Linear Regression, Clearly Explained!!!

Summary

This video, part of the 'StatQuest' series, offers a detailed explanation of linear regression, a powerful statistical method used to fit a line to data and predict outcomes. The video covers the key concepts of linear regression, including least squares, R-squared, and the calculation of p-values, which help assess the strength and significance of the model. Through engaging examples, the video illustrates how these concepts work together to quantify relationships in data, making complex statistical ideas accessible and understandable.

Takeaways

• 📊 **Linear Regression Basics**: The video introduces linear regression, explaining that the primary steps involve using least squares to fit a line to data, calculating R-squared, and then determining a p-value for R-squared.
• 📉 **Fitting a Line**: Linear regression uses the least squares method to find the best-fitting line for the data, minimizing the sum of squared residuals, which are the differences between observed and predicted values.
• 🔍 **Understanding R-squared**: R-squared is a statistical measure that represents the proportion of variance for a dependent variable that's explained by an independent variable or variables in a regression model.
• 🐭 **Example with Mice**: The video uses a dataset of mice, where the aim is to predict mouse size based on weight, showcasing how linear regression can be applied to real-world data.
• ⚖️ **Sum of Squares**: The sum of squares around the mean is compared to the sum of squares around the fitted line, which helps calculate R-squared by showing how much variance is explained by the model.
• 🔄 **Multiple Parameters**: When multiple predictors are used (e.g., mouse weight and tail length), linear regression fits a plane rather than a line, showing that more complex relationships can be modeled.
• 📈 **Adjusted R-squared**: The concept of adjusted R-squared is introduced, which accounts for the number of predictors in the model, providing a more accurate measure when multiple variables are involved.
• 🤔 **Limitations of R-squared**: R-squared alone doesn't indicate whether a model is statistically significant, especially with small datasets or when fitting models to random data points.
• 🔬 **Calculating p-values**: The video explains the process of calculating p-values using the F-statistic, which compares the variance explained by the model to the variance not explained, helping determine statistical significance.
• 🧪 **Practical Application**: Linear regression is highlighted as a powerful tool for quantifying relationships in data, but it requires both a high R-squared value and a low p-value to confirm the reliability and significance of results.

Q & A

What is the main topic of the video script?

The main topic of the video script is linear regression, specifically general linear models, also known as multiple regression.

What are the three most important concepts behind linear regression mentioned in the script?

The three most important concepts behind linear regression mentioned in the script are: fitting a line to the data using least squares, calculating the R-squared value, and calculating a p-value for R-squared.

What is the purpose of using least squares in linear regression?

The purpose of using least squares in linear regression is to find the best-fitting line for the data by minimizing the sum of the squares of the vertical distances (residuals) between the observed data points and the fitted line.

What does the R-squared value represent in the context of linear regression?

The R-squared value represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It indicates the strength of the relationship between the variables.

How is the R-squared value calculated in the script's example with mouse size and weight?

The R-squared value is calculated by taking the difference between the variation around the mean and the variation around the fit, then dividing that by the variation around the mean. In the script's example, the R-squared value is 0.6, indicating that 60% of the variation in mouse size can be explained by mouse weight.

What is the significance of calculating a p-value for R-squared in linear regression?

Calculating a p-value for R-squared is important to determine if the observed R-squared value is statistically significant, which helps to assess whether the relationship between the variables is likely due to chance or a true association.

What does the term 'residual' mean in the context of linear regression?

In the context of linear regression, a 'residual' refers to the difference between the observed value and the value predicted by the regression line for a given data point.

How does the script explain the concept of degrees of freedom in the context of calculating the p-value for R-squared?

The script explains that degrees of freedom are used to turn the sums of squares into variances in the context of calculating the p-value for R-squared. They are related to the number of parameters in the fit line and the mean line, and they are used to adjust the sums of squares to account for the number of data points and parameters.

What is the role of the F-distribution in calculating the p-value for R-squared?

The F-distribution is used to approximate the histogram of F-scores that would be obtained if many random datasets were generated and analyzed in the same way. The p-value is then determined by comparing the F-score from the original dataset to this distribution to see how extreme it is relative to the expected distribution of F-scores.

How does the script illustrate the potential issue with adding too many parameters to a regression model?

The script illustrates the potential issue by pointing out that adding more parameters to a model can lead to a higher R-squared value due to random chance, even if those parameters do not have a true relationship with the dependent variable. This is why adjusted R-squared and p-values are important to ensure the model's validity.

