2.03 Regression - Finding the line

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From the course by University of Amsterdam

Basic Statistics

1679 оценки

University of Amsterdam

Basic Statistics

1679 оценки

Course 3 of 5 in the Specialization Methods and Statistics in Social Sciences

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

From the lesson

Correlation and Regression

In this second module we’ll look at bivariate analyses: studies with two variables. First we’ll introduce the concept of correlation. We’ll investigate contingency tables (when it comes to categorical variables) and scatterplots (regarding quantitative variables). We’ll also learn how to understand and compute one of the most frequently used measures of correlation: Pearson's r. In the next part of the module we’ll introduce the method of OLS regression analysis. We’ll explain how you (or the computer) can find the regression line and how you can describe this line by means of an equation. We’ll show you that you can assess how well the regression line fits your data by means of the so-called r-squared. We conclude the module with a discussion of why you should always be very careful when interpreting the results of a regression analysis.

2.03 Regression - Finding the line3:45

2.04 Regression - Describing the line7:33

2.05 Regression - How good is the line?5:35

Meet the Instructors

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

A recent study shows that it might be a good idea to eat a lot of chocolate.

This scatterplot shows that a country's annual chocolate consumption per person,

so how much chocolate someone eats in a year, is positively related to the number

of Nobel prize winners with ten million people in a country.

Notice that in this scatterplot chocolate consumption is displayed

as the independent variable and

the number of Nobel prize winners as the dependent variable.

The units of analysis in this scatterplot are countries.

And you can see that the correlation is pretty high.

In fact, the Pearson's r correlation coefficient here is 0.93.

This suggests that although eating a lot of chocolate might make you fat,

it also makes you smart.

The Pearson's r tells you how strong the linear correlation between the two

continuous variables is.

This linear correlation can be displayed by a straight line.

In our case that's this line.

This is what we call the regression line, and

in this video, I'll tell you how we can find the regression line.

It is important to know how we find the line, not only because the regression line

shows you, at a glance, how two variables are related, but

also because it forms the basis of many statistical analyses.

So how do we find the regression line?

Imagine that you draw every possible straight line through this scatterplot.

So, you draw this one, this one, this one, this one, and every other possible line.

That's a huge number of lines, so

in practice it will be almost impossible to do it.

However, for now imagine that you have superhuman powers and

that you are able to do it.

Next you measure for every possible line the distance from the line to every case.

So in this case to every flag in the scatterplot.

Let me give you an example based on a random line, say, this one.

You measure the vertical distance between Japan and

the line, the distance between Spain and the line and so on.

Until you know the distance to the line of every case in your study.

Every distance is called a residual and you end up with positive residuals,

the distances from cases above the line to the line displayed in blue.

And negative residuals,

distances from cases below the line to the line displayed in red.

You measure these residuals for every possible line through the scatterplot.

So not only for this line, but also for this line, for this line, and

for this line.

And for every other possible line through the scatter plot.

Eventually, you choose the line for

which the sum of the squared residuals is the smallest, and that's this one.

Why the squared residuals?

Because positive and negative residuals cancel each other out.

The sum of the length of the positive residuals, the blue lines,

is exactly as big as the sum of the length of the negative residuals, the red lines.

The best fitting line is called a regression line and

the name of the method of analysis is called ordinary least squares regression,

which refers to the way we have found the line.

In practice, it almost impossible to draw every possible line and

to compute for every single possible line all the residuals.

Luckily, mathematicians have found a trick to find the regression line.

I won't explain how this trick works here because it is pretty complicated.

For now it suffices to know that it's based on minimizing the sum of the squared

residuals.

To summarize, you have learned two essential things.

First, you now know how the computer finds the regression line.

And second,

you have learned that eating chocolate might well help you to pass this course.

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