Large mixed & crossed-effect model looking at educational spending on crime rates with error messages

Tue, Sep 10, 2019 5:45 AM

Dear James,

Not centring year would require a much stronger random intercept. This
makes the model harder to fit. It will change the random intercept variance
and the correlation between random intercept and random slope. In theory it
shouldn't change the estimates for the fixed effects. But if your model is
unstable / doesn't converge, then how certain can you be on those estimates.

I don't know of any rule of thumb about the overlap in partially crosses
designs. If the model doesn't converge then see how you can simplify it.
E.g. if your model has county_id, then you could drop the state_id as the
county will model any effect at state level (if state level is not in the
model).

I wrote a blogpost on using the same variable both as fixed and as random
effect: https://www.muscardinus.be/2017/08/fixed-and-random/ One of the
examples uses year as a random intercept.

We've used the INLA package (r-inla.org) to model the gamma distribution.

Best regards,

ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE AND
FOREST
Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
thierry.onkelinx at inbo.be
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be

///////////////////////////////////////////////////////////////////////////////////////////
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than asking him to perform a post-mortem examination: he may be able to say
what the experiment died of. ~ Sir Ronald Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer does not
ensure that a reasonable answer can be extracted from a given body of data.
~ John Tukey
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<https://www.inbo.be>

Op di 10 sep. 2019 om 08:51 schreef Ades, James <jades at ucsd.edu>:

Thanks for the comment to center the year variable. I hadn?t thought about
that, and I changed my model to incorporate a centered year. Just
wondering?would I be right in saying that while centering the year might
affect convergence, and the interpretation of the intercept as the average
city in an average year, it wouldn?t affect the coefficient estimates?

I have a model where education spending is averaged (when one city attends
multiple school districts) to do away with district_id. That model does
run. But you?re doubtful that the mixed-effect model could handle the
crossed effects given the large overlap of district_id and city? Is there
any rule of thumb for when a crossed effects model can be used vs when a
variable must be averaged?or is it merely that if it converges then it
converges?

I use year as fixed-effect in my non-null model. Your other suggestion is
to possibly add year as a random intercept. Wouldn?t these two choices
ultimately be looking at different things?

Thanks for the gamma suggestion. Are you suggesting using a gamma
distribution in glmer? Does the gamma distribution in glmer default to the
Erlang distribution? Would that still have to be as a rate rather than a
discrete variable, and in that case, how would that vary from fitting the
model in lmer with crime as a rate and getting rid of population as a fixed
effect?

Much thanks, Thierry et al.!

James

With regard to Douglas? question, I should?ve replied all, but here was my
answer:

Especially in the early ?00s crime was offered as two datasets?one on
metropolitan/city areas with population > 10,000 and the other as
population < 10,000. For 2004, the only data available is the dataset for
cities > 10,000. I?ve contacted the FBI regarding whether they can track
down the corresponding <10,000 dataset; they?re looking into it.

On Sep 9, 2019, at 2:58 AM, Thierry Onkelinx <thierry.onkelinx at inbo.be>
wrote:

Dear James,

Since all ID's are unique, you can rely on the implicit nesting of random
effects. (1|STATE_ID/COUNTY_ID) is equivalent to (1|STATE_ID) +
(1|COUNTY_ID) when each COUNTY_ID is unique (not reused among states).

There is a huge overlap between PLACE_ID and full_district_id. The model
will have a hard time separating both effects. Especially since you have
the same random slope.

You should also center the year variable. Now your model is estimating the
random intercept at year 0.

So in terms of basic model I would

- drop either PLACE_ID or full_district_id
- center year
- add year to the fixed effects or add it as a separate random intercept

Furthermore, I'm not sure it the Poisson distribution is the most relevant
distribution for your response. I've seen a case were the Poisson
distribution failed. The response was an area in hectares rounded to an
integer value. The model fitted well with a Gamma distribution.

Best regards,

Thierry

ir. Thierry Onkelinx
Statisticus / Statistician

///////////////////////////////////////////////////////////////////////////////////////////

<https://www.inbo.be/>

Op zo 8 sep. 2019 om 07:40 schreef Ades, James <jades at ucsd.edu>:

Okay, sorry for the delay; I was waiting to hear back from a couple
people at the census regarding a question on school district info in the
dataset.

