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    pdftitle={Modeling Assumptions},
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Exploring Modeling Assumptions with Census Data}




\markboth{Exploring Modeling Assumptions with Census Data}{
Exploring Modeling Assumptions with Census Data}

\author{Jean Marie Linhart \\
Gary William Epp\\
Central Washington University\\
Ellensburg, WA 98926 USA\\
\url{JeanMarie.Linhart@cwu.edu}
%\\ \url{Gary.Epp@cwu.edu}\\
}

\makeStitlePDFLaTex


\begin{abstract}
  Students who have studied models for population are likely to be familiar with the exponential and the logistic population models.  The goal here is to explore the role of modeling assumptions in choosing which model to use.  We will compare and contrast the United States census data and the Guatemala census data, and how these two datasets fulfill or fail to fulfill the modeling assumptions of the exponential and logistic models.  Along the way, we will introduce the concept of the per capita population growth rate, and calculate the per capita population growth rate for the exponential model, the logistic model, and estimate the per capita population growth rate from the two census datasets.
\end{abstract}


\section{INTRODUCTION AND SCENARIO DESCRIPTION}

The United States conducts a census every 10 years, which gives data on the population of the USA from 1790 to the present.  This data is frequently discussed along with the exponential and logistic population models \cite{USCensusHist, LinhartUSPopSIMIODE, MestertonGibbons, USPopData, USPopData2}  

In contrast, the Republic of Guatemala has taken 16 censuses; its first census was taken in 1778 \cite{Squier}. The Guatemala census has not been taken at regular intervals, and it is considerably less ``smooth'' than the data from the United States. There are concerns that some of this census data may not be accurate. For example, according to Squier \cite[p.~45]{Squier} the 1837 census was discredited at the time.\footnote{Statistician Don Jos\'e de la Valle calculated Guatemala had 600,000 inhabitants in 1837.}  Records from the 1940 census were burned; this data is unavailable. Records from 1778, 1880, 1893, and 1921 were used as scrap paper and no longer exist, although the statistical information from these censuses was preserved \cite{Platt}. To the best of the authors' knowledge, the data for the censuses were not compiled in one easily accessed table until author Gary Epp, then an undergraduate at Central Washington University, put this dataset together after searching out and looking through the references cited and many more as part of a class project in Mathematical Modeling. We also added it to the \href{https://en.wikipedia.org/wiki/Guatemala}{Guatemala Wikipedia page} \cite{GuatWiki}.

While the United States data is considered more reliable and certainly appears to be more smooth, there are also concerns about the accuracy of the United States census data. Few native Americans were included in the census until 1900.  They are not identified in the 1790-1840 censuses.  In 1860, native Americans living in the general population were identified for the first time.  Most of the 1890 census was destroyed by fire \cite{CensusNativeAmericans}, although its statistical information is preserved. There is always error in census data.  For example, consider the 2010 Census. The United States Census Bureau estimates a net overcount of 0.01 percent or about 36,000 people, which is low.  However, not every group is counted that precisely.  The non-Hispanic white population was overcounted by an estimated 0.8 percent, whereas the native american and Alaska native population on reservations were undercounted by 4.9 percent, and the black population was undercounted by 2.1 percent \cite{Census2010Estimates}.  While one can argue that these are all small amounts, notice that 0.8 percent is 80 times 0.01 percent, and 4.9 percent is 490 times 0.01 percent. The Urban Institute estimates an overall 0.5 percent net undercount for the 2020 census, roughly 50 times the estimated 0.01 percent error of the 2010 census.  They note that, as with the 2010 census, there is considerable variation in who is undercounted and who is overcounted overall.  The populations of Mississippi and Texas are estimated to be undercounted by 1.3 and 1.28 percent respectively, whereas Minnesota's population was estimated to be overcounted by 0.76 percent. The Urban Institute notes that if these counts were accurate, Texas would receive over \$247 million more and Minnesota would receive \$156 million less in 2021 Federal Medicaid reimbursements alone.  As is typical, the groups hardest to count in recent censuses were again likely undercounted in the 2020 census.  Black and Hispanic/Latinx people have a net undercount of more than 2.45 and 2.17 percent respectively, and households with a noncitizen present were likely undercounted by 3.36 percent overall \cite{Census2020Estimates}.


Even with the known problems, this data is the best we have to estimate the total population of the United States and Guatemala over the past $\sim$200 years.


We will investigate applying two common population growth models to these two datasets.  The models are the exponential model and the logistic model.
\vfill\eject

\vspace*{1 true cm}
\begin{table}[H]
\begin{tabular}{p{1.5 in} p{5 in}}
\vspace*{-3. in}
{\begin{tabular}[b]{|c|r|}
\hline
Census & United States \\
Year & Population \\
\hline
1790 & 3,929,214 \\
1800 & 5,308,483 \\
1810 & 7,239,861 \\
1820 & 9,638,453 \\
1830 & 12,866,020 \\
1840 & 17,069,453 \\
1850 & 23,191,876 \\
1860 & 31,443,321 \\
1870 & 39,818,449 \\
1880 & 50,155,783 \\
1890 & 62,947,714 \\
1900 & 75,994,575 \\
1910 & 91,972,266 \\
1920 & 105,710,620 \\
1930 & 122,775,046 \\
1940 & 131,669,275 \\
1950 & 151,325,798 \\
1960 & 179,323,175 \\
1970 & 203,302,031 \\
1980 & 226,545,805 \\
1990 & 248,718,302 \\
2000 & 281,424,603 \\
2010 & 308,745,538 \\
2020 &  331,449,281 \\
\hline
\end{tabular}}&
\includegraphics[width=4.5in]{USPopData.png} \\
\end{tabular}
\captionlistentry[figure]{USA Population Data}
\captionsetup{labelformat=andfigure}
\caption{USA Population Data \label{USAData}}
\end{table}

