Statistical analysis sample

2014

Demographic Analysis for Davidson County, TN

2020 POPULATION PROJECTIONS JAMES SASSER UNIVERSITY OF CINCINNATI DR. RAINER VOM HOFE PLAN 7005: PLANNING METHODS 9/19/2014

Table of Contents Introduction .................................................................................................................................................. 1 Simple Growth Methods .............................................................................................................................. 4 Ratio Methods .............................................................................................................................................. 6 Linear Population Model.............................................................................................................................. 8 Geometric Population Model .................................................................................................................... 10 Parabolic Population Model ...................................................................................................................... 11 Logistic Population Model ......................................................................................................................... 12 Female Cohort Model................................................................................................................................. 15 Summary..................................................................................................................................................... 16

List of Figures Figure 1.1 Population Scatter Plot ............................................................................................................... 3 Figure 1.2 Population Pyramid .................................................................................................................... 4 Figure 2.1 AAAC & AAPC Projections ........................................................................................................... 5 Figure 2.2 AAAC & AAPC Projections Graph ................................................................................................ 6 Figure 3.1 State and County Population Data ............................................................................................. 6 Figure 3.2 State and County Population Data-Using Share of Growth Method ..................................... 7 Figure 3.3 State and County Population Data-Using Shift-Share Method .............................................. 8 Figure 3.4 2020 Davidson County Population Projections Using Ratio Methods.................................... 8 Figure 4.1 2020 Davidson County Linear Population Model ................................................................. 9 Figure 5.1 2020 Davidson County Geometric Population Growth ....................................................... 11 Figure 6.1 2020 Davidson County Parabolic Population Growth Model .............................................. 12 Figure 7.1 2020 Davidson County Log Of Reciprocal Difference ......................................................... 13 Figure 7.2 2020 Davidson County Logistic Model Population Projections ........................................... 14 Figure 7.3 2020 Davidson County Decrease in Population Growth Per Year........................................ 14 Figure 8.1 2020 Davidson County Total Projected Female Pop. (2015) ............................................... 15 Figure 8.1 2020 Davidson County Total Projected Female Pop. (2015) ............................................... 16

1. Introduction As planners it is our job to provide for future generations. We strive to make resources available today even more accessible in the future. In order to do this, we must make projections on how many people we will be planning for. To plan for the future, we must study the past and present. Population projections are a pivotal element of comprehensive plans and overall growth. According to Research Methods in Urban and Regional Planning underestimating can lead to shortages and reduction in quality of life, while overestimating can result in wasting local resources through “costly oversupply of services� (Vam Hofe, 2007).

Population projections affect virtually every aspect of urban planning from land use to transportation. By analyzing the past and present population we can make informed political decisions based on what we calculate for the future. There are various methods for calculating our projections. This report will deal with the extrapolation method which observes historical trends and project them into the future.

The different approaches under the extrapolation umbrella I will use are based on different mathematical methods of finding the best fit for the observed data.

I chose to project the 2020 population of Davidson County, Tennessee. Davidson County is home to the famous city of Nashville, TN and along with the southeast US has experienced explosive growth in the past decade.

According to the US Census Bureau, the

“music city” is the second largest city in Tennessee behind Memphis and the 5th largest in the southeast region. Based on our calculations I see no reason why the city’s population will not continue to grow.

Before we begin to apply population projection methods we must first look at our population noticing things like size, percent changes and gender/age profiles. Using data that is readily available we must gain a thorough understanding of our chosen population. By looking at available population data we can make general statements, such as whether the population is increasing or declining.

Davidson County-TN Population Scatter Plot Size of Population

640,000 620,000 600,000 580,000 560,000 1998

2000

2002

2004

2006

2008

2010

2012

Year

Figure 1.1 Population Scatter Plot

As you can see in Figure 1.1, our population is quite large to begin with, given that it is a major metropolitan area. You will also observe that Davidson County has experienced a steady growth of residents in the past 10 years. The scatter plot shows a positive trend with population growing incrementally each year with the exception of our last year of data (2010). If you fit this scatter plot with a trend line it is clear that our population is generally increasing.