I?ve uploaded the dataset as a zip and an xs:

https://drive.google.com/file/d/1hyJYVg0x_-dIHfZdjWzdRuQS_H365buK/view?usp=sharing

https://drive.google.com/file/d/1qKU7vvzsqzrD58lcu6HWlMALwy4mm6RF/view?usp=sharing

There is one important thing to note, here, which is that there is more
city overlap between counties than I thought. The easiest way to explain
this is by checking distinct values: using distinct(PLACE_ID, COUNTY_ID,
year), you should get 107,515; however, with distinct(PLACE_ID, year),
you?ll only get 99,832. So essentially, there are ~8000 times that a city
will overlap in two counties (this likely includes the same city repeating
this overlap every year?though it all depends on what years the city/town
police department reports crime to the FBI). I can figure out the
respective populations in each of the two counties where the city overlaps.
It?s a bit of leg work, but it is possible. Still, it seems that it would
not be principled to include the city/town in both counties because there
would be the assumption that the crime occurring in each corresponding
county is tied solely to population. Still, I left in all cities/towns
overlapping with two counties, in case you have some ideas about how a
mixed-model might be used to account for these (I don?t think a
crossed-effect structure would do it). So pretty much any city with
duplicate values where that value lies within the same county should be
counted (these are just separate school districts), but any city with
duplicate values where one of those values happens to be in another county,
yet the school district is still the same, shouldn't be counted; unless, of
course, there is a way for mixed-modeling in lme4 to account for that
(PLACE_ID will be the same, but COUNTY_ID will be different, but
full_district_id will be the same). Does that make sense? I'll soon send an
email to the group with more clarification.

Use IDs vs names. I included names for reference, but city/towns might
not always be recorded exactly the same, so IDs are the safer bet.

I have tried lmer?originally, my intention was to preserve counts of zero
crime (whether it?s weariness or sanity that has changed my view, the
difference between one and zero doesn?t seem so important anymore. I get
the boundary (singular) warning (if a warning is what it can be called). No
doubt there are many here that know more than me, but my understanding of
this message is that it mostly warns against overfitting. To me,
controlling for time through a fixed effect and as a random slope seems
warranted?different cities/towns have different rates of change in crime
from year to year.

I still think fitting with glmer would be ideal, but I also lack an
understanding of the nuances on the backend with regard to how the two
functions differ. It seems that lmer is faster and more efficient in
modeling, though I?m not sure why?mathematically or programmatically?that?s
the case. If you can point me to an explanation, I?d appreciate that.

Thanks much for your messages and your help.

James

On Sep 4, 2019, at 12:54 AM, Thierry Onkelinx <thierry.onkelinx at inbo.be>
wrote:

Dear James,

Your mail contains only a part of the dput() output. Given the size of
your data, I suggest that you place it somewhere online and send us a link
to the data.

Best regards,

ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE
AND FOREST
Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
thierry.onkelinx at inbo.be
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be

///////////////////////////////////////////////////////////////////////////////////////////

<https://www.inbo.be/>

Op di 3 sep. 2019 om 21:34 schreef Ades, James <jades at ucsd.edu>:

I posted my question at Stack Overflow, where it didn?t get much of a
response, and I was pointed in this direction by Ben Bolker. I?m happy to
send the whole dataset to anyone who wants but thought that it would be
presumptuous to include an enormous dput() here.

I?m looking at the effects of education spending per school district on
crime rate (FBI crime data/UCR) within the cities and towns those school
districts serve over a fifteen year period. The DV now has 203,410
observations of city/town crime data over those fifteen years. (I use that
figure with some reticence, because there are so many moving parts and
things to account for, but having employed over 100 datasets and hours
passing through the code again, I think that figure is correct.)

Cities are technically crossed with school district, in that one city
might attend multiple school districts. This means that one city could have
multiple values for expenditure per student. School districts, however,
also overlap with counties. As if things weren?t complicated enough, cities
are mostly nested within county (though there are cities that exist in two
counties, but it?s not often, and it?s usually by a small amount). Given
that each city/town has a distinct PLACE_ID, my understanding is that this
could be represented as (1|PLACE_ID) + (1|STATE/COUNTY_ID) or
(1|STATE/COUNTY_ID/PLACE_ID).