\begin{table}[H]
\begin{tabular}{p{1.5 in} p{5 in}}
\vspace*{-3. in}
{\begin{tabular}{|c|c|}
    \hline
    Census & Guatemala\\
	Year & Population \\
	\hline
	1778 & 430,859 \cite{Squier}\\
	\hline
	1825 & 507,126 \cite{Squier}\\
	\hline
	1837 & 490,787 \cite{Squier}\\
	\hline
	1852 & 787,000\cite{Squier}\\
	\hline
	1880 & 1,224,602 \cite{Taeuber}\\
	\hline
	1893 & 1,364,678 \cite{Loveman}\\
	\hline
	1914 & 2,183,166  \cite{McCreery}\\
	\hline
	1921 & 2,004,900  \cite{McCreery}\\
	\hline
	1950 & 2,870,272  \cite{McCreery}\\
	\hline
	1964 & 4,287,997  \cite{Monaghan}\\
	\hline
	1973 & 5,160,221  \cite{Monaghan}\\
	\hline
	1981 & 6,054,227  \cite{Monaghan}\\
	\hline
	1994 & 8,321,067 \cite{Monaghan}\\
	\hline
	2002 & 11,183,388 \cite{Holbrock}\\
        \hline
        2018 & 14,901,286  \cite{Guat2018}\\
	\hline
\end{tabular}}
&  \includegraphics[width=4.5 in]{GuatPopData.png}\\
\end{tabular}
\captionlistentry[figure]{Guatemala Census Data}
\captionsetup{labelformat=andfigure}
\caption{Guatemala Population Data \label{GuatData}}
\end{table}




\section{EXPONENTIAL AND LOGISTIC MODELS}\label{sec:ModelIntro}

The two models for population we are discussing are the exponential and the logistic model. Note that the two models are very similar at the beginning, but one obvious difference between these two models is that the logistic model ``levels off'' to a maximum population called the carrying capacity, whereas the exponential model grows without bound.  This is pictured in Figure \ref{fig:ModelCompare}, below.  




One place to start to determine whether one or both models are appropriate for modeling our two sample populations is to look at the graphs of the United States and Guatemala census data in Figures \ref{USAData} and \ref{GuatData} and the graphs of the exponential and logistic curves in Figure \ref{fig:ModelCompare}, and answer the following questions

\begin{enumerate}[1)]
\item Does the shape of the exponential curve approximate the shape of the United States census data?  Does it approximate the shape of the Guatemala census data?  If it does not, describe how the shape of the data differs from the shape of the exponential curve.
  \item Does the shape of the logistic curve approximate the shape of the United States census data?  Does the shape of the logistic curve approximate the shape of the Guatemala census data?  If it does not, describe how the shape of the data differs from the shape of the logistic curve.
  \newcounter{QuestionCounter}
  \setcounter{QuestionCounter}{\theenumi}
\end{enumerate}
Discuss your response with your classmates before continuing.

Note: having a curve or function resemble the data is not enough to conclude that a model is a good one in a given situation.  What we really want is for the model to capture important mechanics of the physical situation that gave rise to the data.  Next we will look at the equations for these two models and discuss how they came about.



\begin{figure}[H]
\begin{center}
  \includegraphics[width=4 in]{PopModels1.png}
\end{center}
\caption{A graphical comparison of the exponential and the logistic population models\label{fig:ModelCompare}}
\end{figure}


\subsection{THE EXPONENTIAL MODEL}

The exponential model is defined by the differential equation and initial value
\begin{equation} \dfrac{dP}{dt} = rP, \qquad P(0) = P_0. \label{eq:ExpDE}\end{equation}
It has solution
\begin{equation}P(t) = P_0e^{rt}. \label{eq:ExpSoln}\end{equation}
 Note that $\frac{dP}{dt}$ is the change in the population with respect to time, and we have this equal to $rP$ where $r$ is a constant.  So the main operational assumption of the exponential model is that the change in the population with respect to time is directly proportional to the population.  We assume the number of births is proportional to the population, as is the number of deaths. If a population of 100 people had 2 births and 1 death in a year, then we would expect a population of 1000 to have 20 births and 10 deaths per year, and we would expect a population of 600 to have 12 births and 6 deaths per year.  The change in population with respect to time is the births minus the deaths, so for a population of 100 the change is 1 per year, for a population of 600 the change is 6 per year, and for a population of 1000 the change is 10 per year.  The only thing that affects the change is the size of the population.

This is a simple and sensible assumption for population change. But is reality really this simple?  Consider the following question.

\begin{enumerate}[1)]
    \setcounter{enumi}{\theQuestionCounter}
\item Do you expect the same proportionality constant to work on populations throughout every historical era?  Do you expect the same proportionality constant to work for every human population across the globe?  Why or why not?
  \setcounter{QuestionCounter}{\theenumi}
\end{enumerate}
Discuss your response with your classmates before moving on.

When we first create a mathematical model, we try to start with the {\em simplest model\/}, and so we make simplifying assumptions.  We change assumptions and complicate our model only when we need to.  We might complicate our model in order to better match our data or to better answer a question.  The exponential model is the simplest population model that is based on a mechanism of population change.


