How to explore/model this problem?

Question

I have data extracted from a state machine. Each row of data looks like this:

label1,label2,label3,impedance,did_state_change

The column data types are described as follows:

label1,label2,label3: These are nominal (categorical) values
impedance: This is a ratio value (float number)
did_state_change: This is a categorical value (boolean)

What I'm trying to do is to come up with a model that will help to estimate the probability of a state change, given a 4-tuple of (label1,label2,label3,impedance).

I am not sure which statistical model to use and will appreciate some help in choosing the correct analysis/model.

Answer 1

First, is it appropriate to assume that the trials are independent of each other? Let's assume that.

Hopefully the predictors label, label2, label3 do not have too many categories. If a categorical variable has $n$ possible values, then the model will have to estimate $n-1$ different parameters. If each has relatively few categories, that is fine, but if there are a large number of categories you may want to think whether they can be grouped together in some way to make the analysis more intuitive.

You have said that impedance is a ratio value. It may turn out that you don't get the best fit using impedance as a linear coefficient because its true relationship to the fit is non-linear. You might try fitting one model with impedance and one with log(impedance), and see which one has a better fit overall.

Luckily, your response variable is binary-valued, and you want to estimate the probability of a state change. You should try a binomial logistic regression, so that you are estimating $ln\left(\frac{p}{1-p}\right)$ rather than $p$ directly (this is called "logit" function). If you used the glm package in R, you would input the model approximately like this:

fitted.model <- glm(did_state_change ~ impedance + label1 + label2 + label3, family = binomial(link="logit"), data = yourDataFrame)

If there is reason to believe that the variance will not be uniform across trials, you can choose quasibinomial as the family rather than binomial.

Consider looking at some example problems in a textbook on the topic, e.g. Gelman and Hill (2007)

Let's say that you are also interested in finding a cutoff point for impedence, above which the probability of a state change, for given values of label_i, exceeds a certain value $x$. When you fit a basic binary logistic regression, you will get parameters specifying an intercept $\alpha$ and $\beta_i$ (coefficients of the predictors). Supposing that we do a dumbed down model, where only impedence is used as a predictor (call it $I$), then the model will return parameters $\alpha$ and $\beta_1$ for an equation having the form:

$$ln\left(\frac{p}{1-p}\right) = \alpha + \beta_1I$$

If $\beta_1$ is positive, then $p$ will increase with increasing $I$. In that case, you can plug in your cutoff point, $x$, and solve for $I$, getting:

$$I(p=x) = \frac{1}{\beta_1}\left\{ln\left(\frac{x}{1-x}\right) - \alpha\right\}$$

You can simplify some of this by programming a little logit function in R:

logit <- function(x){log(x/(1-x))}

Then if you type in logit(x) it will return $ln\left(\frac{x}{1-x}\right)$.

Answer 2

I would recommend a binomial regression model. It gives the probability of taking one of two values based on a matrix of explanatory variables. http://en.wikipedia.org/wiki/Binomial_regression The R command is glm(family=binomial)

Logit models remove much of the interpretation on the coefficients, because you start getting into relative utility theory. Very valuable in lots of applications, but not really when you're describing processes that don't involve valued behavior.