Abstract
Many important questions and theories in demography focus on changes over time, and on how those changes differ over geographic and social space. Space-time analysis has always been important in studying fertility transitions, for example. However, demographers have seldom used formal statistical methods to describe and analyze time series of maps. One formal method, used widely in epidemiology, criminology, and public health, is Knox’s space-time interaction test. In this article, we discuss the potential of the Knox test in demographic research and note some possible pitfalls. We demonstrate how to use familiar proportional hazards models to adapt the Knox test for demographic applications. These adaptations allow for nonrepeatable events and for the incorporation of structural variables that change in space and time. We apply the modified test to data on the onset of fertility decline in Brazil over 1960–2000 and show how the modified method can produce maps indicating where and when diffusion effects seem strongest, net of covariate effects.
The text of this article is only available as a PDF.
References
Baker, R.D. (
2004
). Identifying Space-Time Disease Clusters
. Acta Tropica
, 91
, 291
–99
. 10.1016/j.actatropica.2004.05.007Baller, R.D., Anselin, L., Deane, G., Messner, S., & Hawkins, D. (
2001
). Structural Covariates of U.S. County Homicide Rates: Incorporating Spatial Effects
. Criminology
, 39
, 561
–90
. 10.1111/j.1745-9125.2001.tb00933.xBaller, R.D., & Richardson, K. (
2002
). Social Integration, Imitation, and the Geographic Patterning of Suicide
. American Sociological Review
, 67
, 872
–88
. 10.2307/3088974Balabdaoui, F., Bocquet-Appel, J.P., Lajaunie, C., & Rajan, S.I. (
2001
). Space-Time Evolution of Fertility Transition in India (1961–91)
. InternationalJournal of Population Geography
, 7
, 129
–48
. 10.1002/ijpg.213Barton, D.E., & David, F.N. (
1966
). The Random Intersection of Two Graphs
. In David, F.N. (Ed.), Research Papers in Statistics: Festschrift for J. Neyman
(pp. 445
–59
). New York
: Wiley
.Bhopal, R.S., Diggle, P.J., & Rowlingson, B.S. (
1992
). Pinpointing Clusters of Apparently Sporadic Legionnaire’s Disease
. British Medical Journal
, 304
, 1022
–27
. 10.1136/bmj.304.6833.1022Birch, J.M., Alexander, F.E., Blair, V., Eden, O.B., Taylor, G.M., & McNally, R.J.Q. (
2000
). Space-Time Clustering Patterns in Childhood Leukaemia Support a Role for Infection
. British Journal of Cancer
, 82
, 1571
–78
. 10.1054/bjoc.1999.1072Bocquet-Appel, J.P., & Jakobi, L. (
1998
). Evidence for a Spatial Diffusion of Contraception at the Onset of the Fertility Transition in Victorian Britain
. Population: An English Selection, Special Issue New Advances in Social Sciences
, 10
(1
), 181
–204
.Bocquet-Appel, J.P., Rajan, S.I., Bacro, J.N., & Lajaune, C. (
2002
). The Onset of India’s Fertility Transition
. European Journal of Population
, 18
, 211
–32
.Coale, A.J. (
1973
). The Demographic Transition Reconsidered
International Population Conference
(pp. 53
–72
). Liège, Belgium
: International Union for the Scientific Study of Population
.Coale, A.J., & Treadway, R. (
1986
). A Summary of the Changing Distribution of Overall Fertility, Marital Fertility, and the Proportion Married in the Provinces of Europe
. In Coale, A.J., & Watkins, S.C. (Eds.), The Decline of Fertility in Europe
(pp. 31
–181
). Princeton, NJ
: Princeton University Press
.Coale, A.J., & Watkins, S.C. (
1986
). The Decline of Fertility in Europe
. Princeton, NJ
: Princeton University Press
.Cox, D.R. (
1972
). Regression Models and Life-Tables
. Journal of the Royal Statistical Society, Series B
, 34
, 187
–220
.Diggle, P.J. (
1990
). A Point Process Modelling Approach to Raised Incidence of a Rare Phenomenon in the Vicinity of a Prespecified Point
. Journal of the Royal Statistical Society, Series A
, 153
, 349
–62
.Guinnane, T.W., Okun, B.S., & Trussell, J. (
1994
). What Do We Know About the Timing of Fertility Transitions in Europe?
