Abstract
Recent criminological research has used latent class growth analysis (LCGA), a form of group-based trajectory analysis, to identify distinct terrorism trends and areas of high terrorism activity at the country-level. The current study contributes to the literature by assessing the robustness of recent findings generated by one type of group-based analysis, LCGA, to changes in measurement and statistical methodology. Using data from the Global Terrorism Database (GTD), we consider the challenges and advantages of applying group-based analysis to macro-level terrorism data. We summarize and classify country-level patterns of domestic and transnational terrorism using two types of group-based analyses, LCGA and an alternative yet similar modeling approach, general mixture modeling (GMM). We evaluate the results from each approach using both substantive and empirical criteria, highlighting the similarities and differences provided by both techniques. We conclude that both group-based models have utility for terrorism research, yet for the purposes of identifying hot spots of terrorist activity, LCGA results provide greater policy utility.
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Notes
Unless otherwise noted, for the sake of parsimony we use interchangeably the terms country, nation, and territory.
This is true even if only countries with non-zero counts of terrorist activities in a given year are considered.
Arguably, some countries may have zero counts because of lack of reporting opportunity or media restrictions.
Counts are used instead of rates for two reasons. It is not clear what the appropriate denominator should be for computing the rates because both people and infrastructure can be the target of attacks. Second, population counts are not available for many of the countries in the dataset for the entire study period and therefore using rates would substantially decrease our sample size.
Countries with less than four years of data (n = 2) were excluded from the analysis. The analysis also excludes data from 1993. Original data for 1993 were lost by Pinkerton Global Intelligence Service (PGIS) and have never been recovered (LaFree and Dugan 2007). Fortunately, both LCGA and GMM can accommodate missing data (Muthén and Muthén 1998–2010; Nagin 2005) and because the 1993 data are missing completely at random, our parameter estimates should not be affected by systematic bias (Allison 2002).
The following countries were coded to reflect their changing geographic boundaries, with the time span in parenthesis representing the period in which the country had valid data values: Soviet Union, Yugoslavia (1970–1991); Armenia, Azerbaijan, Belarus, Estonia, Georgia, Kazakhstan, Kyrgyzstan, Latvia, Lithuania, Moldova, Russia, Tajikistan, Turkmenistan, Ukraine, Uzbekistan (1992–2007); Bosnia-Herzegovina, Croatia, Macedonia, Slovenia, Serbia-Montenegro also known as Federal Republic of Yugoslavia (1992–2007); Czechoslovakia (1970–1992); Czech Republic, Slovak Republic, Eritrea (1993–2007); Namibia (1990–2007); North Yemen, South Yemen (1970–1989); Yemen (1990–2007); and East Germany (1970–1989). Sample sizes for analyses change due to changes in countries over time.
It is important to note that the NB estimates more parameters than the ZIP or Poisson models. This is because NB models estimate a dispersion parameter for each of the 37 outcome variables. Thus it is possible that a ZIP model with fewer constraints might better fit the data. However, exploratory analysis indicated this was not the case. The BIC values for a ZIP model that allowed the threshold values to be freely estimated was still greater than the BIC values obtained by the NB models.
A number of efforts were undertaken to obtain convergence with the Poisson and ZIP models. For example, the data were top-coded at lower values for which convergence could be obtained, and then through an iterative process, parameter estimates produced by the more heavily truncated data were used as start values in the models estimated using less truncated data. In addition, the convergence criteria were relaxed. Still, the Poisson and ZIP models with more than four classes did not converge.
We also estimated zero inflated negative binomial models (ZINB), although these models are not commonly used in the literature. These models follow the same logic as the ZIP, but allow for over dispersion in the count portion of the model. The NB and ZINB produced identical log likelihoods and Mplus fixed the values of the threshold parameters in all classes to −15. This occurs when the probability of being a structural zero is virtually zero and suggests that the inflation parameters are not needed. This was further confirmed by comparing the BICs for the NB and the ZINB models. The NB produced a lower BIC value than the ZINB because the BIC levies a penalty for more complicated models that estimate more parameters. These results are available upon request.
In Mplus the BIC is defined as −2 log likelihood + p log n, where p is the number of parameters and n is the sample size (Kreuter and Muthén 2008). This differs from the computation used by Proc Traj in SAS.
Jeffries (2003) has criticized the LMR-LRT on the basis of its theoretical proof; however, it is unclear to what extent this mistake affects the use of this test in practice (Muthén and Muthén 1998–2010). Although there is some evidence that the computationally intensive Bootstrapped Likelihood Ratio Test (BLRT) provides the best assessment of model fit (Nylund et al. 2007), this test never reached convergence with our data.
