AUSTR ALIAN POPULATI ON STUDIES 2017 | Volume 1 | Issue 1 | pages 26–40 © Raymer, Biddle and Guan 2017. Published under the Creative Commons Attribution-NonCommercial licence 3.0 Australia (CC BY-NC 3.0 AU). Journal website: www.australianpopulationstudies.org A multiregional sources of growth model for school enrolment projections James Raymer* The Australian National University Nicholas Biddle The Australian National University Qing Guan The Australian National University *Corresponding author. Email: james.raymer@anu.edu.au. Address: School of Demography, ANU College of Arts and Social Sciences, The Australian National University, 9 Fellows Road, Acton ACT 2601, Australia Paper received 9 May 2017; accepted 17 July 2017; published 20 November 2017 Abstract Background Education planning requires accurate and efficient projection models. Current projection models either do not make use of all available information and are reliant on idiosyncratic expert judgement, or are too complex to be maintained and explained. Aims To test whether a multiregional projection model performs better than current methodology in explaining and projecting school enrolments in a school system with student mobility. Data and methods A multiregional cohort model was developed for projecting enrolments for multiple schools or districts simultaneously. For illustration, data were obtained for all government schools in the Australian Capital Territory (ACT) for the years 2008–2016. Multiregional projections were compared with a cohort transition model and the ACT Education Directorate’s own projections. Results (i) There is great diversity in the sources of school enrolment growth that need to be accommodated in enrolment projections; and (ii) multiregional projections perform slightly better than traditional methods with less effort and more transparency. Conclusion A sources of growth approach guides the understanding of enrolment change, which is critical for making informed projections. Key words Enrolment projections; cohort transition; multiregional demography; Australian Capital Territory; Australia. http://www.australianpopulationstudies.org/ mailto:james.raymer@anu.edu.au Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 27 1. Introduction School districts require information about the expectation of future school enrolments so that they can plan the supply of classrooms, teachers, materials and facilities. School enrolments are also important for understanding the education system within a larger social and economic context (Stone 1965). Without accurate projections, education may be disrupted because of inadequate facilities or a lack of teachers or other resources. Alternatively, education may be over-budgeted in the provision of teachers, infrastructure and resources, thus causing a misallocation of public funds. While projections are never entirely certain due to unobserved factors influencing student mobility decisions (Morrison 2000), there are a variety of demographic tools that can be applied to increase projection certainty and help better judgements to be made. Surprisingly, in our review we find that most education jurisdictions neglect investments in testing or improving projection methodology and instead rely on relatively crude projection models with ad-hoc assumptions built in. In this paper, we assess a method to project future enrolments for multiple schools or school districts simultaneously. Our method is illustrated with data obtained for the Australian Capital Territory (ACT) from the ACT Education Directorate for the period 2008–2016. Our aim is to provide better understanding of the demographic drivers enrolment change, and to develop more efficient projection systems. The multiregional projections are designed to study transitional changes that have occurred over time across school levels, as well as between schools and school regions. 2. Review of methods used to project school enrolments Sweeney and Middleton (2005, p. 366) list several distinct approaches used to project school enrolments. These include cohort survival and regression methods that use past enrolment data and birth records, as well as extended versions of these methods which incorporate other information such as migration and student drop-out rates. They also list the modifiable spatial filter method (Rushton, Armstrong and Lolonis 1995), which combines cohort survival methods with residential address data to determine the size of future school attendance areas. These methods can incorporate birth records, past and future housing unit data and the number of students each housing type yields. Webster (1971) provides a similar list of projection approaches but also includes those based directly on age-specific population counts. Johnstone (1974) further distinguishes between models that are deterministic (including Markov Chain models), stochastic and constrainable. One key issue is whether projections should be focused at the school level or at the broader district level. This also applies to projections for subgroups in heterogeneous populations (Grip 2009). A