Meaning of p-values in regression When I perform a linear regression in some software packages (for example Mathematica), I get p-values associated with the individual parameters in the model.  For, instance the results of a linear regression that produces a result $ax+b$ will have a p-value associated with $a$ and one with $b$.


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*What do these p-values mean individually about those parameters? 

*Is there a general way to compute parameters for any regression model?

*Can the p-value associated with each parameter be combined into a p-value for the whole model?
To keep this question mathematical in nature, I am seeking only the interpretation of p-values in terms of probabilities. 
 A: *

*The p-value for $a$ is the p-value in a test of the hypothesis "$\alpha = 0$" (usually a 2-sided $t$-test).  The p-value for $b$ is the p-value in a test of the hypothesis "$\beta = 0$" (also usually a 2-sided $t$-test) and likewise for any other coefficients in the regression.  The probability models for these tests are determined by the one assumed in the linear regression model. For least-squares linear regression, the pair ($a,b$) follows a bivariate normal distribution centered on the true parameter values ($\alpha, \beta$), and the hypothesis test for each coefficient is equivalent to $t$-testing whether $\alpha = 0$ (resp. $\beta=0$) based on samples from a suitable normal distribution [of one variable, i.e., the distribution of $a$ or $b$ alone].  The details of which normal distributions appear are somewhat complicated and involve "degrees of freedom" and "hat matrices" (based on the notation $\hat{A}$ for some of the matrices that constantly appear in the theory of OLS regression).

*Yes. Usually it is done (and defined) by Maximum Likelihood Estimation.  For OLS linear regression and a small number of other models there are exact formulas for estimating the parameters from the data. For more general regressions the solutions are iterative and numerical in nature.  

*Not directly.  A p-value is calculated separately for a test of the whole model, that is, a test of the hypothesis that all the coefficients (of the variables presumed to actually vary, so not including the coefficient of the "constant term" if there is one). But this p-value cannot usually be calculated from knowledge of the p-values of the coefficients.
A: wrt your first question: this depends on your software of choice. There are really two types of p-values that are used frequently in these scenarios, both typically based upon likelihood ratio tests (there are others but these are typically equivalent or at least differ little in their results).
It is important to realize that all of these p-values are conditional on (part of) the rest of the parameters. That means: Assuming (some of) the other parameter estimates are correct,  you test whether or not the coefficient for a parameter is zero. Typically, the null hypothesis for these tests is that the coefficient is zero, so if you have a small p-value, it means (conditionally on the value of the other coefficients) that the coefficient itself is unlikely to be zero.
Type I tests test for the zeroness of each coefficient conditionally on the value of the coefficients that come before it in the model (left to right). Type III tests (marginal tests), test for the zeroness of each coefficient conditional on the value of all other coefficients.
Different tools present different p-values as the default, although typically you have ways of obtaining both. If you don't have a reason outside of statistics to include the parameters in some order, you will generally be interested in the type III test results.
Finally (relating more to your last question), with a likelihood ratio test you can always create a test for any set of coefficients conditional on the rest. This is the way to go if you want to test for multiple coefficients being zero at the same time (otherwise you run into some nasty multiple testing issues).
