p-values in multiple linear regression depend on number of predictors?

So I am wondering if there usually is any "correction" being made to p-Values (of single predictors) if you add more predictors to the model? Does anybody know how SPSS handles this?

I am well aware that change of a predictors p-value after including another variable could occur due to 1) the predictors effect (regression coefficient) having changed and 2) because the sample size has decreased (if some of the previously included cases have a missing value at the newly included predictor and we use listwise exclusion). But could a change also be due to 3) the model taking into account that when I test a higher number of predictors, the likeliness of one attaining an effect of a given strength rises?

• if you add more predictors to the model? I do not think that it should affect p--value! Why - correction is invoked. – Subhash C. Davar Jul 24 '19 at 14:31

First, the p value reported for an individual coefficient in a standard linear regression is based on a t-test. As that test uses an estimate of the standard error taken from the data, the reliability of the standard error estimate depends on both the number of observations and the number of parameters that are estimated. The degrees of freedom in the t-test decrease as the number of estimated parameters increases: $$(N-P)$$ for $$P$$ parameters estimated from $$N$$ observations. That's the level of correction for numbers of parameters in the p-values reported by standard statistical software from the t-tests for regression coefficients. Note that if the number of parameters estimated is much smaller than the number of observations, this adjustment won't make much of a difference.