I then looked at the gender ration in age cohorts within our county. In order to do this I collected demographic data from the Census Bureau’s American Fact Finder and created a population pyramid graph. This graph uses age cohorts in 5 year intervals (with the exception of the 85+ cohort) to accurately reflect the data, resulting in a double histogram of the sex-age structure in Figure 1.2. There are a total of

303,540 males and 323,141 females in our population, yielding a ratio of 1.06. This means that for every 100 males, there are approximately 106 females. The population pyramid bulges around the 25-29 age cohort, meaning that largest proportion of the population in Davidson County is in their twenties and thirties.

Age

Davidson County-TN Population Pyramid 85+ 80 - 84 75 - 79 70 - 74 65 - 69 60 - 64 55 - 59 50 - 54 45 - 49 40 - 44 35 - 39 30 - 34 25 - 29 20 - 24 15 - 19 10 - 14 5-9 0-4

Male Female

40000 30000 20000 10000

0

10000 20000 30000 40000

Population

Figure 1.2 Population Pyramid

2. Simple Growth Methods Now that we have a basic understanding of our population we can begin calculating projections. The first method I will utilize is using simple growth methods to calculate our future population including: total absolute change, total percent change, average annual absolute change (AAAC) and average annual percent change (AAPC). These methods are appealing to municipalities when projects have low data requirements, low costs, and easy require easy application. By assuming that trends continue we can observe past population trends and project them into the future. In order to calculate total absolute change I subtracted our 2000 population (base year) from our 2010 population (most recent year). I then calculate the AAAC by taking that value and dividing it by the number of years in our data set, in this case we would divide by 10. The equation can be seen as:

(Pt+n – Pt)

(Pt+n – Pt) / n = (P2010 – P2000) / 10

= (P2010 – P2000)

= 57,637 / 10 = (627,957 – 570,320)

AAAC= 5,764

Absolute Change = 57,637 The data is saying that from 2000 to 2010 the population has increased by a total of 57,637 people. Because it is a single aggregated data series and based on population statistics at two points in time we can say that the population has grown by 5,764 people each year. Using the same methodology we can calculate the total percent change and AAPC. We subtract our base year from our most recent year and divide that value by our base year. This gives us a percentage which in this case equates to 10.11%.

To find the AAPC we apply the geometric growth formula and solve for the growth rate. The equation can be seen as: Average annual percent change (AAPC)*:

Percent change: [(Pt+n – Pt) / Pt] x 100

(Pt+n / Pt)1/n - 1 = [(P2010 – P2000) / P2000] x 100

= (P2010 / P2000)1/10 - 1 = (627,957/ 570,320)0.1 - 1

= [627,957 / 570,320] x 100

= 0.0097 = 10.11%

= 0.97%

The data is saying that from 2000 to 2010 the population has increased by a total of 10.11, and a total of .97% each year. Assuming this trend will continue we can use these simple growth methods to project future populations.

Year 2020

AAAC Projection 685,594

AAPC Projection 700,737

Figure 2.1 AAAC & AAPC 2020 Projections

Population

Davidson County-TN Simple Growth Method2020 Projections 710,000 700,000 690,000 680,000 670,000 660,000 650,000 640,000 630,000 2010

2012

2014

2016

2018

2020

2022

Year AAAC

AAPC

Figure 2.2 AAAC & AAPC 2020 Projections Graph

By using these two simple growth methods I can project our population as far into the future as I would like. Our AAAC projection is smaller than our AAPC projection. The reason why our AAAC projection is lower is because the population declined in 2010. The AAPC isn’t just looking at the last data point, but all of the data points, therefore is not significantly affected by the population decline in 2010.

