Freshman hordes slightly more godless than ever
This article is an update to my annual series on one of the most under-reported stories of the decade: the fraction of college freshmen who report no religious preference has tripled since 1985, from 8% to 24%, and the trend is accelerating.
In last year's installment, I made the bold prediction that the trend would continue, and that the students starting college in 2013 would again, be the most godless ever. It turns out I was right -- just barely. The number of students reporting no religious preference increased to 24.6%, slightly higher than the previous record, 24.5% in 2011. Of course, that "difference" is not statistically meaningful. More valid conclusions are
1) This year's data point is consistent with previous predictions, and
2) Data since 1990 support the conclusion that the number of incoming college students with no religious preference is increasing and probably accelerating.
1) This year's data point is consistent with previous predictions, and
2) Data since 1990 support the conclusion that the number of incoming college students with no religious preference is increasing and probably accelerating.
This analysis is based on survey results from the Cooperative Institutional Research Program (CIRP) of the Higher Education Research Insitute (HERI). In 2013, more than 165,000 students at 234 colleges and universities completed the CIRP Freshman Survey, which includes questions about students’ backgrounds, activities, and attitudes.
In one question, students select their “current religious preference,” from a choice of seventeen common religions, “Other religion,” or “None.”
Another question asks students how often they “attended a religious service” in the last year. The choices are “Frequently,” “Occasionally,” and “Not at all.” Students are instructed to select “Occasionally” if they attended one or more times.
The following figure shows the fraction of Nones over more than 40 years of the survey
The blue line shows actual data through 2012; the red line shows a quadratic fit to the data. The dark gray region shows a 90% confidence interval, which quantifies sampling error, so it reflects uncertainty about the parameters of the fit.
The light gray region shows a 90% confidence interval taking into account both sampling error and residual error. So it reflects total uncertainty about the predicted value, including uncertainty due to random variation from year to year.
We expect the new data point from 2013, shown as a blue square, to fall within the light gray interval, and it does. In fact, at 24.6% it falls only slightly below the fitted curve.
Here is the corresponding plot for attendance at religious services:
Again, the new data point for 2013, 27.3%, falls comfortably in the predicted range and slightly ahead of the long term trend.
Predictions for 2014
Using the new 2013 data, we can generate predictions for 2014. Here is the revised plot for "Nones":
The prediction for next year is that the fraction of Nones will hit a new all-time high at 25.8% (from 24.6%). If so, it is likely to match or exceed the fraction of students whose preference is Roman Catholic.
And here is the prediction for "No attendance":
The prediction for 2014 is a small increase to 27.5% (from 27.3%). I'll be back next year to check on these predictions.
Comments
1) As always, more males than females report no religious preference. The gender gap decreased this year, but still falls in the predicted range, as shown in the following plot:
2) I notice that the number of schools and the number of students participating in the Freshman Survey has been falling for several years. I wonder if the mix of schools represented in the survey is changing over time, and what effect this might have on the trends I am watching. The percentage of "Nones" is different across different kinds of institutions (colleges, universities, public, private, etc.) If participation rates are changing among these groups, that would affect the results.
3) Obviously college students are not representative of the general population. Data from other sources indicate that the same trends are happening in the general population, but I haven't been able to make a quantitative comparison between college students and others. Data from other sources also indicate that college graduates are slightly more likely to attend religious services, and to report a religious preference, than the general population.
Data Source
The American Freshman: National Norms Fall 2013
Eagan, K., Lozano, J.B., Hurtado, S., & Case, M.H.
ISBN: 978-1-878477-26-2 187 pages.
Mar 2014
This and all previous reports are available from the HERI publications page.
In class last week, I was talking about correlation and linear regression, and I made the outrageous claim that correlation is evidence of causation. One of my esteemed colleagues, who is helping out with the class, was sitting in the back of the room, and immediately challenged my claim. It wasn't a good time for a long discussion of the question, so I didn't elaborate.
The point I was trying to make (and will elaborate here) is that the usual mantra, "Correlation does not imply causation," is true only in a trivial sense, so we need to think about it more carefully. And as regular readers might expect, I'll take a Bayesian approach.
It is true that correlation doesn't imply causation in the mathematical sense of "imply;" that is, finding a correlation between A and B does not prove that A causes B. However, it does provide evidence that A causes B. It also provides evidence that B causes A, and if there is a hypothetical C that might cause A and B, the correlation is evidence for that hypothesis, too.
In Bayesian terms, a dataset, D, is evidence for a hypothesis, H, if the probability of H is higher after seeing D. That is, if P(H|D) > P(H).
For any two variables, A and B, we should consider 4 hypotheses:
A: A causes B
B: B causes A
C: C causes A and B
N: there are no causal relationships among A, B, and C
And there might be multiple versions of C, for different hypothetical factors. If I have no prior evidence of any causal relationships among these variables, I would assign a high probability (in the sense of a subjective degree of belief) to the null hypothesis, N, and low probabilities to the others. If I have background information that makes A, B, or C more plausible, I might assign prior probabilities accordingly. Otherwise I would assign them equal priors.
