November 9, 2010
Readings:
Chapter 12 Bivariate Data Analysis (Johnson and Reynolds) (pp 477490, 490498)
Chapter 8 Correlation and Linear Regression (Pollock) (pp. 170173)
COURSE LEARNING OBJECTIVES
Students will learn the research methods commonly used in behavioral sciences and will be able to interpret and explain empirical data.
Students will achieve competency in conducting statistical data analysis using the SPSS software program
Bivariate Regression
Lets Try a correlation from a 1990's State of America's Cities
Which correlations are insignificant and why?
Which Relationships are probably spurious and why?
What is the strongest correlation
What is the PRE value
What is the Practical Significance
Bivariate Regression
Bivariate linear regression is an important statistical technique in Political Science. It allows us to measure the effects of an independent variable on a dependent variable. Conducting this in SPSS is simple, but understanding can be difficult.
When you can use regression
Back to Scatterplots and how they Relate to Regression
The Dataset, and use the previous lecture notes to create a scatterplot
Y= Grade on Final Exam
X= Average Grade on First and Second Exams
What the Line is all about
Eyeballing the data, what might a grade be on the final, if he/she received a 80 on the midterm?
at what value does the line cross the yaxis?
In Bivariate regression, our model is
Y= a + b(X) +e
The Concept of Linear Regression is based on the slopeintercept model of Algebra
Where
Y= the value of the dependent variable. It can also be the predicted score
a(lpha)= the constant, or the point at which the line crosses the yaxis..
This number can be positive or negative and does not have to take on a realistic value based on your data!
This is the value of Y (the dependent variable) if the value of the independent variable is 0
X= the value of our independent variable. This is the score being used as the predictor.
b(eta)= the slope and direction of the line
which expresses the number of units of change in Y (the d.v.) accompany one unit of change in X.
The higher the value of beta, the steeper the slope.
Positive slopes mean the line goes up, Negative slopes mean the line goes down.
This is also called the regression coefficient. This tells us how much Y changes with each unit change in X.
If the regression coefficient is positive, x and y go up together. If it is negative, x goes up while y goes down.
e= The error term, what is missing from your regression.
The worse your independent variables explain your dependent variable, the larger the error term.
The farther your dots are from the line, the larger your error term.
You will not need to compute this!
Here is the information from the Example above (using the nonlegacy scatterplot function).
Y= Final Exam
a= 16.24 the constant (the value in where our line crosses the yaxis)
b= .780 (the slope of the line). It is positive, so that as the score on the early exams goes up, the score on the final exam goes up. It also means that every point the early exams increase, the final exams increase .780 points!
X (examavg)= oh, lets say a grade of 80 from the example above
Pop it into the formula
Y= a +b(x)
Y = 16.24 +.780(80)
Y=16.24 + 62.4
Y= 78.64
So the Predicted value of Y also called (Y prime) would be 78.64 if the student averaged an 80 on the first and second exams!
TO COMPUTE A BIVARIATE REGRESSION MODEL IN SPSS
Lets try out another dataset
You must have a ratiolevel dependent variable
You place your dependent and independent variables in the appropriate places and select ok.
READING THE OUTPUTS
Regression Outputs are very simple to understand.
Constant this says the value of the dependent variable, if the independent variable is zero.
INDEPENDENT VARIABLES (in this case Vehicle Weight) This is our independent variable.
For Every lb a car weighs, fuel efficiency decreases by .008 miles per gallon!
TStatistic= computed by dividing the beta coefficient by the standard error.
Lets Try an Example
Null Hypothesis There is no relationship between a state's Union population and the per capita income
Y= What would the predicted per capita income be for a state with a union % of 25?

Lets look at some examples about Health Care
Minority Population and Access to Health Care
For Each Example, What is the
Alpha Value what does it mean
beta value
direction
size what does it mean
Significance of Relationship and Why
Strength of Relationship (standardized beta coefficient)
Practical Significance
Median Family Income and Health Insurance
For Each Example, What is the
Alpha Value what does it mean
beta value
direction
size what does it mean
Significance of Relationship and Why
Strength of Relationship (standardized beta coefficient)
Practical Significance
WHAT ABOUT HERE?: The influence of female legislators or global defense spending?
What is the Constant?
Is our independent variable significant, why or why not?
Is it a positive or negative relationship?
How much change in the military budget results from a change in 1% of parliamentary seats held by females.
What is the potential practical significance of this?
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William Smith
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