More Measles: Vaccination Rates and School Funding

I took a look at California's personal belief exemption rate (PBE) for kindergarten vaccinations in Part I. California also provides poverty information for public schools through the Free or Reduced Price Meals data sets, both of which conveniently include California's school codes. Cleaned versions of these data sets and my R code are in my vaccination GitHub.

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.

	library(lme4)
	rates <- read.csv("data/cakd1314.csv", header=TRUE, stringsAsFactors=FALSE)
	poverty <- read.csv("data/frpm1314.csv", header=TRUE, stringsAsFactors=FALSE)
	dim(rates)
	dim(poverty)
	joined <- merge(rates, poverty, by="School.Code", all.x=TRUE)
	dim(joined)
	joined$School <- as.factor(paste(joined$County, ":" , joined$City, ":", joined$School.Name.x, sep=""))
	joined$City <- as.factor(paste(joined$County, ":" , joined$City, sep=""))
	joined$County <- as.factor(joined$County)
	joined$Funding <- joined$Charter.Funding.Type
	joined[is.na(joined$Funding),]$Funding <- "Private"
	joined[(joined$Funding==""),]$Funding <- "aPublic"
	joined$Funding <- as.factor(joined$Funding)
	joined$Public.Private <- as.factor(joined$Public.Private)
	summary(joined$Funding)
	summary(joined$Public.Private)
	model0 <- cbind(PBE.,Enrollment-PBE.) ~ (1|County) + (1|City) + (1|School)
	model1 <- cbind(PBE.,Enrollment-PBE.) ~ (1|County) + (1|City) + (1|School) + Funding
	fit0 <- glmer(model0, data=joined, family="binomial")
	fit1 <- glmer(model1, data=joined, family="binomial")
	anova(fit0)
	anova(fit1)
	anova(fit0,fit1)
	summary(fit1)
	fixef(fit1)
	ranef(fit1)

view raw pbe_frpm.R hosted with ❤ by GitHub

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: $$\begin{align*} \mathrm{PBE}_{\text{public}} &= \frac{0.022}{1+0.022} &= 0.022\\ \mathrm{PBE}_{\text{private}} &= \frac{0.047}{1+0.047} &= 0.045\\ \mathrm{PBE}_{\text{charter-local}} &= \frac{0.063}{1+0.063} &= 0.059\\ \mathrm{PBE}_{\text{charter-direct}} &= \frac{0.085}{1+0.085} &= 0.078 \end{align*}$$

	> joined$Funding <- as.factor(joined$Funding)
	> joined$Public.Private <- as.factor(joined$Public.Private)
	>
	> summary(joined$Funding)
	aPublic Directly funded Locally funded
	5153 382 172
	Not in CS funding model Private
	7 1667
	> summary(joined$Public.Private)
	PRIVATE PUBLIC
	1649 5732
	>
	> model0 <- cbind(PBE.,Enrollment-PBE.) ~ (1|County) + (1|City) + (1|School)
	> model1 <- cbind(PBE.,Enrollment-PBE.) ~ (1|County) + (1|City) + (1|School) + Funding
	>
	> fit0 <- glmer(model0, data=joined, family="binomial")
	> fit1 <- glmer(model1, data=joined, family="binomial")
	>
	> anova(fit0)
	Analysis of Variance Table
	Df Sum Sq Mean Sq F value
	> anova(fit1)
	Analysis of Variance Table
	Df Sum Sq Mean Sq F value
	Funding 4 523.87 130.97 130.97
	> anova(fit0,fit1)
	Data: joined
	Models:
	fit0: cbind(PBE., Enrollment - PBE.) ~ (1 | County) + (1 | City) +
	fit0: (1 | School)
	fit1: cbind(PBE., Enrollment - PBE.) ~ (1 | County) + (1 | City) +
	fit1: (1 | School) + Funding
	Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
	fit0 4 25964 25991 -12978 25956
	fit1 8 25515 25570 -12749 25499 456.94 4 < 2.2e-16 ***
	---
	Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
	>
	> summary(fit1)
	Generalized linear mixed model fit by maximum likelihood (Laplace
	Approximation) [glmerMod]
	Family: binomial ( logit )
	Formula: cbind(PBE., Enrollment - PBE.) ~ (1 | County) + (1 | City) +
	(1 | School) + Funding
	Data: joined
	AIC BIC logLik deviance df.resid
	25514.8 25569.7 -12749.4 25498.8 6974
	Scaled residuals:
	Min 1Q Median 3Q Max
	-1.4256 -0.6272 -0.1870 0.2439 2.0957
	Random effects:
	Groups Name Variance Std.Dev.
	School (Intercept) 1.1152 1.0560
	City (Intercept) 0.8674 0.9314
	County (Intercept) 0.8289 0.9104
	Number of obs: 6982, groups: School, 6978; City, 919; County, 58
	Fixed effects:
	Estimate Std. Error z value Pr(>|z|)
	(Intercept) -3.81965 0.13730 -27.819 <2e-16 ***
	FundingDirectly funded 1.34812 0.07698 17.513 <2e-16 ***
	FundingLocally funded 1.04913 0.11150 9.410 <2e-16 ***
	FundingNot in CS funding model 0.98645 0.61510 1.604 0.109
	FundingPrivate 0.75254 0.04815 15.630 <2e-16 ***
	---
	Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

view raw pbe_funding.txt hosted with ❤ by GitHub