Logistic Regression: Variable total covid cases per 1MM is not significant but total covid cases (without population adjustment) is significant I am working on a university paper regarding the effects of COVID severity (total cases per 1MM, total cases) on whether a party in an acquisition (company transaction) is more likely to be the Acquirer (=1) or the Target in that transaction.
I am using logistic regression on data from March 2020 up to March 2021. My dependent variable is Acquirer/Target and my independent is COVID severity on the date when the acquisition was announced in the country of the acquirer.
Initially I wanted to use total COVID cases per 1 million population as my independent variable. I ran the regression, but I was getting a highly insignificant p-value. Out of curiosity, I ran a regression with total cases without the correction for population. This time I got a significant p-value. Then I thought, maybe population plays a role here, so I used population as independent variable, again no significant p-value.
Do you have any idea what I am missing?
I am just starting with statistics in university, so I am unfortunately not an expert yet.
Below you can find the R outputs.



 A: If I'm understanding the question right, the concern is basically "cases per capita as a sole predictor is non-significant, and population as a sole predictor is non-significant, but total cases as a sole predictor is significant, how can this be?"
Total cases is the product of cases per capita and population (with some constant scaling). Thus, in the broadest terms, the question is simply "can the product of two individually non-significant predictors be statistically significant?". The answer is certainly yes.
In particular, imagine we have two inputs, $x_1, x_2$, and we generate data with the true underlying model $y = x_1 * x_2 + \epsilon$, where $\epsilon$ is our noise. Thus, the product of the inputs is the "true model". However, imagine that in the data we observe, the two inputs are negatively correlated with each other.
In that case, while $x_1$ has some theoretical positive association with $y$, in the observed data, it might be non-significant (and the same with $x_2$. However, of course their product will be significant.  Turning this into a simple toy example in R:
x1 <- 1:10
x2 <- 10:1
y <- x1*x2 + rnorm(10)

summary(lm(y~x1))
summary(lm(y~x2))
summary(lm(y~x1*x2))

The first two models are non-significant, but the full model is highly statistically significant. Intuitively, due to the negative correlation between $x_1$ and $x_2$ in the observed data, a model based solely on $x_1$ tells us little useful about the response, even though the true data generating model has theoretical positive association.
Obviously, this is linear, not logistic, regression, but if I am interpreting your question correctly, the answer is the same. The product of two individually non-significant predictors might be highly statistically significant. This does not mean that the model itself is insightful or valid, it's just answering what I interpret to be your question.