I?m pretty familiar with mixed-effect models, and I?ve looked through
clear and informative posts such as this one:
https://stats.stackexchange.com/questions/228800/crossed-vs-nested-random-effects-how-do-they-differ-and-how-are-they-specified.
I believe things would remain sane to include school district
(full_district_id) as another crossed effect, as below:
glmer.total <- glmer(CRIME_TOTAL ~ 1 + (year|PLACE_ID) +
(1|STATE/COUNTY_ID) + (year|full_district_id), family = "poisson", control
= glmerControl(optimizer = "nloptwrap", calc.derivs = FALSE), REML = FALSE,
total.years, na.action = "na.omit")

Variables (not included in this model, to keep things null and simple)
are centered and logged: pop per city, pop.dens per city, year,
unemployment rate per county, proportion children living in poverty per
school district, per capita income per county, difference in those who
voted democrat in presidential elections per county, log enforcement per
city/town, centered expenditure per student/ 1000 (per school district).
PLACE_ID corresponds to cities and towns, COUNTY_ID to counties,
full_district_id to school districts, and state.

First, if I try to run the full model, using UCSD?s supercomputer, I get
the error that the job was killed, presumably because it got to a point
where it consumed too much ram (I think 125mb).

I then tried to create a small subsection of data with arrange(STATE,
COUNTY_ID, PLACE_ID) and then slicing by the first ten states (up through
Delaware), so that I have 26,599 values. If I run this null model with the
above code, I get the following error:

```
Error in getOptfun(optimizer) :
  optimizer function must use (at least) formal parameters ?fn?, ?par?,
?lower?, ?control?
```

Then I tried with the optimx, with these configurations: control =
glmerControl(optimizer = "optimx?,
optCtrl = list(method = "nlminb?,
maxit=10000,
iter.max=10000,
eval.max=10000,
lower = c(0,0,0),
upper = c(Inf,10,1)))

and I received the following warning?since this is a null model, there
aren?t any variables to really rescale.\

```
Warning messages:
1: In nlminb(start = par, objective = ufn, gradient = ugr, lower =
lower,  :
  unrecognized control elements named ?lower?, ?upper? ignored
2: In nlminb(start = par, objective = ufn, gradient = ugr, lower =
lower,  :
  unrecognized control elements named ?lower?, ?upper? ignored
3: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,
:
  Model failed to converge with max|grad| = 0.00102386 (tol = 0.001,
component 1)
4: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,
:
  Model is nearly unidentifiable: very large eigenvalue
 - Rescale variables?
```

I then tried more values (to 92,486, through Missouri). First, I tried
the optimizer nloptr, and then I tried the optimix. I still received the
same above errors.

I?ve checked and rechecked everything, so I wanted to solicit advice,
either for where I might be going wrong, or for what I could do to resolve
these error messages.

I?ve provided a brief snippet of the data below (randomly pulling a
number of cities within counties of Arkansas, Arizona, and Alabama, as a
dput.

Thanks!