. Demography
, 31
, 1
–20
. 10.2307/2061905Ihaka, R., & Gentleman, R. (
1996
). R: A Language for Data Analysis and Graphics
. Journal of Computational and Graphical Statistics
, 5
, 299
–314
. 10.2307/1390807Kalbfleisch, J.D., & Prentice, R.L. (
1973
). Marginal Likelihoods Based on Cox’s Regression and Life Model
. Biometrika
, 60
, 267
–78
. 10.1093/biomet/60.2.267Knox, E.G. (
1964
). The Detection of Space-Time Interactions
. Applied Statistics
, 13
(1
), 25
–30
. 10.2307/2985220Knox, E.G. (
1964
). Epidemiology of Childhood Leukaemia in Northumberland and Durham
. British Journal of Preventative and Social Medicine
, 18
, 17
–24
.Knox, E.G. (
2002
). An Epidemic Pattern of Murder
. Journal of Public Health Medicine
, 24
, 34
–37
. 10.1093/pubmed/24.1.34Kulldorff, M., & Hjalmars, U. (
1999
). The Knox Method and Other Tests for Space-Time Interaction
. Biometrics
, 55
, 544
–52
. 10.1111/j.0006-341X.1999.00544.xLand, K., & Deane, G. (
1992
). On the Large-Sample Estimation of Regression Models With Spatial- or Network-Effects Terms: A Two-Stage Least Squares Approach
. In Marsden, P.V. (Ed.), Sociological Methodology
(pp. 221
–48
). Washington, DC
: American Sociological Association
.Land, K., Deane, G., & Blau, J. (
1991
). Religious Pluralism and Church Membership: A Spatial Diffusion Model
. American Sociological Review
, 56
, 237
–49
. 10.2307/2095782Machado-Coelho, G.L., Assunção, R., Mayrink, W., & Caiaffa, W.T. (
1999
). American Cutaneous Leishmaniasis in Southeast Brazil: Space-time Clustering
. International Journal of Epidemiology
, 28
, 982
–89
. 10.1093/ije/28.5.982McKenzie, N., Landau, S., Kapur, N., Meehan, J., Robinson, J., Bickley, H., Parsons, R., & Appleby, L. (
2005
). Clustering of Suicides Among People With Mental Illness
. British Journal of Psychiatry
, 187
, 476
–80
. 10.1192/bjp.187.5.476McNally, R.J.Q., Cairns, D.P., Eden, O.B., Alexander, F.E., Taylor, G.M., Kelsey, A.M., & Birch, J.M. (
2002
). An Infectious Aetiology for Childhood Brain Tumours? Evidence From Space-Time Clustering and Seasonality Analyses
. British Journal of Cancer
, 86
, 1070
–77
. 10.1038/sj.bjc.6600228Montgomery, M.R., & Casterline, J.B. (
1996
). Social Learning, Social Influence, and New Models of Fertility
. Population and Development Review
, 22
, 151
–75
. 10.2307/2808010Morris, J.K., Alberman, E., & Mutton, D. (
1998
). Is There Evidence of Clustering in Down Syndrome?
. International Journal of Epidemiology
, 27
, 495
–98
. 10.1093/ije/27.3.495Morrison, A.C., Getis, A., Santiago, M., Rigau-Perez, J.G., & Reiter, P. (
1998
). Exploratory Space-Time Analysis of Reported Dengue Cases During an Outbreak in Florida, Puerto Rico, 1991–1992
. American Journal of Tropical Medicine and Hygiene
, 58
, 287
–98
.Potter, J.E., Schmertmann, C.P., & Cavenaghi, S.M. (
2002
). Fertility and Development: Evidence From Brazil
. Demography
, 39
, 739
–61
. 10.1353/dem.2002.0039Samuelsson, U., Johansson, C., Carstensen, J., & Ludvigsson, J. (
1994
). Space-Time Clustering in Insulin-Dependent Diabetes Mellitus (IDDM) in South-East Sweden
. International Journal of Epidemiology
, 23
, 138
–42
. 10.1093/ije/23.1.138Tobin, P.C. (
2007
). Space-Time Patterns During the Establishment of a Nonindigenous Species
. Population Ecology
, 49
, 257
–63
. 10.1007/s10144-007-0043-7Tolnay, S.E. (
1995
). The Spatial Diffusion of Fertility: A Cross-Sectional Analysis of Counties in the American South, 1940
. American Sociological Review
, 60
, 299
–308
. 10.2307/2096388Tolnay, S.E., Deane, G., & Beck, E.M. (
1996
). Vicarious Violence: Spatial Effects on Southern Lynchings, 1890–1919
. American Journal of Sociology
, 102
, 788
–815
. 10.1086/230997Watkins, S.C. (
1986
). Conclusions
. In Coale, A.J., & Watkins, S.C. (Eds.), The Decline of Fertility in Europe
(pp. 420
–49
). Princeton, NJ
: Princeton University Press
.© Population Association of America 2010
2010
