pols 7000x statistics in political science class 10 brooklyn college-cuny shang e. ha

POLS 7000X STATISTICS IN POLITICAL SCIENCE

CLASS 10BROOKLYN COLLEGE-CUNYSHANG E. HALeon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society© 2012 SAGE Publications

Chapter 9: Regression and Correlation Overview The Scatter Diagram Linear Relations and Prediction Rules Methods for Assessing the Accuracy for

Predictions Calculating r2

Pearson’s Correlation Coefficient r


Overview

Int

erva

l

Nom

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Depe

nden

tVa

riabl

eIndependent Variables

Nominal Interval

Considers the distribution of one variable across the categories of another variableConsiders the difference between the mean of one group on a variable with another group

Considers how a change in a variable affects a discrete outcome

Considers the degree to which a change in one variable results in a change in another


Overview

Int

erva

l

Nom

inal

Depe

nden

tVa

riabl

eIndependent Variables

Nominal Interval

Logistic Regression

RegressionCorrelation

Lambda

Confidence Intervals

T-Test

Overview


General Examples

Does a change in one variable significantly affect another variable?Do two scores tend to co-vary positively (high on one score high on the other, low on one, low on the other)?Do two scores tend to co-vary negatively (high on one score low on the other; low on one, hi on the other)?

Inte

rval

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Dep

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ntV

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Independent Variables

Nominal Interval

Inte

rval

Nom

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Dep

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ntV

aria

ble


Nominal Interval

Inte

rval

Nom

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Dep

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ntV

aria

ble


Nominal Interval


Specific Examples

Does getting older significantly influence a person’s political views?Does marital satisfaction increase with length of marriage?How does an additional year of education affect one’s earnings?

Inte

rval

Nom

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Dep

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Nominal Interval

Inte

rval

Nom

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Dep

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aria

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Nominal Interval

Inte

rval

Nom

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Dep

ende

ntV

aria

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Nominal Interval


Scatter Diagrams Scatter Diagram (scatterplot)—a

visual method used to display a relationship between two interval-ratio variables.

Typically, the independent variable is placed on the X-axis (horizontal axis), while the dependent variable is placed on the Y-axis (vertical axis.)


Scatter Diagram Example


A Scatter Diagram Example of a Negative Relationship


Linear Relationships Linear relationship – A relationship

between two interval-ratio variables in which the observations displayed in a scatter diagram can be approximated with a straight line.

Deterministic (perfect) linear relationship – A relationship between two interval-ratio variables in which all the observations (the dots) fall along a straight line. The line provides a predicted value of Y (the vertical axis) for any value of X (the horizontal axis.


Graph the data below and examine the relationship:


The Seniority-Salary Relationship


Example: Education & PrestigeDoes education predict occupational prestige? If so, then the higher the respondent’s level of education, as measured by number of years of schooling, the greater the prestige of the respondent’s occupation.Take a careful look at the scatter diagram on the next slide and see if you think that there exists a relationship between these two variables…


Take your best guess?

The mean age for U.S. residents.Now if I tell you that this person owns a skateboard, would you change your guess? (Of course!)With quantitative analyses we are generally trying to predict or take our best guess at value of the dependent variable. One way to assess the relationship between two variables is to consider the degree to which the extra information of the second variable makes your guess better. If someone owns a skateboard, that is likely to indicate to us that s/he is younger and we may be able to guess closer to the actual value.

If you know nothing else about a person, except that he or she lives in United States and I asked you to his or her age, what would you guess?


Take your best guess? Similar to the example of age and

the skateboard, we can take a much better guess at someone’s occupational prestige, if we have information about her/his years or level of education.


Equation for a Straight LineY= a + bX

where a = interceptb = slopeY = dependent variableX = independent variable

X

Y

a

riserunrise run = b


Bivariate Linear Regression Equation

Y = a + bX Y-intercept (a)—The point where the

regression line crosses the Y-axis, or the value of Y when X=0.

Slope (b)—The change in variable Y (the dependent variable) with a unit change in X (the independent variable.)

The estimates of a and b will have the property that the sum of the squared differences between the observed and predicted (Y-Y)2 is minimized using ordinary least squares (OLS). Thus the regression line represents the Best Linear and Unbiased Estimators (BLUE) of the intercept and slope.

ˆ

^


Now let’s interpret the SPSS output...

SPSS Regression Output (GSS)Education & Prestige


The Regression Equation

Prediction Equation:

Y = 13.874 + 1.384(X)

This line represents the predicted values for Y for any and all values of X

ˆ


• If a respondent had zero years of schooling, this model predicts that his occupational prestige score would be 13.874 points.

• For each additional year of education, our model predicts a 1.384 point increase in occupational prestige.

Interpreting the regression equation

Y = 13.874 + 1.384(X)


Ordinary Least Squares

Least-squares line (best fitting line) – A line where the errors sum of squares, or e2, is at a minimum.

Least-squares method – The technique that produces the least squares line.


Estimating the slope: b• The bivariate regression coefficient

or the slope of the regression line can be obtained from the observed X and Y scores.

)(

)()(

1

)(1

)()(

222 XX

YYXX

N

XXN

YYXX

SSbX

YX


Covariance =

Variance of X =

Covariance of X and Y—a measure of how X and Y vary together. Covariance will be close to zero when X and Y are unrelated. It will be greater than zero when the relationship is positive and less than zero when the relationship is negative.Variance of X—we have talked a lot about variance in the dependent variable. This is simply the variance for the independent variable

Covariance and Variance

1

)()(

N

YYXX

1)(

1))(( 2

N

XXN

XXXX


Estimating the Intercept

XbYa

The regression line always goes through the point corresponding to the mean of both X and Y, by definition.

So we utilize this information to solve for a:


Summary: Properties of the Regression Line

Represents the predicted values for Y for any and all values of X.

Always goes through the point corresponding to the mean of both X and Y.

It is the best fitting line in that it minimizes the sum of the squared deviations.

Has a slope that can be positive or negative; null hypothesis is that the slope is zero.


Coefficient of Determination


• Pearson’s Correlation Coefficient (r) — The square root of r2. It is a measure of association between two interval-ratio variables:

• Symmetrical measure—No specification of independent or dependent variables.

• Ranges from –1.0 to +1.0. The sign () indicates direction. The closer the number is to 1.0 the stronger the association between X and Y.

The Correlation Coefficient


r = 0 means that there is no association between the two variables.


Y

X

r = 0




Y

X

r = 1

r = 1 means a perfect positive correlation.




Y

X

r = –1

r = –1 means a perfect negative correlation.


pols 7000x statistics in political science class 10 brooklyn college-cuny shang e. ha

Documents

essentials of statistics

sage publications chapter

variable results

score low

score high

intervalratio variables

political science class

persons political views