What is the adjusted R-squared and why is it used in regression analysis?

Adjusted R-squared is a modified version of R-squared that adjusts for the number of predictors in the model. It is used to provide a more accurate measure of the model's goodness of fit, especially when comparing models with different numbers of predictors, as it penalizes the addition of unnecessary predictors.

Outlines

00:00

🚢 Introduction to Linear Regression and Its Core Concepts

This section introduces the topic of linear regression and provides a brief overview of the StatQuest video series. The speaker begins by setting the stage with a metaphorical journey to StatQuest and then delves into the essentials of linear regression, focusing on fitting a line to data using the least squares method, calculating R-squared, and determining the p-value for R-squared. The importance of understanding these concepts for linear regression is emphasized, with a promise of detailed explanations and examples throughout the video.

05:02

📏 Understanding Variance and Sum of Squares in Linear Regression

This paragraph explains the concept of variance in the context of linear regression, particularly how variance is calculated around the mean and the fitted line. It introduces the terms SS (Sum of Squares) and variance, explaining how they relate to the overall variation in a data set. The section emphasizes the importance of understanding these concepts as they are fundamental to assessing the fit of a regression line and calculating key metrics like R-squared.

10:06

🔄 Examples of R-squared in Linear Regression

Here, the speaker provides three distinct examples to illustrate the concept of R-squared in linear regression. Each example shows different levels of predictive power based on mouse weight and size. The first example demonstrates a 60% reduction in variance, the second shows a perfect prediction with 100% R-squared, and the third illustrates a scenario where mouse weight does not help in predicting mouse size, resulting in an R-squared of 0%. These examples help in understanding how R-squared quantifies the proportion of variation explained by the model.

15:07

📊 The Impact of Adding Parameters and Adjusted R-squared

This section explores the effects of adding more parameters to a regression model, emphasizing that while adding parameters can improve R-squared, it may not always be meaningful. The concept of adjusted R-squared is introduced, which accounts for the number of parameters, thus providing a more accurate measure of model performance. The discussion also touches on the potential for random factors to influence the sum of squares, leading to the need for adjusted metrics.

20:08

📉 Degrees of Freedom and F-statistic in Linear Regression

This paragraph explains the concepts of degrees of freedom and the F-statistic, which are crucial for determining the significance of an R-squared value. The degrees of freedom are related to the number of parameters in the model, and the F-statistic compares the explained variance to the unexplained variance. This section breaks down the calculation of the F-statistic and how it is used to determine the p-value, which indicates the reliability of the regression model.

25:09

✅ Final Review and Importance of R-squared and P-values

In this final section, the speaker reviews the key concepts of R-squared and p-values in linear regression. The importance of having both a high R-squared and a low p-value for a meaningful and reliable model is stressed. The section concludes by summarizing the process of calculating R-squared and p-values, and the overall significance of these metrics in evaluating the strength and reliability of a linear regression model.

Keywords

💡Linear Regression

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In the video, it is the primary focus, with the process of fitting a line to data points using the least squares method being a core concept. The script describes how linear regression is used to predict mouse size based on weight, illustrating the technique's application in a real-world context.

💡Least Squares

Least squares is a mathematical procedure used to find the line of best fit by minimizing the sum of the squares of the vertical distances (residuals) of the data points from the line. The script explains that this method is used to determine the best-fitting line for a set of data, which is crucial for linear regression analysis.

💡R-squared

R-squared, or the coefficient of determination, 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 discusses how to calculate R-squared to determine the goodness of fit for the regression model, using it to quantify how much of the variation in mouse size can be explained by mouse weight.

💡P-value

A p-value is the probability that the observed results of a statistical hypothesis test occurred by chance. In the context of the video, the p-value for R-squared is calculated to assess the statistical significance of the relationship between mouse weight and size. The script explains that a small p-value indicates strong evidence against the null hypothesis, suggesting that the relationship is statistically significant.

💡Residual

In regression analysis, a residual is the difference between the observed values and the values predicted by a model. The script mentions residuals as the distances from the data points to the fitted line, which are squared and summed to find the best-fitting line using the least squares method.

💡Variance

Variance is a measure of the dispersion of a set of data points around their mean. The script uses variance to describe the spread of the data points and how much they vary from the mean or the fitted line. It explains how variance is calculated and used in determining R-squared and the p-value.