231, 206, 935, 1070, 974, 1108, 1244, 1095, 1131, 1151, 1420,
1316, 1321, 1414, 1484), full_district_id = c("0100240", "0100240",
"0100240", "0100240", "0100240", "0100240", "0100240", "0100240",
"0100240", "0100240", "0100240", "0100240", "0100240", "0100240",
"0100240", "0100240", "0100240", "0100240", "0100240", "0100240",
"0100240", "0100240", "0100240", "0100240", "0100240", "0100240",
"0100240", "0100240", "0100240", "0100240", "0100240", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0100270", "0100270", "0100270", "0100270",
"0100270", "0100270", "0407130", "0407130", "0407130", "0407130",
"0407130", "0407130", "0407130", "0407130", "0407130", "0407130",
"0407130", "0407130", "0407130", "0407130", "0408080", "0408080",
"0408080", "0408080", "0408080", "0407130", "0407130", "0407130",
"0407130", "0407130", "0407130", "0407130", "0407130", "0407130",
"0407130", "0407130", "0407130", "0407130", "0400212", "0408020",
"0400212", "0408020", "0400212", "0408020", "0400212", "0408020",
"0400212", "0408020", "0400212", "0408020", "0401180", "0401180",
"0401180", "0401180", "0401180", "0401180", "0401180", "0401180",
"0401180", "0402530", "0402530", "0402530", "0402530", "0402530",
"0402530", "0402530", "0402530", "0408600", "0408600", "0408600",
"0408600", "0408600", "0408600", "0408600", "0408600", "0408600",
"0403150", "0401460", "0403150", "0401460", "0403150", "0401460",
"0403150", "0401460", "0403150", "0401460", "0403150", "0401460",
"0401460", "0403150", "0401460", "0403150", "0401460", "0403150",
"0401460", "0401460", "0401460", "0401460", "0408600", "0408600",
"0408600", "0408600", "0409250", "0409250", "0409250", "0409250",
"0409250", "0409250", "0409250", "0409250", "0409250", "0409250",
"0409250", "0409250", "0409250", "0402860", "0402860", "0402860",
"0402860", "0402860", "0402860", "0402860", "0402860", "0402860",
"0402860", "0402860", "0402860", "0402860", "0402860", "0402860",
"0403080", "0403080", "0403080", "0403080", "0403080", "0403080",
"0403080", "0403080", "0403080", "0403080", "0403080", "0403080",
"0403080", "0403080", "0405820", "0403080", "0405820", "0403080",
"0405820", "0403080", "0405820", "0403080", "0405820", "0403080",
"0405820", "0403080", "0405820", "0403080", "0405820", "0403080",
"0405820", "0403080", "0405820", "0402860", "0402860", "0402860",
"0402860", "0402860", "0402860", "0402860", "0402860", "0402860",
"0402860", "0402860", "0409310", "0409310", "0409310", "0409310",
"0409310", "0409310", "0409310", "0409310", "0409310", "0409310",
"0409310", "0403500", "0405030", "0403500", "0405030", "0403500",
"0405030", "0403500", "0405030", "0403500", "0405030", "0403500",
"0405030", "0403500", "0405030", "0403500", "0405030", "0403500",
"0405030", "0403500", "0405030", "0403500", "0405030", "0403730",
"0405030", "0405030", "0405030", "0405030", "0405030", "0405030",
"0405030", "0405030", "0405030", "0405030", "0405030", "0405030",
"0405030", "0406070", "0406070", "0406070", "0406070", "0406070",
"0406070", "0406070", "0406070", "0406070", "0406070", "0406070",
"0406070", "0406440", "0406440", "0406440", "0406440", "0406440",
"0406440", "0406440", "0406440", "0406440", "0406440", "0406440",
"0406440", "0407240", "0408410", "0407240", "0408410", "0407240",
"0408410", "0407240", "0408410", "0407240", "0408410", "0407240",
"0408410", "0407240", "0408410", "0407240", "0408410", "0407240",
"0408410", "0407240", "0408410", "0407240", "0408410", "0407240",
"0408410", "0407240", "0408410", "0407240", "0408410", "0407240",
"0408410", "0407240", "0408410", "0407240", "0408410", "0407240",
"0408410", "0402110", "0405320", "0402110", "0405320", "0402110",
"0405320", "0402110", "0405320", "0402110", "0405320", "0402110",
"0405320", "0405320", "0402110", "0405320", "0402110", "0405320",
"0405980", "0405980", "0405980", "0405980", "0405980", "0405980",
"0405980", "0509000", "0509000", "0509000", "0509000", "0509000",
"0509000", "0509000", "0509000", "0509000", "0509000", "0509000",
"0509000", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0509000", "0511850", "0509000", "0511850", "0509000",
"0511850", "0509000", "0511850", "0509000", "0511850", "0509000",
"0511850", "0509000", "0511850", "0509000", "0511850", "0509000",
"0511850", "0509000", "0511850", "0509000", "0511850", "0509000",
"0511850", "0509000", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0511850", "0511850", "0511850", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0510680", "0511850", "0510680", "0511850", "0510680", "0511850",
"0500009", "0500009", "0500009", "0500009", "0500009", "0500009",
"0500009", "0500009", "0500009", "0500009", "0509270", "0509270",
"0509270", "0513080", "0513080", "0513080", "0513080", "0513080",
"0513080", "0513080", "0513080", "0513080", "0513080", "0513080",
"0513080", "0513080", "0511610", "0511610", "0511610", "0511610",
"0511610", "0511610", "0511610", "0511610", "0511610", "0511610",
"0512480", "0512480", "0512480", "0512480", "0506270", "0506270",
"0506270", "0506270", "0506270", "0506270", "0506270", "0506270",
"0506270", "0506270", "0508040", "0508040", "0508040", "0508040",
"0508040", "0508040", "0503690", "0503690", "0502790", "0502960",
"0503690", "0507320", "0502790", "0502960", "0503690", "0507320",
"0502790", "0502960", "0503690", "0507320", "0502790", "0502960",
"0503690", "0507320", "0502790", "0502960", "0503690", "0507320",
"0502790", "0502960", "0503690", "0507320", "0502790", "0502960",
"0503690", "0507320", "0502790", "0502960", "0503690", "0507320",
"0502790", "0502960", "0503690", "0507320", "0502790", "0502960",
"0503690", "0507320", "0502790", "0502960", "0503690", "0507320",
"0502790", "0502960", "0503690", "0507320", "0502790", "0502960",
"0503690", "0507320", "0503690", "0503690", "0503690", "0503690",
"0503690", "0503690", "0503690", "0503690", "0503690", "0503690",
"0503690", "0506420", "0506420", "0506420", "0506420", "0506420",
