Abstract
In much demographic analysis, it is important to know how occurrence-exposure rates or transition probabilities vary continuously by age or by time. Often we have coarse or fluctuating data so there can be a need for estimation and smoothing. Since the distributions of rates or counts across age or another variable are often curved, a nonlinear model is likely to be appropriate. The main focus of this paper is on the estimation of detailed information from grouped data such as age and income bands; however, the methods we outline could also be applied to other settings such as smoothing rates where the original data are ragged. The ability to carry out curve fitting is a very useful skill for population geographers and demographers. Curve fitting is not well covered in statistics textbooks, and whilst there is a large literature in journals thoroughly discussing the detail of functions which define curves, these texts are likely to be inaccessible to researchers who are not specialists in mathematics. We aim here to make nonlinear modelling as accessible as possible. We demonstrate how to carry out nonlinear regression using SPSS, giving stepped-through hypothetical and research examples. We note other software in which nonlinear regression can be carried out, and outline alternative methods of curve fitting.
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Acknowledgments
Work underpinning this paper was funded by research grants as follows: Paul Norman and Phil Rees by ESRC Research Awards RES-163-25-0032 and RES-189-25-0162, Alan Marshall by ESRC Postdoctoral Fellowship ES/H030328/1, Chris Thompson by ESRC RIBEN CASE studentship with Acxiom, and Lee Williamson by ESRC CASE studentship S42200124001 and Bradford Metropolitan District Council. The case studies reported here have been enabled by Bradford Birth Statistics Database supplied by Bradford Metropolitan District Council; Research Opinion Poll data supplied by Acxiom Ltd; 1991 and 2001 Census data obtained through the MIMAS CASWEB facility, an academic service supported by ESRC and JISC; and midyear estimates and Vital Statistics provided by the Office for National Statistics. The Census and Vital Statistics data are Crown copyright and are reproduced with permission of the Office of Public Sector Information. The authors are grateful to the anonymous referees whose useful comments have helped us to improve this paper.
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Appendix: Using SPSS syntax
Appendix: Using SPSS syntax
(a) Quick guide to SPSS syntax
SPSS syntax is a ‘high level’ programming language which can readily be used to automate tasks without the user being a trained computer programmer. Users of SPSS will be familiar with the Data Editor window (comprising Data View and Variable View) through which you can open and analyse files with the .sav extension. Any analyses you carry out have the results (e.g. tables and graphs) presented in the Output window (which can be saved as .spo or .spv files depending on SPSS version). SPSS syntax files have the .sps extension.
For people who are inexperienced in using syntax, there is no need to type the command instructions from scratch. There are two easy ways to obtain syntax since SPSS will write it for you. If you use dialogue boxes to carry out an analysis, when you have made your variable selections, click on ‘Paste’ and the syntax commands to carry out your mouse click selections will be pasted into a new or previously open syntax file. A better way is to select via the menu File > Options, then the Viewer tab, tick ‘Display commands in the log’ and the click Apply. Any choices you make through ‘point and click’ are then always recorded in the Output window as syntax. You can then copy and paste syntax which you need into a .sps file and save the syntax file for future use. Existing commands can be edited for a new analysis.
(b) Explanation of the SPSS syntax to carry out nonlinear regression
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Norman, P., Marshall, A., Thompson, C. et al. Estimating detailed distributions from grouped sociodemographic data: ‘get me started in’ curve fitting using nonlinear models. J Pop Research 29, 173–198 (2012). https://doi.org/10.1007/s12546-012-9082-9
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DOI: https://doi.org/10.1007/s12546-012-9082-9