We based these decisions on past research on country-level terrorism trends, which also found that altering the polynomial form does little to change the substantive findings (LaFree et al. 2010).
These models relax the assumption that the within class variance for the intercept is zero, but the variances are constrained to be the same across classes. This constraint was put in place for two reasons. First, models that allow growth parameters to vary both within and across classes are computationally demanding and often have problems with convergence. Second, supplemental analyses (not shown) indicated that models that allowed the intercept to vary both across and within groups produced higher BICs and poorer fit than models that constrained the intercept variance to be equal across groups.
The estimation of trajectory models involves an iterative process in which the estimation algorithm attempts to identify the global maximum of the likelihood function. In more complicated models, the algorithm is more likely to converge on a local, versus a global maximum (Feldman et al. 2009). To protect against this possibility, Mplus enables users to estimate models using multiple random start values. The log likelihood values for the best solutions are provided by Mplus and log likelihood values that can be replicated are trust worthy. Here we only present results for models in which the best log likelihood could be replicated.
Group 1’s increase post 2002 may be due to the classification of Iraq within this group and the inclusion of events that followed the United States led invasion of Iraq that began on 20 March 2003. Although the GTD are designed to exclude cases involving open combat between opposing armed forces, it is often difficult to distinguish between acts of terror and warfare. We also conducted the analysis excluding post 2003 Iraq data and the substantive results remain unchanged in terms of general country-level trends and identification of high risk countries; however, the increase in terrorist activity for group 1 is less striking (See also LaFree et al. 2010).
Yugoslavia, which ceased to exist as a political entity in 1991, appears to fit in trajectory Group 1 because they had very low levels of terrorist activity throughout their time series until the very end of their existence (LaFree et al. 2010).
Note, although not shown, all variance parameters are significantly different from zero. This indicates that within each class, the intercepts exhibit significant variability.
Once again, when we reanalyzed the data excluding post 2003 Iraq data the substantive results remain unchanged.
We emphasize Kreuter and Muthén’s (2008) conclusion to be aware that model comparisons will also produce varying conclusions based on the specific data and outcome.
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Acknowledgments
We wish to thank the three anonymous reviewers that provided useful comments and suggestions to the current manuscript. We also thank Nicholas Corsaro and participants attending the Working Group Symposium on Quantitative Methods for Studying Terrorism at John Jay College, NY for their suggestions. We appreciate the thoughtfulness and thoroughness of their comments. Finally, we would like to thank Bengt and Linda Muthén for assistance with MPlus.
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Appendix
Appendix
Countries by Trajectory Group Membership, LCGA Five Group Results
Group 1 (n = 29) | Group 2 (n = 49) | Group 3 (n = 64) | Group 4 (n = 31) | Group 5 (n = 28) | ||||
|---|---|---|---|---|---|---|---|---|
Afghanistan | Andorra | Maldives | Albania | Liberia | Togo | Australia | Venezuela | Argentina |
Algeria | Antigua & Barbuda | Martinique | Armenia | Libya | Trinidad & Tobago | Austria | Zimbabwe | Chile |
Angola | Bahamas | Mauritania | Bahrain | Lithuania | Tunisia | Azerbaijan | Colombia | |