💡Fitting a Line

Fitting a line refers to the process of determining the best line that represents the data in a scatter plot, typically done through linear regression. The script describes the iterative process of rotating the line to minimize the sum of squared residuals, which is essential for finding the least squares fit.

💡Sum of Squares

The sum of squares, particularly the sum of squares around the mean (SS mean) and around the fit (SS fit), is a key component in calculating R-squared. The script explains that these sums are used to measure the total variability of the data and the unexplained variability after fitting the regression line, respectively.

💡Degrees of Freedom

Degrees of freedom in statistics refer to the number of values that are free to vary in a calculation. In the script, degrees of freedom are used in the context of calculating variance and are important for understanding how the sums of squares are converted into variances, which are then used to calculate the F statistic and the p-value.

💡F Statistic

The F statistic is used in hypothesis testing to determine whether there is a significant difference between group means. In the context of the video, the F statistic is derived from the ratio of variances explained by the model to those not explained, and it is used to calculate the p-value for R-squared, helping to assess the significance of the regression model.

Highlights

StatQuest introduces the concept of linear regression, also known as general linear models, as a powerful statistical tool.

The primary objective of linear regression is to fit a line to data using the least squares method.

Calculating R-squared is essential to measure the goodness of fit of the regression line.

R-squared quantifies the percentage of variance explained by the model.

The video explains the concept of residuals, which are the distances between the data points and the regression line.

Least squares fitting involves rotating the line to minimize the sum of squared residuals.

The equation of the regression line is derived from the least squares method, estimating parameters like slope and y-intercept.

R-squared can be calculated using both the sum of squares around the mean and the sum of squares around the fit.

An R-squared value of 1 indicates a perfect prediction model, while 0 suggests no explanatory power.

The video demonstrates how to apply R-squared to more complex models involving multiple variables.

Least squares fitting can ignore irrelevant parameters by setting their coefficients to zero.

Adjusted R-squared is introduced to account for the number of parameters in the model and avoid overfitting.

The concept of degrees of freedom is discussed in the context of turning sums of squares into variances.

The video explains how to calculate the F-statistic, which is used to determine the statistical significance of R-squared.

The F-distribution is used to approximate the histogram of F-statistics for calculating p-values.

A small p-value indicates that the observed R-squared is statistically significant and not due to random chance.

The video concludes by emphasizing the importance of both a high R-squared and a low p-value for a meaningful linear regression model.

