Germán Rodríguez
Demographic Methods Princeton University

The Third Child in Colombia

We will look at the interval from second to third birth using Colombian WFS data, circa 1976. We start from an extract that has the dates of R's birth, date of interview, the birth history, date of first union if any, and current and childhood type of place of residence.

. use http://data.princeton.edu/eco572/datasets/cofertx, clear
(COSR02 extract)

Time at Risk and Event Indicator

We will use intervals that start in the ten years before the interview and exclude twins. (The results are a bit different from my paper with Hobcraft, which used all birth intervals, but I can reproduce the earlier results by removing the period restriction.)

. keep if b022 >= v007-120 & b022 < v007
(4144 observations deleted)

. drop if b032==b022      // only 6
(6 observations deleted)

We construct time at risk starting in the middle of the month of birth of the second child and ending in the middle of the month when the third child is born or at the end of the month before the interview, whichever occurs first

. gen expo = b032 - b022

. replace expo = v007 - b022 -0.5 if v007 <= b032
(496 real changes made)

. gen third = b032 < v007

If we stset the data we can take advantage of Stata's survival analysis facilities. For example it is very easy to obtain a plot a Kaplan-Meier estimate of survival at parity 2:

. gen id=_n

. stset expo, fail(third) id(id)

                id:  id
     failure event:  third != 0 & third < .
obs. time interval:  (expo[_n-1], expo]
 exit on or before:  failure

------------------------------------------------------------------------------
     1228  total obs.
        0  exclusions
------------------------------------------------------------------------------
     1228  obs. remaining, representing
     1228  subjects
      732  failures in single failure-per-subject data
    34388  total analysis time at risk, at risk from t =         0
                             earliest observed entry t =         0
                                  last observed exit t =     115.5

. sts graph

         failure _d:  third
   analysis time _t:  expo
                 id:  id

. graph export co3rdkm.png, replace
(file co3rdkm.png written in PNG format)

Segments of Exposure

We will take advantage of Stata's facilities to split the exposure into 3 month segments. The cutpoints have the form 0 3.5 6.5 9.5 ... 57.5 60.5 120.

. stsplit segment, at(0 3.5(3)60.5 120)
(9844 observations (episodes) created)

Stata's built-in variables _t and _t0 have the start and end of each segment and _d has the 'death' indicator. We use these to compute events and exposure. This is not a bad time to save the data.

. replace expo = _t-_t0
(9844 real changes made)

. gen births = _d

. save temp,replace
file temp.dta saved

We then collapse to obtain total events and exposure by duration (segment) and childhood place of residence.

. drop if v703 > 3 // a few missing values
(36 observations deleted)

. collapse (sum) births (sum) expo , by(segment v703)

At this point we can easily calculate life tables.

The Birth Function

I will compute the hazard dividing births by exposure and then use the constant hazard assumption to estimate the cumulative hazard and survival (or rather its complement, the birth function). You can get very similar results assuming that events occur half-way through each three-month interval. (Even more similar if we had used single months.)

. gen h = births/expo

. bysort v703 (segment): gen H = 3*sum(h)

. gen B = 1-exp(-H)       // the birth function

. separate B, by(v703)

              storage  display     value
variable name   type   format      label      variable label
-------------------------------------------------------------------------------
B1              float  %9.0g                  B, v703 == Rural
B2              float  %9.0g                  B, v703 == Town
B3              float  %9.0g                  B, v703 == City

. line B1 B2 B3 segment, lpat(solid longdash dash) ///
>         title("The Third Child in Colombia, 1976") ///
>         subtitle(Birth Functions by Childhood Residence) ///
>         ytitle(Proportion having a 3rd birth) /// 
>         xlabel(12 24 36 48 60) xtitle(months since 2nd birth) ///
>         legend(order(1 "Rural" 2 "Town" 3 "City") ///
>         col(1) ring(0) pos(5))

. graph export co3rdbirth.png, replace
(file co3rdbirth.png written in PNG format)

We see that the proportions moving to parity three are higher for women of rural origins than for others at every duration. Women who grew up in towns are a bit less likely to have a third child, and when they do they have it later. Women who grew up in cities are much less likely to have a third child in the long run, but they are also more likely than townfolk to have very short intervals, with a cross-over around 18 months.

