2.04 Regression - Describing the line

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Basic Statistics

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University of Amsterdam

Basic Statistics

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Cours 3 sur 5 dans 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.

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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

Rencontrer les enseignants

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

Does eating chocolate make you smart?

A recent study indicates that this might be the case.

In countries where citizens eat more chocolate there are more Nobel Prize

winners per 10 million people.

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

positively correlates with the number of Nobel Prize winners

per 10 million in a country.

The regression line is the straight line that describes the linear relationship

between the two variables best.

But how can we describe what this line looks like?

This is a very important question, because by describing the line with a formula

we can easily communicate a regression analysis to other people,

predict the number of Nobel Prize winners in other countries, and

identify countries that do not fit the pattern.

Based on the regression line in this scatterplot, we would predict that

a country with an annual chocolate consumption of 6 kilograms per year

per capita would have about 11 Nobel prize winners per 10 million people.

Similarly, based on the same line, we would predict that a state with an annual

chocolate consumption of 11 kilograms per year per capita would have about 25 prize

winners per 10 million people.

For most countries this prediction will not be completely correct.

After all, most countries are not exactly on the line.

However, it is the best prediction that we can make

based on the information that we have.

There is one simple formula with which we can describe the regression line.

And that's this one.

Y hat equals a plus b times x.

Y hat is not the actual value of y but it represents the predicted value of y.

For example, when x equals 12, y hat equals 28.

Notice that the actual value of y in this case is 33.

However the predicted value of y is the value of y on the regression line.

This means that all the values exactly on the regression line are y hats.

A is what we call the intercept, or the constant.

It is the predicted value of y when x equals 0.

It is in other words, the predicted value of y

with the regression line crosses the y axis and x does equal 0.

In our case, that's -5.63.

Notice that this value has no substantive meaning.

It is impossible to have -5.63 Nobel prize winners per 10 million people.

It only has a mathematical purpose to describe the regression line.

B is what we call the regression coefficient or the slope.

It is the change in y hat when x increases with one unit.

In our case we see that when x increases with one unit for example,

from 4 to 5, the predicted value of y increases with 2.80 units.

Because we have a straight line the slope of the regression line is

the same everywhere.

So also if we look at what happens when x increases from 8 to 9,

y hat increases with 2.80 units.

The regression coefficient in our example is 2.80.

This leads to the following regression equation.

y hat equals 5.63 plus 2.80 times x.

Take a look at these two regression lines.

They have the same regression coefficients, or b values.

When x increases with one unit, the predicted y value of line one and

line two increases with the same amount.

These lines have different intercepts, or a values, however.

After all, they cross the y-axis on different positions.

These two regression lines have different regression coefficients.

When x increases with one unit,

y hat of line one increase more than y hat of line two.

Yet the intercepts of these two lines are the same,

because they cross the y-axis at the same spot.

I have already shown you that we can use the regression line to predict

y values based on given x values.

We can also use the regression formula to make predictions.

Let's take our regression formula.

y hat = -5.63 + 2.80x.

We can predict our y values by using the formula.

What if x equals 3.5?

We get -5.63 + (2.80 x 3.5).

That makes 4.17.

So y hat equals 4.17 here.

And what if x equals 10.21?

Then you get -5.63 + (2.80 x 10.21).

That makes y hat equals 22.96.

We get the same values when we just look at our regression line.

For an x value of 3.5 we get a predicted y value of about four.

And for an x value of 10.21 we get a y hat value of about 23.

You can already see that there was a huge advantage of working with the formula.

You can make much more precise predictions.

Usually, the computer finds the regression line for you so

you don't have to compute it yourself.

However, when you know the means and

standard deviations of your variables and also the corresponding Pearson's r,

You can compute the regression equation by means of two formulas.

This formula and this formula.

The first formula computes a regression coefficient by multiplying Pearson's

r with the standard deviation of y divided by the standard deviation of x.

This shows that the regression coefficient

is in fact an unstandardized version of the Pearson's r.

When the Pearson's r equals 0, the regression coefficients equal 0.

When the Pearson's r is a positive number, so is the regression coefficient, and

when the Pearson's r is negative,

the regression coefficient is negative as well.

These are the means, standard deviations and Pearson's r of our study.

So to find the regression coefficient we multiply 0.93 with (11.87 divided by 3.95).

The outcome is 2.79.

The second formula computes the intercept by multiplying the computive regression

coefficient with the mean of x and

then by subtracting the outcome from the mean of y.

So 13.17 minus (2.79 times 6.71).

That makes -5.55.

The regression equation is -5.55 + 2.79x.

The difference with the regression equation found by the computer which was

this one is due to rounding error.

I worked with the roundup means, standard deviations, and Pearson's r.

This has led to a less precise regression equation.

So when working with these formulas try to round off as little as possible.

Congratulations!

You are now able to do a regression analysis and calculate predicted values.

Understanding the basics of regression is of essential importance to be able to

understand inferential regression procedures later on.

So watch this video a couple of times.

If independent variable x is the number of times you watch this video, and

dependent variable y is your knowledge of regression analysis, when you do

a regression analysis the regression slope will be a positive number.

If you don't understand what that means, re-watch this video immediately.

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