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I would like understand more about the interpretation of coefficients in these models. Anyone can help me?

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It really depends on the particular model you are fitting, how you have fitted it and what the purpose of the analysis is. It can be difficult to interpret the coefficients directly, but you can still use them to test hypotheses in the normal way, provided you know what it being tested. As always, "Question 1: What is the Question?"

If you have used a Generalised Linear Models approach and have a simple model with one continuous x variable, then you can think of it as a simple linear regression but on the transformed response scale. Any inferences/predictions that you produce using the model coefficients need to be back-transformed through the logit or probit function to be meaningful.
I am interested to model a binary dependent variable with a number of independent variables, either continuous or categorical and I used logit model for establishing the relationship between probability and independent variables. The odds ratio of a variable , for example x4 was found to be 0.65. How can I interpret this coefficient ?. If the probit model producesd the coefficient of x4 as -0.53, how does it different from odds ratio in logit?.
As Matt states below, Generalized Linear Model techniques transform a response into a continuous range, so the response and coefficients on the right side look like a linear regression model--a linear combination of coefficients multiplied by the predictors that adds up to the prediction response. The prediction response is somewhere in the range of continous negative and positive numbers. A one unit change in any predictor, holding all other constant, will change the response by the amount of the coefficient. So if the model is y=3x, and x goes up 1 unit, then y goes up 3 units. While the original responses are 0s and 1s in the data, the logit or probit response can be transformed into a probability between 0 and 1.

These probabilities can be obtained from the prediction response via an "S curve" that approaches 0 on the bottom goes up, and then approaches 1. The curve accelerates as it moves away from 0, then has a linear portion, and flattens out on the top. So in the middle of the curve the coefficients act linear, have more impact at low values of the predictors, and less at higher levels. Also when a predictor are not continuous but 0 or 1, they can have interesting interpretations for the response. Contact me at my website and I can discuss the details.


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