\subsection{THE LOGISTIC MODEL}
The logistic model is defined by the differential equation and initial value
\begin{equation}\dfrac{dP}{dt} = rP\left(1-\dfrac{P}{L}\right), \qquad P(0) = P_0. \label{eq:LogisticDE}\end{equation}
It has solution
\begin{equation}P(t)  = \dfrac{L}{1 + \dfrac{(L- P_0)}{P_0}e^{-rt}} \label{eq:LogisticSoln}\end{equation}
The logistic differential equation comes about by starting with the modeling assumption that births are proportional to the size of the population, as are some deaths.  This is the same starting assumption as we had for the exponential model.  Now we consider another cause of death: deaths that are due to two-person interactions.  Two people can, of course, have a fight that ends in a death, and one person can murder another.  Wars occur.  We also consider interactions where people compete for the same resources, which might eventually lead to the death of some of the competitors.
Starting from an exponential model (using $a$ for the proportionality constant)
$$\dfrac{dP}{dt} = aP$$
where births and deaths are proportional to the size of the population, we include a component of deaths that are proportional to the number of possible two person interactions.  If there are $P$ people in the population, each one can interact with $P-1$ other people, thus there are $P(P-1)$ possible two person interactions.  So to include deaths proportional to this we subtract $kP(P-1)$, where $k$ is the proportionality constant.  The equation becomes
\begin{align}
  \dfrac{dP}{dt} &= aP - kP(P-1)\\
  &= aP - kP^2 + kP\\
  &= (a+k)P - kP^2 \\
  &= (a+k)P\left(1-\dfrac{k}{a+k}P\right)
\end{align}
We can identify this with our logistic differential equation \eqref{eq:LogisticDE},
by letting $a+k = r$ and $\frac{k}{a+k} = \dfrac{1}{L}$.
For an accessible history of the logistic model check out \cite{RefractModel}.

These assumptions for the logistic model should also make sense.  While it is more complicated than the exponential model, this model is also a vast simplification of population change. Consider the following question.
\begin{enumerate}[1)]
    \setcounter{enumi}{\theQuestionCounter}
  \item What other factors influence the size of human populations like the population of the United States or the population of Guatemala that we have omitted from both the exponential and the logistic model?
  \setcounter{QuestionCounter}{\theenumi}
\end{enumerate}
Discuss your response with your classmates and make a list of other factors that influence populations that have been omitted from this model.

We see that the exponential model and the logistic model are simplifications of a more complicated reality, but that both make sense and capture some important mechanics of population change.  This is why these two models are frequently used to model populations.

\section{TROUBLE\label{sec:Trouble}}

As a class we used nonlinear least squares estimation to find parameter values for the United States census data with both the exponential and logistic model as described in \cite{LinhartUSPopSIMIODE}.  When Gary tried to repeat these same steps with the Guatemala data, he ran into trouble.  Visually, the exponential model and the logistic model were identical when graphed with the data, see Figure \ref{fig:ModelResults}.  Upon further reflection, we discovered that the carrying capacity for Guatemala estimated for the  logistic model was $2.6\times10^8$ million people, or 260 trillion people, compared to an estimate of 501 million people for the United States.  The high carrying capacity for Guatemala seems preposterous.  We also noticed a big discrepancy in the covariance matrices that measure goodness of fit.  We noted the variance for the carrying capacity United States estimation was $8.15\times10^2$ giving us a carrying capacity of $501\pm 29$ million where we take the estimate plus or minus 1 standard deviation of error. In contrast the variance for the carrying capacity of Guatemala was $1.00\times 10^{29}$, so for Guatemala we estimate the carrying capacity as $2.6\times 10^8 \pm 3.2\times10^{14}$ million.  In other words, the error in the parameter estimation with the Guatemala data for the carrying capacity was larger than the carrying capacity itself.

We know that the logistic model starts out looking like an exponential model when $P << L$, and with the high estimate for $L$, this is precisely what we saw.  In contrast, the parameter estimation and the resulting logistic model for the United States data was good, and its results were an improvement on the corresponding exponential model.  See Figure \ref{fig:ModelResults}.

We wondered, why was this?  To uncover the reason, we needed to investigate the per-capita population growth rate.


\begin{figure}[H]
\begin{center}
  \includegraphics[width=2.5 in]{USModels.png} \includegraphics[width=2.5in]{GuatemalaModels.png}
  \end{center}
\caption{ United States and Guatemala with exponential and logistic model curve fits.  Note that the Guatemala logistic curve fit exactly overlays the exponential curve fit.\label{fig:ModelResults}}
\end{figure}

\section{PER-CAPITA POPULATION GROWTH RATE \label{sec:PPGR}}

A commonly used concept in discussing population change is the {\em Per-capita Population Growth Rate} which we will refer to as {\em PPGR}.  Given a population as a function of time $P(t)$, we already know that
$$\dfrac{dP}{dt}$$
is the change in population with respect to time.  Assuming the population is growing, this is what is meant by the {\em population growth rate}.  To get the {\em per-capita} population growth rate, all we have to do is to divide by the total population:
\begin{equation}
  \text{\em PPGR} = \dfrac{1}{P} \dfrac{dP}{dt} \label{eq:PPGRDef}\end{equation}
This is the formal mathematical definition of PPGR.  It measures the average change in the population due to one person.  For example, let's say that in a year, we have a population of $1000$ adult heterosexual cisgender people made up of 500 men and 500 women, and each one of the women has one baby, and no one dies.  With 500 people added to an initial population of $1000$, we see on average, over the past year, one person contributed 1/2 a person to the population growth.  From another angle, notice we have $P(t) = 1000$, $P(t+1) = 1500$. We approximate the derivative using the forward difference formula
\begin{equation}
  \dfrac{dP}{dt} \approx \dfrac{P(t+h) - P(t)}{h}, \label{eq:FwdDif}
\end{equation}
and we see this agrees with our calculation of PPGR in this scenario:
\begin{equation}
  PPGR = \dfrac{1}{P(t)}\dfrac{dP}{dt} \approx \dfrac{1}{1000}\dfrac{\left(1500 - 1000\right)}{1} = \dfrac{500}{1000} = 1/2.\end{equation}