3. Ratio Methods The ratio methods I have employed in this research consist of the share of growth and shift-share methods. These two models are very popular amongst planners mainly because they are among the simplest extrapolation methods.

2000 (by) 2010 (ly) 2020 (ty)

Tennessee (n)

Davidson (m)

5,703,203 6,346,105 6,860,231

569,889 627,957

Figure 3.1 State and County Population Data

Here we have our data for the state as well as our county. Using the projections for the state we can determine what amount of growth will be shared in Davidson County.

The Share of growth method observes the smaller areas share of population growth for a past time period- the base period. “Assuming that this observed share of growth remains constant and knowing the larger areas projected population for the future target year, we can project the smaller areas future population” (Vom Hofe, 2007). In other words the share of growth method dictates how much a smaller region accounts for the total larger regions share of growth. We add that share of growth to the base year for the smaller county and project the growth of the smaller region. This equation can be seen as:

Pop m,ty

 Pop m,ly  Pop m,by   Pop m,ly    Pop n,ty  Pop n,ly   Pop n,ly  Pop n,by  

Using the share of growth equation we are able to project the 2020 population for Davidson County, TN.

2000 (by) 2010 (ly) 2020 (ty)

Tennesse (n)

Davidson (m)

5,703,203 6,346,105 6,860,231

569,889 627,957 674,394

Figure 3.2 State and County Population Data-Using Share of Growth Method

Rather than using shares of growth, the shift-share method uses the smaller areas share of total population in the base year and in the launch year. It than uses the larger regions target year data in order to project the smaller regions target year population by employing this equation.

Popm,ty

 Popm,ly   years pp  Popm,ly Popm,by      Popn,ty         Popn,ly   yearsbp  Popn,ly Popn,by 

Using the shift-share equation we are able to project the 2020 population for Davidson County, TN.

2000 (by) 2010 (ly) 2020 (ty)

Tennessee (n)

Davidson (m)

5,703,203 6,346,105 6,860,231

569,889 627,957 672,157

Figure 3.3 State and County Population Data-Using Shift-Share Method

As you can see, both methods are similar in result. Ratio methods like share of growth and shiftshare are considered to be very accurate in determining population projections. It is important to note that a good knowledge of the smaller and larger areas population trends from past and present will be useful in interpreting projections. Ratio methods are useful in making sure your results are reasonable. By looking at base years and determining a steady growth or decline, reasonable results are clear.

"Davidson County�

Share of Growth:

674,394

2020

Shift-Share:

672,157

2020

Figure 3.4 2020 Davidson County Population Projections Using Ratio Methods

4. Linear Model In the Linear Population Growth Model, it is assumed that the population will grow by the same amount of people each year. The projected change from year to year is calculated by fitting the data from 2000 to 2010 with a trend line to capture the average population trend. Using the linear equation of the

trend line seen below, the future populations can be calculated, where alpha is the y-intercept, beta is the slope (or in this case, the annual population change) and Tn is the index number for any given year, ‘n.’

Popn = α + β(Tn)

Popn = 564,303 + 6725.2(Tn)

The trend line for the Davidson County population data can be seen in Figure 4.1 below.

DAVIDSON COUNTY, TN LINEAR POPULATION MODEL 725,000

POPULATION

700,000 675,000 650,000 625,000 600,000 575,000 550,000 2000

2005

2010

2015

2020

YEAR

Figure 4.1 2020 Davidson County Linear Population Model

The trend line produces a beta of 6725.2, which means the population in the linear model is projected to grow by about 6725 people each year. Using the formula above with the alpha and beta from the trend line, I can project what the population would be in 2020. The index number for year 2020 is 21.

Popn = 564,303 + 6725.2(Tn) = 564,303 + 6725.2(21) = 705,532 This simple calculation yield gives us a projected population for Davidson County of 705,532 people in 2020.

5. Geometric Model The Geometric Population Model is another form of extrapolating data using a trend line, similar to the linear model. However in this method, the population is assumed to grow at a constant rate as opposed to a constant amount of people. This assumption gives us an exponential trend line when the projection is graphed. The formula for this exponential relationship is given below.