bottom-up (school level) approach can be tailored to fit each school’s composition and needs; however, this approach may not consider wider demographic changes in the region around the school. A top-down (district level) approach, on the other hand, is often considered pragmatic because projections for large areas tend to be more accurate and less prone to rapid population changes (Berk & Associates 2008; Grip 2009; Schellenberg and Stephens 1987, p. 13; Stronge and Schultz 1981; Swanson and Tayman 2012, p. 281). It can also incorporate wider demographic changes which are occurring in the region of interest. The main disadvantage is that a top-down approach may produce unrealistic results for schools which have trends or student profiles that are substantially different from other schools within their district. 28 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 In order for projections to be understood and utilised, the methodology and assumptions need to be transparent and replicable. The projection approach must also be flexible and able to be adapted to a variety of situations and changes that occur to enrolment patterns if the time-consuming process of developing specific projection models for each school or school district is to be avoided (Gould 1993; Johnstone 1974; Stone 1965). This includes schools in established neighbourhoods, as well as those in areas that are currently being developed or planned. At times, it may be necessary to incorporate expert judgements or scenarios to account for future change possibilities (Morrison 2000). Furthermore, the projection methodology should be regularly assessed in terms of accuracy (how well it predicts the truth) and efficiency (amount of time and effort required to produce results) (Swanson and Tayman 2012). In most developed countries, including Australia, there tends to be very high rates of grade progression (close to 100 per cent) in most schools. This implies the expectation that, for example, nine years from now nearly all students enrolled in academic level 1 today will be enrolled somewhere in level 10. Moreover, there tends to be high retention of students at particular schools. The exception, of course, is for students transitioning between primary and secondary school or, in jurisdictions such as the ACT, from a government high school (years 7 to 10) to a secondary college (years 11 and 12). Current bulges or dips in student numbers can be expected to move through the academic levels over time. If primary school enrolments are lower in one year, secondary or high school enrolments will follow at predictable times in the future. Projections for new schools require a relational model where auxiliary information such as births, population projections, household composition and anticipated migration is used to estimate the long-term stable size and composition of future student enrolments. There are other situations where auxiliary information is also needed. For example, newly developed areas in middle or outer growth precincts may be expected to first attract young families with preschool and primary school aged children due to the relative affordability of land or housing for newly established household units. Over time, these children will progress through the school system, with the school eventually reaching a steady state in respect to the number of new and continuing enrolments (Herrick 1952). One approach which has not been considered widely in projecting future education requirements is a multiregional or multistate population model, which is an extension of the cohort transition model. Multiregional population models provide a general and flexible platform for modelling and analysing subnational population change over time (Rogers 1975, 1995; Land and Rogers 1982; Schoen 1988). They enable the combination of all the main components of population change by age with various transitions that each population group may experience over the life course. These transitions may be between academic levels (including primary to secondary or high school or high school to a secondary college) or between government and non-government schools. Despite the many theoretical and analytical advantages, multiregional models have been relatively unexplored because of the large amount of input data needed and requirement for matrix calculations to perform the projections and analyses. One exception is Sweeney and Middleton (2005), who applied a multiregional cohort enrolment projection method to better understand intra- district school mobility and evaluate existing enrolment projections in Santa Barbara, California. Their aim was to understand intra-district school transfers in an open enrolment system for a heterogeneous population. In this paper, we build on their ideas to improve school enrolment projections in the ACT. Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 29 3. Analysing the sources of enrolment change The population of the ACT has grown substantially over the past decade and a half from 322,000 in 2001 to 398,000 in 2016 (ACT Government 2017). This growth has been driven both by natural increase and net internal migration. Net internal migration contributed only a small amount despite large flows as levels of in-migration tended to offset out-migration. The ACT’s average annual population growth rate between 2001 and 2014 was 1.4 per cent with a peak of 2.2 per cent occurring in the 2006–2007 period. The ACT has seven population regions: Belconnen, Gungahlin, North Canberra, South Canberra, Weston Creek, Woden Valley and Tuggeranong.1 The relatively new region of Gungahlin, located in Canberra’s north, experienced the most rapid growth during this time, increasing from 25,000 in 2001 to 62,000 in 2014 with an average annual population growth rate of 7.4 per cent. In Australia, like in many other federal systems, school planning is undertaken at the state or territory level. The ACT is considered a geographically small system by Australian standards with four main levels of public school-based education: • preschool – the level before full-time schooling, denoted as level P • primary school – the first seven years of full-time schooling, referred to as level K (Kindergarten) to level 6 • high school – the next four years of full-time schooling, or levels 7 to 10 • secondary college – the last two years of full-time schooling, or levels 11 and 12. The percentage of students in the ACT attending a government school increased from 57.5 per cent in 20082 to 59.6 per cent in 20163. The remainder were in the non-government school sector, with transitions between both systems as documented in this paper. The ACT Education Directorate is responsible for 88 public schools in the ACT comprising six early childhood schools (i.e., preschool to level 2), 51 primary schools (some with preschools attached), 10 high schools, nine combined primary/high schools, eight secondary colleges and four specialist schools. 4 Record-level data from 2008 to 2016 taken at each year’s census in February were provided for this study. The data represents 611,674 student observations and contains the following variables: student ID (anonymised); census date; school; academic level (levels P–12); suburb; state. From these data, the transitions between academic levels were calculated, as well as movements amongst schools for all students in the ACT. Following Rees and Willekens (1986), data were obtained that could be used to identify the main sources of enrolment change. This accounting framework is outlined in Table 1. The variable gij represents internal migration events from one school region i to another school region j. Internal migration events are excluded when i = j. Instead, terms ri are entered, which are accounting balances that capture the result of subtracting all possible exit events from the starting enrolment population. 1 Note, for our analyses, the Canberra Central region is divided into North Canberra and South Canberra. A map of the schools and regions is available from http://www.education.act.gov.au/__data/assets/pdf_file/0005/73319/160210-ACT- Public-Schools-Map-2016.pdf 2 http://www.abs.gov.au/AUSSTATS/abs@.nsf/allprimarymainfeatures/FFBE2CE6D8296D21CA2576EA0011F617?opendocument 3 http://www.abs.gov.au/AUSSTATS/abs@.nsf/DetailsPage/4221.02016?OpenDocument 4 http://www.education.act.gov.au/publications_and_policies/publications_a-z/annual_report/annual-report-2015- 2016/section-b/b.1-organisational-overview. Specialist schools are excluded from this study. http://www.education.act.gov.au/__data/assets/pdf_file/0005/73319/160210-ACT-Public-Schools-Map-2016.pdf http://www.education.act.gov.au/__data/assets/pdf_file/0005/73319/160210-ACT-Public-Schools-Map-2016.pdf http://www.abs.gov.au/AUSSTATS/abs@.nsf/allprimarymainfeatures/FFBE2CE6D8296D21CA2576EA0011F617?opendocument http://www.abs.gov.au/AUSSTATS/abs@.nsf/DetailsPage/4221.02016?OpenDocument http://www.education.act.gov.au/publications_and_policies/publications_a-z/annual_report/annual-report-2015-2016/section-b/b.1-organisational-overview http://www.education.act.gov.au/publications_and_policies/publications_a-z/annual_report/annual-report-2015-2016/section-b/b.1-organisational-overview 30 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 This number represents students who transition to the next academic level while remaining in the school region. Total internal out-migration from each school region are denoted by gi+ and total internal in-migration to each school region are denoted by g+j . The ii variable signifies the number of migration events from outside the system of interest and oi tabulates the corresponding number of persons leaving the system of interest. m i and ni denote transfers from and to non-government schools, respectively. The graduation events, x i, and preschool, bi, complete the flows in the table. Note, in this table, persons who leave the school system without graduating are included in oi. Table 1: A sources of growth accounting framework for government school enrolments across regions Destination Region Region (t+1) (t) 1 2 3 4 5 6 7 NG MIG GRAD Total 1 r1 g 12 g 13 g 14 g 15 g 16 g 17 m 1 o 1 x 1 te1 2 g 21 r2 g 23 g 24 g 25 g 26 g 27 