Considering a new variable in a regression model If you've been given results for a regression analysis and asked how you would extend the analysis to find whether a new variable also affects the response variable how do you do it? Is it by considering a model with the new variable as a full model and testing whether a reduced model is adequate or is an added variable plot enough?
 A: No strategy can be predicted confidently to be "enough"!
The key questions include not only 


*

*Is it worth adding a new predictor to the model? 


but also 


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*Does the new predictor permit simplifying the original model because one or more of the original predictors now look less useful? 


and also 


*

*Should the new predictor be added "as itself" or in transformed form (e.g. logarithm of predictor, or whatever is appropriate)?


and also 


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*Is there is a case for adding interaction terms to the model? 


and also 


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*Does the new model make more (less) scientific or practical sense? 


This is not to be treated as a full list. 
Certainly, added-variable plots are among the useful tools here. 
A: This depends greatly on a good number of things: is the model for prediction or inference? Are we talking about an exploratory data analysis or multivariate modeling for confirmatory studies?
In prediction, we are less restricted by the choice of variables, except in the case of overfitting (which can be assessed using cross-validation). Adding a variable with no structural purpose will still be structurally "correct" as long as we reliably train the model. The only problem involved is whether collecting data on this factor is expensive, invasive, or unethical. It's a slightly extraneous issue, but imagine having an testicular cancer risk assay which involves extracting serum from the testicles via syringe to screen for testicular cancer risk when you can just examine testicles for lumps in the shower.
In inference, exploratory data analyses can often use data driven approaches to identifying important confounders, mediators, and precision variables. Forward, backward model selection are commonly implemented approaches to this. You might add a factor to the model because it has a large effect size (as is the case in LASSO), or because it improves some information criterion (like AIC or BIC in stepwise model selection), or because it changes the main effect by a significant amount (as in epidemiologic modeling).
In confirmatory inferential models, variables should be prespecified and a documented or plausible causal link with either the outcome (precision) or the exposure and outcome of interest (confounding). Effect measures from adjusted models having unique interpretations depending on which variables were chosen for adjustment, so should not really be compared. For instance, the smoking status vs. height difference is a completely different measure from the age adjusted smoking status vs. height difference. You can't compare them due to the confounding effects of age.
A: If you are using Adjusted R-squared as a model of fit for the OLS model, there is a rule of thumb to decide whether adding extra variable improves the fit (and hence to include in the model) or not: 
Adding an extra variable will increase adjusted R-squared if the absolute value of the t-statistic related with that variable is greater than one in value. The reverse is also true. There is a proof here if you are curious. 