Now suppose I find a correlation between A and B, with p-value=0.01. I would compute the likelihood of this result under each hypothesis:
L(D|A) ≈ 1: If A causes B, the chance of finding a correlation is probably high, depending on the noisiness of the relationship and the size of the dataset.
L(D|B) ≈ 1, for the same reason.
L(D|C) ≈ 1, or possibly a bit lower than the previous likelihoods, because any noise in the two causal relationships would be additive.
L(D|N) = 0.01. The probability of seeing a correlation with the observed strength, or more, under the null hypothesis, is the computed p-value, 0.01.
When we multiply the prior probabilities by the likelihoods, the probability assigned to N drops by a factor of 100; the other probabilities are almost unchanged. When we renormalize, the other probabilities go up.
In other words, the update takes most of the probability mass away from N and redistributes it to the other hypotheses. The result of the redistribution depends on the priors, but for all of the alternative hypotheses, the posterior is greater than the prior. That is
P(A|D) > P(A)
P(B|D) > P(B)
P(C|D) > P(C)
Thus, the correlation is evidence in favor of A, B and C. In this example, the Bayes factor for all three is about 100:1, maybe a bit lower for C. So the correlation alone does not discriminate much, if at all, between the alternative hypotheses.
If there is a good reason to think that A is more plausible than B and C, that would be reflected in the priors. In that case the posterior probability might be substantially higher for A than for B and C.
And if the resulting posterior, P(A|D), were sufficiently high, I would be willing to say that the observed correlation implies causation, with the qualification that I am using "imply" in the sense of strong empirical evidence, not a mathematical proof.
People who have internalized the mantra that correlation does not imply causation might be surprised by my casual (not causal) blasphemy. But I am not alone. This article from Slate makes a similar point, but without the Bayesian mumbo-jumbo.
And the Wikipedia page on "Correlation does not imply causation" includes this discussion of correlation as scientific evidence:
Much of scientific evidence is based upon a correlation of variables – they are observed to occur together. Scientists are careful to point out that correlation does not necessarily mean causation. The assumption that A causes B simply because A correlates with B is often not accepted as a legitimate form of argument. However, sometimes people commit the opposite fallacy – dismissing correlation entirely, as if it does not suggest causation. This would dismiss a large swath of important scientific evidence.
In conclusion, correlation is a valuable type of scientific evidence in fields such as medicine, psychology, and sociology. But first correlations must be confirmed as real, and then every possible causative relationship must be systematically explored. In the end correlation can be used as powerful evidence for a cause-and-effect relationship between a treatment and benefit, a risk factor and a disease, or a social or economic factor and various outcomes. But it is also one of the most abused types of evidence, because it is easy and even tempting to come to premature conclusions based upon the preliminary appearance of a correlation.
I think this is a reasonable conclusion, and hopefully not too shocking to my colleagues in the back of the room.
UPDATE February 21, 2014: There is a varied and lively discussion of this article on reddit/r/statistics.
One of the objections raised there is that I treat the hypotheses A, B, C, and N as mutually exclusive, when in fact they are not. For example, it's possible that A causes B and B causes A. This is a valid objection, but we can address it by adding additional hypotheses for A&B, B&C, A&C, etc. The rest of my argument still holds. Finding a correlation between A and B is evidence for all of these hypotheses, and evidence against N.
One of my anonymous correspondents on reddit added this comment, which gives examples where correlation alone might be used, in the absence of better evidence, to guide practical decisions:
This [meaning my article] is not too different from the standard view in medicine, though usually phrased in more of a discrete "levels of evidence" sense than a Bayesian sense. While direct causal evidence is the gold standard in medicine, correlational studies are still taken as providing some evidence that is sometimes worth acting on, in the absence of better evidence. For example, correlations with negative health outcomes are sometimes taken as reasons to issue recommendation to avoid certain behaviors/drugs/foods (pending further data), and unexpected correlations are often taken as good justification for funding further studies into a relationship.
In general, one of the nice things about Bayesian analysis is that it provides useful inputs for decision analysis, especially when we have to make decisions in the absence of conclusive evidence.
The American Freshman: National Norms Fall 2013
Eagan, K., Lozano, J.B., Hurtado, S., & Case, M.H.
ISBN: 978-1-878477-26-2 187 pages.
Mar 2014
This and all previous reports are available from the HERI publications page.
In class last week, I was talking about correlation and linear regression, and I made the outrageous claim that correlation is evidence of causation. One of my esteemed colleagues, who is helping out with the class, was sitting in the back of the room, and immediately challenged my claim. It wasn't a good time for a long discussion of the question, so I didn't elaborate.
The point I was trying to make (and will elaborate here) is that the usual mantra, "Correlation does not imply causation," is true only in a trivial sense, so we need to think about it more carefully. And as regular readers might expect, I'll take a Bayesian approach.
