Simple Linear Regression in R | R Tutorial 5.1 | MarinStatsLectures

Summary

In this educational video, Mike Marin introduces 'simple linear regression' using R programming language. He demonstrates how to model the relationship between age and lung capacity with lung capacity as the dependent variable. The video covers creating a scatter plot, calculating Pearson's correlation, and fitting a linear regression model with the 'lm' command. It also explains interpreting the model summary, extracting coefficients, adding regression lines to plots, and generating confidence intervals. The tutorial concludes with a preview of regression diagnostic plots for the next video.

Takeaways

• 📚 The video introduces 'simple linear regression' using R, a statistical method for modeling the relationship between two numeric variables.
• 📈 Simple linear regression can be applied even when the explanatory variable is categorical, but this is discussed in a later video.
• 🗂️ The video uses lung capacity data, focusing on the relationship between 'Age' and 'Lung Capacity', with 'Lung Capacity' as the dependent variable.
• 📊 A scatter plot is created to visualize the relationship, with 'Age' on the x-axis and 'Lung Capacity' on the y-axis.
• 🔍 Pearson's correlation is calculated to assess the linear association between 'Age' and 'Lung Capacity', showing a positive correlation.
• 📝 The 'lm' command in R is used to fit a linear regression model, with the formula structured as Y ~ X.
• 🔍 The summary of the model provides insights into residuals, intercept, slope, and their statistical significance.
• 📊 The 'abline' function in R adds a regression line to the scatter plot, allowing for visual interpretation of the model fit.
• 📊 The 'coef' command is used to extract model coefficients, which are crucial for understanding the model's parameters.
• 📐 The 'confint' command provides confidence intervals for the model coefficients, indicating the precision of the estimates.
• 📊 The 'anova' command generates an ANOVA table, offering a statistical test for the overall model fit.
• 🔍 The video concludes with a mention of regression diagnostic plots to be discussed in the next video, focusing on regression assumptions.

Q & A

What is the main topic of the video presented by Mike Marin?

The main topic of the video is introducing 'simple linear regression' using R programming language.

What is the purpose of simple linear regression in data analysis?

Simple linear regression is used to examine or model the relationship between two numeric variables.

Can simple linear regression be applied using a categorical explanatory variable?

While it is possible to fit a simple linear regression using a categorical explanatory variable, the video mentions that this topic will be covered in a later video.

What data set is used in the video to demonstrate simple linear regression?

The lung capacity data set is used to demonstrate the relationship between age and lung capacity in the video.

Which variable is considered the outcome or dependent variable in the lung capacity example?

In the lung capacity example, Lung Capacity is considered the outcome or dependent variable.

How is a scatter plot created in the video to visualize the data?

A scatter plot is created by plotting Age on the x-axis and Lung Capacity on the y-axis, with a title added for clarity.

What statistical measure is used to quantify the linear association between Age and Lung Capacity?

Pearson's correlation is used to quantify the linear association between Age and Lung Capacity.

How is a linear regression model fitted in R, as demonstrated in the video?

A linear regression model is fitted in R using the 'lm' command, with the dependent variable entered first, followed by the independent variable.

What does the summary output of a linear regression model in R provide?

The summary output provides information about the residuals, estimates of the intercept and slope, their standard errors, test statistics, p-values, residual standard error, r-squared, and adjusted r-squared, among other things.

How can the coefficients of the regression model be extracted in R?

The coefficients of the regression model can be extracted using the 'coef' function or by using the dollar sign ($) followed by the attribute name 'mod$coefficients'.

What command is used in R to add a regression line to a plot?

The 'abline' command is used in R to add a regression line to a plot, with options to customize color and line width.

How can confidence intervals for the model coefficients be produced in R?

Confidence intervals for the model coefficients can be produced using the 'confint' command, with the 'level' argument specifying the confidence level.

What does the 'anova' command in R generate for a linear regression model?

The 'anova' command in R generates an ANOVA (Analysis of Variance) table for the linear regression model, which includes the F-test and associated p-value.

How are regression diagnostic plots mentioned in the video related to the assumptions of regression?

Regression diagnostic plots, such as residual plots and QQ plots, are used to examine the assumptions of regression, such as linearity, homoscedasticity, and normality of residuals.

What will be the focus of the next video in the series according to the transcript?

The next video in the series will discuss how to produce regression diagnostic plots to examine the regression assumptions.