m 2 o 2 x 2 te2 3 g 31 g 32 r3 g 34 g 35 g 36 g 37 m 3 o 3 x 3 te3 4 g 41 g 42 g 43 r4 g 45 g 46 g 47 m 4 o 4 x 4 te4 5 g 51 g 52 g 53 g 54 r5 g 56 g 57 m 5 o 5 x 5 te5 6 g 61 g 62 g 63 g 64 g 65 r6 g 67 m 6 o 6 x 6 te6 7 g 71 g 72 g 73 g 74 g 75 g 76 r7 m 7 o 7 x 7 te7 P b 1 b 2 b 3 b 4 b 5 b 6 b 7 0 0 0 b + NG n 1 n 2 n 3 n 4 n 5 n 6 n 7 0 0 0 n + MIG i 1 i 2 i 3 i 4 i 5 i 6 i 7 0 0 0 i + Total 11 +te 12 +te 13 +te 14 +te 15 +te 16 +te 17 +te m + o + x + Notes: (1) Definitions of variables and subscripts: e = school enrolment population; r = balancing terms; g = internal migrations (within the ACT); P (b) = preschool enrolments; NG (m, n) = transfer to/from non-government schools; MIG (o, i) = migration to/from other states within Australia or overseas; GRAD (x) = exits due to graduation from government schools; 0 = structural zeroes; t = time; i = subscript for region; and + = summation over regions. (2) Assumes zero deaths. The sum of the numbers in the rows of Table 1 add up to the enrolments at the beginning of the time interval, tie . The balancing term is obtained by subtracting the total number of migrations to other regions within the ACT, transfers to non-government schools, migrations to places outside the ACT, and graduations from the population at the beginning of the time interval, i.e., 𝑟𝑟𝑖𝑖 = 𝑒𝑒𝑖𝑖 𝑡𝑡 − 𝑔𝑔𝑖𝑖+ − 𝑜𝑜𝑖𝑖 − 𝑚𝑚𝑖𝑖 − 𝑥𝑥𝑖𝑖 Similarly, the variables in the columns of Table 1 add up to the enrolments at the end of the time interval. We can compute these by adding to the balancing term the total in-migrations, migrations from outside the ACT, transfers from non-government schools and preschool entrants, i.e., 𝑒𝑒𝑖𝑖 𝑡𝑡+1 = 𝑟𝑟𝑖𝑖 + 𝑔𝑔+𝑖𝑖 + 𝑖𝑖𝑖𝑖 + 𝑛𝑛𝑖𝑖 + 𝑏𝑏𝑖𝑖 If we combine these two equations, the balancing term cancels out and we obtain the components of school enrolment change: 𝑒𝑒𝑖𝑖 𝑡𝑡+1 = 𝑒𝑒𝑖𝑖 𝑡𝑡 − 𝑔𝑔𝑖𝑖+ − 𝑜𝑜𝑖𝑖 − 𝑚𝑚𝑖𝑖 − 𝑥𝑥𝑖𝑖 + 𝑔𝑔+𝑖𝑖 + 𝑖𝑖𝑖𝑖 + 𝑛𝑛𝑖𝑖 + 𝑏𝑏𝑖𝑖 Thus, the information described in Table 1 provides the basis for understanding the sources of enrolment change from one year to the next at the region level. Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 31 Consider Table 2, which contains the sources of growth and cohort transitions occurring across school regions in the ACT for levels P–11 in 2015 to levels K–12 in 2016. The row sums equal the P–11 enrolments for schools in each of the seven ACT regions. The column sums equal the corresponding K–12 enrolments one year later. Between 2015 and 2016, 72,694 students made a transition. This included 5,393 students who moved out of the ACT and 6,387 students who moved to the ACT. Over 80 per cent of the transitions occurred within the seven regions, with non-government schools having the highest retention. The proportion of students leaving government schools between 2015 and 2016 in the ACT ranged from 5.6 per cent in Weston Creek to 10.8 per cent in Woden Valley. In-migration was greatest to non-government schools (n=2,142), followed by Gungahlin (n=823), Belconnen (n=788) and North Canberra (n=696). Table 2: Sources of enrolment change and cohort transition proportions for school regions in the ACT: levels P–11 (2015) to levels K–12 (2016) Region Enrolment Region 2016 2015 GUN BEL SCAN TUGG WOD WEST NCAN NG MIG Total Enrolment change GUN 5,635 84 21 8 9 6 55 332 628 6,778 BEL 124 8,582 27 18 20 12 124 331 722 9,960 SCAN 12 17 3,455 32 145 3 14 129 392 4,199 TUGG 33 32 22 6,010 146 38 32 346 706 7,365 WOD 14 11 47 98 3,177 21 132 152 441 4,093 WEST 8 4 10 33 121 1,612 22 77 112 1,999 NCAN 56 88 89 21 19 2 4,319 144 558 5,296 NG 190 133 140 214 125 30 128 23,823 1,834 26,617 MIG 823 788 576 526 609 169 754 2,142 0 6,387 Total 6,895 9,739 4,387 6,960 4,371 1,893 5,580 27,476 5,393 72,694 Transition proportions GUN 0.831 0.012 0.003 0.001 0.001 0.001 0.008 0.049 0.093 1.000 BEL 0.012 0.862 0.003 0.002 0.002 0.001 0.012 0.033 0.072 1.000 SCAN 0.003 0.004 0.823 0.008 0.035 0.001 0.003 0.031 0.093 1.000 TUGG 0.004 0.004 0.003 0.816 0.020 0.005 0.004 0.047 0.096 1.000 WOD 0.003 0.003 0.011 0.024 0.776 0.005 0.032 0.037 0.108 1.000 WEST 0.004 0.002 0.005 0.017 0.061 0.806 0.011 0.039 0.056 1.000 NCAN 0.011 0.017 0.017 0.004 0.004 0.000 0.816 0.027 0.105 1.000 NG 0.007 0.005 0.005 0.008 0.005 0.001 0.005 0.895 0.069 1.000 MIG 823 788 576 526 609 169 754 2,142 0 6,387 Notes: GUN = Gungahlin; BEL = Belconnen; SCAN = South Canberra; TUGG = Tuggeranong; WOD = Woden Valley; WEST = Weston Creek; NCAN = North Canberra; NG = ACT non-government schools; MIG = migration to/from other states within Australia or overseas. Next, consider the cohort transitions presented in Table 3 (next page) from level 10 to level 11. The diagonal elements represent those who remained in schools within the region and progressed from level 10 to level 11. Most regions retained the majority of their students in the transition from high school to secondary college, except Weston Creek (0 per cent, where no secondary college is available) and Woden Valley (46%). Tuggeranong had the highest retention (83%). The data clearly show that substantial movements occurred across school regions in the transition from high school to secondary college. 