Religiosity vs. Time

In reference to a Pew study on intergenerational religiosity, Scott writes:

Contra compelling anecdotes, only ~5% of people raised very religious end up atheist later in life (X). Most people are about as religious as their parents; most exceptions are only slightly less religious, and most families that secularize do it over several generations.

Okay, but those generations add up.

Here is the Pew data as a matrix. I have reversed the row order such that both row and column indicies increase with religiosity. I have also subtracted 1% from element (4,4) such that the table sums to 1.0.

PEW =

0.0600	0.0100	0.0100	0
0.0400	0.0800	0.0300	0.0100
0.0300	0.0900	0.1600	0.0300
0.0200	0.0500	0.1400	0.2400

Note these are joint probabilities. To find the religious probabilities of the parent generation, we must sum all columns in each row (all code is MATLAB):

P_parent = sum(PEW,2);

P_parent'

ans =

0.0800 0.1600 0.3100 0.4500

To find the religiosity of each subsequent generation, we need the conditional probabilities, which we can derive from Bayes' Theorem:

P_child_given_parent = PEW./repmat(P_parent,1,4) 

P_child_given_parent =

0.7500	0.1250	0.1250	0
0.2500	0.5000	0.1875	0.0625
0.0968	0.2903	0.5161	0.0968
0.0444	0.1111	0.3111	0.5333

Finally, we construct a table showing the generational change over time:

for gen = 1:10

    R_gen(gen,:) = P_parent' * P_child_given_parent^(gen-1);

end

R_gen =

0.0800	0.1600	0.3100	0.4500
0.1500	0.2300	0.3400	0.2800
0.2153	0.2636	0.3245	0.1966
0.2675	0.2748	0.3050	0.1527
0.3056	0.2763	0.2899	0.1281
0.3321	0.2748	0.2795	0.1137
0.3498	0.2727	0.2726	0.1048
0.3616	0.2709	0.2682	0.0993
0.3693	0.2695	0.2653	0.0959
0.3743	0.2686	0.2635	0.0937

Which we can plot:

So, clearly, the Very Religious are the long-term demographic losers. My first thought was that we could make this up in volume, but then I realized that the initial table confounds our greater on-average fertility with the starting percentages; this is a poll of the children, so the children of very religious families were already more numerous among the respondents, assuming the poll was representative. My calculation of the conditional probability assumes fertility is equal, but I'm pretty sure I would need additional fertility data and a way to map it on to these categories in order to correct this.

My second thought is that these statistics might be biased with respect to "family" religiosity: it is plausible that children's characterization of their family's religiosity may be colored by their own. For instance, a child from a "somewhat" religious family who is personally "not at all" religious, might exaggerate family religiosity to "very". OTOH, I'm not sure this matters for predicting the trend.