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Simple v. Multiple regression
- Simple: 1 axis
- Multiple: more than 1
- Assuming the relationship is linear
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Smoothing curve v. Regression line
- Smoothing curve: Not straight, fits a curve using a given percentage of the points
- Regression Line: straight line using a statistical method
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r2 value (coefficient of multiple determination)
- how well the model fits
- how much of the variation is y is explained by x
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adjusted r2
adjusts r2 by dividing each sum of squares by its associated degrees of freedom
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residual
observed-predicted (with the line)
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Why do we use simple linear regression?
- understand cause and effect relationships
- make decisions on cost
- to predict outcomes
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case or data point
observed pairs of explanatory x, response y variables
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Yi = β0 + β1Xi + εi
for i (1 to n)
- Simple linear regression model
- β0 = intercept
- β1 = slope
- ε = independent, normally distributed random errors with mean 0 and variance σ2
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ε ~iid N(0,σ2)
iid = independent
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how to find estimates for beta 1 and beta 0
- from plotting the regression line or
- Analytical procedure:
- b1 = Sum(Xi - mean X)(Yi - mean Y) / sum(Xi - mean X)2
- b0 = mean Y - b1 times mean X
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Maximum Likelihood
A way to find estimators b1 and b0 but they are the same for simple linear regressions
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to calc b1 by hand with points
- b1 = sum(ki times Yi)
- Ki = (Xi - mean X)/sum(Xi - mean X)2
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