Quintums and Trimeans

To compute quintums we need to interpolate in the category 57.5-60.5. Linear interpolation would probably do, but with constant hazard a more exact calculation based on S(60) = S(57.5) exp{-2.5 h(57.5)} gives Q = 1-(1-B(57.5))exp{-2.5 h(57.5)}. Note that the way we calculated things, the survival and birth functions pertain to the end of each segment, which is why the code below looks at 54.5

. gen Q =1-(1-B[_n]) * exp(-2.5*h[_n+1]) if  segment==54.5
(60 missing values generated)

. list v703 segment h B Q if segment > 54 & v703 < 9, sep(3)

     +--------------------------------------------------+
     |  v703   segment          h          B          Q |
     |--------------------------------------------------|
 19. | Rural      54.5   .0207612   .8118494   .8230233 |
 20. | Rural      57.5   .0244898   .8251771          . |
 21. | Rural      60.5   .0086022    .829631          . |
     |--------------------------------------------------|
 40. |  Town      54.5   .0363636   .7815775   .7930105 |
 41. |  Town      57.5   .0215054   .7952242          . |
 42. |  Town      60.5   .0059435   .7988432          . |
     |--------------------------------------------------|
 61. |  City      54.5   .0137931   .6706035   .6766252 |
 62. |  City      57.5   .0073801   .6778163          . |
 63. |  City      60.5   .0223537   .6987138          . |
     +--------------------------------------------------+

So the quintums are 82.3, 79.3 and 67.7 for women who grew up in rural areas, towns and cities. (These are a bit lower than in my paper with Hobcraft, reflecting the fact that they pertain to more recent births.)

To obtain the trimean we use a similar interpolation strategy for computing the quartiles and then apply Tukey's formula. I will illustrate with rural women. A quarter of the quintum of .823 is .206, so we need the durations by which 20.6, 41.2 and 61.7% of the women have moved to parity 3. For the first value we need to interpolate in category 9.5-12.5. Write 1 - 0.206 = (1-B(9.5)) exp{-2.5 h(9.5)(Q1-9.5)} and solve for Q1 to obtain Q1 = 9.5 -log((1-0.206)/(1-B(9.5)))/h(9.5). Because the calculations are repetitive I encapsulate them in a little program which takes as argument a category of v703, finds the row where we need to interpolate, and applies the above formula:

. capture program drop trimean

. program define trimean
  1.         args ctpr
  2.         quietly sum Q if v703==`ctpr'
  3.         local q = r(mean)/4
  4.         local T = 0
  5.         tempvar i
  6.         gen `i' = _n
  7.         forvalues k=1/3 {
  8.                 local b = `k'*`q'
  9.                 quietly sum `i' if v703==`ctpr' & B < `b'
 10.                 local n = r(max)
 11.                 local Q = seg[`n']-log((1-`b')/(1-B[`n']))/h[`n'+1]
 12.                 display "Q`k' = `Q'"
 13.                 local T = `T' + `Q'
 14.                 if (`k' == 2) local T = `T' + `Q'
 15.         }
 16.         local T = `T'/4
 17.         display "Trimean: `T'"
 18. end

We now run the computations for the three categories

. trimean 1
Q1 = 13.04612965753713
Q2 = 18.84358957514994
Q3 = 27.81360231842412
Trimean: 19.63672778156528

. trimean 2
Q1 = 14.75915079139585
Q2 = 21.72924126033396
Q3 = 30.01968865537609
Trimean: 22.05933049185996

. trimean 3
Q1 = 13.15086885691454
Q2 = 19.51578799192259
Q3 = 28.77935879083599
Trimean: 20.24045090789893

So the average birth intervals for those who go on to have a third child are 19.6, 22.1 and 20.2 months, respectively, for women who grew up in rural areas, towns and cities. (The intermediate value for the last group was also observed in the analysis using all intervals.)

