What determines to which variable the explained variance is granted? What happens when there is a linear regression of the kind
y ~ x1 + x2 + x3 + x4
What determines which variables will be granted the explained variance rather than the others given a correlation matrix (not assuming independence)?
In case this really is unpredictable, why do we insist on using regression rather than simply looking at the correlation matrix: at least then you know you are not accounting for effects of other variables when looking at the correlation coefficient, while using regression you'd be oblivious to which variable gets the explained variance (and how correctly or wrong this is!).
 A: There are several properties of regression that are directly relevant to questions of the relative "importance" or "impact" of predictors but are widely misunderstood:


*

*First and foremost, all the variables that truly matter must be present in the regression equation. If any important variables are omitted then the results can be misleading unless all the omitted variables are uncorrelated with all the included variables. There 
is no point in attempting to discover the relative importance of 
some predictors for which you have data unless you already know 
that these are the only predictors that matter.

*R^2 can be partitioned into components representing the unique 
contribution of each predictor only when all the predictors are 
mutually uncorrelated. The problem is not partitioning R^2 -- that 
can always be done. The problem is that the results do not always 
represent the unique contribution of each predictor. However 
intuitively straightforward the notion of unique contributions 
may seem, there is no mathematical definition that is entirely 
satisfactory when the predictors are correlated.

*Importance ratings obtained by comparing semipartial correlations 
or changes in R^2 (i.e., squared semipartials) depend on the joint 
distribution of the predictors. Contrary to what is often implicit in 
the importance question, the results are not inherent properties of 
the variables alone, but joint properties of the variables and the 
particular multivariate distribution they happen to have. This is 
especially important when the distribution of the predictors is an 
artifact -- either a direct artifact, because the investigator set 
the values of the predictors; or an indirect artifact, because the 
investigator selected cases or sampled nonrandomly. And even if the 
sample distribution is a valid estimate of some "natural" population 
distribution, if the population distribution changes then the true 
semipartials can also change, even though the mechanism relating the 
predictors to the outcome variable has not changed.

*If the predictors are in the same units (possibly after data-independent unit-equating transformations, which excludes sample-specific standardization), then comparing the raw-score regression 
weights can lead to conclusions of relative importance that are 
inherent properties of the variables alone. However, the definition 
of importance that is implicit in comparisons of the regression 
weights is the expected change in the outcome variable for a unit 
change in the predictor in question, with all other predictors held 
constant. In some situations this definition may be appropriate; in 
others, not. 
So where does this leave us? It may often mean that questions about 
importance can not be answered -- at least not in the sense that 
they were asked. Unfortunately, regression seems to have been "sold" 
to many as a way to answer all such questions. It can't.
