Linear models, as their name implies, relates an outcome to a set of predictors of interest using linear assumptions. Regression models, a subset of linear models, are the most important statistical analysis tool in a data scientist’s toolkit. This course covers regression analysis, least squares and inference using regression models. Special cases of the regression model, ANOVA and ANCOVA will be covered as well. Analysis of residuals and variability will be investigated. The course will cover modern thinking on model selection and novel uses of regression models including scatterplot smoothing.
Introduction to Regression
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From the course by Johns Hopkins University
Regression Models
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From the lesson
Week 1: Least Squares and Linear Regression
This week, we focus on least squares and linear regression.
Meet the Instructors
Brian Caffo, PhDProfessor, Biostatistics
Bloomberg School of Public Health
Roger D. Peng, PhDAssociate Professor, Biostatistics
Bloomberg School of Public Health
Jeff Leek, PhDAssociate Professor, Biostatistics
Bloomberg School of Public Health
Hi, my name is Brian Caffo, and welcome to the introduction to regression
lecture as part of the regression Coursera class.
This is part of our data science specialization,
and this class is co-taught by my co-instructors Jeff Leek and Roger Peng.
We're all in the department
of biostatistics at the Johns Hopkins Bloomberg School of Public Health.
Regression is probably the most fundamental topic for the data scientist.
Before you do any fancy machine learning or anything like that, your go to
procedure is going to be linear regression or its generalization, linear models.
Regression has a rich history and it goes back to this fellow here, Francis Galton,
who invented the, the term and the concept regression, and the term and
the concept correlation, which is intimately tied to linear regression.
And he did many things,
one of which was predicting a child's height from a parent's height is a dataset
that we will investigate a lot just because of its historical significance.
And my friend and co instructor Jeff Leek noted that this
example is actually surprisingly still relevant.
And here's an article from the American Journal of Human Genetics.
Where they compare modern genetic data using high throughput bioinformatics
to old, to the old Victorian Era measurements created by Francis Galton,
and see that it doesn't do too much better.
So anyway, I give you the links there.
You can read them over.
It's a very interesting, and kind of quirky take on modern genetics and
genetic analysis, but it also goes back and reminds us about the importance of
what Francis Galton did and how ahead of his time he really was.
Let's go through a more modern example that's a little bit more similar to what
a data scientist would be doing in their day to day lives.
So, if you don't read the blog Simply Statistics you really should.
It's written by Jeff Leek, Roger Peng, both of who are co-instructors
in the data science specialization and Rafael Irazarry, who has a couple classes
on add X that you should check out who's at Harvard Biostatistics.
And Rafa wrote this post was talking about ball hogging and players.
And he basically, you know, here has a plot of the percentage of
the shots that were taken by Kobe Bryant, and the point differential.
The game specific point differential.
And he fit a linear regression to it, and
he made some of the claims in this, in the blog post.
And you can find it and, and read the full post.
But, for example, one of the claims is data supports the claim that if
Kobe stops ball hogging the Lakers will win more.
But, probably more relevant to this class is he makes this specific work, statement.
Linear regression suggests an increase of 1% in percent of shots taken by Ko,
Kobe results in a drop of 1.16 points, and
he gives a standard error in score differential.
So creating this sentence is what a large part of this class will be about.
How do you create this sentence, how do you interpret it, how do, you know,
Rafa's adhering to good statistical practice in giving a standard error.
And these are the sorts of things we're going to talk about in this class.
But I'd also like, just for this example to think about, you know,
not only how it was done, but do you agree with the analysis?
What are things that maybe Rafa should've thought about or what,
is this a complete analysis.
And maybe read the blog post and some of the comments.
Where some of the topics that the germane digression anals,
regression analysis are covered there.
Let's go back to our example, where Francis Galton was using
the parent's heights to predict the children's heights, and
talk about the kinds of questions we can answer.
So, we might want to use a parent's height, and try and
predict their children's heights.
We might want to find a parsimonious and
easily described mean relationships between the parent's and child's height.
So we don't want anything complicated.
We want the simplest possible relationship, and
that is what regression is best at.
While machine learning and other techniques generate highly elaborate,
in many cases, accurate prediction models, they tend to not be parsimonious.
They tend not to explain the data, and they tend not to generate new
parsimonious knowledge, whereas this is what regression is good at.
This is what regression is in fact best at.
We can talk about variation that's unexplained by the regression model.
The so called residual variation.
We can quantify the impact that things like genotype information has
beyond parent, parental height explained child's height and that's related to this
American Journal of Human, European Journal of Human Genetics paper that
looked at what re, residual variation was left after using parental height,
or what residual variation was left after using genetics.
And then,
very importantly, we're going to connect this back to the subject of inference.
How do we take our data, which is just a sample,
it only talks about that data set, and try to
figure out what assumptions are needed to extrapolate it to a larger population.
This is a deep subject called statistical inference.
We have a whole another class on it in the course area of data science
specialization.
But we're going to apply the tools of inference,
which I'm hoping most of you will have had as a prerequisite.
We're going to apply the tools of inference
to this new subject of regression.
And then here's a final question that Galton answered and
I, it's called regression to the mean or regression to mediocrity.
And the basic question is why do children's of very tall, why do children
of very tall parents tend to be tall, but a little bit shorter than their parents.
And why do the children of very short parents tend to be short, but
a little bit taller than their parents, and this is the famous idea of so
called regression to the mean.
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