In both the exponential and the logistic differential equations, \eqref{eq:ExpDE} and \eqref{eq:LogisticDE} respectively, the left hand side is solved for $\dfrac{dP}{dt}$.  Thus, we can divide through by population, $P$, to find the assumptions about PPGR as a function of population.  The exponential equation becomes
\begin{equation}
  \dfrac{1}{P}\dfrac{dP}{dt} = r.\label{eq:ExpPPGR}\end{equation}
The logistic equation becomes
\begin{equation}
  \dfrac{1}{P}\dfrac{dP}{dt} = r\left(1-\dfrac{P}{L}\right).\label{eq:LogisticPPGR}\end{equation}
The exponential model assumes a constant PPGR (the simplest possible assumption), and the logistic model assumes a linearly decreasing PPGR with increasing population (the next simplest possible assumption), as seen in Figure \ref{fig:ExpLogisticPPGR}.

\begin{figure}[H]
\begin{center}
  \includegraphics[width=4 in]{PPGRModels.png}
  \end{center}
  \caption{A graphical comparison of the exponential and the logistic PPGR models\label{fig:ExpLogisticPPGR}}
\end{figure}

\subsection{CALCULATING PPGR FROM THE CENSUS DATA}
We can use the population data given in Table 1 and Table 2 to estimate the PPGR.  Take a look at the data. It is in the form $\{(t_i,P_i)\}$ where we assume $1\leq i \leq N$, and the data is in chronological order, so $t_1<t_2 < ... < t_N$.  Recall \eqref{eq:PPGRDef} gives the mathematical definition of PPGR.

\begin{enumerate}[1)]
  \setcounter{enumi}\theQuestionCounter
  \item Assuming that population is in units of the number of people, and time is measured in years, what are the units on PPGR?
  \item Give a formula to approximate $\dfrac{dP}{dt}$ from the population data ${(t_i,P_i)}$.  Will your formula calculate $\dfrac{dP}{dt}$ at every data point?  If not, which ones  does it miss?
  \item Is there more than one possible formula you can use to approximate $\dfrac{dP}{dt}$?  If so, what alternatives are available to you?  Is there an advantage to using one over another?
  \item Give a formula to approximate PPGR from the population data. Will this formula calculate PPGR at every data point?  If not, which ones does it miss?
  \item Is there more than one possible formula you can use to approximate the PPGR from the data?  If so, what alternatives are available to you? Is there any advantage to using one PPGR formula over another? 
  \setcounter{QuestionCounter}{\theenumi}
\end{enumerate}



\begin{figure}[H]
  \begin{center}
    \includegraphics[width=4 in]{PPGRPseudoData.png}
    \end{center}
  \caption{Pretend PPGR(P) data.  How would you find the Exponential (horizontal) PPGR line in Figure \ref{fig:ExpLogisticPPGR}, or the logistic line in Figure \ref{fig:ExpLogisticPPGR} from this data? \label{fig:PPGRPseudoData}}
\end{figure}

Now let us assume we have estimated PPGR from the data.  In Figure \ref{fig:PPGRPseudoData}, you see a hypothetical plot of PPGR vs. $P$. Discuss the following questions with your group:


\begin{enumerate}[1)]
 \setcounter{enumi}{\theQuestionCounter}
\item How would you find the (horizontal) Exponential PPGR line (see Figure \ref{fig:ExpLogisticPPGR}) from the data in Figure \ref{fig:PPGRPseudoData}?
\item How would you find the logistic PPGR line (see Figure \ref{fig:ExpLogisticPPGR}) from the data in Figure \ref{fig:PPGRPseudoData}?
  \setcounter{QuestionCounter}\theenumi
  \end{enumerate}

\subsection{USING A SPREADSHEET TO MAKE PPGR CALCULATIONS}
We can use a spreadsheet and simple spreadsheet calculations to perform these calculations for PPGR, and make a graph of $\mbox{PPGR}(P)$.  The data is included with this modeling scenario in two formats: {\tt .csv} format is a text-based file with ``Comma-Separated Values'', and {\tt .xls} is an Excel spreadsheet format.  The {\tt .csv} version can be opened and understood by any spreadsheet (including the free \href{https://www.openoffice.org/product/calc.html}{OpenOffice Calc}) and a variety of other calculational software. We also provide the data from this modeling scenario in Google Sheets:
\begin{itemize}
\item \href{https://docs.google.com/spreadsheets/d/1iAdZCQ39DC1GcExWvg-ru8dGyMS-1bsyanI7mmo7LUY/edit?usp=sharing}{United States Census Data:} \url{https://tinyurl.com/UnitedStatesCensusData}
\item \href{https://docs.google.com/spreadsheets/d/1tPJSQaV28tty2CJ7FJzOGoFBnkiH_48vkyd7Tbvxc4I/edit?usp=sharing}{Guatemala Census Data:} \url{https://tinyurl.com/GuatemalaCensusData}
\end{itemize}
The instructions below are written assuming students are using Google Sheets, which was the spreadsheet the authors found easiest to use. To use the read only data provided with the links above, go to the File menu and ``Make a copy.'' Students should work in small groups of two or three, and start with the United States census data.

