Stochastic Model

Growth of the Church
Stochastic Model

Assume the function

is of the form

Where is normally distributed with a constant variance. If the church is growing at a constant exponential rate, then will be equal to 0. The question is, what are the correct values of and ?

Table 1 shows 18 years of membership data. If we set to the year 1983, then running a least-squares regression results in and . Table 2 shows the ANOVA for the regression and Chart 5 shows the fitted line and the measured delta.

The null-hypothesis is that . This is the scenario where the church is growing at a constant exponential rate of , and the year-to-year variations from that are due to random fluctuations. The F-statistic of 4.65 with 16 and 1 degrees of freedom means that we can reject that hypothesis at the 95% significance level. In other words, we can be 95% certain that the decrease in over this time period has not been to random chance, but rather to an underlying trend.

There are 2 anomalous points on Chart 5, corresponding to the years 1989 and 1990. Why did

almost double in 1989 and then go back down? To answer this, let's look at Chart 6 (Table 3), the 3 components of the annual growth rate (value of formula (4.5) ) over the same time period.

The light blue area is the percentage growth of the church due to children of record being baptized. The purple area (down to the axis) is growth due to convert baptisms. The red area is the percentage of people who leave the church each year for any reason. So, the area from the top of the light blue to the top of the red line is the net-amount that the church grew each year.

During the 18 years on the chart, there were 4 years that the number of decrements was negative(!). For example, on January 1 1989 there were 6.72 million members of the church. 75 thousand children of record joined the church, and 318,940 converts joined. This would lead us to believe that there were 7.114 million members on Dec 31 1989, less that year's decrements. But the reported membership for Dec 31 was 7.3 million. In other words, even though only 394,000 people joined the church that year, the reported number of members grew by 580,000.

There are only 2 possible explanations. One explanation is that 580,000+ people really did get baptized that year but for some reason they weren't all counted in the baptismal statistics. The other explanation is that people that weren't previously counted as members had their status changed to members that year, without the benefit of baptism. The second explanation is more likely. It's possible that the church previously assumed that everyone in a set of long-lost members was dead, but retroactively decided to assume they were alive. Or it's possible that they decided to include un-baptized children as members. In other words, the way members and non-members were delimited was changed.

Be the explanation what it may, it is an anomalous data point that adds no value to the study. There are at least two ways it could be handled. We could try to guess when in the previous years those names were inadvertently removed from the membership count and replace them in those years. Or we could simply remove that observation from the regression. The first method of restating previous membership levels would cause the church growth levels to be higher in each year before 1989 were members would be added. This would show a more radical drop in the growth rates. The problem with this is we don't know how many people should be added to each year, and would thus be distorting the data.

If we remove 1989 as a data point the slope of the regression line is more flat, (-0.000913) but at a lower y-intercept (0.04787). Chart 7 compares the two regression lines. By removing the one datum the F statistic more than doubles to 10.358, giving a strong indication that the underlying growth rate of the church is in fact slowing down.

If you have a question or would like to discuss these topics, I suggest that you go to a Mormon-related bulletin board. If you'd like to <i>contact me</i> with comments or feedback, you may send an email to analytics@lds4u.com.CompanyEmail