The Hazard Functions

The fact that the hazard functions are somewhat noisy is easily verified:

. separate h, by(v703)

              storage  display     value
variable name   type   format      label      variable label
-------------------------------------------------------------------------------
h1              float  %9.0g                  h, v703 == Rural
h2              float  %9.0g                  h, v703 == Town
h3              float  %9.0g                  h, v703 == City

. gen d = segment+1.5

. line h1 h2 h3 d, lpat(solid longdash dash) ///
>         title("The Third Child in Colombia, 1976") ///
>         subtitle(Hazard Functions by Childhood Residence) ///
>         xtitle(duration since 2nd birth) ///
>         ytitle(hazard of having 3rd birth) ///
>         legend(order(1 "Rural" 2 "Town" 3 "City")       row(1))

. graph export co3rdhaz.png, replace
(file co3rdhaz.png written in PNG format)

Fitting a proportional hazards model is quite easy using the Poisson trick. (This will be covered in some detail in wws509.)

Let us construct dummy variables for duration and childhood residence and then run a regression for the first five years using the log of exposure time as an offset.

. gen town = v703==2

. gen city = v703==3

. forvalues d = 12(3)57 {
  2.         gen dur`d' = segment==`d'.5
  3. }

. gen os = log(expo)

. gen touse = seg > 9 & seg < 60 

. poisson births dur12-dur57 town city if touse, offset(os)

Iteration 0:   log likelihood = -126.22561  
Iteration 1:   log likelihood = -125.00294  
Iteration 2:   log likelihood =  -124.9991  
Iteration 3:   log likelihood =  -124.9991  

Poisson regression                                Number of obs   =         51
                                                  LR chi2(18)     =     120.26
                                                  Prob > chi2     =     0.0000
Log likelihood =  -124.9991                       Pseudo R2       =     0.3248

------------------------------------------------------------------------------
      births |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       dur12 |   .8387151   .1677175     5.00   0.000     .5099949    1.167435
       dur15 |   1.030044   .1672236     6.16   0.000     .7022913    1.357796
       dur18 |   .9195063   .1758572     5.23   0.000     .5748326     1.26418
       dur21 |    1.00266   .1794974     5.59   0.000      .650851    1.354468
       dur24 |   1.131769   .1816092     6.23   0.000     .7758211    1.487716
       dur27 |   .7913616   .2053167     3.85   0.000     .3889482    1.193775
       dur30 |   .9017034    .209525     4.30   0.000     .4910419    1.312365
       dur33 |   .3386295   .2624972     1.29   0.197    -.1758555    .8531145
       dur36 |   .3342246   .2728632     1.22   0.221    -.2005775    .8690267
       dur39 |   .3081459   .2853069     1.08   0.280    -.2510455    .8673372
       dur42 |   .7543197   .2579441     2.92   0.003     .2487585    1.259881
       dur45 |  -.1988996   .4022531    -0.49   0.621    -.9873012    .5895019
       dur48 |  -.4680164   .4679286    -1.00   0.317     -1.38514    .4491068
       dur51 |   .3255532   .3448924     0.94   0.345    -.3504236     1.00153
       dur54 |   .3411703   .3606533     0.95   0.344    -.3656972    1.048038
       dur57 |   .0717598   .4308883     0.17   0.868    -.7727656    .9162853
        town |  -.2020243    .088543    -2.28   0.023    -.3755654   -.0284831
        city |  -.3866947   .0977489    -3.96   0.000     -.578279   -.1951103
       _cons |  -3.951717   .1419443   -27.84   0.000    -4.229923   -3.673512
          os |   (offset)
------------------------------------------------------------------------------

. estat gof

         Goodness-of-fit chi2  =  51.44767
         Prob > chi2(32)       =    0.0161

The duration coefficients track an early increase in hazard up to about a year and a half followed by a steady decline thereafter. On average, at any given duration since second birth the risk of having a third child is 18.3% lower for women who grew up in towns and 32.1% lower for women who grew up in cities than for those who grew up in rural areas. The goodness of fit test indicates, however, that the hazards for the three categories of childhood residence are not proportional. This is consistent with the differences in timing between women with town and city origins noted earlier.