32 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 Table 3: Cohort transition proportions for school regions in the ACT: level 10 (2015) to level 11 (2016) Region Enrolment Region 2016 2015 GUN BEL SCAN TUGG WOD WEST NCAN NG MIG Total GUN 0.807 0.048 0.013 0.000 0.003 0.000 0.033 0.008 0.090 1.000 BEL 0.056 0.789 0.016 0.004 0.007 0.000 0.047 0.005 0.074 1.000 SCAN 0.010 0.005 0.829 0.025 0.005 0.000 0.005 0.020 0.101 1.000 TUGG 0.009 0.009 0.018 0.826 0.055 0.000 0.000 0.005 0.078 1.000 WOD 0.006 0.006 0.091 0.360 0.457 0.000 0.000 0.000 0.080 1.000 WEST 0.013 0.000 0.046 0.119 0.722 0.000 0.013 0.007 0.079 1.000 NCAN 0.038 0.077 0.162 0.031 0.198 0.000 0.396 0.018 0.080 1.000 NG 0.048 0.023 0.043 0.042 0.026 0.000 0.016 0.747 0.055 1.000 MIG 67 99 78 95 161 0 72 87 0 659 Notes: GUN = Gungahlin; BEL = Belconnen; SCAN = South Canberra; TUGG = Tuggeranong; WOD = Woden Valley; WEST = Weston Creek; NCAN = North Canberra; NG = ACT non-government schools; MIG = Migration to/from Canberra. Gungahlin Tuggeranong North Canberra Figure 1: Components of enrolment change for schools in Gungahlin, Tuggeranong and North Canberra, 2008–2016 -1200 -800 -400 0 400 800 1200 2008-09 2009-10 2010-11 2011-12 2012-13 2013-14 2014-15 2015-16 -1500 -500 500 1500 2008-09 2009-10 2010-11 2011-12 2012-13 2013-14 2014-15 2015-16 -1000 -500 0 500 1000 2008-09 2009-10 2010-11 2011-12 2012-13 2013-14 2014-15 2015-16 Preschool enrolments Graduates Net migration within ACT Net migration from outside ACT Move to non-government schools Net gain Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 33 Between 2008 and 2016 most school regions in the ACT, except Tuggeranong, experienced growth in enrolments each year. The sources and levels of enrolment change, however, varied considerably across the regions with some regions dependent on several sources of growth (e.g., South Canberra, Woden Valley and non-government schools) and others driven primarily by preschool enrolments (e.g., Belconnen, Tuggeranong and Weston Creek). Consider, for example, the sources of enrolment change during 2008–2016 for schools in the Gungahlin, Tuggeranong and North Canberra regions (Figure 1, previous page). Gungahlin grew the most, with this growth particularly evident from 2011 with 500 to 560 new students added each year. Preschool enrolments were the most significant and increasing component of growth for Gungahlin. However, there was a steady net loss of students to non-government schools each year. Preschool enrolments similarly were the only source of enrolment growth in the Tuggeranong region between 2008 and 2016, declining marginally in 2015–2016. 4. Cohort transition projection models Three models are introduced for projecting enrolments by academic level in this section. The first is the standard cohort transition model based on annual enrolment numbers by academic level (Johnstone and Philp 1973; Gould 1993; Webster 1970). The second projection model is a multiregional cohort transition model that integrates all of the main sources of enrolment change presented in Table 1. The third projection model is that utilised by the ACT Education Directorate, which, at the individual school level, combines cohort transition information with assumptions on household compositional changes over time in relation to planned developments in the school’s priority intake area. Feedback from school principals on the plausibility of the results is also incorporated. To assess the accuracy of the projections, 2008–2012 data are used to predict the observed 2013– 2016 enrolments with the assumption that preschool enrolments are known. The purpose of the analysis is to see how well the two types of data-based projections perform against the ACT Education Directorate’s projections. 4.1 Cohort transition model A simple and effective way to project school enrolments is to calculate the ratios of students progressing from one academic level to the next. While this does not explain changes to enrolment numbers from one year to the next, it does provide a measure of the overall change, which can be compared over longer periods. Consider, for example, the observed enrolment data for schools in the Gungahlin region from 2008– 2016 (Table 4). Here, we see that there were 316 level 1 students in 2008 (shaded grey). In 2009, this cohort of students grew to 325 level 2 students and then to 341 level 3 students in 2010. In 2016, the same cohort represented 385 level 9 students. Note, as mentioned previously, Gungahlin is a relatively new school region and has been growing rapidly due to migration from outside the ACT (level 11 became available in 2011 and led to the first cohort of level 12 students in 2012). Cohort sizes in other school regions remained very similar or declined. 