"0506420", "0506420", "0506420", "0506420", "0506420", "0506420",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0503690", "0511850", "0503690",
"0511850", "0503690", "0511850", "0503690", "0511850", "0500015",
"0500015", "0500015", "0500015", "0500015", "0500015", "0500015",
"0500015", "0500015", "0500015", "0500015", "0500015", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850",
"0511850", "0511850", "0511850", "0511850", "0511850", "0511850"
), COUNTY = c("autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"autauga county", "autauga county", "autauga county", "autauga county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "baldwin county",
"baldwin county", "baldwin county", "baldwin county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "apache county",
"apache county", "apache county", "apache county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"cochise county", "cochise county", "cochise county", "cochise county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "coconino county", "coconino
county",
"coconino county", "coconino county", "gila county", "gila county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "gila county", "gila county", "gila county", "gila
county",
"gila county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "graham county", "graham county", "graham county",
"graham county", "greenlee county", "greenlee county", "greenlee county",
"greenlee county", "greenlee county", "greenlee county", "greenlee
county",
"greenlee county", "greenlee county", "greenlee county", "greenlee
county",
"greenlee county", "greenlee county", "greenlee county", "greenlee
county",
"greenlee county", "greenlee county", "la paz county", "la paz county",
"la paz county", "la paz county", "la paz county", "la paz county",
"la paz county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
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"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"pulaski county", "pulaski county", "pulaski county", "pulaski county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"randolph county", "randolph county", "randolph county", "randolph
county",
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county",
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county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"randolph county", "randolph county", "randolph county", "randolph
county",
"st. francis county", "st. francis county", "st. francis county",
"st. francis county", "st. francis county", "st. francis county",
"st. francis county", "st. francis county", "st. francis county",
"st. francis county", "st. francis county", "st. francis county",
"st. francis county", "st. francis county", "st. francis county",
"st. francis county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
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"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
"saline county", "saline county", "saline county", "saline county",
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"saline county", "saline county", "saline county", "saline county",
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"saline county", "saline county", "saline county", "saline county",
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"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Autauga County School District", "Autauga County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Baldwin County School District",
"Baldwin County School District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "St. Johns Unified District",
"St. Johns Unified District", "St. Johns Unified District", "St. Johns
Unified District",
"St. Johns Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Round Valley Unified District", "Round Valley Unified District",
"Benson Unified School District", "St. David Unified District",
"Benson Unified School District", "St. David Unified District",
"Benson Unified School District", "St. David Unified District",
"Benson Unified School District", "St. David Unified District",
"Benson Unified School District", "St. David Unified District",
"Benson Unified School District", "St. David Unified District",
"Bisbee Unified District", "Bisbee Unified District", "Bisbee Unified
District",
"Bisbee Unified District", "Bisbee Unified District", "Bisbee Unified
District",
"Bisbee Unified District", "Bisbee Unified District", "Bisbee Unified
District",
"Douglas Unified District", "Douglas Unified District", "Douglas Unified
District",
"Douglas Unified District", "Douglas Unified District", "Douglas Unified
District",
"Douglas Unified District", "Douglas Unified District", "Tombstone
Unified District",
"Tombstone Unified District", "Tombstone Unified District", "Tombstone
Unified District",
"Tombstone Unified District", "Tombstone Unified District", "Tombstone
Unified District",
"Tombstone Unified District", "Tombstone Unified District", "Fort
Huachuca Accommodation District",
"Sierra Vista Unified District", "Fort Huachuca Accommodation District",
"Sierra Vista Unified District", "Fort Huachuca Accommodation District",
"Sierra Vista Unified District", "Fort Huachuca Accommodation District",
"Sierra Vista Unified District", "Fort Huachuca Accommodation District",
"Sierra Vista Unified District", "Fort Huachuca Accommodation District",
"Sierra Vista Unified District", "Sierra Vista Unified District",
"Fort Huachuca Accommodation District", "Sierra Vista Unified District",
"Fort Huachuca Accommodation District", "Sierra Vista Unified District",
"Fort Huachuca Accommodation District", "Sierra Vista Unified District",
"Sierra Vista Unified District", "Sierra Vista Unified District",
"Sierra Vista Unified District", "Tombstone Unified District",
"Tombstone Unified District", "Tombstone Unified District", "Tombstone
Unified District",
"Willcox Unified District", "Willcox Unified District", "Willcox Unified
District",
"Willcox Unified District", "Willcox Unified District", "Willcox Unified
District",
