2.08 Example Pearson's r and regression

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统计基础

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University of Amsterdam

统计基础

1926 個評分

課程 3（共 5 門，Specialization 社会科学的方法和统计）

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.

從本節課中

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.

2.06 Correlation is not causation5:22

2.07 Example contingency table3:26

2.08 Example Pearson's r and regression8:27

與講師見面

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

Social scientists have shown that a leader's physical height is related to his

or her success.

Suppose you want to test if you can replicate this result.

To do that, you look at the heights and

average approval ratings of the four most recent presidents of the United States.

You employ this data matrix, and your goal is to answer four related questions.

One, is there a linear relationship between the two variables?

Two, what is the size of Pearson's r correlation coefficient?

Three, what do the regression equation and the regression line look like?

And four, what is the size of r-squared?

Let's start with the first question.

Is there a linear relationship between the two variables?

To answer that question, we make a scatterplot.

To make a scatterplot, you must first decide what's the dependent variable and

what's the independent variable.

In this case, it's more likely that the leader's physical height influences his or

her approval ratings than that approval ratings affect the leader's height.

After all, it would be silly to expect the leader to become taller once his or

her approval ratings get better.

So, the independent variable, height, goes on the x axis, and

the dependent variable, approval rating, on the y axis.

Based on the minimum and maximum values of our variables, we scale our axis.

Our independent variable, height, ranges from 182 centimeters to 188 centimeters.

We, therefore, use a scale from 180 to 190 centimeters.

Our dependent variable ranges from 47 through 60.9, we,

therefore, scale this axis from 45 through 65.

Next, we decide based on our data matrix

where we should position the four presidents.

Obama is 185cm tall and has an approval rating of 47,

so he should be positioned here.

Bush Jr. has a physical height of 182 centimeters and

an average approval rating of 49.9, so we position him here.

Clinton and Bush Sr. are located here.

Now we can answer the first question, yes, there seems to be

linear relationship between a leader's height and his approval rating.

The line describing this relationship goes up,

which means the correlation between the two variables is positive.

The second question is what the value of Pearson's r is.

To compute Pearson's r, we need this formula.

To start with, we need to compute all the z-scores of both our independent and

our dependent variable.

To do that, we need the means and standard deviations of these variables.

I assume that you know how to compute them, so I will just give them to you.

The mean of the independent variable, height, is 185.75 centimeters,

and the standard deviation is 2.87 centimeters.

The mean approval rating, the dependent variable in the study,

is 53.23, and the standard deviation is 6.12.

First, we compute the z-scores for

our independent variable by subtracting the mean from every original score and

then dividing the outcome by the standard deviation.

We do that here.

185 minus 185.75 divided by 2.87,

that makes -0.26132.

We also do that for the other scores, here are the results.

We then repeat that for the dependent variable.

47minus 53.23 divided by 6.11 makes -1.01964.

And we do that for the other cases, too.

The next step is to multiply the z-scores of every case with each other.

For the first case, this results in -0.26132 multiplied

with -1.01964, that makes 0.266456, and so on.

We have now finished this part of the formula.

Next, we have to add up all these values, that makes 2.202649.

Finally, we have to divide by (n- 1).

The n is 4, so n minus 1 equals 4, minus 1, is 3.

The result, rounded up, is 0.73, that's our Pearson's r.

It indicates that there's a rather strong and positive linear correlation between

a leader's body height and his average approval rating.

The next step is to find the regression equation.

The computer finds the regression line by looking for

the line that minimizes the sum of the squared residuals.

You do not have to do this yourself.

Luckily, the complicated procedure boils down to two rather simple formulas.

One formula to compute the regression coefficient, that's this one, and

one formula to compute the intercept, that's this one, and

together these formulas give you your regression line.

We already have all our necessary ingredients, so

now we can use the formulas.

The regression slope is 0.73 multiplied with

6.12 divided by 6.87, that makes 1.56.

The intercept is 53.23 minus 1.56

multiplied with 185.75,

that makes -237.11.

The regression equation is y hat minus 237.11 plus 1.56 times x.

The intercept indicates the predicted y value is -237.11 when x is zero.

This number has no substantive meaning because a physical height of 0 meter

is impossible.

The intercept only serves mathematical purposes.

It makes it possible to draw the line.

With the regression equation found, we can predict the value of our dependent

variable when our independent variable equals 182 centimeters,

the minimum value in the sample.

That's -237.11 plus 1.56 times 182,

that makes 46.81, that's here.

We can also do that for our maximum value, that's -237.11 plus

1.56 multiplied with 188, that's 56.17, and that's here.

We can now draw the regression line.

This line is the straight line that best represents the linear relationship

between x and y.

It is the line for which the sum of the squared residuals is the smallest.

We can, of course, predict y-values for every possible x-value.

All of these predicted y-values,

or y-hats, are located on the regression line.

The fourth question we want to answer is, what the value of r-squared is.

That's easy, it's Pearson's r squared,

so 0.73 multiplied with 0.73 equals 0.53, but

how should we interpret this number?

Well, we can say that the prediction error is 53% smaller when we

use the regression line than when we employ the mean of the dependent variable.

We an also say that 53% of the variation, or

the variance in the dependent variable, is explained by our independent variable.

So, what have we done in this video?

First, we determined the straight line that describes the relationship

between our two variables best.

Second, we have predicted values of our dependent variable based on the line and

the corresponding regression equation.

And third, by means of Pearson's r and r-squared, we have investigated how

well the line fits our data, but what have we learned substantively?

Well, that tall leaders are more successful than short leaders.

However, this conclusion is based on a sample of only four American presidents,

who don't differ much from each other when it comes to their physical height.

It is up to you to decide if this warrants far reaching inferences about

relationship between height and approval ratings.

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