34 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 Table 4: Observed enrolment data by academic level for schools in Gungahlin, 2008–2016 Level 2008 2009 2010 2011 2012 2013 2014 2015 2016 P 589 543 579 619 633 736 815 888 898 K 361 402 379 432 516 538 613 699 773 1 316 373 423 397 460 549 580 586 693 2 326 325 346 429 428 471 531 570 598 3 335 331 341 378 433 443 485 553 553 4 296 350 360 316 401 459 468 494 575 5 309 318 357 350 350 423 461 497 521 6 309 303 319 349 390 380 440 485 502 7 242 273 284 264 339 339 332 387 458 8 229 259 266 295 278 354 360 367 392 9 204 231 260 271 284 298 365 380 385 10 203 204 245 261 289 321 316 399 415 11 0 0 0 308 369 413 502 479 570 12 0 0 0 0 303 311 396 489 458 Notes: P = Preschool; K = Kindergarten. Transition ratios calculated from the enrolment numbers in Table 4 can be used to produce projections of school enrolments. The projection model is specified, in matrix form, as 𝐄𝐄𝑡𝑡+1 = 𝐆𝐆𝐄𝐄𝑡𝑡 + 𝐁𝐁 where 𝐄𝐄𝑡𝑡 are vectors denoting enrolments at time t and in academic level x, G is the transition matrix and B is a vector including the projected preschool enrolments. In detailed form, the matrix equation looks like ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑒𝑒𝑃𝑃 𝑡𝑡+1 𝑒𝑒𝐾𝐾 𝑡𝑡+1 𝑒𝑒1 𝑡𝑡+1 𝑒𝑒2 𝑡𝑡+1 𝑒𝑒3 𝑡𝑡+1 𝑒𝑒4 𝑡𝑡+1 𝑒𝑒5 𝑡𝑡+1 𝑒𝑒6 𝑡𝑡+1 𝑒𝑒7 𝑡𝑡+1 𝑒𝑒8 𝑡𝑡+1 𝑒𝑒9 𝑡𝑡+1 𝑒𝑒10 𝑡𝑡+1 𝑒𝑒11 𝑡𝑡+1 𝑒𝑒12 𝑡𝑡+1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔𝑃𝑃 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔𝐾𝐾 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑔𝑔11 0⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑒𝑒𝑃𝑃 𝑡𝑡 𝑒𝑒𝐾𝐾 𝑡𝑡 𝑒𝑒1 𝑡𝑡 𝑒𝑒2 𝑡𝑡 𝑒𝑒3 𝑡𝑡 𝑒𝑒4 𝑡𝑡 𝑒𝑒5 𝑡𝑡 𝑒𝑒6 𝑡𝑡 𝑒𝑒7 𝑡𝑡 𝑒𝑒8 𝑡𝑡 𝑒𝑒9 𝑡𝑡 𝑒𝑒10 𝑡𝑡 𝑒𝑒11 𝑡𝑡 𝑒𝑒12 𝑡𝑡 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ + ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡𝑏𝑏 𝑡𝑡–4𝑔𝑔0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ where 𝑒𝑒𝑥𝑥𝑡𝑡 denotes enrolments at time t, 𝑔𝑔𝑥𝑥 denotes transitions between school levels with the subscript marking the origin academic level, 𝑏𝑏𝑡𝑡–4 denotes births and g0 is the ratio of births at time t–4 to preschool enrolments. This model could be adapted further to include repetition rates as in Johnstone and Philp (1973). Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 35 4.2 Multiregional cohort transition model The enrolment transition model presented above is designed for projecting enrolments for single regions/schools independent of other regions/schools. To allow multiple regions/schools to be projected simultaneously, the matrix projection model needs to be reworked to include transitions amongst schools as well as academic levels. This matrix projection model is specified as: Et+1,x+1 = GE t,x+I x+1 where E t,x is a vector of enrolments of regions or schools at time t and academic level x, G contains the transition probabilities between levels and schools (gij ), and I x+1 is a vector of in-migration counts of enrolments at academic level x+1. In detailed form, the matrix equation for an eight school region system looks like ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡𝑒𝑒1,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒2,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒3,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒4,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒5,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒6,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒7,𝑥𝑥+1 𝑡𝑡+1 𝑒𝑒8,𝑥𝑥+1 𝑡𝑡+1 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑟𝑟1 𝑔𝑔21 𝑔𝑔31 𝑔𝑔41 𝑔𝑔51 𝑔𝑔61 𝑔𝑔71 𝑔𝑔81 𝑔𝑔12 𝑟𝑟2 𝑔𝑔32 𝑔𝑔42 𝑔𝑔52 𝑔𝑔62 𝑔𝑔72 𝑔𝑔82 𝑔𝑔13 𝑔𝑔23 𝑟𝑟3 𝑔𝑔43 𝑔𝑔53 𝑔𝑔63 𝑔𝑔73 𝑔𝑔83 𝑔𝑔14 𝑔𝑔24 𝑔𝑔34 𝑟𝑟4 𝑔𝑔54 𝑔𝑔64 𝑔𝑔74 𝑔𝑔84 𝑔𝑔15 𝑔𝑔25 𝑔𝑔35 𝑔𝑔45 𝑟𝑟5 𝑔𝑔65 𝑔𝑔75 𝑔𝑔85 𝑔𝑔16 𝑔𝑔26 𝑔𝑔36 𝑔𝑔46 𝑔𝑔56 𝑟𝑟6 𝑔𝑔76 𝑔𝑔86 𝑔𝑔17 𝑔𝑔27 𝑔𝑔37 𝑔𝑔47 𝑔𝑔57 𝑔𝑔67 𝑟𝑟7 𝑔𝑔87 𝑔𝑔18 𝑔𝑔28 𝑔𝑔38 𝑔𝑔48 𝑔𝑔58 𝑔𝑔68 𝑔𝑔78 𝑟𝑟8 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡𝑒𝑒1,𝑥𝑥 𝑡𝑡 𝑒𝑒2,𝑥𝑥 𝑡𝑡 𝑒𝑒3,𝑥𝑥 𝑡𝑡 𝑒𝑒4,𝑥𝑥𝑡𝑡 𝑒𝑒5,𝑥𝑥 𝑡𝑡 𝑒𝑒6,𝑥𝑥 𝑡𝑡 𝑒𝑒7,𝑥𝑥𝑡𝑡 𝑒𝑒8,𝑥𝑥 𝑡𝑡 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ + ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑖𝑖1,𝑥𝑥+1 𝑖𝑖2,𝑥𝑥+1 𝑖𝑖3,𝑥𝑥+1 𝑖𝑖4,𝑥𝑥+1 𝑖𝑖5,𝑥𝑥+1 𝑖𝑖6,𝑥𝑥+1 𝑖𝑖7,𝑥𝑥+1 𝑖𝑖8,𝑥𝑥+1⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ , where the diagonal elements of the G matrix is equal to the retention proportions (r i) and the off- diagonal elements capture the movements between schools or regions (gij ). This model is similar to the one used by Sweeney and Middleton (2005) to study enrolment transfers in the Santa Barbara Elementary School District in California. 5. School enrolment projections 5.1 School regions To provide an assessment of different projection models, we first focus on school regions. This allows us to make broad comparisons regarding expectations of change across a range of schools, and to work with data that tend to be more stable over time and less prone to specific situations that require expert assumptions or local knowledge. For the assessment, we assume that preschool enrolments are known. Normally these are estimated using observed or projected birth data (t–4). In addition to our comparison of the two projection models against the ACT Education Directorate’s projections and the observed data, we show the effects of different assumptions concerning the transitions used for the projections. The enrolment transition projection model uses the average annual transitions from 2008–2012. Three different transitions assumptions are used in the multiregional projection model. The first keeps 2011–2012 transitions constant. The second uses average 2008–2012 transitions and holds them constant. The third incorporates trends based on linear regressions of the observed transitions from 2008–2009 to 2011–2012. 