Describing Relationships - Chapter 15

# Describing Relationships - Chapter 15 - Describing...

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Samuel Clark Department of Sociology, University of Washington Institute of Behavioral Science, University of Colorado at Boulder Agincourt Health and Population Unit, University of the Witwatersrand Describing Relationships: Regression, Prediction, and Causation Chapter 15

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March 6, 2007 2 Regression lines We can summarize a “straight-line” relationship between two variables with a regression line A regression line summarizes the relationship between two variables: When one of the variables helps explain or predict the other When the relationship has the shape of a straight line Regression describes a relationship between a response variable and one (or more) explanatory variable(s)
March 6, 2007 3 Regression lines …

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March 6, 2007 4 Example 1: Fossil bones Previously we saw that the lengths of two bones in fossils of Archaeopteryx closely follow a straight-line pattern The next slide shows a plot of the relationship for existing complete fossils with a line drawn to summarize the relationship Suppose we have another incomplete fossil with a 50cm femur but no humerus Given the relationship we have identified with the complete fossils, can we predict the length of the missing humerus?
March 6, 2007 5 Predicted humerus Length: ~56 cm

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March 6, 2007 6 Example 1: Fossil bones … The straight-line pattern we have identified between humerus length and femur length is very strong, and we feel comfortable predicting humerus length from femur length To do this, find the 50cm point along the horizontal axis, draw a vertical line straight up to the line diagonal line that summarizes the relationship, and from the point where they cross, draw a horizontal line straight to the left until it intersects with the vertical axis. The intersection point on the vertical axis is the length of the humerus that goes with a 50cm femur: about 56cm This is the “up and over” method
March 6, 2007 7 Example 1: Fossil bones … ~56cm is the length of the humerus of this fossil if its humerus-femur point lay exactly on the line that summarizes the relationship for the existing fossils Because all the points for the other fossils are very close to the summary line, we can be pretty sure that the point for this fossil will be too In other words, we think the prediction of ~56cm is pretty accurate!

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March 6, 2007 8 Example 2: Presidential elections Republican Ronald Reagan was elected president twice in 1980 and 1984 The plot on the next slide shows the percentage of voters in each state who voted for Reagan’s Democratic opponents in 1984 (vertical axis) and 1980 (horizontal axis) There is a positive, straight-line relationship between these percentages some states tend to vote Democrat and other Republican A regression line is drawn to summarize this relationship
March 6, 2007 9

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March 6, 2007 10 Example 2: Presidential elections … We can use this line to predict a state’s 1984 vote from its 1980 vote There is a lot more scatter about this regression line than the fossil regression line: r = 0.994 for the fossils
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