Outlines

00:00

📊 Introduction to Simple Linear Regression in R

In this segment, Mike Marin introduces the concept of simple linear regression using the R programming language. The focus is on how to model the relationship between two numeric variables, specifically Age and Lung Capacity, using lung capacity data from a previous series. The process begins with creating a scatter plot to visualize the data and calculate Pearson's correlation to assess the linear association. The lm function in R is then utilized to fit a linear model, with the first variable entered as the dependent variable (Lung Capacity) and the second as the independent variable (Age). The summary of the model provides insights into the residuals, intercept, slope, and their statistical significance. The summary also includes the residual standard error, r-squared, and adjusted r-squared values, which are essential for evaluating the model's fit and predictive power.

05:03

📈 Analyzing Model Coefficients and Diagnostics

This paragraph delves into the analysis of the model coefficients and diagnostic measures for the simple linear regression model. It explains how to extract and interpret the coefficients using the 'coef' command in R. The paragraph also discusses how to add a regression line to the scatter plot using the 'abline' function, with options to customize the line's appearance. The importance of the residual standard error is highlighted, showing its equivalence to the square root of the mean squared error from the ANOVA table. The paragraph concludes with a mention of future content, which will cover regression diagnostic plots to assess the assumptions of the regression model, including residual and QQ plots. The viewer is encouraged to explore additional instructional videos for further learning.

Keywords

💡Simple Linear Regression

Simple Linear Regression is a statistical method used to model the relationship between two variables by fitting a straight line. In the context of the video, it is used to examine the relationship between age and lung capacity. The script mentions fitting a linear regression using the 'lm' command in R, which stands for 'linear model', and the model is used to predict lung capacity based on age.

💡Numeric Variables

Numeric variables are data points that can be represented by numbers and are used in statistical analysis. In the video, age and lung capacity are numeric variables that are being analyzed to understand their linear relationship. The script specifically mentions examining the relationship between these two numeric variables using simple linear regression.

💡Categorical Explanatory Variable

A categorical explanatory variable is a type of independent variable that can be grouped into categories. The script mentions that while simple linear regression can technically use a categorical variable, the focus of the video is on numeric variables. This distinction is important for understanding the types of data that can be used in regression analysis.

💡Scatter Plot

A scatter plot is a type of plot that displays the values of two numeric variables, with one variable plotted along each axis. In the video, a scatter plot is produced with age on the x-axis and lung capacity on the y-axis to visualize the relationship between these two variables before fitting the regression line.

💡Pearson's Correlation

Pearson's correlation is a measure of the linear relationship between two variables. It is mentioned in the script as a way to quantify the association between lung capacity and age, indicating whether the relationship is positive, negative, or non-existent, and how strong it is.

💡lm Command

The 'lm' command in R is used to fit linear models. The script describes how to use this command to perform simple linear regression, specifying the dependent variable (lung capacity) and the independent variable (age), and then saving the model in an object named 'mod'.

💡Intercept

In the context of a linear regression model, the intercept is the point where the line crosses the y-axis. The script discusses the estimate of the intercept, its standard error, and the hypothesis test to determine if the intercept is significantly different from zero, although it notes that this is often not of interest.

💡Slope

The slope of a line in a linear regression model represents the change in the dependent variable for a one-unit change in the independent variable. The script explains the estimation of the slope for age and its significance in the model, indicating whether age has a statistically significant effect on lung capacity.

💡Residual Standard Error

Residual standard error is a measure of the average distance that the observed values fall from the regression line. The script mentions this value as a key statistic in the summary of the model, indicating the variability of the data around the fitted line.

💡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 refers to r-squared and adjusted r-squared as metrics to assess the goodness of fit of the regression model.

💡Coefficients

In the context of regression analysis, coefficients are the constants in the equation of a line that represent the relationship between variables. The script describes how to extract the coefficients from the model using the 'coef' command in R, which includes the intercept and slope.

💡Confidence Intervals

Confidence intervals provide a range of values within which the true population parameter is likely to fall, with a certain level of confidence. The script explains how to produce confidence intervals for the model coefficients using the 'confint' command, and how to adjust the level of confidence.

💡ANOVA Table

ANOVA, or Analysis of Variance, is a statistical method used to compare means of two or more groups, and in the context of regression, it tests the null hypothesis that the regression coefficients are equal to zero. The script mentions producing an ANOVA table for the linear regression model to assess the model's overall significance.

💡Regression Diagnostic Plots

Regression diagnostic plots are used to assess the assumptions of a regression model, such as linearity, homoscedasticity, and normality of residuals. The script indicates that the next video will discuss these plots, including residual plots and QQ plots, which are essential for validating the regression model's fit.

Highlights

Introduction to simple linear regression using R.

Simple linear regression is used for modeling relationships between two numeric variables.

The video will use lung capacity data to model the relationship between Age and Lung Capacity.

A scatter plot is created to visualize the data with Age on the x-axis and Lung Capacity on the y-axis.

Calculation of Pearson's correlation to examine the linear association between Age and Lung Capacity.