"Willcox Unified District", "Willcox Unified District", "Willcox Unified
District",
"Willcox Unified District", "Willcox Unified District", "Willcox Unified
District",
"Willcox Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Fredonia-Moccasin Unified District",
"Fredonia-Moccasin Unified District", "Fredonia-Moccasin Unified
District",
"Fredonia-Moccasin Unified District", "Fredonia-Moccasin Unified
District",
"Fredonia-Moccasin Unified District", "Fredonia-Moccasin Unified
District",
"Fredonia-Moccasin Unified District", "Fredonia-Moccasin Unified
District",
"Fredonia-Moccasin Unified District", "Fredonia-Moccasin Unified
District",
"Fredonia-Moccasin Unified District", "Fredonia-Moccasin Unified
District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Fredonia-Moccasin Unified District", "Page Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
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"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Flagstaff
Unified District",
"Flagstaff Unified District", "Flagstaff Unified District", "Williams
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"Williams Unified District", "Williams Unified District", "Williams
Unified District",
"Williams Unified District", "Williams Unified District", "Williams
Unified District",
"Williams Unified District", "Williams Unified District", "Williams
Unified District",
"Williams Unified District", "Globe Unified District", "Miami Unified
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"Globe Unified District", "Miami Unified District", "Globe Unified
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"Miami Unified District", "Globe Unified District", "Miami Unified
District",
"Globe Unified District", "Miami Unified District", "Globe Unified
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"Miami Unified District", "Globe Unified District", "Miami Unified
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"Globe Unified District", "Miami Unified District", "Globe Unified
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"Miami Unified District", "Globe Unified District", "Miami Unified
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"Globe Unified District", "Miami Unified District", "Hayden-Winkelman
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"Miami Unified District", "Miami Unified District", "Miami Unified
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"Miami Unified District", "Miami Unified District", "Miami Unified
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"Miami Unified District", "Miami Unified District", "Miami Unified
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"Miami Unified District", "Payson Unified District", "Payson Unified
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"Payson Unified District", "Payson Unified District", "Payson Unified
District",
"Payson Unified District", "Payson Unified District", "Payson Unified
District",
"Payson Unified District", "Payson Unified District", "Payson Unified
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"Payson Unified District", "Pima Unified District", "Pima Unified
District",
"Pima Unified District", "Pima Unified District", "Pima Unified
District",
"Pima Unified District", "Pima Unified District", "Pima Unified
District",
"Pima Unified District", "Pima Unified District", "Pima Unified
District",
"Pima Unified District", "Safford Unified District", "Thatcher Unified
District",
"Safford Unified District", "Thatcher Unified District", "Safford
Unified District",
"Thatcher Unified District", "Safford Unified District", "Thatcher
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"Safford Unified District", "Thatcher Unified District", "Safford
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"Thatcher Unified District", "Safford Unified District", "Thatcher
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"Thatcher Unified District", "Safford Unified District", "Thatcher
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"Safford Unified District", "Thatcher Unified District", "Safford
Unified District",
"Thatcher Unified District", "Clifton Unified District", "Morenci
Unified District",
"Clifton Unified District", "Morenci Unified District", "Clifton Unified
District",
"Morenci Unified District", "Clifton Unified District", "Morenci Unified
District",
"Clifton Unified District", "Morenci Unified District", "Clifton Unified
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"Morenci Unified District", "Morenci Unified District", "Clifton Unified
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"Morenci Unified District", "Clifton Unified District", "Morenci Unified
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"Parker Unified School District", "Parker Unified School District",
"Parker Unified School District", "Parker Unified School District",
"Parker Unified School District", "Parker Unified School District",
"Parker Unified School District", "Little Rock School District",
"Little Rock School District", "Little Rock School District",
"Little Rock School District", "Little Rock School District",
"Little Rock School District", "Little Rock School District",
"Little Rock School District", "Little Rock School District",
"Little Rock School District", "Little Rock School District",
"Little Rock School District", "Pulaski County Special School District",
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"Pulaski County Special School District", "Pulaski County Special School
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"Pulaski County Special School District", "Pulaski County Special School
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"Fountain Lake School District", "Fountain Lake School District",
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Large mixed & crossed-effect model looking at educational spending on crime rates with error messages

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