36 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 Table 5: Average observed and projected student enrolments by school region, 2013–2016 Multiregional School Region Observed ED Cohort Transition 2011–12 2008–12 Trend GUN 6,941 7,195 6,229 6,663 5,971 6,911 BEL 10,464 10,683 10,652 10,340 10,636 10,443 SCAN 4,633 4,766 4,517 4,501 4,471 4,789 TUGG 7,942 7,873 7,756 8,181 7,985 8,304 WOD 4,548 4,280 4,325 4,354 4,247 4,476 WEST 2,063 2,178 2,083 2,071 2,041 2,210 NCAN 5,628 5,724 5,637 5,690 5,511 5,860 Total 42,219 42,699 41,198 41,799 40,861 42,994 Notes: GUN = Gungahlin; BEL = Belconnen; SCAN = South Canberra; TUGG = Tuggeranong; WOD = Woden Valley; WEST = Weston Creek; NCAN = North Canberra; ED = ACT Education Directorate. Table 5 presents the average 2013–2016 results from the school region projections with the best performance shaded grey. Overall, the multiregional projection model with 2011–2012 transitions performed the best, followed closely by the ACT Education Directorate’s projections. However, the performance for each model varied by school region: • The ACT Education Directorate’s projection model did well on average for the Tuggeranong school region (-0.9% error), but poorly for Gungahlin (3.7%), Woden Valley (-5.9%) and Weston Creek (5.6%). • The enrolment transition method performed the best out of all the models for South Canberra (-2.4%) and North Canberra (0.2%), well for Weston Creek (0.9%) but poorly for Gungahlin (9.9%) and Woden Valley (-5.2%). • The multiregional model with 2011–2012 transitions performed well for Weston Creek (0.4%) but poorly for Gungahlin (-4.5%) and Woden Valley (-4.5%). • The multiregional model with 2008–2012 transitions performed well for Tuggeranong (0.5%) and Weston Creek (-1.1%), but very poorly for Gungahlin (-14.6%) and poor for Woden Valley (-6.9%). • The multiregional model with trend transitions did very well for Gungahlin (-0.5%) and Belconnen (-0.2%), but not well for South Canberra (3.5%), Tuggeranong (4.5%), Weston Creek (7.2%) and North Canberra (4.2%). The year-to-year projection results and observed enrolment numbers for Gungahlin, Tuggeranong and North Canberra are presented in Figure 2 (next page). In summary, the multiregional models applied to school regions appear to work best overall (especially considering the objective nature of the information required). Average transition models work well when the school regions are relatively stable but poorly when regions are growing, such as Gungahlin and Belconnen. Here, incorporating trends or auxiliary information on planning developments in the transitions makes sense. The big advantage of the multiregional projection models is that the patterns of change (and error) can be explained and linked to observed trends. There is also consistency in the projection framework for school region moves within the ACT. That is, a departure from one school region must be an entry into another. This is particularly important when considering the transition between primary and high school levels and between high school and secondary college levels. The other factor to consider is efficiency. Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 37 Gungahlin Tuggeranong North Canberra Figure 2: Observed and estimated enrolments for schools in Gungahlin, Tuggeranong and North Canberra, 2008–2016 Notes: ACT ED = ACT Education Directorate’s projection; CT = cohort transition; MR = multiregional projection; Obs = observed values. 3,500 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 2008 2009 2010 2011 2012 2013 2014 2015 2016 ACT ED CT MR 11-12 MR 08-12 MR Trend Obs 7,500 7,700 7,900 8,100 8,300 8,500 8,700 8,900 9,100 9,300 9,500 2008 2009 2010 2011 2012 2013 2014 2015 2016 ACT ED CT MR 11-12 MR 08-12 MR Trend Obs 4,900 5,100 5,300 5,500 5,700 5,900 6,100 6,300 2008 2009 2010 2011 2012 2013 2014 2015 2016 ACT ED CT MR 11-12 MR 08-12 MR Trend Obs % Error MR 11-12 1.1 ACT ED 1.8 MR 8-12 -3.5 CT 0.2 MR Trend 6.9 % Error MR 11-12 3.3 ACT ED -3.2 MR 08-12 -0.3 CT -4.3 MR Trend 6.8 % Error MR 11-12 -7.1 ACT ED 0.9 MR 08-12 -23.4 CT -13.7 MR Trend 0.6 38 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 5.2 School projections for North Canberra The multiregional projection method is particularly useful for capturing students transitioning between primary and high schools and between high schools and secondary colleges. This is further illustrated with schools in North Canberra, an area with a relatively stable total population in terms of age composition and growth. Table 6: Observed and projected enrolments for schools in North Canberra, 2013–2016 2013 2014 2015 2016 School Obs ED MR Obs ED MR Obs ED MR Obs ED MR A. Observed and projected enrolments A 424 436 442 410 432 434 405 443 420 396 480 412 B 315 344 331 327 335 343 350 345 345 350 336 338 C 421 429 426 432 423 432 471 456 446 480 