Transcripts

sailing on a boat headed towards

statquest

join me on this boat let's go to stat

Quest it's super cool

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 going to be talking about

linear regression AKA General linear

models part one there's a lot of parts

to linear models but it's a really cool

and Powerful concept so let's get right

down to it

I promise you I have lots and lots of

slides that talk about all the Nitty

Gritty details behind linear regression

but first let's talk about the main

ideas behind it

the first thing you do in linear

regression is use least squares to fit a

line to the data

the second thing you do is calculate r

squared

lastly calculate a p-value for r squared

there are lots of other little things

that come up along the way but these are

the three most important Concepts behind

linear regression

in the stat Quest fitting a line to data

we talked about

fitting a line to data duh

but let's do a quick review

I'm going to introduce some new

terminology in this part of the video so

it's worth watching even if you've

already seen the earlier stat Quest

that said if you need more details check

that stat Quest out

for this review we're going to be

talking about a data set where we took a

bunch of mice and we measured their size

and we measured their weight

our goal is to use mouse weight as a way

to predict Mouse size

first draw a line through the data

second measure the distance from the

line to the data Square each distance

and then add them up

terminology alert

the distance from the line to the data

point is called a residual

third rotate the line a little bit

with the new line measure the residuals

Square them and then sum up the squares

now rotate the line a little bit more

sum up the squared residuals

etc etc etc we rotate and then sum up

the squared residuals rotate then sum up

the squared residuals just keep doing

that

after a bunch of rotations you can plot

the sum of squared residuals and

corresponding rotation

so in this graph we have the sum of

squared residuals on the y-axis and the

different rotations on the x-axis

lastly you find the rotation that has

the least sum of squares

more details about how this is actually

done in practice are provided in the

stat Quest on fitting a line to data

so we see that this rotation is the one

with the least squares so it will be the

one to fit to the data

this is our least squares rotation

superimposed on the original data

bam now we know why the method for

fitting a line is called least squares

now we have fit a line to the data this

is awesome

here's the equation for the line

least squares estimated two parameters

a y-axis intercept

and a slope

since the slope is not zero it means

that knowing a mouse's weight will help

us make a guess about that Mouse's size

how good is that guess

calculating r squared is the first step

in determining how good that guess will

be

the stat Quest r squared explained talks

about you got it r squared

let's do a quick review I'm also going

to introduce some additional terminology

so it's worth watching this part of the

video even if you've seen the original

stat Quest on r squared

first calculate the average Mouse size

okay I've just shifted all the data

points to the y-axis to emphasize that

at this point we are only interested in

Mouse size

here I've drawn a black line to show the

average Mouse size

bam

sum the squared residuals

just like in least squares we measure

the distance from the mean to the data

point and square it and then add those

squares together

terminology alert we'll call this SS

mean for sum of squares around the mean

note the sum of squares around the mean

equals the data minus the mean squared

the variation around the mean equals the

data minus the mean squared divided by n

n is the sample size in this case n

equals 9.

