Large mixed & crossed-effect model looking at educational spending on crime rates with error messages
I'm having trouble reading this file - it's 50MB to start with, my
Linux system thinks it's gzipped ("gzip compressed data"), then when I
un-gzip it I get a 100+MB file that it thinks is "data" (which just
means "binary file I don't understand"). Is there any way you can
make a **much smaller** reproducible example for us to inspect??
I think the answer to your proximal question about per_race is that
you would need five *different* numerical variables, one for each
race-percentage. However, you can't/shouldn't actually use all five,
since they will be multicollinear (since the sum of all five will be 1
or 100 (or however you've scaled the total).
Also: I wouldn't recommend Nelder-Mead, it's probably the slowest
and least reliable option ...
On Fri, Sep 27, 2019 at 7:05 PM Ades, James <jades at ucsd.edu> wrote:
I?m confused. So I?ve added race to my current model (see below). I?ve made race (latino, white, black, asian, two or more races) into percents for each city/town, such that I have five values for each city/town for each year (2003-2017). I?ve gathered the five variables into the one column ?race? and the values containing the percents into another column named ?per_race.? This is the way that I?ve structured my model to include race: lmer(log(CRIME_TOTAL + 1) ~ year + cent.log.pop + cent.log.pop.dens + per_race + (year|PLACE_ID / race) + (1|COUNTY_ID) + (1|STATE), control = lmerControl(optimizer="Nelder_Mead"), na.action = 'na.omit', dat, REML = FALSE) In the current model, I am only getting one value of race as a regression coefficient ?per_race.? It seems that race would also need to be a fixed effect. How do I restructure my model to receive all five race values as regression coefficients? Would I need to include race as an interaction between per_race and race? ( lmer(log(CRIME_TOTAL + 1) ~ year + cent.log.pop + cent.log.pop.dens + per_race*race + (year |PLACE_ID/race) + ( 1|COUNTY_ID) + (1| STATE) ) This doesn?t seem right. I?ve uploaded the data here: https://drive.google.com/file/d/1oKMpjnh2Y-ONvYufcX5MxF6WKMBw58YO/view?usp=sharing Thanks much! James On Sep 10, 2019, at 5:45 AM, Thierry Onkelinx <thierry.onkelinx at inbo.be<mailto:thierry.onkelinx at inbo.be>> wrote: 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<http://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<mailto:thierry.onkelinx at inbo.be> Havenlaan 88 bus 73, 1000 Brussel www.inbo.be<http://www.inbo.be/> /////////////////////////////////////////////////////////////////////////////////////////// To call in the statistician after the experiment is done may be no more 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 /////////////////////////////////////////////////////////////////////////////////////////// [https://inbo-website-prd-532750756126.s3-eu-west-1.amazonaws.com/inbologoleeuw_nl.png]<https://www.inbo.be/> Op di 10 sep. 2019 om 08:51 schreef Ades, James <jades at ucsd.edu<mailto: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<mailto: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 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<mailto:thierry.onkelinx at inbo.be> Havenlaan 88 bus 73, 1000 Brussel www.inbo.be<http://www.inbo.be/> /////////////////////////////////////////////////////////////////////////////////////////// To call in the statistician after the experiment is done may be no more 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 /////////////////////////////////////////////////////////////////////////////////////////// [https://inbo-website-prd-532750756126.s3-eu-west-1.amazonaws.com/inbologoleeuw_nl.png]<https://www.inbo.be/> Op zo 8 sep. 2019 om 07:40 schreef Ades, James <jades at ucsd.edu<mailto: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<mailto: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<mailto:thierry.onkelinx at inbo.be> Havenlaan 88 bus 73, 1000 Brussel www.inbo.be<http://www.inbo.be/> /////////////////////////////////////////////////////////////////////////////////////////// To call in the statistician after the experiment is done may be no more 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 /////////////////////////////////////////////////////////////////////////////////////////// [https://inbo-website-prd-532750756126.s3-eu-west-1.amazonaws.com/inbologoleeuw_nl.png]<https://www.inbo.be/> Op di 3 sep. 2019 om 21:34 schreef Ades, James <jades at ucsd.edu<mailto: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", 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"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 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"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 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"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 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", "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", "Williams Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Globe Unified District", "Miami Unified District", "Hayden-Winkelman Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami Unified District", "Miami 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 District", "Payson Unified District", "Payson Unified District", "Payson Unified District", "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 Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "Safford Unified District", "Thatcher Unified District", "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 District", "Morenci Unified District", "Morenci Unified District", "Clifton Unified District", "Morenci Unified District", "Clifton Unified District", "Morenci Unified 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", "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", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Little Rock School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "Pulaski County Special School District", "North Little Rock School District", "Pulaski County Special School District", "North Little Rock School District", "Pulaski County Special School District