\begin{enumerate}[1)]
  \setcounter{enumi}{\theQuestionCounter}
  \item Create a new column with an estimate of $dP/dt$ calculated from the census data.
\item Create a new column with the estimated PPGR from the census data.
  \item Calculate the average PPGR, and fill a new column with this value
\item Insert a chart with a  scatter plot of the calculated PPGR and average PPGR estimates on the $y$-axis vs. $P$ on the $x$-axis.  Compare your graph with the graphs of other groups to make sure you've got the correct plot.  If they aren't exactly the same, figure out why.  Some common issues are
  \begin{itemize}
  \item Someone graphed $\mbox{PPGR}(P)$ on the $y$-axis, but $t$ on the $x$-axis instead of $P$.
  \item The groups used different formulas to calculate PPGR.
    \item Specific to the Guatemala PPGR, note that the data points are not evenly spaced, so in calculating a forward difference approximation to the derivative, the denominator is not a constant $h= 10$ as it would be with the United States data.  You have to calculate $h$ from the census year data.
    \end{itemize}
\item Use the Chart editor to add a trendline to your chart for the PPGR data.  Label it with the equation of the line.
\item Add a trendline to your chart for the average of the PPGR data.  Label it with the average value of the PPGR data
\item Add descriptive axes labels and a title for your plot.
\setcounter{QuestionCounter}{\theenumi}
\end{enumerate}

Using forward differences to approximate the derivatives, your charts from Google Sheets should look like those in Figure \ref{fig:LineAverage}.

\begin{figure}[H]
  \begin{center}
    \includegraphics[width=4 in]{USPPGR_line_average.png}
    
\medskip
    \includegraphics[width=4 in]{GuatemalaPPGR_line_average.png}
  \end{center}
  \caption{United States and Guatemala PPGR, Trendline and Average
      \label{fig:LineAverage}}
    \end{figure}

\subsection{CONNECTING THE PPGR GRAPHS AND THE POPULATION MODELS\label{sec:Connect}}

Compare and contrast the results in the graphs from Figure \ref{fig:LineAverage}.  With those graphs in hand,


\begin{enumerate}[1)]
  \setcounter{enumi}\theQuestionCounter
\item Discuss what you see.  Are the pictures in Figure \ref{fig:LineAverage} consistent with Figure \ref{fig:ExpLogisticPPGR} for both the United States and the Guatemala census data?
    \item Use your results to discuss the appropriateness of both the exponential model and the logistic model in modeling the census data for the United States and for Guatemala.
  \setcounter{QuestionCounter}{\theenumi}
\end{enumerate}


\begin{thebibliography}{98}
%\bibitem{R2Paper} Anderson-Sprecher, Richard. 1994.  Model Comparison and $R^2$. {\it The American Statistician}. 48:2, May 1994. pp. 113-117.

%\bibitem{Stat2} Cannon, Ann R., et. al. 2013. {\it STAT2: Building Models for a World of Data}. New York: W. H. Freeman and Company.

  %\bibitem{LifeCourse} Cohan, C. L. and Cole, S. W. 2002. Life course transitions and natural disaster: marriage, birth and divorce following Hurricane Hugo. {\it Journal of Family Psychology}. 16:1, Mar 2002. pp. 14-25.


 \bibitem{Census2020Estimates} Elliott, Diana et al. 2021. Simulating the 2020 Census: Miscounts and the Fairness of Outcomes. \url{https://www.urban.org/research/publication/simulating-2020-census-miscounts-and-fairness-outcomes}. Accessed 22 November 2021

\bibitem{USCensusHist} Gibson, Campbell and Jung, Kay. 2002. {\it Population Division: Historical Census Statistics on Population Totals by Race 1790-1990...}. U.S. Census Bureau, Washington DC 20233. \url{https://www.census.gov/content/dam/Census/library/working-papers/2002/demo/POP-twps0056.pdf}. Accessed 10 December 2021.

\bibitem{Holbrock} Holbrock, M.J. 2016. \textit{Mayan Literacy Reinvention in Guatemala}. Albuquerque, NM: University of New Mexico Press. \url{https://muse.jhu.edu/book/48245}. Accessed 10 December 2021.
 
  
\bibitem{RefractModel} Kingsland, Sharon. 1982. The Refractory Model: The Logistic Curve and the History of Population Ecology.  {\it The Quarterly Review of Biology}. 57:1, Mar 1982. pp. 29-52.

\bibitem{LinhartUSPopSIMIODE} Linhart, Jean Marie. 2017. {\it 1-066-T-USCensusModeling} and {\it 1-066-S-USCensusModeling}.

\bibitem{Loveman} Loveman, M. 2014. \textit{National Colors: Racial Classification and the State in Latin America}. Oxford, UK: Oxford University Press.

\bibitem{McCreery} McCreery, D. 1994. \textit{Rural Guatemala, 1760-1940}. Stanford, CA: Stanford University Press.

\bibitem{MestertonGibbons} Mesterton-Gibbons, Mike. 2007. {\it A Concrete Approach to Mathematical Modeling}. Hoboken, NJ: John Wiley \& Sons, Inc.

  \bibitem{Monaghan} Monaghan, J.D. and Edmonson, B.W. 2000. \textit{Supplement to the Handbook of the Middle American Indians, Volume 6: Ethnology. Handbook of the Middle American Indians}. Austin, TX: University of Texas Press.

\bibitem{Guat2018} National Institute of Statistics (Guatemala), United Nations Population Fund (UNFPA). {\it Guatemala Population and Housing Census 2018.\/} \url{http://ghdx.healthdata.org/record/guatemala-population-and-housing-census-2018}. Accessed 30 September 2021.

