Are regression coefficients in a model with interactions ALL made conditional, or just those involved in the interaction? In every example I've seen describing the interpretation of interactions in multiple regression (I know logistic regression is different), they use an example of two regression coefficients and the interaction between them. I want to know how the rest of a model is affected when interaction terms are included, and if it is more appropriate to first present a model without an interaction and then with it (like I've seen in some articles and presentations). 
For example, let's say I have 10 predictors, plus an interaction term between two of them, in my regression model. I understand that the coefficients for the interacting terms are now conditional effects instead of main effects. What I don't understand is how it affects the interpretation of the other 8 predictors in my model. 
Thoughts?
 A: The interpretation is motivated by considering how the model predictions change when controlled, simple changes are induced in the original variables. 
Let's frame this a little abstractly because it doesn't make the situation any more complicated while revealing the essence of the matter.  If we denote those variables by $u=(u_1, \ldots, u_m)$, say, then we may write the regressors--by which I mean the variables that actually are involved in the regression--as specified functions $f_1,f_2, \ldots, f_p$ of $u$. For example, $m=3$ numerical variables plus an interaction between the first two, would produce $p=4$ regressors; namely, $$\eqalign{x_1 &= f_1(u) = u_1,\\ x_2 &= f_2(u) = u_2,\\ x_3&=f_3(u)=u_3, \text{ and}\\x_4 &= f_4(u)=u_1u_2\  (\text{the interaction}).\tag{*}}$$
The fitted model based on estimated parameters $b=(b_0, b_1, b_2, \ldots, b_p)$ (an "intercept" is hereby included as $b_0$) fits or "predicts" the response $y$ associated with regressor values $x_1, \ldots, x_p$ to be
$$\hat y = b_0 + b_1 x_1 + b_2 x_2 + \cdots + b_p x_p.$$
One common way of interpreting this asks how $\hat y$ would change when, say, the original set of variables $u$ is changed by adding a fixed amount $\delta$ to just one variable $u_j$, becoming $u_j + \delta$, thereby creating a new set of values $u^\prime$.  Those not appealing to the Calculus will compute the difference directly, as follows.  Let $\hat y$ be the fitted value for regressors associated with $u^\prime$.  Subtracting off $\hat y$ and organizing the result by the index $1, 2, \ldots, p$ exhibits the change in response as a linear combination of changes in the regressors:
$$\hat y^\prime - \hat y = b_1(f_1(u^\prime)-f_1(u)) + \cdots + b_p(f_p(u^\prime)-f_p(u)).\tag{**}$$
The form of this expression highlights the (obvious) fact that changing $u_j$ affects only the terms for which $f_i(u)$ actually depends on $u_j$.  (This is the one aspect of this analysis you might want to commit to memory.)  In example $(*)$, for instance, if $j=3$ then only $x_3$ is changed when $u_3$ is changed, thereby becoming
$$x_3^\prime - x_3 = f_3(u^\prime) - f_3(u) = u^\prime_3 - u_3 = (u_3 + \delta) - u_3 = \delta.$$
In all other cases $i\ne 3$, $x_i^\prime-x_i=f_i(u^\prime) - f_i(u)=0$: there is no change. 
 Plugging these changes into $(**)$ simplifies it right down to
$$\hat y^\prime - \hat y = b_3\,\delta.$$
Interpretation: "changing $u_3$ by $\delta$ (while fixing all the other $u_i$) changes the response $y$ by $b_3$ times $\delta$."  Most readers of this site will appreciate that this is intended only as an English description of the foregoing mathematical relationships; in particular, it is not a causal claim.  It says nothing about what will happen in the world to $y$ if somehow an observation could be altered to change $u_3$ to $u_3+\delta$ (if that is even possible).
Note that $b_3$ does not depend on whatever values the $u_i$ might have: it is a "constant."  This makes the interpretation particularly simple.
Continuing the example, suppose instead that $u_1+\delta$ is used in the model instead of $u_1$.  Now two of the $x_i$ in $(*)$ are affected: $x_1$ increases by $\delta$ while $x_4$ increases by $\delta u_2$.  Consequently $(**)$ yields
$$\hat y^\prime - \hat y = b_1\delta + b_4 \delta u_2  = (b_1 + b_4 u_2)\delta.$$
Interpretation: "changing $u_1$ by $\delta$ (while fixing all the other $u_i$) changes the response $y$ by $b_1 + b_4 u_2$ times $\delta$."  There is the interaction: the change in predicted value depends on the values of the underlying variable $u_2$. Notice that $u_3$ is not involved.
The answer to the original question should now be clear from the very forms of $(*)$ and $(**)$.  This method of analysis applies not only to interactions, but--by virtue of the abstract specification of the functions $f_j$--for all other models that combine the $u_i$ in any manner whatsoever.  (This includes polynomial models, higher-order interactions, and other nonlinear models that might involve exponential growth, sinusoidal variation, and more.)  In particular, the interpretation of any interaction does not depend on any variables that are not directly involved in that interaction.
A: Using your example - including an interaction term won't affect the interpretation of the other 8 coefficients, but it may change the coefficient itself. For instance, if you have a model y ~ a*age + b*gender + c*age:gender + d*sex + ... + j*race, the interpretation of sex would be: "holding age, gender, ... , race constant, increasing sex by one unit (i.e. moving from female to male), increases y by d. This interpretation holds regardless of whether there's an interaction between age and gender. Though, the value of d may be different depending on if an interaction term is in the model or not.  
