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education/statistics/Correlation and Regression.md
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@ -99,6 +99,13 @@ $$ \hat{y} = \frac{x-\bar{x}}{\sigma_x} * r * \sigma_y + \bar{y} $$
- For a positive association, for every $\sigma_x$ above average we are in $x$, the line predicts $y$ to be $\sigma_y$ standard deviations above y.x - For a positive association, for every $\sigma_x$ above average we are in $x$, the line predicts $y$ to be $\sigma_y$ standard deviations above y.x
- There are two separate regression lines, one for predicting $y$ from $x$, and one for predicting $x$ from $y$ - There are two separate regression lines, one for predicting $y$ from $x$, and one for predicting $x$ from $y$
- Do not extrapolate outside of the graph - Do not extrapolate outside of the graph
(Ch 12, stat 1040)
Predicting a y value for a given x value can be calculated when given the regression equation.
$$ y = mx + b $$
Where $y$ is the predicted value, $m$ is the slope, x is the given val and the $b$ is the intercept.
$$ slope = \frac{r * \sigma_y}{\sigma_x} $$
### The Regression Effect ### The Regression Effect
- In a test-retest situation, people with low scores tend to improve, and people with high scores tend to do worse. This means that individuals score closer to the average as they retest. - In a test-retest situation, people with low scores tend to improve, and people with high scores tend to do worse. This means that individuals score closer to the average as they retest.
- The regression *fallacy* is contributing this to something other than chance error. - The regression *fallacy* is contributing this to something other than chance error.
@ -114,9 +121,13 @@ $$ \sqrt{1-r^2}(\sigma_y) $$
- To approximate the R.M.S error for a scatter diagram, take a high value and a low value for a given $x$ coordinate, and divide by 4, because r.m.s error is within $2\sigma$ of either side of the line. - To approximate the R.M.S error for a scatter diagram, take a high value and a low value for a given $x$ coordinate, and divide by 4, because r.m.s error is within $2\sigma$ of either side of the line.
- 68% = $2\sigma$, 95% = $4\sigma$ - 68% = $2\sigma$, 95% = $4\sigma$
- RMS can help determine which observations are outliers. Typically if a value is more than *2 r.m.s* away from the prediction estimate, it is considered to be an outlier - RMS can help determine which observations are outliers. Typically if a value is more than *2 r.m.s* away from the prediction estimate, it is considered to be an outlier
- The RMS error is only appropriate for homoscedastic scatter diagrams (football shape)
- Heteroscedastic scatter diagrams should not be used to make a prediction, because they do not follow a football shape
- Homoscedastic scatter diagrams can be used to make predictions, because they follow a football shape
- Heteroscedastic scatter diagrams should not be used to make a prediction
- homoscedastic scatter diagrams can be used to make predictions, because they follow
--- ---
# Terminology # Terminology
| Term | Definition | | | Term | Definition | |