*! version 0.3 14mar2021
program distcomp, rclass sortpreserve byable(recall)
  version 11
  syntax varname(numeric default=none) [if] [in] , BY(varname) [Alpha(real 0.10) Pvalue noplot GROPTLINE0(string asis) GROPTLINE1(string asis) GROPTTWOWAY(string asis) GROPTREJ(string asis) SAVING(string)]
  *syntax varname(numeric default=none) [if] [in] , BY(varname) [Alpha(real 0.10) Pvalue noplot]
  if (`"`groptrej'"'=="") {
    local groptrej lwidth(thick) lcolor(black) lpattern(solid)
  }
  * Basic setup and check for problems before calling mata function
  if (`alpha'!=0.10 & `alpha'!=0.05 & `alpha'!=0.01) {
    display as error "Need to choose alpha equal to 0.10 or 0.05 or 0.01 (or else use the R version)"
    exit
  }
  marksample touse
  qui replace `touse'=0 if missing(`by')
  qui replace `touse'=0 if missing(`varlist')
  qui count if `touse'
  if !r(N) { //taken from ksmirnov.ado
    error 2000
  }
  qui tab `by' if `touse' , matrow(vals)
  if r(r) != 2 {
    di as error "`by' takes on " r(r) " values, not 2"
    exit 450
  }
  scalar pflag = ("`pvalue'"=="pvalue") // compute global p-value?

  * Main subroutine call
  mata: distcompmain("`varlist'", "`by'", `alpha', "`touse'", GoldmanKaplan2sLookup, "vals", "pflag")

  * Store r() values
  return scalar alpha = `alpha'
  return scalar alpha_sim = distcompalphasim // simulated maximum alpha (if applicable)
  if (pflag) { // global/GOF p-value
    if (distcompglobalpvalue>0) {
      return scalar p_gof = distcompglobalpvalue
    }
    else {
      return scalar p_gof = 1/1e4
    }
  }
  return scalar rej_gof01 = (distcompglobalrej1>0) //  1% level: 1 (reject global H0:F0()=F1()) or 0 (don't)
  return scalar rej_gof05 = (distcompglobalrej5>0) //  5% level: 1 (reject global H0:F0()=F1()) or 0 (don't)
  return scalar rej_gof10 = (distcompglobalrej10>0) // 10% level: 1 (reject global H0:F0()=F1()) or 0 (don't)
  matrix colnames distcompN = Ntotal Ngroup1 Ngroup2
  return matrix N = distcompN // number of observations
  matrix colnames distcomprejranges = from to

  * Display some results
  * Global/GOF results
  di as text "Comparing distribution of `varlist' when `by'=" vals[2,1] " vs. `by'=" vals[1,1]
  di as text " "
  di as text "Global test of equality of two CDFs:"
  if (pflag) {
    if (distcompglobalpvalue>0) {
      di as result "    Simulated p-value = " distcompglobalpvalue
    }
    else {
      di as result "    Simulated p-value < " 1/1e4
    }
  }

  * Global/GOF test results for alpha=1%,5%,10%
  foreach a in 10 5 1 {
    local rejtxt "do not reject"
    if (distcompglobalrej`a'>0) {
      local rejtxt "reject"
    }
    if (distcompglobalrej`a'==1) {
      local rejtxt "reject (<)"
    }
    else if (distcompglobalrej`a'==2) {
      local rejtxt "reject (>)"
    }
    else if (distcompglobalrej`a'==3) {
      local rejtxt "reject (< and >)"
    }
    if (`a'<10) {
      di as result "    At a  `a'% level: `rejtxt'"
    }
    else {
      di as result "    At a `a'% level: `rejtxt'"
    }
  }

  * Rejected ranges (MTP results)
  local levperc `=100*`alpha''
  if (distcompglobalrej`levperc'>0) {
    di as text " "
    di as text "With strong control of FWER at a `levperc'% level,"
    di as text "CDF equality is rejected at all points in the following"
    di as text "ranges of `varlist' (in favor of first CDF < or >):"
    // matlist distcomprejranges , colorcoleq(res) lines(none) names(columns)
    matlist distcomprejranges , colorcoleq(res) lines(none) twidth(5)
  }
  tempname rejrows
  local `rejrows' = rowsof(distcomprejranges)

