[This page explains how BirthChoiceUK has estimated the national normal birth rate for England for years prior to 1995, using spontaneous onset and spontaneous delivery data. If you want to know about normal birth in general, or find out normal birth rates for maternity units in England or Scotland, please visit the BirthChoiceUK Normal Birth Pages.]
For 1995 to 2001 the data is readily available to calculate normal birth rates. Prior to then, rates for spontaneous onset and delivery are not available. What is available however, is spontaneous onset and spontaneous delivery rates separately.
What needs to be determined is the proportion of women who start labour spontaneously and then carry on to give birth spontaneously. Let us designate this proportion by r(t). The (t) implies an explicit time dependence, i.e. the number is expected to change from one year to the next.
So we can write the spontaneous onset and delivery rate as:-
Sod(t)=r(t)*So(t)
Where So(t) is the spontaneous onset rate.
We designate the proportion of women who have been induced and yet carry on to have a spontaneous delivery as q(t).
Now by simple arithmetic
Sd(t)= r(t)*So(t)+q(t)*I(t)
Where Sd(t) is the spontaneous delivery rate and I(t) the induction rate.
To proceed we need to make some simplifying assumptions. Let us assume that the proportion of women induced who go on to have a spontaneous delivery is unchanged year on year relative to the not-induced women who have a spontaneous delivery.
So we can write
q(t)=p*r(t)
where "p" is a constant which does not change from year to year.
This implies that the combined Caesarean and Instrumental rates for induced women increases at the same rate as for not-induced women. A priori there is no reason to doubt this assumption.
After a little algebra we can write:-
Sod(t)= So(t)*Sd(t) / (So(t)+p*I(t))
Where the only unknown is the constant "p". There are however, 7 years of data which we can use to estimate p.
Using maximum likelihood "p" comes out to be 0.86 =/- 0.01. (The expected value of the second derivative of the log likelihood function is used to derive the error term. "p" has been assumed to be a normally distributed random variable, technically it can not be of course, as it is bounded, however this assumption will suffice.)
As there is only the one unknown it is possible to calculate "p" for every year. Doing this confirms the assertion that "p" is stationary.
The quantity "p" is interesting in its own right as it is related to the increased propensity of induced women to succumb to further complications during labour.
Now we have a model, an estimate of its parameters and sufficient data to make a good estimate of the normal birth rate for the 5 previous years, 1990-1994.
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