According to one study by Wang and colleagues (2013), smoking cigars has “a positive and independent” effect on testosterone. What is an independent effect and what is the significance of one?
I have seen this terminology used in medical journals. A quick search yields similar claims made by Svartberg and Jorde (2006) in the International Journal of Andrology, though they discuss cigarette smoking more generally.
Note that cigar smoking, either its general usage (e.g., "ever smoked" versus "never smoked") or the amount of cigars smoked per unit time (e.g., 6 cigars per day/year), is the independent variable in a multiple linear regression equation predicting total or free testosterone levels in men. The authors of the study more than likely adjusted for (controlled for) other variables (e.g., age, medical history, waist circumference, etc.) which might confound this relationship. By adjusting for these factors, the authors then make claims of the independent effect of smoking on testosterone. In statistical applications, the authors are merely stating that this effect is independent of these other confounding variables.
To put this in the context of the aforementioned study, the authors claim the effect of smoking is a positive and independent predictor of testosterone. In other words, smoking is a positive predictor because the linear increase in the number of cigarettes smoked per day/year is associated with an "increase" in testosterone. Put simply, if smoking increases, then testosterone increases. And smoking is also an independent predictor because the effect is now "independent of" or "apart from" the influence of the other factors. Put another way, think of it as the effect of smoking on testosterone after eliminating its correlation with the other variables in the model.
It is worth noting that the "postive" effect of smoking on testosterone is probably short-lived, especially given the plethora of evidence suggesting smoking's long-term, negative effects on overall health.