  * Plot ECDFs and indicate rejected ranges graphically
  if ("`plot'"!="noplot") {
    tempvar F0 F1
    cumul `varlist' if `touse' & `by'==vals[2,1] , g(`F1')
    cumul `varlist' if `touse' & `by'==vals[1,1] , g(`F0')
    sort `touse' `by' `varlist' `F0' `F1'
    * OVER 32-CHARACTER LIMIT =( local gname "distcomp_`varlist'_by_`by'_alpha`levperc'_`=_byindex()'"
    local gname "distcomp_`=_byindex()'"
    capture graph drop "`gname'"
    graph twoway (line `F1' `varlist' if `touse' & `by'==vals[2,1] , connect(stairstep) `groptline1') ///
                 (line `F0' `varlist' if `touse' & `by'==vals[1,1] , connect(stairstep) `groptline0') ///
          , name("`gname'") legend(lab(1 "`by'=`=vals[2,1]'") lab(2 "`by'=`=vals[1,1]'")) `gropttwoway'
    if (distcompglobalrej`levperc'>0) {
      forvalues r = 1/``rejrows'' {
        graph addplot function y=0 , range(`=distcomprejranges[`r',1]' `=distcomprejranges[`r',2]') `groptrej' , legend(order(1 "`by'=`=vals[2,1]'" 2 "`by'=`=vals[1,1]'" 3 "rejected ranges (`levperc'% FWER)"))
        *see https://www.stata.com/statalist/archive/2006-05/msg00249.html
      }
    }
    if ("`saving'"!="") {
      graph save `gname' `saving'
    }
  }
  return matrix rej_ranges = distcomprejranges // rows=ranges with level-alpha stat. sig. difference
end
*
*
*
mata:
void distcompmain(string scalar yname, string scalar catname, real scalar alpha, string scalar touse, real matrix LOOKUP, string scalar vals, string scalar pvalueflagname) {
  real scalar i, cnt, a, p, n0, n1 // counters, sample sizes
  transmorphic scalar val0, val1 // category variable values
  real colvector y // from dataset
  transmorphic colvector cat // from dataset
  real colvector y0, y1, y0uniq, y1uniq // y when cat=val0 and y when cat=val1
  real colvector aret
  real matrix CI0raw, CI1raw, CI0, CI1, rejranges, tmpmat0, tmpmat1
  string matrix rowstr
  tmp = st_matrix(vals)
  val0 = tmp[1]
  val1 = tmp[2]
  pvalueflag = st_numscalar(pvalueflagname)
  st_view(y, ., yname, touse)
  st_view(cat, ., catname, touse)
  y0 = J(colsum(cat:==val0),1,.)
  y1 = J(colsum(cat:==val1),1,.)
  n0 = 0
  n1 = 0
  for (i=1; i<=length(y); i++) {
    if (cat[i]==val0) {
      n0++
      y0[n0] = y[i]
    }
    else if (cat[i]==val1) {
      n1++
      y1[n1] = y[i]
    }
    else {
      cat[i]
       _error(sprintf("group variable %s takes a third value; should only have two",catname))
    }
  }
  if (n0/n1<0.05 | n1/n0<0.05) {
    printf("{err}warning: one group's size is <5%% of the other group; performance has not been studied thoroughly in such cases.\n")
  }
  st_matrix("distcompN",(n0+n1,n0,n1))
  //sort y values within group
  y0 = sort(y0,1)
  y1 = sort(y1,1)
  for (i=1;i<=3;i++) {
    a = i^2 + (i>1) // a=1,5,10
    // get alpha.tilde (and alpha.sim)
    aret = distcompalpha(n0, n1, a, LOOKUP)
    // compute CIs for y0 and y1
    CI0 = distcompCI(n0,aret[1])
    CI1 = distcompCI(n1,aret[1])
    // If duplicate values in y0 or y1, remove them,
    //    and adjust CIs accordingly
    tmpmat0 = distcompuniqCI(y0,CI0)
    tmpmat1 = distcompuniqCI(y1,CI1)
    // compute ranges of rejection (and 3rd column = which direction)
    rejranges = distcomprejranges(tmpmat0[.,1],tmpmat1[.,1],tmpmat0[.,2..3],tmpmat1[.,2..3])

