Random graphs (27): Between and within regression

// Case A
clear
set seed 1
set obs 300

generate  e = 0 + (200 - 0) * runiform()  // To generate random variates over the
generate  x = 0 + (600 - 0) * runiform()  // interval [a,b), a+(b-a)*runiform()

generate  y = e // Relationship between X and Y random
generate id = 1
replace   y = (e + 200) if x > 200 & x <= 400
replace  id = 2         if x > 200 & x <= 400
replace   y = (e + 400) if x > 400
replace  id = 3         if x > 400

bys id: egen ym = mean(y)  // Generate cluster means for Y and X
bys id: egen xm = mean(x)

twoway (scatter y x if id == 1, msymbol(oh)) ///
       (scatter y x if id == 2, msymbol(dh)) ///
       (scatter y x if id == 3, msymbol(sh)) ///
       (scatter ym xm,          msymbol(S)) ///
       (lfit ym xm,             lpattern(solid)) ///
       (lfit y x if id == 1,    lpattern(dash)) ///
       (lfit y x if id == 2,    lpattern(dash)) ///
       (lfit y x if id == 3,    lpattern(dash)) ///
      , /*legend(order(1 "A" 2 "B" 3 "C") title(Region))*/  legend(off) ///
        xtitle("Component 1") ytitle("Component 2") ///
        title("Case A:" "Regional correlation," "individual orthogonality") ///
        xlabel(none) ylabel(none) name(casea, replace)

// Case B
clear
set seed 1

set obs 4 // Generate four clusters

generate id = _n
generate u_i = 300 if id == 1  // Generate cluster-specific intercepts
replace u_i = 0 if id == 2
replace u_i = 0 if id == 3
replace u_i = -300 if id == 4

expand 100  // Generate units in clusters

bysort id: generate x = -100 + (600 + 100) * runiform()
generate e_ij = rnormal(0,70)                // Generate level-1 error term

generate y = x + u_i + e_ij
drop if y < 0 | x < 0
drop if id == 2 & x >= 300
drop if id == 3 & x  < 300

bys id: egen ym = mean(y)
bys id: egen xm = mean(x)

twoway (scatter ym xm, msymbol(S)) ///
       (scatter y  x  if id == 1, msymbol(oh)) ///
       (scatter y  x  if id == 2, msymbol(dh)) ///
       (scatter y  x  if id == 3, msymbol(sh)) ///
       (scatter y  x  if id == 4, msymbol(th)) ///
       (lfit y x if id == 1,    lpattern(dash)) ///
       (lfit y x if id == 2,    lpattern(dash)) ///
       (lfit y x if id == 3,    lpattern(dash)) ///
       (lfit y x if id == 4,    lpattern(dash)) ///
       (lfit ym xm, lpattern(solid)) ///
      , legend(order(2 "A" 3 "B" 4 "C" 5 "D" 1 "Means") size(small) title(Region:, size(small)) row(1)) ///
        xtitle("Component 1") ytitle("Component 2") ///
        title("Case B:" "Regional orthogonality," "individual correlation") ///
        xlabel(none) ylabel(none) name(caseb, replace)


// Case C
clear
set seed 1
set obs 3

generate  id = _n
generate u_i = 300 if id == 1
replace u_i = 0 if id == 2
replace u_i = -300 if id == 3

expand 150

bysort id: generate x = -100 + (600 + 100) * runiform()
generate e_ij = rnormal(0,70)                // Generate level-1 error term

generate y = x + u_i + e_ij
drop if y < 0 | x < 0

bys id: egen ym = mean(y)
bys id: egen xm = mean(x)

twoway (scatter y  x  if id == 1, msymbol(oh)) ///
       (scatter y  x  if id == 2, msymbol(dh)) ///
       (scatter y  x  if id == 3, msymbol(sh)) ///
       (scatter ym xm, msymbol(S)) ///
       (lfit y x if id == 1,    lpattern(dash)) ///
       (lfit y x if id == 2,    lpattern(dash)) ///
       (lfit y x if id == 3,    lpattern(dash)) ///
       (lfit ym xm, lpattern(solid)) ///
       , /*legend(order(1 "A" 2 "B" 3 "C") title(Region))*/ legend(off) ///
       xtitle("Component 1") ytitle("Component 2") ///
       title("Case C:" "Negative regional correlation," "positive individual correlation") ///
       xlabel(none) ylabel(none) ///
       name(casec, replace)

grc1leg casea caseb casec, legendfrom(caseb) span pos(6) row(1) name(combined, replace)

Random graphs (23): Between and within regression

clear

input j i xij x_j yij y_j
// Data from Snijders and Bosker (1999, Table 3.2):
1 1 1 2 5 6
1 2 3 2 7 6
2 1 2 3 4 5
2 2 4 3 6 5 
3 1 3 4 3 4
3 2 5 4 5 4
4 1 4 5 2 3
4 2 6 5 4 3
5 1 5 6 1 2
5 2 7 6 3 2
end


// Plot as Figure 3.4 of Snijders ans Bosker (1999):
twoway (scatter yij xij) ///
       (lfit yij xij) ///
       (lfit y_j x_j) /// 

       (lfit yij xij if j == 1, lpattern(dash_dot)) ///

       (lfit yij xij if j == 2, lpattern(dash_dot)) ///
       (lfit yij xij if j == 3, lpattern(dash_dot)) ///
       (lfit yij xij if j == 4, lpattern(dash_dot)) ///
       (lfit yij xij if j == 5, lpattern(dash_dot)) ///
     , legend(label(2 "Total regression") ///
       label(3 "Between regression") ///
       label(4 "Within regression") ///
       order(2 3 4) ///
       pos(1) ring(0)) ///
       xtitle(X) ytitle(Y) ///
       title("Within, between, and total relations") /// 

       xlabel(none) ylabel(none) ///
       name(wbt_legend, replace)
    

// Lines directly labeled
twoway (scatter yij xij) ///
       (lfit yij xij) ///
       (lfit y_j x_j) ///
       (lfit yij xij if j == 1, lpattern(dash_dot)) ///
       (lfit yij xij if j == 2, lpattern(dash_dot)) ///
       (lfit yij xij if j == 3, lpattern(dash_dot)) ///
       (lfit yij xij if j == 4, lpattern(dash_dot)) ///
       (lfit yij xij if j == 5, lpattern(dash_dot)) ///
     , legend(off) ///
       text(1.8 6.5 "Between") ///
       text(3.0 7.5 "Total") ///
       text(7.0 3.5 "Within") ///
       text(5.0 5.5 "Within") ///
       xtitle(X) ytitle(Y) ///
       title("Within, between, and total relations") ///
       xlabel(none) ylabel(none) ///
       name(wbt_nolegend, replace)

graph combine wbt_legend wbt_nolegend, col(1) ysize(8) xcommon ycommon
// ysize(4) and xsize(5.5) are defaults

Reference

 Snijders, Tom, and Roel Boskers. 1999. Multilevel Analysis. An Introduction to Basic and Advanced Multilevel Modeling. Sage.