\bibitem{Platt} Platt, L. D. 1998. \textit{Census Records for Latin America and the Hispanic United States}. Baltimore, MD: Genealogical Publishing Company.

  
\bibitem{Squier} Squier, E. G. 1858. \textit{The States of Central America: Compromising Chapters on Honduras, San Salvador, Nicaragua, Costa Rica, Guatemala, Belize, the Bay Islands, the Mosquito Shore, and the Honduras Inter-Oceanic Railway}. New York, NY: Harper \& Brothers.

\bibitem{Taeuber} Taeuber, Irene B. 1943. \textit{General Censuses and Vital Statistics in the Americas: An Annotated Bibliography of the Historical Censuses and Current Vital Statistics of the 21 American Republics, the American Sections of the British Commonwealth of Nations, the American Colonies of Denmark, France, and the Netherlands, and the American Territories and Possessions of the United States}. Washington, DC: United States Government Printing Office.
  
\bibitem{USPopData} U. S. Census Bureau. 2002. {\it Measuring America: The Decennial Censuses 1790 to 2000}. U. S. Census Bureau, Washington DC 20233.  \url{https://www.census.gov/prod/2002pubs/pol02marv.pdf}.  Accessed 29 September 2021.

\bibitem{CensusNativeAmericans} U.S. Census Bureau. Censuses of American Indians. \url{https://www.census.gov/history/www/genealogy/decennial_census_records/censuses_of_american_indians.html}. Accessed 22 November 2021

\bibitem{Census2010Estimates} U.S. Census Bureau. 2012. Census Bureau Releases Estiimates of Undercount and Overcount in the 2010 Census. \url{https://www.census.gov/newsroom/releases/archives/2010_census/cb12-95.html}. Accessed 22 November 2021.

\bibitem{USPopData2} U. S. Census Bureau. 2021. {\it Decennial Census of Housing and Population: By Decade}. \url{https://www.census.gov/programs-surveys/decennial-census/decade.html}. Accessed 29 September 2021.
  
%\bibitem{USPopData} U. S. Census Bureau. 2010. {\it Resident Population Data (Text Version)}. U. S. Census Bureau, Washington DC 20233.  Available on the world wide web at \url{https://www.census.gov/2010census/data/apportionment-pop-text.php}
  
\bibitem{GuatWiki} Wikipedia Contributors. 2021. {\it Guatemala}. Wikipedia, The Free Encyclopedia, 29 September 2021. \url{https://en.wikipedia.org/wiki/Guatemala#Demographics}.  Accessed 29 September 2021.

\bibitem{USPopWiki} Wikipedia Contributors. 2021. {\it Demography of the United States}. Wikipedia, The Free Encyclopedia, 21 December 2021. \url{https://en.wikipedia.org/wiki/Demography_of_the_United_States}. Accessed 21 December 2021.


%\bibitem{Gause1934} Gause, G.~F. 1971. {\it The Struggle for Existence}. New York:  Dover Publications, Inc. First published in 1934 by The Williams \& Wilkins Company and available completely on the world wide web at \url{http://www.ggause.com/Contgau.htm}. Accessed  31 March 2008.








  

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%begin teacher section

\teacher{
  \section{NOTES FOR TEACHERS}
  This scenario is meant for use early in a differential equations class or possibly in a mathematical modeling class. We envision it being used to help students connect real data, functions, derivatives, and how derivatives can be used in a modeling context.  We hope it helps students understand why we study differential equations.  Differential equations mean something in the real world and help us model and understand the real world. 

  While you can avoid having your students do the spreadsheet work from Section \ref{sec:PPGR},  we highly recommend it.  Upper level mathematics majors, including secondary teaching candidates, should know the basics of how to use a spreadsheet; this is a basic for employers. We've found that many students do not know how to start and are anxious when asked to do any work with a spreadsheet.  Even if you skip the graph making and simply have them calculate estimates for the derivatives and PPGR from the data, your students will learn a lot.  We think that working in a group, even with one person operating the computer and the others looking on and making helpful comments is a good way to learn.  It is especially helpful to put the person who is least confident in charge of running the computer, and forcing others to help, suggest, explain, and guide.

  We used Google Sheets in writing up this scenario because we found that making the graphs was more intuitive with Sheets than it was with Microsoft Excel, and access to Google Sheets is free.

  It is helpful if students have some familiarity with the least squares regression line to data, either through a linear algebra class or a statistics class.
  
  The idea of featuring the Guatemala population in this modeling scenario came about when co-author Gary Epp was taking a senior level Mathematical Modeling course at Central Washington University.  The class was working on a project based on \cite{LinhartUSPopSIMIODE}. Gary's chosen second population dataset was Guatemala, and this turned out to be a much bigger project for him than what was intended.  The first challenge was gathering the data.  Most students start by finding something that interests them with data on Wikipedia.  Guatemala's census data was not on Wikipedia at the time (we have since added it), and Gary put many hours into searching for information on the Guatemala census.  Of course, after getting the data (which was intended to be a relatively easy task!), he then still had to do all the parameter estimation.  This was not easy either; see Section \ref{sec:Trouble} about the trouble he ran into.  Taking a look at the PPGR for the Guatemala data and getting the least squares regression line  told a clear story of why we were seeing the results we were.  The data really isn't suited for a logistic model which assumes decreasing PPGR.

  If you want a deeper dive into modeling with census data and parameter estimation, explore \cite{LinhartUSPopSIMIODE}.

