Interpretation of the squared term coefficient in the presence of interaction

kiton

New Member
#1
Hello dear form members,

Please consider the following regression model:

y = a + b1x1 + b2x1^2 + b3x2 + b4x1*x2 + e,

Note that x1 is (1) used as a moderator, therefore both main effect (b1x1) and interaction (b4x1*x2) are included; and (2) also included as a squared term (b2x1^2) to test for curvilinearity.

I understand that the effect of x2 is interpreted as “conditional” effect at the mean value of x1 — no problem here.

But what about the squared term -- how shall I interpret the b2 coefficient in the presence of interaction?

Thank you in advance!
 

hlsmith

Omega Contributor
#4
Maarten, are you interpreting this as logistic regression. If so, what would the interpretation by in a multiple linear model?
 

rogojel

TS Contributor
#5
hi,
I believe that for all except the simplest models the best would be to not provide an interpretation of the individual coefficients except to say, that they in ensemble describe the model. Any practical questions will be answered by the model, anyway.
 

maartenbuis

TS Contributor
#6
hlsmith: I was talking about linear regression. So lets take a concrete example:

Code:
. // some example data
. sysuse nlsw88, clear
(NLSW, 1988 extract)

. 
. // create indicator (dummy) variable black
. gen byte black = race ==2 if race < 3
(26 missing values generated)

. label define black 0 "white" 1 "black"

. label value black black

. 
. // center grade at 12 years of education 
. // (= high school)
. replace grade = grade - 12
(2,244 real changes made)

. 
. // linear regression with interactions and quadratic
. reg wage c.grade##c.grade i.black c.grade#i.black

      Source |       SS           df       MS      Number of obs   =     2,218
-------------+----------------------------------   F(4, 2213)      =     67.57
       Model |   8016.8557         4  2004.21393   Prob > F        =    0.0000
    Residual |   65642.939     2,213  29.6624216   R-squared       =    0.1088
-------------+----------------------------------   Adj R-squared   =    0.1072
       Total |  73659.7947     2,217  33.2249863   Root MSE        =    5.4463

---------------------------------------------------------------------------------
           wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
          grade |   .5850541   .0698755     8.37   0.000     .4480256    .7220825
                |
c.grade#c.grade |   .0288827   .0126564     2.28   0.023     .0040631    .0537022
                |
          black |
         black  |  -.8270054   .2798517    -2.96   0.003    -1.375805    -.278206
                |
  black#c.grade |
         black  |   .2230768   .1067155     2.09   0.037     .0138038    .4323498
                |
          _cons |   7.092939   .1588648    44.65   0.000     6.781399    7.404479
---------------------------------------------------------------------------------
So a year increas in education for someone who is white (black = 0) and starts with highschool (grade = 0) is associated with an 59 cents increase in hourly wage. This effect education increases by 3 cents for every year more education.

A black person with highschool can expect 83 cents less in hourly wage than a with person with highschool.

The effect of a year more education for someone who is black and starts with highschool is 22 cents larger than that effect for a white person. So a year more education if a black person starts with high school is associated with a 59+22 = 81 cents increase in hourly wage. Since there I included no interaction effect between age^2 and black, this effect of education increases by 3 cents for every year increase in education, just like it did for white persons.

A white person (black =0) with highschool (grade=0) can expect 7.09 dollars an hour.

I do look at the coefficients and interpret them, but I also look at graphs.