I have a logistic regression like this:
Y = a1 + b1*(number positive scores) + b2*(number negative scores) + b3*Z + b4*Z*(number positive scores) + b5*Z*(number negative scores) + additional non-interaction terms
Y is the probability of an outcome that takes on binary values (0,1).
Z is a continuous variable; I am trying to determine if it is significant
Number of positive and negative scores take on integer values between (0,5)
When I run the regression using
matlab, I find that all coefficients are significant except for b3; b1 and b4 are positive whereas b2 and b5 are negative. I would like to draw conclusions about whether Z is in fact a significant factor in the outcome variable via interactions b4 and b5, but I understand that in a logistic regression all coefficients and particularly interactions need to be evaluated in the context of specific values of the independent variable x. So this is what I have done so far:
Let's say bhat is the estimated vector of coefficients
xhat is the sample mean of
x=(num pos scores, num neg scores,Z,Z*num pos scores, Z*num neg scores). Also say that bi_sigma is the estimated standard error on the ith element of bhat and
xi_sigma is the sample standard deviation of the ith element of x. Let's say
L(bx) is the logistic cdf evaluated at bx.
Am I right to evaluate the odds ratios
exp(bhat*xhat+bi_sigma*xi_hat), then determine whether each element bi_hat is significant based on whether the range of these odds ratios is strictly greater than or less than one? In other words, my thinking is that if the odds ratios don't include one then they significantly improve or decrease the odds. For instance an odds ratio range of (1.3,2) reflects a bi that improves the odds for mean levels of x. Yes?
Secondly, am I correct to evaluate the first differences,
L(bhat*xhat+bi*xi_sigma)-L(b*xhat) as two measures of the size of the impact of each
xi. Can I do this for the interaction also?
Thanks very much for any advice or help. If there are references that you would suggest for this, I would appreciate that too. All the examples I've found online involve binary or categorical variables only, and no continuous variables.