  \subsection{EXPONENTIAL AND LOGISTIC MODELS}

  We want the students to come away with an understanding that the differential equations for the exponential and the logistic models come from assumptions about how populations change.  We present the logistic model as incorporating proportional births and deaths and two-person interactions. See \cite{MestertonGibbons} for an alternative derivation of the logistic model for plant biomass, and an accessible history is available in \cite{RefractModel}.

  We also want students to realize that these models make simplifying assumptions and have them think about where these assumptions break down. The proportionality constant in the exponential model that takes into account human births and deaths need not be constant in time or location.  For example, the ongoing COVID19 pandemic has substantially increased the death rate in the United States, which would decrease the constant.  Medical and agricultural advances might increase the constant.  Differences in wealth, culture, and lifestyle all influence the birthrate and death rate, and hence the proportionality constant. If you look closely at the United States data in Figure \ref{USAData} you will see a slight dip in the 1940 census caused by declining birthrates during the Great Depression, not deaths from WWII as many surmise. In addition to the factors taken into account in the exponential and logistic models, we also would expect immigration or emigration to affect the populations of the United States and Guatemala.

  Students often evaluate the suitability of a model based on the shape of the function as compared to the shape of the data.  We expect students to identify that both datasets have an exponential shape, but students may disagree based on the bumpiness in the early Guatemala data. This might be a good opportunity for a teachable moment discussing that we would only expect a simple model to match the general trend of data.

  Students might note that the logistic curve becomes horizontal as population increases, a feature we really don't see in either of our two datasets.  Another teachable moment might be to note that since the early part of the logistic curve looks a lot like the exponential model, we can argue that both datasets look to be approximated by a logistic curve. It can be very difficult to make a clean evaluation ``by eye'' which is why we need mathematics to help us with this.

  \subsection{PER CAPITA POPULATION GROWTH RATE}

  Note that PPGR is in units of 1/time, so in the example where we have a population of 500 heterosexual men and 500 women where each woman has one baby over the course of a year, and no one dies, the PPGR is 1/2 and the unit is ``per year''.  This is an extraordinarily high PPGR; note that in Figure \ref{fig:LineAverage} you can see that PPGR never gets higher than about 0.05. We apologize for leaving out our LGBTQ and nonbinary friends in this simplistic scenario.   We gave the forward difference formula \eqref{eq:FwdDif} to approximate the derivative.  We could also approximate the derivative with a a backward difference formula or centered difference formula.  The forward difference formula will not calculate a derivative at the most recent data point, the backward difference formula will miss the earliest data point, and the centered difference formula will miss both the earliest and the most recent data points.  The centered difference formula is usually more accurate than either the forward difference or the backwards difference formula, that said, it isn't a huge difference, and any formula will work for the approximation.  To get PPGR, we simply divide the difference approximation by the population at the given time, and we will miss the same points missed by our difference formula.  Having these formulas thought out and written out will make it easier for students to do spreadsheet work in the next section.  We also hope this exercise helps reemphasize the importance of the definition of the derivative and use the definition to connect approximating the change in the data with calculating the derivative of a function.

  Once PPGR is calculated from the data, the horizontal line representing the exponential PPGR uses one PPGR value to stand in for all of the data. While there are many values you could pick, the average of all the calculated PPGR values is the best representative.  The average value gives you the $y$-coordinate of the horizontal exponential line in Figure \ref{fig:ModelCompare}.  The logistic line can be calculated by linear least squares regression line of PPGR vs. $P$.


  \subsection{USING A SPREADSHEET TO MAKE PPGR CALCULATIONS}

  We used forward difference formulas to calculate the derivatves and the PPGR. Read-only copies of our spreadsheets are available here:
  \begin{itemize}
  \item United States Data: \url{https://tinyurl.com/JMLUSPop}%\url{https://docs.google.com/spreadsheets/d/1MEBvqqi1LicVfvuforZ4LtZPOcqwOcXyuI0fzTuMRt8/edit?usp=sharing}
  \item Guatemala Data: \url{https://tinyurl.com/JMLGuatPop} % \url{https://docs.google.com/spreadsheets/d/1_Z1O_hLi7kjZzeleLPf9VxQdYTZkiisE0LHV1XrSHqc/edit?usp=sharing}
  \end{itemize}

If you are usng a forward difference to approximate $dP/dt$ with the United States data, the denominator is always 10 since the census is taken every 10 years.  A common error when transitioning to the Guatemala data is to keep this same denominator.  The Guatemala census points are not evenly spaced in time, and so the denominator must be calculated as a difference between the next census year and the current one in the forward difference formula. A common error in making the graph is that the $x$-coordinate will either not be specified and so will be an index count of the datapoints, or the $x$-coordinate will default to the census year.  Note that the $x$-coordinate should be the population count.
  
 Changing the difference formula used to calculate the derivative will change these graphs slightly.  The general trend of the data will be similar, and regardless, your students will find that the best fit line to the United States PPGR data is decreasing, whereas the best fit line to the Guatemala PPGR data is increasing.

  We provide nicely labeled final graphs of the PPGR data and trendlines so that if class time is running short, students can either be sent home to reproduce the graphs shown, or students can simply be given the final graphs and asked to consider what these graphs mean for the appropriateness of using the two proposed models with the data. 


  \subsection{CONNECTING THE PPGR GRAPHS AND THE POPULATION MODELS}

  We see in Figure \ref{fig:ExpLogisticPPGR} that the exponential model approximates PPGR with a constant value, and the logistic model approximates PPGR with a line that decreases as PPGR increases.  The average value of the PPGR will always let us approximate the exponential model, but the best fit line doesn't aways approximate a logistic model.  We see this with the Guatemala PPGR data and its trendline, which is increasing. Since we are seeing an increasing trend with the Guatemala PPGR, we shouldn't expect the logistic model to do work well with the Guatemala data.  This is exactly what Gary found out when he tried to do parameter estimation.  

