Of course the intrinsic randomness might have a relatively small impact in terms of variability in our outcome. In this case, a value of will never be attainable. Because of this, it will never be possible to predict with almost 100% certainty whether a new subject will have Y=0 or Y=1. From this perspective, the definition of seems quite appropriate – the gold standard value of 1 corresponds to a situation where we can predict whether a given subject will have Y=0 or Y=1 with almost 100% certainty.Īn alternative perspective says that there is, at some level, intrinsic randomness in nature – parts of quantum mechanics theory state (I am told!) that at some level there is intrinsic randomness. In this case, our stochastic probability models are models which include randomness which is caused by our imperfect knowledge of predictors or our inability to correctly model their effects on the outcome. From one perspective, we might think of nature (or whatever it is we’re investigating and trying to predict) as deterministic. The definition of also raises (I think) an interesting philosophical point. Of course in most empirical research typically one could not hope to find predictors which are strong enough to give predicted probabilities so close to 0 or 1, and so one shouldn’t be surprised if one obtains a value of which is not very large. The log of 1 is 0, and so the log-likelihood value will be close to 0. This means that the likelihood value for each observation is close to 1. If this is the case, the probability of seeing when is almost 1, and similarly the probability of seeing when is almost 1. How would this happen? Remembering that the logistic regression model’s purpose is to give a prediction for for each subject, we would need for those subjects who did have, and for those subjects who had. Next, suppose our current model explains virtually all of the variation in the outcome, which we’ll denote Y. Therefore the ratio of the two log-likelihoods will be close to 1, and will be close to zero, as we would hope. If the model has no predictive ability, although the likelihood value for the current model will be (it is always) larger than the likelihood of the null model, it will not be much greater. For individual binary data, the likelihood contribution of each observation is between 0 and 1 (a probability), and so the log likelihood contribution is negative. To try and understand whether this definition makes sense, suppose first that the covariates in our current model in fact give no predictive information about the outcome. Where denotes the (maximized) likelihood value from the current fitted model, and denotes the corresponding value but for the null model – the model with only an intercept and no covariates. McFadden’s R squared measure is defined as
![multiple r squared xlstat multiple r squared xlstat](http://cameron.econ.ucdavis.edu/excel/mregression3.gif)
the parameter estimates are those values which maximize the likelihood of the data which have been observed. Logistic regression models are fitted using the method of maximum likelihood – i.e. There are certain drawbacks to this measure – if you want to read more about these and some of the other measures, take a look at this 1996 Statistics in Medicine paper by Mittlbock and Schemper. In this post I’m going to focus on one of them, which is McFadden’s R squared, and it is the default ‘pseudo R2’ value reported by the Stata package.
![multiple r squared xlstat multiple r squared xlstat](https://i.ytimg.com/vi/WJ6Zxg-1ObI/hqdefault.jpg)
Over the years, different researchers have proposed different measures for logistic regression, with the objective usually that the measure inherits the properties of the familiar R squared from linear regression. How is R squared calculated for a logistic regression model? Well it turns out that it is not entirely obvious what its definition should be. Perhaps the second most common type of regression model is logistic regression, which is appropriate for binary outcome data. Of course not all outcomes/dependent variables can be reasonably modelled using linear regression. In previous posts I’ve looked at R squared in linear regression, and argued that I think it is more appropriate to think of it is a measure of explained variation, rather than goodness of fit.