Ron Heck, Summer 2012 Seminars
Multilevel Regression Models and Their Applications Seminar
1
A Note on Investigating Random Slopes in the HTT (2010) Textbook (Chapter 3)
In Chapter 3 of the text (Multilevel and Longitudinal Modeling with IBM SPSS), please note that
we are making a change in the upcoming second edition regarding how we specify the random
slope equations in the level-2 slope model.
For example, if you look at Eq. 3.15 (p. 99) the predictors within the slope equations area
specified in SPSS by adding them as cross-level interactions with the random level-1 predictor
slope (in this case, the student SES-achievement slope). On p. 99, we specify the three level-2
predictors in the current random SES-achievement slope equation ( 1 j ) as follows:
1 j   10   11ses _ mean j * sesij   12 per 6 yrc j * sesij   13 public j * sesij  u1 j
In formulating the slope equation in this manner, we wanted to emphasize that the analyst has to
create these cross-level interactions in SPSS as she or he is adding the predictors to the slope
model.
However, we should have actually illustrated for readers that the level-2 predictors actually
become the cross-level interactions slope equation for 1 j (Eq. 3.15) when the Level 2 slope
equation is substituted into the level-1 equation (Eq. 3.7). At that time, we also substitute the
level-2 intercept equation (Eq. 3.12) into Eq. 3.7.
The three equations should look like the following:
Yij  0 j  1 j sesij   ij
(3.7)
0 j =  00 +  01ses _ mean j + 02 per 4 yrc j + public j +u0 j
(3.12)
1 j   10   11ses _ mean j   12 per 6 yrc j   13 public j  u1 j
(3.15)
Substituting equations 3.12 and 3.15 into 3.7 creates the following combined equation. You can
see the three cross-level interactions are now there in the combined equation. Moreover, the
variance term ( u1 j ) in Eq. 3.15 now reflects that it is a level-2 random effect (representing the
residual variance of the level-1 ses-achievement slope).
Summer 2012 Quantitative Methods Series at Portland State University
A Note on Investigating Random Slopes in the Textbook
More specifically, the first line reflects the substitution of Eq. 3.12 for
second line reflects the substitution of Eq. 3.15 for
2
 0 j in Eq. 3.7. The
1 j in Eq. 3.7.
Yij   00   01ses _ mean j   02 per 4 yrc j   03 public j  u0 j   ij 
 10 sesij   11sesij * ses _ mean j   12 sesij * per 4 yc j  sesij * public j  u1 j sesij
We can then rearrange the error terms to appear at the end of the equation.
Yij   00   01ses _ mean j   02 per 4 yrc j   03 public j 10 sesij   11sesij * ses _ mean j 
 12 sesij * per 4 yc j  sesij * public j  u1 j sesij  u0 j   ij
Another example of this is in Chapter 1 (see Eq. 1.14 on p. 14). Equation 1.14 is actually the
following:
1 j   10   11resources j  u1 j
The cross-level interaction is then built by substituting Eq. 1.14 into the level-1 equation (Eq. 1.5
on p. 11), along with the level 2 intercept equation (Eq. 1.10). The combined model will then
show the cross-level interaction and the more complex random error term for u1 j .
Yij   00   01resources j   10attitudeij   11resources j * attitudeij  u1 j attitudeij  u0 j
In the second edition of the text, we are actually going to show these combined equations for
readers. Currently, there is a place or two in each chapter where we mention predictors in
random- slope models where we feel we did not make it clear for readers we were showing how
the cross-level interactions must be built before actually substituting them into the appropriate
level-1 equation to created the combined single-equation model.
We hope this helps clarify the substitution process to create the combined equation.