Bangladesh | Barbados | Mauritius | Bulgaria | Luxembourg | United Arab | Belgium | Corsica France | |
Bosnia-Herzegovina | Belize | North Korea | Cameroon | Macao | Emirates | Bolivia | El Salvador | |
Burundi | Benin | Qatar | Cen.African Rep | Madagascar | Uzbekistan | Brazil | France | |
Cambodia | Bermuda | Romania | Chad | Malaysia | Zambia | Canada | Germany | |
China | Bhutan | Seychelles | Congo | Mali | Costa Rica | Greece | ||
Egypt | Botswana | Singapore | Cuba | Malta | Croatia | Guatemala | ||
Georgia | Brunei | Solomon Islands | Czech Republic | Moldova | Cyprus | India | ||
Haiti | Burkina Faso | South Vietnam | Czechoslovakia | Morocco | Dominican Republic | Iran | ||
Indonesia | Belarus | South Yemen | Denmark | Namibia | East Germany | Israel | ||
Iraq | Cayman Islands | St Kitts and Nevis | Djibouti | New Caledonia | Ecuador | Italy | ||
Kenya | Comoros | Tanzania | East Timor | New Zealand | Ethiopia | Lebanon | ||
Macedonia | Dominica | Vanuatu | Estonia | Niger | Honduras | Mexico | ||
Myanmar | Equatorial Guinea | Vietnam | Fiji | North Yemen | Ireland | Nicaragua | ||
Nepal | Eritrea | Virgin Islands Us | Ghana | Norway | Japan | Northern Ireland | ||
Nigeria | Falkland Islands | Wallis & Futuna | Guadeloupe | Papua New Guinea | Jordan | Pakistan | ||
Rwanda | Finland | Western Samoa | Guinea | Paraguay | Kuwait | Palestine | ||
Serbia-Montenegro | French Guiana | Western Sahara | Guyana | Poland | Mozambique | Peru | ||
Somalia | French Polynesia | Hong Kong | Saudi Arabia | Netherlands | Philippines | |||
Soviet Union | Gabon | Hungary | Senegal | Panama | Russia | |||
Sudan | Gambia | Ivory Coast | Sierra Leone | Portugal | South Africa | |||
Tajikistan | Gibraltar | Jamaica | Slovak Republic | Puerto Rico | Spain | |||
Thailand | Grenada | Kazakhstan | Slovenia | Sweden | Sri Lanka | |||
Uganda | Guinea-Bissau | Kyrgyzstan | South Korea | Switzerland | Turkey | |||
Yemen | Iceland | Laos | Suriname | Syria | United Kingdom | |||
Zaire | Isle of Man | Latvia | Swaziland | Ukraine | United States | |||
Yugoslavia | Malawi | Lesotho | Taiwan | Uruguay | ||||
Countries by Trajectory Group Membership, LCGA Four Group Results
Group 1 (n = 50) | Group 2 (n = 51) | Group 3 (n = 69) | Group 4 (n = 31) | |||||
|---|---|---|---|---|---|---|---|---|
Afghanistan | Nepal | Andorra | Maldives | Albania | Laos | Suriname | Algeria | United Kingdom |
Angola | Nigeria | Antigua and Barbuda | Isle of Man | Armenia | Latvia | Sweden | Argentina | United States |
Australia | Panama | Bahamas | Martinique | Azerbaijan | Lesotho | Taiwan | Bangladesh | |
Austria | Portugal | Barbados | Mauritania | Bahrain | Liberia | Togo | Chile | |
Belgium | Puerto Rico | Belize | Mauritius | Bulgaria | Libya | Trinidad and Tobago | Colombia | |
Bolivia | Rwanda | Benin | North Korea | Cameroon | Lithuania | Tunisia | Corsica France | |
Bosnia-Herzegovina | Serbia-Montenegro | Bermuda | Qatar | Canada | Luxembourg | Ukraine | Egypt | |
Brazil | Somalia | Bhutan | Romania | Central African Republic | Macao | United Arab Emirates | El Salvador | |
Burundi | Sudan | Botswana | Seychelles | Chad | Madagascar | Uzbekistan | France | |
Cambodia | Switzerland | Brunei | Singapore | Congo | Malaysia | Zambia | Germany | |
China | Syria | Burkina Faso | Solomon Islands | Croatia | Mali | Yugoslavia | Greece | |
Costa Rica | Tajikistan | Belarus | St Kitts and Nevis | Cuba | Malta | Guatemala | ||
Cyprus | Thailand | Cayman Islands | Swaziland | Czech Republic | Moldova | India | ||
Dominican Republic | Uganda | Comoros | Tanzania | Czechoslovakia | Morocco | Iran | ||
East Germany | Uruguay | Dominica | Vanuatu | Denmark | Namibia | Israel | ||
Ecuador | Venezuela | Equatorial Guinea | Vietnam | Djibouti | New Caledonia | Italy | ||
Ethiopia | Yemen | Eritrea | Virgin Islands US | East Timor | New Zealand | Lebanon | ||
