Application of the Limited Enthusiasm Model with Reversion, Written in 2022.

The Union of Welsh Independents is a congregational church with about 400 congregations in Wales. Also known as the Annibynwyr, most churches hold serves in Welsh. Its first congregation was formed in 1639. The first few generations of members were firmly Puritan in their convictions. With the passage of time, the church has liberalised and declined. This analysis investigates the likelihood of decline leading to denominational extinction.

Estimating Model Parameters

I will use the Welsh Independents’ membership data to estimate the parameters of the Limited Enthusiasm Model with Reversion. Using this cut-down version of the Demographics Model avoids estimating values for child retention among church members, hardening rates etc. There is no information about these values for this church. The reversion model has only five parameters:

  1. Reproduction Potential
  2. Duration Enthusiastic
  3. Loss Rate
  4. Fraction of Converts who Become Enthusiasts
  5. Initial Fraction of Enthusiasts in the Church

The key parameter is the reproduction potential. If this potential is less than one, then the church’s decline is to extinction.

Calibration Strategy

The best fit between data and model is found using least squares. Avoiding systematic deviations from the data can help reduce the number of best fits. To help find the most suitable parameter values, I vary parameters 3-5 above and optimise for 1-2, the reproduction potential and duration enthusiastic. Out of this optimisation set, I count how many indicate the church will become extinct. Although these parameter values cannot be determined with certainty, the likelihood of extinction is more robust.

Membership 2000-2020

I took the membership data of the Presbyterian Church of Wales from Brierley (2008-2020). The data from 2000 to 2004 was inconsistent with later data. Thus, I started the simulation in 2005. Likewise, I needed to remove data from 2017 onwards, which was estimated yet indicating a slower decline. I will use this data to set an alternative extinction date.

Every optimisation indicates that the Welsh Independents will end up extinct.

One such optimisation for 2005-2017 is given in figure 1. The reproduction potential chosen was an average value of 0.55, well under the extinction threshold of 1. I set the fraction of new converts who become enthusiasts at 10%. If I had used a more optimistic figure of 50%, then the average reproduction potential increases to 0.87, still indicating extinction.

Data fot of limited enthusiasm model to data for the union of Wlesh Independents
Figure 1: Data Fit for the Union of Welsh Independents Membership and the Limited Enthusiasm with Reversion Model 2005–2017

Range of Optimised Parameters 2005–2017

The optimisations for 2005–2017 indicate a large range of possible parameter values:

Reproduction Potential0.103–0.840
Duration Enthusiasticabout 1 year
Loss Rate
(Reversion and Deaths)
Fraction of Converts who Become Enthusiastsabout 10%
Initial Fraction of Enthusiasts in the Church0.5%–3%
Model Parameters for Union of Welsh Independents

If a case could be made for 50% of the converts becoming enthusiasts, the reproduction potential has a range is 0.367–0.972. Thus, all possible scenarios indicate extinction for the denomination.

Membership 1980–2020

Because of the difficulty with the data above, I also tried data fitting over 1980-2020. The range for the reproduction potential narrows to 0.228-0.787 where only 10% of converts become enthusiastic. One such optimisation for 1980-2020 is given in figure 2 using a reproduction potential of 0.505. The rate of decline of the church steepens in the 1990s then slows from in the 2000s. It is possible that aging effects will prevent the decline from slowing.

Figure 2: Data Fit for the Union of Welsh Independents Membership and the Limited Enthusiasm with Reversion Model 1980–2020.

Possible Extinction Date

Although the model predicts extinction, it does not indicate an extinction date. The model contains exponential decline because it does not capture the effects of ageing. By 2060 it forecasts a church of just 2,500 members.

Fitting a straight line to the data from 2005 to 2017 gives a more realistic estimate of the extinction date, as decline through ageing is a linear process. This linear fit forecasts an extinction date of 2035 for The Union of Welsh Independents, figure 3. Nevertheless, it is possible the church decline may slow down, delaying extinction. Although the data for 2018–2020 is estimated, the estimates indicate a slowing of decline. Using a linear fit of 2015–2020 gives an extinction date of 2052. More reliable data is needed to narrow down this date.

Figure 3: Linear Fit to Welsh Independents Membership Data

Long–Term View

Figure 4 shows the rise and fall of the Welsh Independents from from 1900. I could not find data before 1900. One of my sources, Currie et al. combines the independents with the English Congregationalist in Wales. This combined data indicates the Welsh Independents rose throughout the 19th century.

Figure 4: Long-term growth and decline of the Union of Welsh Independents

The church declined from 1940 onwards with a very steep phase during the 70s and 80s, figure 4. The decline of the Welsh language and population movements away from rural areas may have contributed to this church’s decline.