Logistic Regression Review Cheat Sheet (DRAFT) by Chrish0204

Beta-0
- log(odds) when X1 = 0..., X-i = 0
- Often not intepr­etable (outside range of data)
- Sometimes can be thought of as background odds

Beta-1
- Difference in log(odds) for two groups differing in their level of X-1 by one unit, but otherwise similar for all other X-i (same as the log of the OR comparing these two groups)
- Because of the properties of logoar­thms, beta-1 is also the log of the odds ratio for two groups differing in their level of X-1 by one unit, but otherwise similar for all other X-i
- (beta-­1)(1) is the log odds ratio between two groups differing in X-1 by one unit, while (beta-­1)(5) is the log odds ratio between two groups differing in X-1 by five units

e^beta-1
- Odds ratio for two groups differing in their level of X-1 by one unit, but otherwise agreeing in their level of all other X-i
- Similarly: (beta-­1)(5) is the log(odds) between two groups different in their value of X-1 by 5, and e^(bet­a-1)(5) is the odds ratio between two such groups
- (e^{beta-1)(1) is OR comparing one unit apart, while (e}beta-1­)*(5) is OR comparing five units apart

- Relati­onship between X and Y is moderated through Z
- This means the OR for Y between two groups that differ on X varies with Z

log(od­ds(­Y|X,Z)) = beta-0 + beta-X(X) + beta-Z(Z) + beta-X­Z(X*Z)

Coeffi­cient Interp­ret­ation
- Beta-0 still log(odds) of Y, given all X-i are zero
- Beta-X is difference in log(odds) of Y between two groups differing by one unit of X, when Z = 0
- Beta-Z is difference in log(odds) of Y between two groups differing by one unit of Z, when X = 0
- Possible that X and/or Z = 0 outside of range of data, but still need to include this

Intera­ction Term
- BetaXZ is change in slope per 1 unit difference in X, comparing 1 unit differ­ences in Z
- e^beta­(in­ter­action) is ratio of the OR when intera­cting variable =1 compared to when intera­cting variable = 0
- "The intera­ction term is the difference in log(OR) comparing situations where the intera­cting variable differs by one unit."
- Note that on the log scale, this is a differ­ence, whereas on the OR scale, it is a ratio

- Model coeffi­cients estimated by achieving "­minimum devian­ce"
- No general formula exists for this; software is needed to do this
- The beta-hats identified through this process are called the maximum likelihood estimates (MLEs) of the true beta-0 and beta-1

Likelihood Theory
- Provides tools for converting modeling assump­tions into SE estimates
- Assumes that in popula­tion, Y and X really do have logistic relati­onship; however, can still get "best estima­tes­" with minimum deviance (just no SEs/CIs)

Estimation Theory
- Use robust estimates to obtain SEs/CIs of coeffi­cients in the model, even when true population is not a logistic relati­onship
- Point estimates will be same as model-­based, but CIs are slightly different