Suppose I have a simple dataset of numerous observations, each with a continuous numerical variable $x$ and a binary numerical variable $y$ (with values 0 for unsatisfactory, 1 for satisfactory).
How can I predict how many of many of my observations would satisfy $y=1$ when the average $x$ in my dataset increases, say by 50%?
I was thinking of starting with a logistic regression model, but I'm not sure how to proceed.
Any guidance would be much appreciated.
You can fit a model and use it to predict the probability of any given instance to be $1$, conditional on your predictor value.
As you write, one possible model is a logistic regression.
Note that the effect of increasing the predictor by 50%, or by any other increment, will depend on what you increase the value from. Here is some R code that illustrates this (note the type="response" part in the predict()s that ensure we gat probability predictions):
set.seed(1)
xx <- runif(100)
prob <- 1/(1+exp(-(xx+0.5)^2-2*(xx-0.5)))
obs <- runif(length(xx))<prob
model <- glm(obs~xx,family="binomial")
xx_plot <- seq(min(xx),max(xx),by=0.01)
plot(xx,obs,pch=19)
lines(xx_plot,
predict(model,newdata=data.frame(xx=xx_plot),type="response"),
col="red")
predict(model,newdata=data.frame(xx=c(0.6,0.9)),type="response")
# 0.7800048 0.9282456
predict(model,newdata=data.frame(xx=c(0.2,0.3)),type="response")
# 0.3869597 0.4928357
If you can now assume that your instances are independent, you can calculate the expected number of $y=1$ for each predictor value. Or use the binomial distribution to get a handle on the entire distribution of the number of $1$s for a constant predictor value, or possibly generalizations of the binomial for different probabilities coming from different predictor values.
