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Start with all predictors, remove the predictor with the smallest (k-1)th order squared semi-partial correlation (based on the number of predictors in the equation). If a significant change in R2 occurs, stop and use model with the last removed predictor.
When a variable affects two or more predictors, example age is a confounding variable when examining the relationship between math scores and height.
Semi-partial Correlations (r1(2.3))
The PPMCC between two variables from which the effects of one (or more) other variables have been removed (or partialled) from only one of the two variables. AKA The Uniqueness Index when squared (a predictor's unique contribution to R2)
The product with the largest Beta has the highest semi-partial.
a correlation between two variables that does not result from any direct relation between them but from their relation to other variables. E.g. Height and math scores - the correlation appears because of the third variable, age. But when age is controlled for, the correlation disappears.
Partial Correlation (r12.3)
The PPMCC between two variables from which the effects of one or more other variables have been removed (i.e. partialled) from the two variables. Tells you the magnitude of the correlation.
- Notation is symmetric: r12.3 = r21.3
- ***Must take the square root from the SAS printout. ***
A variable that is uncorrelated with one of the original variables, yet when it is partialled out, the apparent relationship between the two original variables increases.
E.g. Math computational scores vs. Math Word Problem Scores, reading is a suppressor variable because it affects the relationship between comp and word problem scores.
Tolerance: Lower values=higher multicollinearity (≤.2)
VIF: Higher values=higher multicollinearity (≥4)
When a predictor variable has a perfect or almost perfect relationship with other predictor variables, while ignoring the dependent variable.
- Actual Value - Predicted Value
- Also known as SS error
Combination of Forward and Stepwise regression. Start with the single best predictor, add the best available predictor given what is already in the equation, if there is a significant change in R2, remove only the non-contributing predictors. Predictors need to be significant to get into and stay in model.
- Based on Theory
- The process to find the least complicated and best fitting model
Indicates change in b if observation is deleted.
Cut-off is 2/√n
Indicates influence by taking into account both the size of the error and the leverage.
Should be <1
Outliers: Studentized Resdiuals
t-value obtained by dividing error by its standard error
Cut off is >2
All of the y "hats" or prediceted y values fall on the line
Residual SS or Unexplained SS
Deviation between observed and predicted value
- Explained SS or RegressionSS
- Deviation between predicted value and average of dependent value
- Deviation between observed and average dependent value
- SStotal = Σ(y-ý)2
Assumptions of LR
- *The true conditional probabilities are a logistic function of the independent variables
- *No important variables are ommitted
- *No extraneous variables are included
- *The independent variables are measured without error
- *The observations are independent
- *The independent variables are not linear combinations of each other
pi (range: 0 to 1)
Model Effect Size
- R2L = (-2LLnull)-(-2LLk)/(=2LLnull)
- Explains proportion of null deviance accounted for by predictors
- χ2 = -2LLsmall-(-2LLlarge) <- Also critical value
- -2LL also known as deviance or misfit
***Want to see a drop in deviance***
predicted odds = e.2+.5x1+.1x2
- Replace x1 and x2 w/ given values
LR: Odds Ratios
predicted odds = e.2+.5x1+.1x2
(1-OR)*100 = % Differences
LN(Odds) (range: -infinity to +infinity)
LR: Odds Ratio
Odds for Group A/Odds for Group B
pi/1-pi (range: 0 to +infinity)
Seeking the predictor with the biggest correlation coefficient. Start w/ the single best predictor and add the best available predictor given what is already in the equation to achieve a significant change in R2.