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Copyright 2009, The Johns Hopkins University and John McGready. All rights reserved. Use of these materials permitted only in accordance with license rights granted. Materials provided “AS IS”; no representations or warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for obtaining permissions for use from third parties as needed.
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John McGready Johns Hopkins University
Simple Linear Regression
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Review: The Equation of a Line
Section A
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The Equation of a Line
Recall, from algebra, there are two values which uniquely define any line - Y-intercept—where the line crosses the y-axis (when x = 0) - Slope—the “rise over the run”—how much y changes for each
one unit change in x
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The Equation of a Line
Recall, from algebra, there are two values which uniquely define any line
y = mx + b - b = y-intercept - m = slope
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The Equation of a Line
Of course statisticians must have their own notation!
y = bo + b1x - bo = y-intercept - b1 = slope
y = βo + β1x - βo = y-intercept - β1 = slope
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The Intercept, βo
The intercept βo is the value of y when x is 0 - It is the point on the graph where the line crosses the y
(vertical) axis, at the coordinate (0, βo)
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βo
y = βo + β1x
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x
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y = βo + β1x
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x
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β1 y = βo + β1x
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x
Another interpretation: β1 is difference in y-values for x+1 compared to x
This change/difference is the same across the entire line
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x
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β1
β1
β1
y = βo + β1x
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x: β1 is difference in y-values for x+1 compared to x
All information about the difference in the y-value for two differing values of x is contained in the slope!
For example: two values of x three units apart will have a difference in y values of 3* β1
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The Slope, β1
For example: two values of x three units apart will have a difference in y values of 3* β1
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β1
β1
β1
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The Slope, β1
For example: two values of x three units apart will have a difference in y values of 3×β1 (3β1 )
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β1
β1
β1
3β1
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x: β1 is difference in y-values for x+1 compared to x - If slope β1 = 0, this indicates that there is no association:
(i.e., the values of y are the same regardless of the values of x) - If slope β1 > 0, this indicates that there is a positive association:
(i.e., the values of y increase with increasing values of x) - If slope β1 < 0, this indicates that there is a negative
association: (i.e., the values of y decrease with increasing values of x)
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The Slope, β1
The slope β1 is the change in y corresponding to a unit increase in x: β1 is difference in y-values for x+1 compared to x
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β1 = 0
β1 > 0
β1 < 0
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The Equation of a Line
In linear regression situations, points don’t fit exactly to a line
We estimate a line that relates the mean of an outcome y to a predictor x
- E[y] = estimated “expected” (mean) value of y - = estimated y-intercept - = estimated slope
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The Equation of a Line
and are called estimated regression coefficients
These two quantities are estimated using the data - Line estimated is line that “fits the data best”
Many times the equation is just written as the following:
or
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The Equation of a Line
and are called estimated regression coefficients
We will see that in a regression context, is nothing more than estimated mean difference in y between two groups who differ by one unit in x - i.e., how much the mean of y changes for a one-unit increase
in x
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