Fitting a linear regression model using the 'lm' command in R.

The importance of entering the Y variable first followed by the X variable in the 'lm' function.

Explanation of the summary output from the linear regression model, including residuals, intercept, and slope.

Use of stars in the summary to denote significant coefficients.

Understanding the residual standard error as a measure of variation around the regression line.

Introduction to the 'attributes' command to explore the stored attributes of the regression model.

Extracting specific attributes like coefficients from the regression model using the dollar sign.

Adding a regression line to a scatter plot using the 'abline' command with customization options.

Calculating and displaying confidence intervals for the model coefficients with the 'confint' command.

Adjusting the level of confidence for the confidence intervals using the 'level' argument.

Generating an ANOVA table for the linear regression model to examine the F-test.

Correlation between residual standard error and the square root of the mean squared error from the ANOVA table.

Upcoming discussion on regression diagnostic plots for examining regression assumptions in the next video.

Transcripts

hi! I am Mike Marin and in this video

we'll introduce "simple linear regression" using R.

Simple linear regression is useful for examining

or modelling the relationship between two numeric variables;

well in fact, we can fit a simple linear regression

using a categorical explanatory or X variable,

but we'll save that topic for a later video. We will be working

with the lung capacity data that was introduced earlier

in these series of videos. I have already gone ahead and imported the data into R

and attached it. We will model the relationship between

Age and Lung Capacity, with Lung Capacity

being our outcome, dependent, or Y variable.

We can begin by producing a scatter plot of the data

plotting Age on the x-axis and Lung Capacity on the y-axis

and we'll add a title here. We may also want to go ahead

and calculate Pearson's correlation between Lung Capacity

and Age. We can see

that there's a positive, fairly linear association between Age and Lung Capacity.

We can fit a linear regression in R using the

"lm" command. To access the Help menu you can type "help"

and in brackets the name of the command or simply place a question mark (?)

in front of the command name. Let's go ahead and fit a linear regression to this data

and save it in the object: mod. To do so

we'll fit a linear model predicting Lung Capacity

using the variable Age; it's important to note here

that the first variable we enter should be our Y variable

and the second variable the X variable. We can then ask for a summary

of this model. Here

we can see that we are returned a summary for the residuals

or errors, we can see the estimate of the intercept,

its standard error as well as the test statistic

and p-value for a hypothesis test that the intercept is zero.

it's worth noting that a test if the intercept is 0

is often not of interest. We can also see

the estimate the slope for Age, its standard error

and the test statistic and p-value for the hypothesis test

that the slope equal 0. You'll also notice that stars are used to identify

significant coefficients. here we can see the residual standard error

of 1.526, which is a measure of the variation of observations around the

regression line.

This is the same as the square root of the mean squared error

or Root-MSE. We can also see the r-squared

and the adjusted r-squared, as well as the hypothesis test

and p-value for a test that all the coefficients in the model are zero.

Recall in earlier videos

we saw the "attributes" command. Here we can ask for the attributes for our model,

and this will let us know which particular attributes are stored in this

object mod.

We can extract certain attributes using the dollar sign ($);

for example we may want to pull out the coefficients

from our model. it's worth noting

that we'll only need to type "coef" here

and R will know that these are the coefficients we're asking for.

We may also extract certain attributes in the following way:

here we'll ask for the coefficients of our model. Now let's go ahead and produce

that plot we had earlier.

If we would like to add

the regression line to this plot we can do so using the

"abline" command. Here we would like to add the line

for our regression model; and as we've seen earlier

we can add colours to this line as well as change

the line width using these commands. It's worth noting

that we will need to do something slightly different to add regression

lines for multiple linear regressions with multiple variables.

We've already seen the "coef" command to get our model coefficients.

We can produce confidence intervals for these coefficients using the "confint" command.

here we would like the confidence interval for model coefficients.

If you would like to change the level of confidence for these, we can do so

using the "level" argument within the "conf.int" command.

Here let's go ahead and have ninety-nine percent (99%) confidence intervals.

You recall that we can ask for summary of the model using the "summary" command.

We can also produce the ANOVA table for the linear regression model

using the "anova" command. Here we like the ANOVA table for this model.

You'll note that this ANOVA table corresponds to the f-test

presented in the last row of the linear regression summary.

One final thing to note is that the residual standard error

of 1.526 presented in the linear regression summary

is the same as the square root of the mean squared error

or mean squared residual from the ANOVA table.

We can see if we take the square root of the 2.3,

we get the same value as the residual standard error,

the slight difference is due to rounding error.

In the next video in this series we'll discuss how to produce

some regression diagnostic plots to examine the regression assumptions:

these include residual plots and QQ plots among a few others.

Thanks for watching this video and make sure to check out my other instructional videos