the shorthand notation is the variation

around the mean equals the sum of

squares around the mean divided by n the

sample size

another way to think about variance is

as the average sum of squares per Mouse

now go back to the original plot and sum

up the squared residuals around our

least squares fit

we'll call This Ss fit for the sum of

squares around the least squares fit

the sum of squares around the least

squares fit is the sum of the distances

between the data and the line squared

just like with the mean the variance

around the fit

is the distance between the line and the

data squared divided by n the sample

size

the shorthand is the variation around

the fitted line equals the sum of

squares around the fitted line divided

by n the sample size

again we can think of the variation

around the fit as the average of the sum

of squares around the fit for each Mouse

in general the variance of something

equals the sum of squares divided by the

number of those things

in other words it's an average of sum of

squares

I mentioned this because it's going to

come in handy in a little bit so keep it

in the back of your mind

okay let's step back a little bit this

is the raw variation in Mouse size

and this is the variation around the

least squares line

there's less variation around the line

that we fit by least squares that is to

say the residuals are smaller

as a result we say that some of the

variation in Mouse size is explained by

taking mouse weight into account

in other words heavier mice are bigger

lighter mice are smaller

r squared tells us how much of the

variation in Mouse size can be explained

by taking mouse weight into account

this is the formula for r squared it's

the variation around the mean minus the

variation around the fit divided by the

variation around the mean

let's look at an example

in this example the variation around the

mean equals 11.1 and the variation

around the fit equals 4.4

so we plug those numbers into the

equation

the result is that r squared equals 0.6

which is the same thing as saying 60

percent

this means there is a sixty percent

reduction in the variance when we take

the mouse weight into account

alternatively we can say that mouse

weight explains 60 percent of the

variation in Mouse size

we can also use the sum of squares to

make the same calculation

this is because when we're talking about

variation everything's divided by n the

sample size since everything's scaled by

n we can pull that term out and just use

the raw sum of squares

in this case the sum of squares around

the mean equals one hundred

and the sum of squares around the fit

equals 40. plugging those numbers into

the equation gives us the same value we

had before r squared equals 0.6 which

equals 60 percent

60 percent of the sums of squares of the

mouse size can be explained by mouse

weight

here's another example we're also going

to go back to using variation in the

calculation since that's more common

in this case knowing mouse weight means

you can make a perfect prediction of

mouse size

the variation around the mean is the

same as it was before 11.1

but now the variation around the fitted

line equals zero because there are no

residuals

plugging the numbers in gives us an r

squared equal to one which equals one

hundred percent

in this case mouse weight explains 100

percent of the variation in Mouse size

okay one last example

in this case knowing mouse weight

doesn't help us predict Mouse size

if someone tells us they have a heavy

Mouse well that Mouse could either be

small or large with equal probability

similarly if someone said they had a

light Mouse well again we wouldn't know

if it was a big mouse or a small Mouse

because each of those options is equally

likely

just like the other two examples the

variation around the mean is equal 11.1

however in this case the variation

around the fit is also equal 11.1

so we plug those numbers in and we get r

squared equals 0 which equals zero

percent

in this case mouse weight doesn't

explain any of the variation around the

mean

when calculating the sum of squares

around the mean we collapse the points

onto the y-axis just to emphasize the

fact that we were ignoring mouse weight

but we could just as easily draw a line

y equals the mean Mouse size and

calculate the sum of squares around the

mean around that

in this example we applied r squared to

a simple equation for a line Y equals

0.1 plus 0.78 times x

this gave us an r squared of 60 percent

meaning 60 percent of the variation in

Mouse size could be explained by mouse

weight

but the concept applies to any equation

no matter how complicated

first you measure square and sum the

distance from the data to the mean

then measure square and sum the distance

from the data to the complicated

equation

once you've got those two sums of

squares just plug them in and you've got

r squared

let's look at a slightly more

complicated example

imagine we wanted to know if mouse

weight and tail length did a good job

predicting the length of the mouse's

body

so we measure a bunch of mice

to plot this data we need a

three-dimensional graph

we want to know how well weight and tail

length predict body length

the first Mouse we measured had weight

equals 2.1

tail length equals 1.3 and body length

equals 2.5

so that's how we plot this data on this

3D graph

here's all the data in the graph the

larger circles are points that are

closer to us and represent mice that

have shorter tails

the smaller circles are points that are

further from us and represent mice with

longer tails

now we do a least squares fit

since we have the extra term in the

equation representing an extra Dimension

we fit a plane instead of a line

here's the equation for the plane

the Y value represents body length

least squares estimates three different

parameters

the first is the y-intercept that's when

both tail length and mouse weight are

equal to zero

the second parameter 0.7 is for the

mouse weight

the last term

0.5 is for the tail length

if we know a mouse's weight and tail

length we can use the equation to guess

the body length

for example given the weight and tail

length for this mouse

the equation predicts this body length

just like before we can measure the

residuals Square them and then add them

up to calculate r squared

now if the tail length or the z-axis is

useless and doesn't make the sum of

squares fit any smaller than least

squares will ignore it by making that

parameter equal to zero in this case

plugging the tail length into the

equation would have no effect on

predicting the mouse size

this means equations with more

parameters will never make the sum of

squares around the fit worse than

equations with fewer parameters

in other words this equation Mouse size

equals 0.3 plus mouse weight plus flip

of a coin plus favored color plus

astrological sign plus extra stuff will

never perform worse than this equation

Mouse size equals 0.3 plus mouse weight

this is because least squares will cause

any term that makes sum of squares

around the fit worse to be multiplied by

zero and in a sense no longer exist

now due to random chance there is a

small probability that the small mice in

the data set might get heads more

frequently than large mice

if this happened then we'd get a smaller

sum of squares fit and a better r

squared

here's the frowny face of sad times

the more silly parameters we add to the

equation the more opportunities we have

for random events to reduce sum of

squares fit and result in a better r

squared

thus people report an adjusted r squared

value that in essence scales are squared

by the number of parameters

r squared is awesome

but it's missing something

what if all we had were two measurements

we'd calculate the sum of squares around

