Logistic model output when all inputs are zero

Question

Consider a case where I have developed a predictive model using logistic regression. Now the logistic models gives a probability even when all the inputs are zero (because of the intercept). Now consider the case of predicting if a subject has disease or not. Here the probability carries a lot of significance. In this case, even when all the inputs are zero, there is a probability that the subject has disease. Is there a way we can remove this? Can we make a condition in the output that when all the inputs are zeros then the probability is zero, or can we subtract the probability with the probability when all inputs are zero?

To explain further, consider this example from Wikipedia https://en.wikipedia.org/wiki/Logistic_regression#Example:_Probability_of_passing_an_exam_versus_hours_of_study

The logistic model is

             Coefficient    Std.Error   z-value    P-value (Wald)
Intercept    −4.0777        1.7610      −2.316     0.0206
Hours         1.5046        0.6287       2.393     0.0167

And the output of the model is

Hours of study  Probability of passing exam
1               0.07
2               0.26
3               0.61
4               0.87
5               0.97

Here when the "Hours of study" is 0, the output is 0.02. Now how can we remove this bias?

Answer 1

If you know a priori that the probability of the event $p_i$ must be zero when a covariate $x_i$ is zero you can model this by including $\ln x_i$ instead of $x_i$ in your model. You then have $$ \operatorname{logit}p_i = \ln \frac{p_i}{1-p_i} = \beta_0 + \beta_1\ln x_i. \tag{1} $$ This implies that the relationship between the odds of the event and $x_i$ is described by the power law $$ \frac{p_i}{1-p_i} = e^{\beta_0} x_i^{\beta_1}, \tag{2} $$ such that the odds is directly proportional to $x_i$ when $\beta_1=1$ which might be sensible but this of course depends very much on the underlying mechanisms generating the data.

If the data includes observations for which both the covariate $x_i=0$ and the response $y_i=0$, special care must be taken if fitting the model using for example glm in R. Under the model specified by (2), such observations have probability $P(y_i=0)=1$ for any value of $\beta_1>0$ and thus contribute a constant equal to $\ln 1=0$ to the total log likelihood (for $\beta_1\le 0$ such observations would be impossible). Provided that $\beta_1>0$, the overall likelihood can thus be maximised by maximising the likelihood contribution from the remaining observations for which $x_i>0$, that is, by removing observations for which $x_i=0$ before fitting the model as in the following example. Hence, we do not need to work with logarithm of zero at any point.

# Simulate some data from the model
n <- 100
x <- seq(0,10,len=n)
eta <- .1+.9*log(x)
p <- 1/(1+exp(-eta)) # this gives zero for eta = -Inf
y <- rbinom(n, size=10, prob=p)

# Plot the observed data
plot(x,y/10)

# Fit the model ommiting observations for which x == 0
data <- data.frame(x,y)
model <- glm(cbind(y,10-y)~log(x), binomial, data[x>0,])

# Compute predicted probabilities using the fitted model (x==0 don't give problems here)
xx <- seq(0,10,len=100)
lines(xx,predict(model, newdata=data.frame(x=xx), type="response"))