Can you relate in multiple ways? Multiple linear regression and stepwise regression - Improving the User Experience through Practical Data Analytics - page 245

Interpreting these results involves a bit of a subtle point. When a t-test is per-
formed in a multiple regression, what is being tested is whether the particular X adds
incremental value to the prediction of Y. The formal hypotheses for a given X vari-
able in a multiple regression are:
H0: Variable X is NOT helping you predict Y, above and beyond (i.e., incremen-
tal to) the other variables in the regression equation.
H1: Variable X is, INDEED, helping you predict Y, above and beyond (i.e.,
incremental to) the other variables in the regression equation.
In other words, we are not testing simply whether the X variable helps us pre-
dict Y in a vacuum. Whether a particular X is found signiicant (i.e., when we reject
H0, and thus conclude that the X variable does add incremental value) may depend
on which other variables are in the regression equation. Pretty cool (or subtle!!),
right?
So, what this means in the above example is that component tasks 1 and 3
each gives us incremental value in the prediction of Y; each p -value is below 0.05.
Another way to look at it is that each of those two variables is needed for the best
prediction of Y. Furthermore, the coeficient of each variable is positive (1.10 for
component task 1 and 0.90 for component task 3), indicating that for each of these
two component tasks, the higher the time required, the higher the time required for
the Major Task (Y).
However, for component task 2, we ACCEPT H0, and conclude that knowing
the completion time for component task 2 cannot be said to add to our knowledge
base about Y; in other words, once we know the times for component tasks 1 and
3, we do not gain anything additional by knowing the time for component
task 2.
But why is component task 2 not useful? There are two possibilities. First, it is
possible that the time a person takes for component task 2 is simply not related to the
time a person takes for the Major Task.
But, there is another possibility. Perhaps, the time a person takes for component
task 2 is related to the time a person takes to complete the Major Task, and would
help us predict the time a person takes for the Major Task, if it were the only infor-
mation we had . But once we know the time the person takes to perform component
tasks 1 and 3, the knowledge of the time the person takes to complete component
task 2 is redundant.
As we've said, the time for component task 2 does not add incrementally to our
knowledge base about how long it takes for the person to complete the Major Task.
Thus, we do not need X2 in the equation. As a consequence, we are better off drop-
ping that variable and performing a multiple regression with only component tasks 1
and 3.
If there are a larger number of variables, and more than one of them were not
signiicant (as was P-task 2), determining which variables are saying what about Y
is somewhat challenging. However, there is a special technique, called “stepwise
regression,” that we shall discuss in a later section and which is extremely useful.