the mean in this case that would be 10

then we'd calculate the sum of squares

around the fit which equals zero

the sum of squares around the fit equals

zero because you can always draw a

straight line to connect any two points

what this means is when we calculate r

squared by plugging the numbers in we're

going to get 100 percent

100 percent is a great number we've

explained all the variation but any two

random points will give us the exact

same thing it doesn't actually mean

anything

we need a way to determine if the r

squared value is statistically

significant

we need a p-value

before we calculate the p-value let's

review the main Concepts behind r

squared one last time

the general equation for r squared is

the variance around the mean minus the

variance around the fit divided by the

variance around the mean

in our example this means the variation

in the mouse size minus the variation

after taking weight into account divided

by the variation in Mouse size

in other words r squared equals the

variation in Mouse size explained by

weight divided by the variation in Mouse

size without taking weight into account

in this particular example r squared

equals 0.6 meaning we saw a 60 reduction

in variation once we took mouse weight

into account

now that we have a thorough

understanding of the ideas behind r

squared let's talk about the main ideas

behind calculating a p-value for it

the p-value for r squared comes from

something called f

f is equal to the variation in Mouse

size explained by weight divided by the

variation in Mouse size not explained by

weight

the numerators for r squared and for f

are the same

that is to say it's the reduction in

variance when we take the weight into

account

the denominator is a little different

these dotted lines the residuals

represent the variation that remains

after fitting the line

this is the variation that is not

explained by weight

so together we have the variation in

Mouse size explained by weight divided

by the variation in Mouse size not

explained by weight

now let's look at the underlying

mathematics

just as a reminder here's the equation

for r squared

this is the general equation that will

tell us if r squared is significant

the meat of these two equations are very

similar and rely on the same sums of

squares

like we said before the numerators are

the same

in our Mouse size and weight example the

numerator is the variation in Mouse size

explained by weight

and the sum of squares around the fit is

just the residuals squared and summed up

around the fitted line so that's the

variation that the fit does not explain

these numbers over here are the degrees

of freedom

they turn the sums of squares into

variances

I'm going to dedicate a whole stat quest

to degrees of freedom but for now let's

see if we can get an intuitive feel for

what they're doing here

let's start with these

P fit is the number of parameters in the

fit line

here's the equation for the fit line in

a general format we just have the

y-intercept plus the slope times x

the y-intercept and the slope are two

separate parameters

that means P fit equals two

p mean is the number of parameters in

the mean line

in general that equation is y equals the

y-intercept

that's what gives us a horizontal line

that cuts through the data

in this case the y-intercept is the mean

value

this equation just has one parameter

thus p mean equals one

both equations have a parameter for the

y-intercept

however the fit line has one extra

parameter the slope in our example this

slope is the relationship between weight

and size

in this example P fit minus p mean

equals 2 minus 1 which equals one

the fit has one extra parameter mouse

weight

thus the numerator is the variance

explained by the extra parameter in our

example that's the variance in Mouse

size explained by mouse weight

if we had used mouse weight and tail

length to explain variation in size

then we would end up with an equation

that had three parameters and P fit

would equal three

thus P fit minus p mean would equal

three minus 1 which equals two

now the fit has two extra parameters

mouse weight and tail length

with the fancier equation for the fit

the numerator is the variance and mouse

size explained by mouse weight and tail

length

now let's talk about the denominator for

our equation for f

denominator is the variation in Mouse

size not explained by the fit

that is to say it's the sum of squares

of the residuals that remain after we

fit our new line to the data

y divide sum of squares fit by n minus P

fit instead of just n

intuitively the more parameters you have

in your equation the more data you need

to estimate them for example you only

need two points to estimate a line but

you need three points to estimate a

plane

if the fit is good

then the variation explained by the

extra parameters in the fit will be a

large number and the variation not

explained by the extra parameters in the

fit will be a small number

that makes f a really large number

now that question we've all been dying

to know the answer to how do we turn

this number into a p-value

conceptually

generate a set of random data

calculate the mean and the sum of

squares around the mean

calculate the fit in the sum of squares

around the fit

now plug all those values into our

equation for f

and that will give us a number in this

case that number is 2.

now plot that number in a histogram

now generate another set of random data

calculate the mean and the sum of

squares around the mean

then calculate the fit and the sum of

squares around the fit

plug those values into our equation for

f

and in this case we get f equals three

so we then plug that value into our

histogram

and then we repeat with yet another set

of random data in this case we got f

equals one that's plotted on our

histogram

and we just keep generating more and

more random data sets calculating the

sums of squares plugging them into our

equation for f and plotting the results

on our histogram

now imagine we did that hundreds if not

millions of times

when we're all done with our random data

sets we return to our original data set

we then plug the numbers into our

equation for f in this case we got f

equals 6.

the p-value is the number of more

extreme values divided by all of the

values

so in this case we have the value at f

equals 6 and the value at f equals 7

divided by all the other randomizations

that we created originally

if this concept is confusing to you I

have a stat Quest that explains p-values

so check that one out

bam

you can approximate the histogram with a

line in practice rather than generating

tons of random data sets people use the

line to calculate the p-value

here's an example of one standard F

distribution that people use to

calculate p-values the degrees of

freedom determine the shape

the red line represents another standard

F distribution that people use to

calculate p-values

in this case the sample size used to

draw the red line is smaller than the

sample size used to draw the blue line

notice that when n minus P fit equals 10

the distribution tapers off faster

this means that the p-value will be

smaller when there are more samples

relative to the number of parameters in

the fit equation

triple bam

hooray we finally got our p-value now

let's review the main ideas

given some data that you think are

related

linear regression quantifies the

relationship in the data

this is r squared this needs to be large

it also determines how reliable that

relationship is

this is the p-value that we calculated

with f

this needs to be small

you need both to have an interesting

result

hooray we've made it to the end of

another exciting stat Quest wow this was

a long one I hope you had a good time

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