    // store return values
    // st_numscalar(sprintf("distcompglobalrej%f",a), rows(rejranges)>0)
    st_numscalar(sprintf("distcompglobalrej%f",a), anyof(rejranges[.,3],1)+2*anyof(rejranges[.,3],0))
    if (abs(100*alpha-a)<0.000001) { // store rej ranges only for desired alpha level
      st_numscalar("distcompalphasim",aret[2])
      if (rows(rejranges)==0) {
        st_matrix("distcomprejranges", J(1,2,.))
        // rejranges = J(1,2,.)
      }
      else {
        rowstr = (J(rows(rejranges),1,""), J(rows(rejranges),1,"<"))
        rowstr[selectindex(rejranges[.,3]:==0), 2] = J(sum(rejranges[.,3]:==0),1,">")
        st_matrix("distcomprejranges",rejranges[. , 1..2])
        st_matrixrowstripe("distcomprejranges", rowstr)
      }
    }
  }

  // compute global p-value if desired
  if (pvalueflag) {
    p = distcompgofp(y0,y1)
    st_numscalar("distcompglobalpvalue", p)
  }
}


// compute confidence intervals at each order statistic
real matrix distcompCI(real scalar n, real scalar at) {
  real matrix ret
  ret = J(n,2,.)
  for (k=1; k<=n; k++) {
    ret[k,1] = invibeta(k, n+1-k,   at/2)
    ret[k,2] = invibeta(k, n+1-k, 1-at/2)
  }
  return(ret)
}


// remove duplicate values from y and CI (i.e., y0,CI0 or y1,CI1)
real matrix distcompuniqCI(real colvector y, real matrix CI) {
  // return matrix (ret) columns: y, CIlow, CIhigh
  real matrix ret
  real scalar i, cnt
  ret = (y[1], CI[1,.])
  cnt = 1 // row index for ret
  for (i=2; i<=length(y); i++) {
    if (y[i]==y[i-1]) { // duplicate y...
      ret[cnt,3] = CI[i,2] // ...only replace upper CI endpoint
    }
    else { // new y value, so make new row
      cnt++
      ret = ret \ (y[i], CI[i,.])
    }
  }
  return(ret)
}


// check global/GOF rejection
real scalar distcompgofrej(real colvector y0, real colvector y1, real matrix CI0, real matrix CI1) {
  real rowvector ind, n
  real scalar xy, tmp, n0, n1
  real matrix y, L, H
  n0 = length(y0)
  n1 = length(y1)
  n = (n0,n1)
  tmp = 1+max(n) // need 1+ to get infinity @ end of both
  y = J(tmp,2,.) // rem: . is like +Infinity
  L = J(tmp,2,.)
  H = J(tmp,2,.)
  y[1..n0,1] = y0;  L[1..n0,1] = CI0[.,1];  H[1..n0,1] = CI0[.,2]
  y[1..n1,2] = y1;  L[1..n1,2] = CI1[.,1];  H[1..n1,2] = CI1[.,2]
  ind = (1,1) // index counters for y0,y1
  xy = 1 // 1=y0, 2=y1
  // ensure y[1,xy]>y[1,3-xy]
  if (y[1,1] < y[1,2]) {
    xy = 2
  }
  while (ind[xy]<=n[xy]) {
    xy = 3-xy
    while (y[ind[xy]+1,xy] < y[ind[3-xy],3-xy]) {
      ind[xy] = ind[xy]+1
    }
    if (L[ind[xy],xy] > H[ind[3-xy],3-xy]) {
      return(1)
    }
    ind[xy] = ind[xy]+1
  }
  return(0) // if made it this far, don't reject
}


