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

The Presbyterian Church of Wales is a protestant church based in Wales, UK. It originated in the Methodist revivals under the leadership of Daniel Rowlands, Howell Harris and William Williams, Pantycelyn, from 1735 onwards. Originally known as the Calvinist Methodists, they finally broke from the Anglican church in 1811, becoming a separate denomination. They changed their name to the current one in 1928 and shortly afterwards changed their constitution. Decline set in after the 1904/5 revival. This analysis investigates the likelihood of decline leading to denominational extinction.

Estimating Model Parameters

I will use the Presbyterian Church of Wales’ 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). Even when checked with the church’s own estimates, the data is inconsistent over time. The change from 2010 to 2011 is too small compared with the previous few years. I think this may be due to a change in the definition of membership. Whatever the reason, it made data fitting to the whole period less robust than I would have liked. Thus, I also optimised just for the period 2012-2020.

Every optimisation indicates that the Presbyterian Church of Wales will end up extinct.

One such optimisation for 2000-2020 is given in figure 1. The reproduction potential chosen was an average value of 0.62, well under the extinction threshold of 1. The date fit for 2012-202 is more convincing, figure 2. It has a similar reproduction potential of 0.63. Despite the difference in data, the story behind each optimisation is the same – extinction.

Figure 1: Data Fit for Welsh Presbyterian Church Membership and the Limited Enthusiasm with Reversion Model 2000–2020
Figure 2: Data Fit for Welsh Presbyterian Church Membership and the Limited Enthusiasm with Reversion Model 2012–2020

Range of Optimised Parameters

The optimisations for 2012–2020 indicate a large range of possible parameter values:

Reproduction Potential0.019–0.85
Duration Enthusiastic0.7–1.4 years
Loss Rate
(Reversion and Deaths)
Fraction of Converts who Become Enthusiastsabout 10%
Initial Fraction of Enthusiasts in the Church0.5%–1%
Model Parameters for Welsh Presbyterians

The wide range of reproduction potentials is matched with the various options for the loss rate. Losses include deaths and are undoubtedly high because of the church’s older age profile. A middle value for both parameters is more likely.

I repeated this optimisation strategy for different total population values, starting with the full England/Wales value, and then using lower values. The Welsh Presbyterians may have a restricted influence in society and have contact with only part of the population. The model might give a more optimistic reproduction potential with a lower population. Extinction occurred in all such optimisations.

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 1,000 members.

Fitting a straight line to the data from 2000 to 2020 gives a more realistic estimate of the extinction date, as decline through ageing is a linear process. The Presbyterian Church of Wales is forecast to be extinct by 2033, figure 3. Nevertheless, it is possible the church’s decline may slow down, delaying extinction. Although the 2000–2020 decline shows no sign of slowing, the current decline rate is less than during the 60s and 70s, as the next section will show.

Figure 3: Linear Fit to Welsh Presbyterian Membership Data

Long–Term View

The rise and fall of the Welsh Presbyterian from its beginnings until now displays the classic institutional lifecycle, figure 4[1]. The church experienced repeated revivals that drove its growth from 1735 to 1905. Once the revivals stopped, the church declined. It experienced very steep decline after the First World War, only slowing after 1980.

Figure 4: Long-term growth and decline of the Presbyterian Church of Wales

The decline from 1960 to 1980 was very steep. Had this data been used to predict the Presbyterian’s extinction date, it would have yielded 2004. However, this did not happen. Instead, the church’s decline slowed, and now the predicted extinction date is 2033.

Decline in the Welsh Presbyterians is more advanced than other denominations, such as the Welsh Independents and Church in Wales. Perhaps the Presbyterians’ behaviour indicates that decline slows near the end. If true, this pattern may give hope to the other declining churches that their extinction may take longer – long enough to give time to aid recovery.

Why Did Presbyterian Church of Wales Decline?

Why has decline slowed in the Presbyterian Church of Wales? Firstly, there is a reason that is unique to Wales. At one time, the church was predominantly Welsh-speaking. Thus, the steeper decline is partly caused by the decline in the Welsh language. The language has seen a mild recovery, helping the church resist decline.

Secondly, because of the need to provide for both Welsh and English speakers, more congregations were formed than needed by numbers alone. As the decline set in, many churches closed over a short period. The few people remaining after a congregation closed often moved to a different denomination to find a church according to their language.

Thirdly, in the twentieth century, the church embraced liberalism in a large way. Many people left to join more evangelical congregations, some within the emerging Evangelical Movement of Wales. Others went to Baptist and Pentecostal churches. For example, the congregation I started my life in had nearly 300 members in the 1950s. Due to the ministry of its newly-appointed liberal minister, there were only 20 members left by the end of the 1960s. My father had moved to the local Methodist Church. My mother went to the afternoon service of the Salvation Army. I was sent to a Baptist Sunday school! We were Presbyterian no longer.


[1] I have written previously on the connection between the Welsh Presbyterians’ growth and decline and the phenomenon of revival.