Can you relate in multiple ways? Multiple linear regression and stepwise regression - Improving the User Experience through Practical Data Analytics - page 254

Now, note that of the 15 independent (X) variables, 7 of them are signiicant. We
know this by examining the p -values. The signiicant variables are highlighted in the
p -value column of Figure 10.9 ; see vertical arrow in Figure 10.9 . Each highlighted
p -value is less than 0.05.
The signiicant variables (i.e., those with a p -value < 0.05) are (in the order listed
in the output):
Ability to search by job title
Ability to search by years of experience
Ability to search by location
Ability to search candidates by education level
Ability to search by skills
Ability to search candidates by companies in which they have
worked
Ability to perform a Boolean search
This means that these seven variables provide incremental/unique predictive value
about Y, the respondent's likelihood to adopt the search engine , beyond a reason-
able doubt. Let us take the liberty, for the moment, of calling these variables “the big
7.” This does not mean that the other variables of the 15 are unrelated to Y, but rather,
that none of the other variables, by themselves, add incremental (unique) predictive
value about Y. However, there can be subtlety that is discussed in the “Nonsigniicant
but Useful Variables” sidebar.
Now, if we run a regression using only “the big 7” X variables, the regression
analysis yields an r 2 value of 0.469 (46.9%), only slightly under the 0.493 (49.3%)
we had with all 15 variables. This might surprise you, but this slight reduction always
occurs. That is, even though the other eight variables are not signiicant, each does
produce at least a tiny bit of r 2 . Indeed, we have eight variables (all not signiicant)
which are adding only about 2.4% in total to the overall r 2 value, which is an average
of three-tenths of a percent each (i.e., 0.3%)!!
SIDEBAR: WHEN ZERO IS NOT ZERO
Remember that even if the r 2 of an X with Y is really zero, or incrementally adds zero, given it is
real data and (of course) not ininite data, a variable's r 2 will not come out exactly zero, but will be
some small value (e.g., 0.3%).
If we pick two variables that we know are unrelated, and, hence, have a true r 2 value of zero,
with real data (and, obviously, less than ininite data!!), we will not get an r 2 of exactly zero when
we analyze the data. It will be some (very likely small) positive value, since a squared value cannot
be negative. This logic harkens back to the logic of hypothesis testing in general, as discussed way
back in Chapter 1, where we noted that means and other values from a sample do not come out
exactly the respective true value.