It is true that correlation doesn't imply causation in the mathematical sense of "imply;" that is, finding a correlation between A and B does not prove that A causes B. However, it does provide evidence that A causes B. It also provides evidence that B causes A, and if there is a hypothetical C that might cause A and B, the correlation is evidence for that hypothesis, too.
In Bayesian terms, a dataset, D, is evidence for a hypothesis, H, if the probability of H is higher after seeing D. That is, if P(H|D) > P(H).
For any two variables, A and B, we should consider 4 hypotheses:
A: A causes B
B: B causes A
C: C causes A and B
N: there are no causal relationships among A, B, and C
And there might be multiple versions of C, for different hypothetical factors. If I have no prior evidence of any causal relationships among these variables, I would assign a high probability (in the sense of a subjective degree of belief) to the null hypothesis, N, and low probabilities to the others. If I have background information that makes A, B, or C more plausible, I might assign prior probabilities accordingly. Otherwise I would assign them equal priors.
Now suppose I find a correlation between A and B, with p-value=0.01. I would compute the likelihood of this result under each hypothesis:
L(D|A) ≈ 1: If A causes B, the chance of finding a correlation is probably high, depending on the noisiness of the relationship and the size of the dataset.
L(D|B) ≈ 1, for the same reason.
L(D|C) ≈ 1, or possibly a bit lower than the previous likelihoods, because any noise in the two causal relationships would be additive.
L(D|N) = 0.01. The probability of seeing a correlation with the observed strength, or more, under the null hypothesis, is the computed p-value, 0.01.
When we multiply the prior probabilities by the likelihoods, the probability assigned to N drops by a factor of 100; the other probabilities are almost unchanged. When we renormalize, the other probabilities go up.
In other words, the update takes most of the probability mass away from N and redistributes it to the other hypotheses. The result of the redistribution depends on the priors, but for all of the alternative hypotheses, the posterior is greater than the prior. That is
P(A|D) > P(A)
P(B|D) > P(B)
P(C|D) > P(C)
Thus, the correlation is evidence in favor of A, B and C. In this example, the Bayes factor for all three is about 100:1, maybe a bit lower for C. So the correlation alone does not discriminate much, if at all, between the alternative hypotheses.
If there is a good reason to think that A is more plausible than B and C, that would be reflected in the priors. In that case the posterior probability might be substantially higher for A than for B and C.
And if the resulting posterior, P(A|D), were sufficiently high, I would be willing to say that the observed correlation implies causation, with the qualification that I am using "imply" in the sense of strong empirical evidence, not a mathematical proof.
People who have internalized the mantra that correlation does not imply causation might be surprised by my casual (not causal) blasphemy. But I am not alone. This article from Slate makes a similar point, but without the Bayesian mumbo-jumbo.
And the Wikipedia page on "Correlation does not imply causation" includes this discussion of correlation as scientific evidence:
Much of scientific evidence is based upon a correlation of variables – they are observed to occur together. Scientists are careful to point out that correlation does not necessarily mean causation. The assumption that A causes B simply because A correlates with B is often not accepted as a legitimate form of argument. However, sometimes people commit the opposite fallacy – dismissing correlation entirely, as if it does not suggest causation. This would dismiss a large swath of important scientific evidence.
In conclusion, correlation is a valuable type of scientific evidence in fields such as medicine, psychology, and sociology. But first correlations must be confirmed as real, and then every possible causative relationship must be systematically explored. In the end correlation can be used as powerful evidence for a cause-and-effect relationship between a treatment and benefit, a risk factor and a disease, or a social or economic factor and various outcomes. But it is also one of the most abused types of evidence, because it is easy and even tempting to come to premature conclusions based upon the preliminary appearance of a correlation.
I think this is a reasonable conclusion, and hopefully not too shocking to my colleagues in the back of the room.
UPDATE February 21, 2014: There is a varied and lively discussion of this article on reddit/r/statistics.
One of the objections raised there is that I treat the hypotheses A, B, C, and N as mutually exclusive, when in fact they are not. For example, it's possible that A causes B and B causes A. This is a valid objection, but we can address it by adding additional hypotheses for A&B, B&C, A&C, etc. The rest of my argument still holds. Finding a correlation between A and B is evidence for all of these hypotheses, and evidence against N.
One of my anonymous correspondents on reddit added this comment, which gives examples where correlation alone might be used, in the absence of better evidence, to guide practical decisions:
This [meaning my article] is not too different from the standard view in medicine, though usually phrased in more of a discrete "levels of evidence" sense than a Bayesian sense. While direct causal evidence is the gold standard in medicine, correlational studies are still taken as providing some evidence that is sometimes worth acting on, in the absence of better evidence. For example, correlations with negative health outcomes are sometimes taken as reasons to issue recommendation to avoid certain behaviors/drugs/foods (pending further data), and unexpected correlations are often taken as good justification for funding further studies into a relationship.
In general, one of the nice things about Bayesian analysis is that it provides useful inputs for decision analysis, especially when we have to make decisions in the absence of conclusive evidence.
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