Popn = ι * βTn

Popn = 565,197 * (1.0111)Tn

To determine the alpha and beta for our data, I took the log of the population of Davidson County for each year, graphed it on a scatter plot, and fit a linear trend line to it. I then extracted the beta and alpha from the equation of this trend line and took the antilog of these two values. The result was a beta of 1.0111, meaning the annual growth rate for Davidson County is projected to be 1.11%. Using the alpha and beta in the above equation, I was then able to project the population by plugging in the index for the next ten years. To demonstrate, I will show the calculation for the year 2020.

Popn = 565,197 * (1.0111)Tn = 565,197 * (1.0111)21 = 712,853 Using the Geometric Population model, the projected population for Davidson County, TN in 2020 is 712,853. The projection results for years 2011 through 2020 can be seen as a graph in Figure 4.1 below.

POPULATION

DAVIDSON COUNTY, TN GEOMETRIC POPULATION GROWTH 725,000 700,000 675,000 650,000 625,000 600,000 575,000 550,000 2000

2005

2010

2015

2020

YEAR

Figure 5.1 2020 Davidson County Geometric Population Growth

This projection might be preferred over others if planners believe Davidson County will experience more rapid growth for the foreseeable future. This model predicts a constant annual growth rate, but the population being added would increase each year without slowing.

6. Parabolic Population Growth In the Parabolic Population Growth model, population is not expected to follow a linear nor exponential growth pattern. Instead, the growth rate changes over time, while in the Geometric Model growth is assumed to be constant. The model involves a linear and non-linear component to allow for this flexibility. The formula used is seen below.

Popn = α + β1Tn + β2Tn2 To calculate the alpha and the two betas, I used a website provided by Rainer vom Hofe to run a regression analysis of the data - www.xuru.org/rt/MLR.asp#CopyPaste. The regression uses three parameters (index year, index year squared, and the population data) in order to determine the best fit parabolic equation. From this equation I extracted the alpha, beta1 and beta2, seen below. I then used the index year 21 in order to calculate the projected population in 2020.

= 558668.6545 + 9325.555245(Tn) – 216.6993007(Tn)2 = 558668.6545 + 9325.555245(21) – 216.6993007(21)2 = 658,941 In figure 6.1 below, I have graphed the projected population for the next 10 years using the same equation from above. This graph again shows the population growth slowing down as the years progress.

Davidson County, TN Parabolic Population Growth Model 660,000

Population

655,000 650,000 645,000 640,000 635,000 2010

2012

2014

2016

2018

2020

2022

Year

Figure 6.1 2020 Davidson County Parabolic Population Growth Model

7.

Logistic Population Model This model is the only model out of our methods that requires predetermining an upper

population boundary. Choosing a population “ceiling” is arbitrary, however, choose wisely because it can largely affect your data and therefore projection. With the population declining in our most recent year (2010) I decided to be modest when choosing a population ceiling. I felt 750,000 would be a good cap to an existing 630,000 or so population. The logistic population model is especially effective in populations that experience rapid fluctuations in change. The logistic population model can be written as:

In our case this will look like:

750,000/(1+.00000046*750,000*.94406)^20 Now we will take it step by step. This applies an S shaped logistic curve. In order to transform the logistic curve to the linear form I will use the equation on the bottom of the above image. Using logarithms allows us to use regression analysis in order to calculate our population projection. After we determine our reciprocal population value by dividing our population for that year by 1 we can determine our reciprocal difference which can be calculated by subtracting that number from 1 over our population ceiling.

