June 10, 2009
ROSTOCK, Germany. I’m here to teach one day of a summer school at the Max Planck Institute for Demographic Research, assisted by Niveen Abu-Rmeileh of the Institute for Community and Public Health at Birzeit University in Ramallah, Palestine.
Our topic is “Time-Plotting, Quantum and Tempo of Life Cycle Events.” During the 3 hour morning session we did time-plotting (you can read more about this in earlier posts), combining presentations and exercises.
In the afternoon I did a brief presentation on fertility (mainly) and mortality (a little bit) tempo. After the session Niveen pointed out that it would have been helpful to show some numbers and calculations to help students understand the material.
She is right, of course, so I decided to get up a spreadsheet and post it here for the benefit of the summer school students and anyone else interested in understanding the method described in the 1998 PDR paper by Bongaarts and Feeney “On the Quantum and Tempo of Fertility.”
The spreadsheet shows the calculation only for first births, but the calculation for higher order births is the same. The input data is a table of annual, single year (15-49) age-order-specific first birth rates for 1917-1991. Download/view the file fertility-tempo-calculations-for-first-births.xls from docstoc.com.
The tempo calculations are at the bottom of the “Data” sheet, beneath the first birth rates for age 49. TFR1 is the “total first birth rate,” which is simply the sum of the age-order-specific first birth rates over all ages. MAC1 is the mean age at first birth, calculated by multiplying each rate by the midpoint of the age summing over all births, and dividing by TFR1.
The adjustment for each year requires the rate of change in MAC1 for the year. To calculate this we average the MAC1 values for 1917 and 1918 (for example) to get a MAC for the beginning of 1918, average the values for 1918 and 1919 to get a MAC for the end of 1918, and subtract the former from the later to get r, the rate of change in the mean age at childbearing during calendar year 1918. Note that we loose the first and the last year of the series as a result of this calculation.
The adjusted TFR1 is calculated by dividing TFR1 by (1-r. That’s it, that’s all there is to it. You can of course verify this description by downloading the file and looking at the formulas in the “Data” sheet.” Other sheets in the file show plots of TFR1, original and adjusted, MAC1,and r.
The final sheet shows normalized cumulative distributions for each year. Here’s the rationale for looking at these—this incidentally goes beyond what’s in the 1998 paper. The assumptions on which the validity of the adjustment rests imply that the shape of the age-schedule of first births does not change from one year to the next, that it moves to older or younger ages, and is increased or decreased by a constant factor, but that the shape is fixed.
If this is the case, the normalized cumulative distributions should all be simple translations of each other, backward or forward on the age axis. This is true of the distribution functions as well, but it will be easier to see when looking at the cumulative functions because the cumulative functions will not, if the assumption is valid, intersect each other.
So the idea was to plot these and inspect to get an idea of how well the assumption holds. Unfortunately, the shifts are so small that we can’t see anything on the scale of even a large computer screen. (This could have been thought through without doing the plot, but plots are so easy that this kind of thinking is discouraged). You can get some idea by changing the scales of the plot to show very small time and age intervals, but this is too klutzy to pursue very far.
Some further graphic and perhaps transformational ingenuity is needed to follow through on this idea. It is well worth doing, for there are clearly circumstances in which the assumption doesn’t hold and the adjustment should be discounted or omitted.