

Interpretation of Coefficients
Beta0
 log(odds) when X1 = 0..., Xi = 0
 Often not intepretable (outside range of data)
 Sometimes can be thought of as background odds
Beta1
 Difference in log(odds) for two groups differing in their level of X1 by one unit, but otherwise similar for all other Xi (same as the log of the OR comparing these two groups)
 Because of the properties of logoarthms, beta1 is also the log of the odds ratio for two groups differing in their level of X1 by one unit, but otherwise similar for all other Xi
 (beta1)(1) is the log odds ratio between two groups differing in X1 by one unit, while (beta1)(5) is the log odds ratio between two groups differing in X1 by five units
e^beta1
 Odds ratio for two groups differing in their level of X1 by one unit, but otherwise agreeing in their level of all other Xi
 Similarly: (beta1)(5) is the log(odds) between two groups different in their value of X1 by 5, and e^(beta1)(5) is the odds ratio between two such groups
 (e^{beta1)(1) is OR comparing one unit apart, while (e}beta1)*(5) is OR comparing five units apart 


Interaction Terms
 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(YX,Z)) = beta0 + betaX(X) + betaZ(Z) + betaXZ(X*Z)
Coefficient Interpretation
 Beta0 still log(odds) of Y, given all Xi are zero
 BetaX is difference in log(odds) of Y between two groups differing by one unit of X, when Z = 0
 BetaZ 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 Fitting
 Model coefficients estimated by achieving "minimum deviance"
 No general formula exists for this; software is needed to do this
 The betahats identified through this process are called the maximum likelihood estimates (MLEs) of the true beta0 and beta1
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 modelbased, but CIs are slightly different 