// compute global/GOF/goodness-of-fit two-sided p-value
//   y0 = sorted observed data, group #1
//   y1 = sorted observed data, group #2
real scalar distcompgofp(real colvector y0, real colvector y1) {
  real scalar ret // the global p-value to return
  real scalar n0, n1, TOL, alo, ahi, acur, rej, EPS // sample sizes; tolerance; alpha(tilde) bounds; etc.
  real matrix tmpmat0, tmpmat1
  n0 = length(y0)
  n1 = length(y1)
  // search for smallest alpha.tilde such that Reject
  TOL = 1.01 // stop when a.hi/a.lo<TOL
  EPS = 0.00000001 // a very very small number
  alo = 0.100
  ahi = 0.900
  // if necessary, decrease alo until fail to reject w/ alphatilde=alo
  rej = 1
  while (rej==1 & alo>EPS) {
    alo = alo/10
    CI0 = distcompCI(n0,alo)
    CI1 = distcompCI(n1,alo)
    tmpmat0 = distcompuniqCI(y0,CI0)
    tmpmat1 = distcompuniqCI(y1,CI1)
    rej = distcompgofrej(tmpmat0[.,1],tmpmat1[.,1],tmpmat0[.,2..3],tmpmat1[.,2..3])
  }
  if (alo<=EPS) { // alo is really close to zero, so just return
    return(distcompverify(alo,n0,n1))
  }
  // if necessary, increase ahi until reject w/ alphatilde=ahi
  rej = 0
  while ((rej==0) & (ahi < (1-EPS))) {
    ahi = 1-(1-ahi)/10
    CI0 = distcompCI(n0,ahi)
    CI1 = distcompCI(n1,ahi)
    tmpmat0 = distcompuniqCI(y0,CI0)
    tmpmat1 = distcompuniqCI(y1,CI1)
    rej = distcompgofrej(tmpmat0[.,1],tmpmat1[.,1],tmpmat0[.,2..3],tmpmat1[.,2..3])
  }
  if (ahi >= 1-EPS) { // ahi really close to 1, so just return
    return(distcompverify(ahi,n0,n1))
  }
  // at this point: don't reject w/ alo, do w/ ahi
  // next, increase alo and decrease ahi while maintaining these conditions
  // eventually, alo and ahi are very close, showing the alphatilde where
  // "barely" rejects; can compute the p-value from that alphatilde
  while (ahi/alo > TOL) {
    acur = (alo+ahi)/2
    CI0 = distcompCI(n0,acur)
    CI1 = distcompCI(n1,acur)
    tmpmat0 = distcompuniqCI(y0,CI0)
    tmpmat1 = distcompuniqCI(y1,CI1)
    rej = distcompgofrej(tmpmat0[.,1],tmpmat1[.,1],tmpmat0[.,2..3],tmpmat1[.,2..3])
    if (rej==0) {
      alo = acur
    }
    else {
      ahi = acur
    }
  }
  // simulate overall type I error rate for alphatilde found above
  // err on "conservative" side (ahi instead of alo; slightly higher p-value)
  ret = distcompverify(ahi,n0,n1)
  return(ret)
}


