14 Jan 2026 11:47:46
These notes represent a brief discussion of centering with cross sectional data. Since so much of my current work focuses on cross national work on parenting and child development, I use these ideas as my substantive example.
Consider a cross-national data set where we are attempting to understand predictors of behavior problems as a function of per capita income and parental use of physical punishment.
. clear all
. set obs 100 Number of observations (_N) was 0, now 100.
. generate income = runiform(10000, 70000)
. generate physical_punishment = rbinomial(1,.3)
. generate country = int(_n/10) + 1
. generate e = rnormal(0,1) // individual error
. generate u = country - 5 // random intercept
. generate behavior_problems = 110 + -.0001 * income + 10 * physical_punishment > + u + e // plausible regression relationship
. drop e u // drop error terms
. list in 1/10, table // list out some data
┌──────────────────────────────────────────┐
│ income physic~t country behavi~s │
├──────────────────────────────────────────┤
1. │ 20148.49 0 1 103.3466 │
2. │ 18292.35 0 1 105.1445 │
3. │ 20029.46 0 1 104.3675 │
4. │ 49927.27 0 1 103.4318 │
5. │ 34768.54 0 1 102.9267 │
├──────────────────────────────────────────┤
6. │ 47199.07 0 1 101.1913 │
7. │ 44796.36 1 1 111.0747 │
8. │ 50592.45 0 1 99.76586 │
9. │ 37461.37 0 1 101.8378 │
10. │ 55994.71 1 2 111.1749 │
└──────────────────────────────────────────┘
\[\text{behavior problems} = \beta_0 + \beta_1 \text{income} + \beta_2 \text{physical punishment} + u_{\text{country}} + e_{ij}\]
. twoway (scatter behavior_problems income if physical_punishment ==0) ///
> (scatter behavior_problems income if physical_punishment == 1), ///
> legend(order(1 "no physical punishment" 2 "physical punishment") pos(6)) ///
> title("Behavior Problems by Income and Physical Punishment") ///
> xtitle("Per Capita Income") ///
> ytitle("Behavior Problems") ///
> scheme(michigan)
. graph export myscatter.png, width(1000) replace
file
/Users/agrogan/Desktop/GitHub/multilevel/centering-in-cross-sectional-dat
> a/myscatter.png saved as PNG format
. mixed behavior_problems income physical_punishment || country:
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -157.07646
Iteration 1: Log likelihood = -157.07646
Computing standard errors ...
Mixed-effects ML regression Number of obs = 100
Group variable: country Number of groups = 11
Obs per group:
min = 1
avg = 9.1
max = 10
Wald chi2(2) = 2230.63
Log likelihood = -157.07646 Prob > chi2 = 0.0000
─────────────┬────────────────────────────────────────────────────────────────
behavior_p~s │ Coefficient Std. err. z P>|z| [95% conf. interval]
─────────────┼────────────────────────────────────────────────────────────────
income │ -.0001021 5.25e-06 -19.44 0.000 -.0001124 -.0000918
physical_p~t │ 9.859842 .2315382 42.58 0.000 9.406036 10.31365
_cons │ 110.9004 .9343151 118.70 0.000 109.0691 112.7316
─────────────┴────────────────────────────────────────────────────────────────
─────────────────────────────┬────────────────────────────────────────────────
Random-effects parameters │ Estimate Std. err. [95% conf. interval]
─────────────────────────────┼────────────────────────────────────────────────
country: Identity │
var(_cons) │ 8.878869 3.878162 3.771943 20.90019
─────────────────────────────┼────────────────────────────────────────────────
var(Residual) │ .8283655 .1242951 .6173058 1.111587
─────────────────────────────┴────────────────────────────────────────────────
LR test vs. linear model: chibar2(01) = 183.09 Prob >= chibar2 = 0.0000
We note that -1.021 is the effect of every additional $10,000 of per capita income. 9.860 is the effect of physical punishment.
Notably, for this handout, 110.900 is the level of behavior problems for a child who did not receive physical punishment living in a family with $0 income.
Grand mean centering helps us to have more meaningful intercepts of our continuous variables.
Essentially, we are going to create \(\text{income}_{\text{grand mean centered}} = \text{income} - \overline{\text{income}}\)
\[\text{behavior problems} = \beta_0 + \beta_1 \text{income}_{\text{grand mean centered}} + \beta_2 \text{physical punishment} + u_{\text{country}} + e_{ij}\]
. egen m_income = mean(income) // grand mean of income
. generate c_income = income - m_income // grand mean centered income
. twoway (scatter behavior_problems c_income if physical_punishment ==0) ///
> (scatter behavior_problems c_income if physical_punishment == 1), ///
> legend(order(1 "no physical punishment" 2 "physical punishment") pos(6)) ///
> title("Behavior Problems by Income and Physical Punishment") ///
> caption("Income is Grand Mean Centered") ///
> xtitle("Per Capita Income") ///
> ytitle("Behavior Problems") ///
> scheme(michigan)
. graph export myscatter2.png, width(1000) replace
file
/Users/agrogan/Desktop/GitHub/multilevel/centering-in-cross-sectional-dat
> a/myscatter2.png saved as PNG format
In a graph, we see that grand mean centering has transformed the intercept so now the \(\beta_0\) term is the level of behavior problems for the average child who did not recieve physical punishment.