471 439 D 567 632 591 585 662 604 598 675 614 600 656 621 E 483 425 443 554 436 447 556 456 466 611 462 466 F 77 84 79 76 84 78 70 85 74 77 84 72 G 545 587 541 553 588 544 562 600 532 573 622 515 H 718 744 711 741 797 721 723 823 719 721 842 710 I 1,068 1,064 1,059 1,054 1,064 1,082 1,050 1,064 1,073 1,065 1,064 1,084 J 848 854 809 866 841 805 919 837 824 871 833 835 Total 5,466 5,599 5,433 5,598 5,662 5,490 5,704 5,784 5,510 5,744 5,850 5,491 B. Difference between projected and observed A 12 18 22 24 38 15 84 16 B 29 16 8 16 -5 -5 -14 -12 C 8 5 -9 0 -15 -25 -9 -41 D 65 24 77 19 77 16 56 21 E -58 -40 -118 -107 -100 -90 -149 -145 F 7 2 8 2 15 4 7 -5 G 42 -4 35 -9 38 -30 49 -58 H 26 -7 56 -20 100 -4 121 -11 I -4 -9 10 28 14 23 -1 19 J 6 -39 -25 -61 -82 -95 -38 -36 Total 133 -33 64 -108 80 -194 106 -253 Notes: Obs = observed values; ED = ACT Education Directorate projection; MR = multiregional projection. The school level projections for North Canberra focus on the level and between-school movements across seven primary schools (schools A–G), two high schools (schools H–I) and one secondary college (school J). In this illustration, only schools in North Canberra are included, but the model framework could be extended to include any set of schools. Also, for illustration purposes, we apply the average 2008–2012 transitions as the basis for estimating transitions for 2013–2016. Australian Population Studies 1 (1) 2017 Raymer J, Biddle N and Guan Q 39 The projection results for schools in North Canberra are presented in Table 6 (panel A). Also included are the corresponding observed values and the projections made by the ACT Education Directorate (panel B) with the closest value to the observed value shown in grey. The multiregional projection model out-performed the ACT Education Directorate’s projections for seven out of the 10 schools in each of the four projection years. These differences, however, are fairly modest. The real benefits of the multiregional model are that: (i) less assumptions are required; and (ii) there is a capacity to disaggregate the projection error. 6. Discussion and conclusion Understanding the demographic drivers of enrolment change is essential for making accurate and informed projections. Variables influencing change include the birth of children in school intake areas, movements between government schools in a city or district, migration into or out of a city or region, and movements to or from non-government schools. The multiregional cohort projection model presented in this paper provides a flexible platform for including the main transitions affecting school enrolment change. By grouping schools in a multiregional model, understanding about student movements is increased and projection bias reduced. In order to make and assess projections, some understanding of the components of enrolment change and their trends is required. There is no single projection model that performs best in all situations. School districts should consider a variety of assumptions when developing projection models and adapt them to meet local differences. They need also to incorporate proposed or new housing developments and land releases which may result in increased migration by young families into the area and rapid population growth impacting education provision for the areas affected. As projections are often heavily scrutinised, the ability to explain trends in terms of the sources of growth provides a stronger evidence base and argument for school infrastructure, personnel and resource requirements. It also allows the projections producer to understand where the sources of error occurred in relation to changing enrolments over time. This is the main motivation underlying the multiregional cohort projections. We have demonstrated the variability in the sources of student enrolment change across the ACT and the relative stability in the different sources of enrolment change and cohort transitions. Movements within and outside the ACT government school system can have large effects and are more difficult to predict. The multiregional school enrolment projection model captures these movements and may be applied to a wide array of situations. However, the transition data must be available. Key messages • Accurate projections of school enrolments and understanding of sources of error are essential for good school planning. • The multiregional projection model used in our study performed as well or better than a cohort transition model and current projection model used by the ACT Education Directorate. • A multiregional model that interrogates enrolment change can be used for short- to medium- term enrolment projections in the ACT. 40 Raymer J, Biddle N and Guan Q Australian Population Studies 1 (1) 2017 Acknowledgements This paper is drawn from a previously commissioned review of school enrolment projection methodology prepared for the ACT Education Directorate. The authors would like to thank the Directorate for their comments and suggestions, and Xujing Bai for her research assistance. 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