterozygote mice the

first thing we do is calculate the sum

of squares around the mean we do this

just like before we calculate an overall

mean value for all of the categories and

then Square the residuals and sum them

up no big

deal the equation for the overall mean

is just y equals mean expression that

equation only has a single parameter the

overall mean so p mean equals

1 now we calculate the sum of squares

around the fitted

lines the equation for the fitted lines

has five parameters one for each mean

therefore P fit equals five here's what

the design Matrix looks like one column

per

category now now that we've calculated

the sum of squares around the mean and

the sum of squares around the fit along

with p mean and P fit we can plug those

values in and calculate F triple bam if

we can calculate F then we've got

ourselves a P

value one last important detail before

we're done the design matrices that I've

shown you are not the standard design

matrices used for doing T tests and a

NOA this is what we used for the T Test

in this stat Quest but this is a more

common design Matrix for the same thing

both design matrices will get the job

done it's just the one on the right is

more commonly used we'll talk about this

one and other more elaborate designs in

the next stat

Quest hooray we've made it to the end of

another exciting stat Quest if you like

this and would like to see more stat

quests like it feel free to subscribe

and if you if you have any suggestions

for future stat quests put them in the

comments below tune in next time for

another exciting stack Quest