// simulate global/GOF type I error rate given alpha-tilde
real scalar distcompverify(real scalar atilde, real scalar n0, real scalar n1) {
  real scalar BLOCKSIZE, MINDRAWS, NDRAWS, irep, i, j, ret
  real colvector rejs
  real matrix CI0, CI1, OS0, OS1
  rseed(112358) // to be reproducible
  MINDRAWS = 1e4
  // constant parameters
  BLOCKSIZE = max((1,round(1e6/(n0+n1))))
  NDRAWS = BLOCKSIZE*ceil(MINDRAWS/BLOCKSIZE)
  // storage variables
  rejs = J(NDRAWS,1,.)
  CI0 = distcompCI(n0,atilde)
  CI1 = distcompCI(n1,atilde)
  for (irep=1;irep<=NDRAWS/BLOCKSIZE;irep++) {
    OS0 = rgamma(BLOCKSIZE,n0+1,1,1)
    for (i=1;i<=BLOCKSIZE;i++) {
      OS0[i,.] = runningsum(OS0[i,.]) / sum(OS0[i,.])
    }
    OS0 = OS0[.,1..n0]
    OS1 = rgamma(BLOCKSIZE,n1+1,1,1)
    for (i=1;i<=BLOCKSIZE;i++) {
      OS1[i,.] = runningsum(OS1[i,.]) / sum(OS1[i,.])
    }
    OS1 = OS1[.,1..n1]
    for (j=1;j<=BLOCKSIZE;j++) {
      rejs[BLOCKSIZE*(irep-1)+j] = distcompgofrej(OS0[j,.]',OS1[j,.]',CI0,CI1)
    }
  }
  ret = mean(rejs)
  return(ret)
}


// given CIs and order statistics (sorted values), determine
//   statistically significantly different ranges of values.
// os0 and os1 should (each) have unique values.
real matrix distcomprejranges(real colvector os0, real colvector os1, real matrix CI0, real matrix CI1) {
  real matrix ret
  real colvector r1a, r3a, r4a, r1, r3, r4, intfrom, intto
  real rowvector tmprow
  string colvector r2a, r2
  real vector tmp0, tmp1
  real scalar i, i2
  ret = J(0,3,.)
  CI0 = (-999, .) \ CI0 \ (., 999)
  CI1 = (-999, .) \ CI1 \ (., 999)
  r1a = os0\os1
  r2a = J(length(os0),1,"y0") \ J(length(os1),1,"y1")
  // sort r1a and r2a by value of (pooled) order statistics
  r2a = r2a[order(r1a,1), .]
  r1a = sort(r1a, 1)
  tmp0 = r2a:=="y0"
  tmp1 = r2a:=="y1"
  r3a = J(length(r1a), 1, .) // store cumsum of tmp0
  r4a = J(length(r1a), 1, .) // store cumsum of tmp1
  r3a[1] = 0+tmp0[1]
  r4a[1] = 0+tmp1[1]
  for (i=2; i<=rows(r3a); i++) {
    r3a[i] = r3a[i-1]+tmp0[i]
    r4a[i] = r4a[i-1]+tmp1[i]
  }
  // deal with duplicates
  v = J(length(r1a),1,1)
  for (i=2;i<=length(r1a);i++) {
    if (r1a[i]==r1a[i-1]) v[i-1]=0
  }
  r1 = r3 = r4 = J(sum(v),1,.)
  r2 = J(sum(v),1,"")
  for (i=i2=1; i<=length(r1a); i++) {
    if (v[i]) {
      r1[i2] = r1a[i]; r3[i2] = r3a[i]; r4[i2] = r4a[i]
      r2[i2] = r2a[i]
      if (i>1) {
        if (v[i-1]==0) r2[i2]="01" //source = both y0 and y1
      }
      i2++
    }
  }
  // set up potential ranges
  intfrom = (r1[1]-999) \ r1
  intto   = r1 \ (r1[length(r1)]+999)
  for (i=1; i<=rows(r3); i++) {
    if ((CI0[r3[i]+1,1]>CI1[r4[i]+1+1,2]) | (CI0[r3[i]+1+1,2]<CI1[r4[i]+1,1])) {
      tmp = (rows(ret)>0)?(ret[rows(ret),2]==intfrom[i+1]):(0)
      if (tmp) {
        ret[rows(ret),2] = intto[i+1]
      }
      else {
        tmprow = (intfrom[i+1], intto[i+1], (CI0[r3[i]+1,1]>CI1[r4[i]+1+1,2]))
        ret = ret \ tmprow
      }
    }
    else if (r2[i]=="01") {
      if ((CI0[r3[i]+1,1]>CI1[r4[i]+1,2]) | (CI0[r3[i]+1,2]<CI1[r4[i]+1,1])) {
        tmp = (rows(ret)>0)?(ret[rows(ret),2]==intfrom[i+1]):(0)
        if (!tmp) {
          tmprow = (intfrom[i+1], intfrom[i+1], (CI0[r3[i]+1,1]>CI1[r4[i]+1,2]))
          ret = ret \ tmprow
        }
      }
    }
  }
  return(ret)
}