. mixed behavior_problems c_income physical_punishment || country:
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -157.07646
Iteration 1: Log likelihood = -157.07646
Computing standard errors ...
Mixed-effects ML regression Number of obs = 100
Group variable: country Number of groups = 11
Obs per group:
min = 1
avg = 9.1
max = 10
Wald chi2(2) = 2230.63
Log likelihood = -157.07646 Prob > chi2 = 0.0000
─────────────┬────────────────────────────────────────────────────────────────
behavior_p~s │ Coefficient Std. err. z P>|z| [95% conf. interval]
─────────────┼────────────────────────────────────────────────────────────────
c_income │ -.0001021 5.25e-06 -19.44 0.000 -.0001124 -.0000918
physical_p~t │ 9.859842 .2315382 42.58 0.000 9.406036 10.31365
_cons │ 106.7117 .9085729 117.45 0.000 104.931 108.4925
─────────────┴────────────────────────────────────────────────────────────────
─────────────────────────────┬────────────────────────────────────────────────
Random-effects parameters │ Estimate Std. err. [95% conf. interval]
─────────────────────────────┼────────────────────────────────────────────────
country: Identity │
var(_cons) │ 8.878869 3.878162 3.771942 20.90019
─────────────────────────────┼────────────────────────────────────────────────
var(Residual) │ .8283655 .1242951 .6173058 1.111588
─────────────────────────────┴────────────────────────────────────────────────
LR test vs. linear model: chibar2(01) = 183.09 Prob >= chibar2 = 0.0000
We see that the \(\beta_1\) and \(\beta_2\) regression coefficients have not changed. However, the intercept, \(\beta_0\) has changed, and is now more meaningful.
In group mean centering, we are doing something slightly different. We are creating a mean for each group, which in this data is country: e.g. \(\text{income}_{\text{group mean centered}} = \text{income} - \overline{\text{income}}_{.j}\), where \(j\) is the index for group or country.
. bysort country: egen m_g_income = mean(income) // GROUP mean of income
. generate c_g_income = income - m_g_income // GROUP mean centered income
. bysort country: egen m_g_physical_punishment = mean(physical_punishment) // G > ROUP mean of physical punishment
. generate c_g_physical_punishment = physical_punishment - m_g_physical_punishm > ent // GROUP mean centered physical punishment
Interestingly, group mean centering has many implications. Here I focus on how employing different variables might provide conceptually or theoretically different results. For the sake of parismony, in the brief discussion below I focus on these conceptual or theoretical differences, and do not provide output. I use the
quietlyprefix to suppress output.
Two versions of the equation are equally appropriate. Both address conceptually or theoretically different questions.
One parameterization of the multilevel model is to enter the covariate and its group level mean i.e. \(x_{ij}\) and \(\overline{x_{.j}}\).
\[\text{behavior problems} = \beta_0 + \beta_1 \text{income} + \beta_2 \text{income}_{\text{group mean}} + \beta_3 \text{physical punishment} + u_{\text{country}} + e_{ij}\]
A second, equally valid, but conceptually different parameterization of the multilevel model is to enter the covariate deviated from its group level mean and the group level mean i.e. \(x_{ij} - \overline{x_{.j}}\) and \(\overline{x_{.j}}\).
\[\text{behavior problems} = \beta_0 + \beta_1 \text{income}_{\text{group mean centered}} + \beta_2 \text{income}_{\text{group mean}} + \beta_3 \text{physical punishment} + u_{\text{country}} + e_{ij}\]
Again, for the sake of parsimony, I use the quietly
prefix to suppress output of the multilevel models.
This first parameterization focuses on individual scores on covariates and their country level means.
What is the effect of income, country level mean income, physical punishment and country level mean of physical punishment on behavior problems?
. quietly: mixed behavior_problems income m_g_income physical_punishment m_g_ph > ysical_punishment || country:
This second parameterization focuses on how individuals differ from their country level means, and country level means.
What is the effect of income deviated from its country level mean, country level mean income, physical punishment deviated from its country level punishment, and country level mean of physical punishment on behavior problems?
. quietly: mixed behavior_problems c_g_physical_punishment m_g_income c_g_physi > cal_punishment m_g_physical_punishment || country: