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We can use the school code as a key to join these two data sets. But remember, the FRPM data set only includes data about public schools, so we'll have to retain the private school data for PBEs by doing what's called a left outer join. This still performs a join on the school code key, but if any school codes included in the left data don't have corresponding entries in the right data set we still retain them. The missing values for the right data set in this case are set to NULL.

We can perform a left outer join in R by using "merge" with the option "all.x=TRUE". I'll start by looking at how the PBE rate varies between charter, non-charter public and private schools, so we'll need to replace those missing values for funding source after our join. If the funding source is missing, it's a private school. The FRPM data also denotes non-charter public schools with funding type "", so I'll replace those with "aPublic" for convenience. For factors, R will by default set the reference level to be the level that comes first alphabetically.

Here's a subset of the output. The addition of the funding source is an improvement over the model that doesn't include it, and the estimates for the odds ratios for funding source is the highest for directly funded charter schools, followed by locally funded charter schools and private schools. Remember, public schools are the reference level, so for the public level \(\log(\text{odds ratio}) = 0\). Everything else being equal, our odds ratio estimates based on funding source would be: \begin{align*}

\mathrm{OR}_{\text{public}} &= e^{-3.820}\times e^{0} &= 0.022\\

\mathrm{OR}_{\text{private}} &= e^{-3.820}\times e^{0.752} &= 0.047\\

\mathrm{OR}_{\text{charter-local}} &= e^{-3.820}\times e^{1.049} &= 0.063\\

\mathrm{OR}_{\text{charter-direct}} &= e^{-3.820}\times e^{1.348} &= 0.085

\end{align*}

Converting to estimated PBE rates, we get: \begin{align*}

\mathrm{PBE}_{\text{public}} &= \frac{0.022}{1+0.022} &= 0.022\\

\mathrm{PBE}_{\text{private}} &= \frac{0.047}{1+0.047} &= 0.045\\

\mathrm{PBE}_{\text{charter-local}} &= \frac{0.063}{1+0.063} &= 0.059\\

\mathrm{PBE}_{\text{charter-direct}} &= \frac{0.085}{1+0.085} &= 0.078

\end{align*}

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