// retrieve alpha.tilde (and alpha.sim) from lookup table
real colvector distcompalpha(real scalar n0, real scalar n1, real scalar alphaperc, real matrix LOOKUP) {
  real scalar alpha
  real colvector ret, n
  alpha = alphaperc / 100
  ret = J(2,1,.)
  n = (min((n0,n1)),max((n0,n1)))'
  // Formula: good if n0==n1 and n0,n1>=200, o/w conservative
  if (alphaperc==1) {
    ret = (exp(0.02576944-3.37109911*sqrt(ln(ln(max(n))))), .)'
  }
  else if (alphaperc==5) {
    ret = (exp(0.6872947-3.1503102*sqrt(ln(ln(max(n))))), .)'
  }
  else if (alphaperc==10) {
    ret = (exp(1.081619-3.125037*sqrt(ln(ln(max(n))))), .)'
  }
  if (n[1]/n[2]<0.5) {
    if (n[1]/n[2]<0.1) {
      ret[1] = ret[1]*(1/2)*((n[1]/n[2])/0.1) // from 0 to 1/2 as n1/n2 from 0 to 0.1
    }
    else {
      ret[1] = ret[1]*((n[1]/n[2]+0.3)/0.8) // from 0.5 to 1 as n1/n2 from 0.1 to 0.5
    }
  }
  if ((max(n)>=1000 & n[1]/n[2]>0.9) | (max(n)>100000 & n[1]/n[2]>0.5)) {
    return(ret)
  }
  // Otherwise use lookup table
  // First filter by alpha
  L = select(LOOKUP, LOOKUP[.,1]:==alpha)
  if (rows(L)==0) {
    _error("Need alpha=0.10 or 0.05 or 0.01 or else use the more functional R code from the author's website.")
  }
  // Second filter by n
  L = select(L, (L[.,2]:>=n[1]) :* (L[.,3]:>=n[2]))
  if (rows(L)==0) {
    return(ret) // sample sizes very large; use formula
  }
  // Third find alpha.tilde
  biggest = max(L[.,4])
  L = select(L, L[.,4] :== biggest)
  if (L[1,4]*1.01>ret[1] | (min(n)<200)) {
    ret[1] = L[1,4]
    ret[2] = L[1,5]
    if (L[1,5]!=L[1,6]) { // err on conservative side
      ret[1] = ret[1] - 0.5*10^(-8)
    }
  }
  return(ret)
}


// "load" lookup table for critical values
// columns: "ALPHA","nx","ny","alpha.tilde","ALPHA.simulated.lo","ALPHA.simulated.hi"
GoldmanKaplan2sLookup = (0.010,      5,     10,0.07883072,0.00999300,0.00999300)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.050,      5,     10,0.13562066,0.03956000,0.05944000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.100,      5,     10,0.21463444,0.08753500,0.12874000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.010,      9,     10,0.05366823,0.00696100,0.01143400)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.050,      9,     10,0.12543036,0.04372000,0.05807000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.100,      9,     10,0.15284111,0.08454000,0.10606000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.190,      9,     10,0.24567187,0.16892500,0.20262000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.200,      9,     10,0.24567187,0.16892500,0.20262000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.010,     10,     10,0.05041165,0.00650100,0.01219400)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.050,     10,     10,0.10937500,0.04419000,0.05253000)
GoldmanKaplan2sLookup =GoldmanKaplan2sLookup \ (0.100,     10,     10,0.18032046,0.08845500,0.13192000)
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end

*EOF