Georgia | Zaire | Falkland Islands | Wallis and Futuna | Estonia | Niger | Mexico | ||
Haiti | Zimbabwe | Finland | Western Samoa | Fiji | Norway | Nicaragua | ||
Honduras | Soviet Union | French Guiana | Western Sahara | Ghana | North Yemen | Northern Ireland | ||
Indonesia | French Polynesia | South Yemen | Guadeloupe | Papua New Guinea | Pakistan | |||
Iraq | Gabon | South Vietnam | Guyana | Paraguay | Palestine | |||
Ireland | Gambia | Hong Kong | Poland | Peru | ||||
Japan | Gibraltar | Hungary | Saudi Arabia | Philippines | ||||
Jordan | Grenada | Ivory Coast | Senegal | Russia | ||||
Kenya | Guinea | Jamaica | Sierra Leone | South Africa | ||||
Macedonia | Guinea-Bissau | Kazakhstan | Slovak Republic | Spain | ||||
Mozambique | Iceland | Kuwait | Slovenia | Sri Lanka | ||||
Myanmar | Malawi | Kyrgyzstan | South Korea | Turkey | ||||
Countries by Trajectory Group Membership, Random Intercept GMM Four Group Results
Group 1 (n = 51) | Group 2 (n = 36) | Group 3 (n = 72) | Group 4 (n = 42) | |||||
|---|---|---|---|---|---|---|---|---|
Afghanistan | Maldives | Albania | Senegal | Andorra | Greece | South Vietnam | Azerbaijan | Man, Isle of |
Algeria | Myanmar | Antigua and Barbuda | Sierra Leone | Argentina | Grenada | South Yemen | Belize | Martinique |
Angola | Namibia | Benin | Slovenia | Armenia | Guyana | Spain | Bolivia | Mauritius |
Bahrain | Nepal | Bermuda | Soviet Union | Australia | Iran | Sweden | Botswana | Moldova |
Bangladesh | Niger | Brunei | St Kitts and Nevis | Austria | Ireland | Switzerland | Burkina Faso | Mozambique |
Bhutan | Nigeria | Burundi | Togo | Bahamas | Israel | Syria | Chile | New Caledonia |
Bulgaria | Pakistan | Cameroon | Vietnam | Barbados | Italy | Tanzania | Croatia | Nicaragua |
Cambodia | Palestine | Cayman Islands | Virgin Islands Us | Belgium | Jamaica | Trinidad And Tobago | Cyprus | Panama |
Central African Rep | Poland | Czechoslovakia | Wallis and Futuna | Bosnia-Herzegovina | Jordan | Tunisia | Denmark | Peru |
Chad | Qatar | Czech Republic | Western Samoa | Brazil | Kuwait | Turkey | Djibouti | Romania |
Congo | Russia | China | Belarus | Latvia | Ukraine | Ecuador | Singapore | |
Cuba | Saudi Arabia | Comoros | Canada | Lebanon | United Arab Emirates | El Salvador | Slovak Republic | |
East Timor | Serbia-Montenegro | Equatorial Guinea | Colombia | Macedonia | United Kingdom | Estonia | South Africa | |
Egypt | Solomon Islands | French Polynesia | Corsica France | Mauritania | United States | Gibraltar | South Korea | |
Fiji | Somalia | Gambia | Costa Rica | Mexico | Uruguay | Guadeloupe | Suriname | |
Finland | Sri Lanka | Ghana | Dominica | Morocco | Vanuatu | Guatemala | Tajikistan | |
Guinea | Sudan | Hungary | Dominican Republic | Netherlands | Venezuela | Honduras | ||
Guinea-Bissau | Swaziland | Laos | East Germany | New Zealand | Western Sahara | Iceland | ||
Haiti | Taiwan | Liberia | Eritrea | Northern Ireland | Zambia | Japan | ||
Hong Kong | Thailand | Macao | Ethiopia | North Yemen | Zimbabwe | Kazakhstan | ||
India | Uganda | Madagascar | Falkland Islands | Norway | Lesotho | |||
Indonesia | Uzbekistan | Malawi | France | Paraguay | Libya | |||
Iraq | Yemen | Mali | French Guiana | Philippines | Lithuania | |||
Ivory Coast | Zaire | North Korea | Gabon | Portugal | Luxembourg | |||
Kenya | Yugoslavia | Papua New Guinea | Georgia | Puerto Rico | Malaysia | |||
Kyrgyzstan | Rwanda | Germany | Seychelles | Malta | ||||
Countries by Trajectory Group Membership, Random Intercept GMM Five Group Results
Group 1 (n = 22) | Group 2 (n = 38) | Group 3 (n = 45) | Group 4 (n = 59) | Group 5 (n = 37) | |||||
|---|---|---|---|---|---|---|---|---|---|
Afghanistan | Albania | Senegal | Angola | Norway | Andorra | Kuwait | South Vietnam | Azerbaijan | Nicaragua |
Algeria | Antigua & Barbuda | Sierra Leone | Bahrain | Palestine | