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

The United Reformed Church (URC) is a UK-based protestant church. It was formed in 1972 through the merger of the Presbyterian Church of England and the Congregational Union of England and Wales (Wikipedia). It has a wide variety of churchmanships. In common with most historic UK Christian denominations, it has steadily declined since 1960. This analysis investigates the likelihood of decline leading to denominational extinction.

Estimating Model Parameters

I will use the URC’s 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 the URC. 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 URC’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 2000-2017

I took the membership data from the URC’s website for recent values and Brierley (2008-2020). I preferred not to go beyond 2017 as I could not confirm those figures from the URC directly.

Every optimisation indicates that the URC will end up extinct.

One such optimisation is given in figure 1. The reproduction potential chosen was an average figure of 0.8, well under the extinction threshold of 1.

United Reformed Church - data fit to Limited enthusiasm model
Figure 1: Data Fit for URC Membership and the Limited Enthusiasm with Reversion Model

Range of Optimised Parameters

The optimisations indicate a range of possible parameter values:

Reproduction Potential0.71–0.84
Duration Enthusiastic0.9–6.5 years
Loss Rate
(Reversion and Deaths)
Fraction of Converts who Become Enthusiastsabout 10%
Initial Fraction of Enthusiasts in the Church0.5%–3%
Model Parameters for URC

The low value of converts who become enthusiasts is consistent with the view I hear from those engaged in evangelism. Similarly, for the fraction of the church who are enthusiasts. The leaving rate is high as, in this model, this includes the difference between deaths and births. This difference is high due to the church’s age profile.

I repeated this optimisation strategy for different values of the total population, starting with the full England/Wales value and then using lower values. It is possible the URC has a restricted influence in society and has 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 6,000 members. This is probably too optimistic.

Fitting a straight line to the data gives a more realistic estimate of the extinction date, as decline through ageing is a linear process. The URC is forecast to be extinct by 2038, figure 2. Its future will likely lie somewhere between linear and exponential decline due to the presence of some city-based strong congregations. By 2030, some slowing down of its decline may be seen, delaying extinction by another 20 years

United Reformed Church - data fit to linea model
Figure 2: Linear Fit to URC Membership Data

The URC’s decline is similar to other historic UK denominations, such as the Church of England, Church of Scotland and the UK Methodist Church.

Long–Term View

The membership of the URC and its constituents, from 1840 to the present day, shows the classic institutional lifecycle, figure 3[1]. The church did well in the latter part of the 19th century but fell back from the Second World War onwards. This decline temporarily halted during the 1950s, like many UK denominations. Some people believe this was a post-war baby boom effect. Since the 1960s, the decline has been relentless. The URC is now very small, 0.07% of the GB population compared to the heyday of its constituents 1840-1940, at just under 1% of the population.

Figure 3: Membership of URC and the combined memberships of the Presbyterian Church of England and the Congregational Union of England and Wales


  1. I added together the membership of the Presbyterian Church of England to that of the Congregational Union of England Wales for the years before 1972, the date of their merger. Data was obtained from Currie 1977 and Brierley 1991-2020.