Log of Reciprocal Difference

Davidson County- TN Log of Reciprocal Difference -6.3000 -6.4000 -6.5000 -6.6000 -6.7000

y = -0.025x - 6.3487 Linear‌ 0

5

10

15

Years

Figure 7.1 2020 Davidson County Log Of Reciprocal Difference

I then go on to take logarithms of that value for each year. After writing our equation for our best fit trend line we now have our beta (-.025) and alpha (-6.3487). Our next step is to take the ant-log of that

alpha and beta. Finally we are ready make projections. We than put all our values into the formula and can project into any year.

Population

Davidson County TN Logistic Model Population Projections 685,000 680,000 675,000 670,000 665,000 660,000 655,000 650,000 645,000 640,000 635,000 2010

2012

2014

2016

2018

2020

2022

Year

Figure 7.2 2020 Davidson County Logistic Model Population Projections

As you can see our population begins to slow in growth as the year’s progress. The population declines by a certain percentage in growth each year show in figure 3.5.

Davidson County Tn Decrease in Population Growth Per Year 6,000

Population

5,000 4,000 3,000 2,000 1,000 0 2010

2012

2014

2016 Year

2018

2020

2022

Figure 7.3 2020 Davidson County Decrease in Population Growth Per Year

If the logistic model is accurate, the population projection should fall somewhere in between the projections of the parabolic and geometric models. The logistic model is a good median between the previous two models. After determining our population ceiling, establishing our beta and alpha, and finding our anti-logs we can employ the previous equation to project our population into the next ten years and determine our Davidson County 2020 projection using the logistic model. The 2020 projection using the logistic population model calculated to 680,369 people.

8. Female Cohort Module Using a demographic balancing equation for Galltain County, KY I can project the project population based on a specific cohort. You can do this for any population composition based on age, sex or ethnicity. Here I used 5 year intervals to show the projected population for Gallatain County, KY in 2015. For this set of data I was able to calculate a fertility rate of women in Gallatin County of 1.811. By including survival, net migration, fertility and adjusted fertility rates we can predict each population by cohort, and by adding those values project a total projected female population. Fertility and mortality are critical factors in the change of the population.

Male Female

Live Births Survival

Projected

2010 - 2015

by Sex*

Rates 5sr2010

Population Child DeathsAge in 2015

by Sex 10 to 15

273

0.9344

255

18

0-4

Total Projected Female Pop. F2015 4,514

Figure 8.1 2020 Davidson County Total Projected Female Pop. (2015)

9. Summary After reviewing the various population projections I used the logistic model in our forecast for Davidson County, TN. I believe the logistic model to be the most accurate in projecting the population for 2020. Due to the decline in population in our most recent year (2010) I believe that the population growth is decelerating. Although the county continues to grow each year, I have selected a model that shows a less dramatic increase. The logistic model incorporates the “high” and “low” of our projections to display a trend that levels in the later years. I believed a moderate projection to be most accurate. Among the various projection models the logistic model is not considered to be an outlier because it is in the middle.

While the county continues to grow, our projections show that it won’t necessarily grow at a rate seen in previous years.

Year (n)

AAAC

AAPC

2011

633,721

2012

Share of Growth

Linear

Geometric

Parabolic

Logistic

635,235

645,005

645,357

639,371

640,033

639,484

642,513

651,730

652,529

643,279

645,326

2013

645,248

649,791

658,455

659,781

646,753

650,403

2014

651,012

657,069

665,180

667,114

649,794

655,271

2015

656,776

664,347

671,905

674,528

652,402

659,934

2016

662,539

671,625

678,631

682,025

654,577

664,397

2017

668,303

678,903

685,356

689,604

656,318

668,666

2018

674,067

686,181

692,081

697,268

657,626

672,747

2019

679,830

693,459

698,806

705,018

658,500

676,646

2020

685,594

700,737

705,531

712,853

658,941

680,369

674,394

Shift-Share

672,157

Figure 8.1 2020 Davidson County Total Projected Female Pop. (2015)

Works Cited Wang X & Rainer Vom Hofe (2007). Research methods in urban and regional planning. Springer-Verlag Berlin and Heidelberg GmbH & Co. KG; 2008-09-02.