# Making the Connection: Broadband Access and Online Course Enrollment at Public Open Admissions Institutions

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## Abstract

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## Notes

1. cf. the Open University and its founding as an autonomous institution.

2. Theoretical justifications notwithstanding, it is important to note that when weakly-informative priors are used in applied analyses, otherwise comparable Bayesian and frequentist models generally produce substantively similar estimates. Prior distributions are discussed later in the section.

3. A Bayesian posterior distribution is properly written as \begin{aligned} p(\theta \mid X)&= \frac{p(\theta )p(X\mid \theta )}{p(X)}, \end{aligned} which includes the evidence, p(X), in the denominator. Because data are assumed fixed in many Bayesian analyses, p(X) may be treated as a normalizing constant and dropped, yielding, $$p(\theta \mid X) \propto p(\theta )p(X\mid \theta )$$.

4. When coding the model, I technically use improper priors. Priors are improper when $$\int p(\theta )\,d\theta \ne 1$$, that is, the probabilities do not sum to one. All regression coefficients are drawn from a uniform distribution with support on $$\theta \in (-\infty , \infty )$$ and all variances from a positive uniform distribution: $$\theta \in (0, \infty )$$. Improper priors may combine with a likelihood function, however, to produce proper posterior distributions (Gelman et al. 2014).

5. In an alternative specification, I use the proportion of students as the outcome of interest. To properly model the proportion, which is bounded by [0,1], I use a beta likelihood function. The results for these models are qualitatively the same as those given by the log outcome/normal likelihood models, so I present the latter for ease of interpretation. More details about the beta likelihood specification as well results from the models are shown in Appendix 1.

6. IPEDS variable: EFDESOM.

7. Though subsequent years of student enrollment data have since been released, the National Broadband Map stopped being updated in June 2014. I have not incorporated the newest enrollment data for this reason.

8. Because the census block group population values are constant, it is not strictly necessary build an entire $$N\times K$$ matrix, which is simply a vector of values of length K repeated N times, or compute the row sums and weights N times. For purposes of computation, however, it is easier to build a full matrix that can be easily combined with the other weighting matrix when computing the final average.

9. Combined weighted broadband equation: \begin{aligned} wbroadband_{s} = \sum _{c = 1}^{C}\frac{ \Big (\frac{pop_{c}}{\sum _{c = 1}^C pop_{c}}\Big ) \Big (\frac{id_{sc}}{\sum _{c = 1}^C id_{sc}}\Big ) \cdot broadband_{c}}{\sum _{c = 1}^{C} \Big (\frac{pop_{c}}{\sum _{c = 1}^C pop_{c}}\Big ) \Big (\frac{id_{sc}}{\sum _{c = 1}^C id_{sc}}\Big )} \end{aligned}

10. All models were estimated using (CmdStan, version 2.18.0), the command line version of Stan’s No-U-Turn Sampler (NUTS), a variant of the Hamiltonian Monte Carlo sampler that may more efficiently explore the parameter space. To reduce the amount of lagged auto-correlation between successive draws in the chains, improving convergence and the number of effective samples, all models were estimated using centered data and a QR reparameterization (Stan Development Team 2018; Lunn et al. 2013).

11. All reported 95% credible intervals represent the middle 95% of the posterior distribution, meaning that the lower and upper bound values are the 2.5% and 97.5% quantile values, respectively.

12. Throughout the rest of the chapter, I will generally use the Bayesian point estimates when referring to the mean of parameter’s posterior distribution. They should be understood, therefore, in their proper context as useful summaries of full probability distributions.

13. The margins in this section are computed using the full posterior distribution of the main and quadratic parameters. They are therefore slightly different than what is computed using only the Bayesian point estimates presented in the results tables.

14. If the mean value presented in the table represented a frequentist point estimate, one could not reject the null that $$\beta = 0$$ under a two-tailed test of significance at conventional levels of significance ($$\alpha = 0.05$$). This is not to say, however, that it would not be jointly significant with its quadratic term.

15. These were computed from the data by taking the fractional value of the minimum upload speed over the minimum download speed, the same fractional value of the relative maximum speeds, and building a sequence of 11 values (1 to 11 on the NBM scale) that range between them.

16. While the focus of this paper has been on public institutions with open admissions policies, I also ran all models using the full census of colleges and universities during this period who reported having any students take some courses online as part of a sensitivity analysis. All models were the same, but with the inclusion of indicators for being private-non-profit, private-for-profit, or having an open admissions policy. Findings were qualitatively similar to those discussed in the paper. Tables showing these results are included in Appendix 2.

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## Acknowledgements

This research was supported in part by the Bonsal Applied Education Research Award. I thank two anonymous reviewers as well as Angela Boatman, Josh Clinton, Stephen DesJardins, Will Doyle, Carolyn Heinrich and participants at the 2015 ASHE annual conference for their helpful comments and suggestions. This article is better for their help. All findings and conclusions remain my own.

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Correspondence to Benjamin T. Skinner.

## Appendices

### Appendix 1: Alternative Specifications

In an alternative specification, I fit equations using the proportion of students who took some online courses as the outcome. To model this outcome, I use a beta likelihood function with dispersion parameter that accurately accounts for the [0,1] bounds of the dependent variable (Gelman et al. 2014):

\begin{aligned} \frac{online}{total}&\sim beta(a,b) \\ a&= \mu \times \phi \\ b&= (1 - \mu ) \times \phi \\ \mu&= \frac{exp({\varvec{X}}\beta )}{1 + exp({\varvec{X}}\beta )} \end{aligned}

As with the primarily analysis models, I fit single level as well as multilevel models in which the intercepts and parameters on broadband measures are allowed to vary at the state level. Results from these models show the marginal effect of broadband on the logged-odds of a percentage increase in the number of students who took some online courses. The results from these models are qualitatively the same as those reported in the main text of the chapter (Tables 5, 6).

### Appendix 2: Sensitivity Analyses

See Tables 7, 8, 9, 10, 11, and 12.

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Skinner, B.T. Making the Connection: Broadband Access and Online Course Enrollment at Public Open Admissions Institutions. Res High Educ 60, 960–999 (2019). https://doi.org/10.1007/s11162-018-9539-6