 \subsection{POSSIBLE EXTENSIONS OF THESE IDEAS}
  Note that these PPGR graphs give us parameter estimation for $r$ in the exponential model for both the United States and Guatemala census data, and for the other parameter $P_0$ you can use any data point $(t_0,P_0)$.  We also get parameter estimates for $r$ and $L$ in the logistic model with the United States data, and again take $P_0$ as any data point.  Here we are using the models in the form
  \begin{align} \mbox{Exponential:}\quad P(t) &= P_0e^{r(t-t_0)} \label{eq:ExpSoln2}\\
    \mbox{Logistic:} \quad P(t) &=  \dfrac{L}{1 + \dfrac{(L- P_0)}{P_0}e^{-r(t-t_0)}} \label{eq:LogisticSoln2}
  \end{align}
  where we have applied a horizontal translation to make $t_0$ correspond to $t=0$ in the original solutions, equations \eqref{eq:ExpSoln} and \eqref{eq:LogisticSoln}.

  Sadly, the models with these parameters don't give a particularly satisfying fit to the data.  To get a really good fit, we need nonlinear least squares parameter estimation, which is the topic of my modeling scenario \cite{LinhartUSPopSIMIODE}.

  Another route for further study is to take a look at the model you get from assuming that the PPGR is linear and increasing. This model is called the superexponential model because the growth is faster than exponential. You can replace $L$ with $-L$ in the solution of the logistic differential equation, or use separation of variables and partial fractions to solve

  \begin{equation} \dfrac{dP}{dt} = aP(1+bP) \end{equation}
  where we assume both $a$ and $b$ are positive constants.  Regardless, while the exponential model has the property that $P(t)$ goes to infinity as $t$ goes to infinity, the superexponential model has the property that $P(t)$ goes to infinity at a finite time $t_\infty$ which can be calculated!  The superexponential model, as we might predict from looking at Figure \ref{fig:LineAverage}, provides an excellent fit to the Guatemala census data, better than the exponential model.
  


  }%end teacher
\end{document}

 

This is a set of COMMENTS about the scenario, for the teacher only. It might include a ``solution" or a  number of solution strategies. However, the main narrative should be about how to use the scenario in teaching, e.g., what are prerequisites, how long might one devote to it, what resources for students could be expected to be used, what  issues which cause students difficult or ease should be expected to come up, discussion of technology use, pointers to other files, etc.  This set of COMMENTS might have its own  pointers to datasets, supplementary files,  videos, etc. for student and teacher  use, the material for students could be something the teacher might use with discretion. There could be an extended bibliography \cite{Gause1934,Hutchinson1978a}  for additional teacher resources, analyses,  data, etc.


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\section*{APPENDIX A}

Gary Epp, then an undergraduate at Central Washington University, collected the Guatemala census data you find here as part of a class project in Mathematical Modeling.  Finding this data was not at all a simple task.  There was no one reference that had all the data points for the population of Guatemala at the time (we've since added it to the \href{https://en.wikipedia.org/wiki/Guatemala#Demographics}{Guatemala Wikipedia page}), and finding this information required searching through many references on Guatemala, and piecing this table together one data point at a time.  Table \ref{tab:GuatemalaCite} contains the references we have for the data points in Table and Figure \ref{GuatData}.


\bigskip
\begin{table}[H]
\begin{tabular}{|c|c|c|}
	\hline
	Year & Possible Citations & Citation Used\\
	\hline
	1778 & \cite{Squier} \cite{McCreery} \cite{Platt} & \cite{Squier}\\
	\hline
	1825 & \cite{Squier} \cite{McCreery} & \cite{Squier}\\
	\hline
	1837 & \cite{Squier} & \cite{Squier}\\
	\hline
	1852 & \cite{Squier} & \cite{Squier}\\
	\hline
	1880 & \cite{Taeuber} \cite{McCreery} \cite{Platt} & \cite{Taeuber}\\
	\hline
	1893 & \cite{Taeuber} \cite{McCreery} \cite{Loveman} \cite{Platt} & \cite{Loveman}\\
	\hline
	1914 & \cite{McCreery} & \cite{McCreery}\\
	\hline
	1921 & \cite{Taeuber} \cite{McCreery} \cite{Platt} & \cite{McCreery}\\
	\hline
	1940 & \cite{Taeuber} \cite{McCreery} \cite{Platt} & \cite{McCreery} \cite{Platt} \\
	\hline
	1950 & \cite{McCreery} & \cite{McCreery}\\
	\hline
	1964 & \cite{Monaghan} & \cite{Monaghan}\\
	\hline
	1973 & \cite{Monaghan} & \cite{Monaghan}\\
	\hline
	1981 & \cite{Monaghan} & \cite{Monaghan}\\
	\hline
	1994 & \cite{Monaghan} & \cite{Monaghan}\\
	\hline
	2002 & \cite{Holbrock} & \cite{Holbrock}\\
        \hline
        2018 & \cite{Guat2018} & \cite{Guat2018}\\
	\hline
\end{tabular}
\caption{References for the Guatemala Census Data \label{tab:GuatemalaCite}}
\end{table}

\bigskip
One interesting note, 
``The census records of Guatemala for 1778, 1880, 1893 and 1921 were used as scrap paper and no longer exist, nor were they preserved in any other format except for their statistical information.  The 1940 census was burned.'' \cite{Platt}

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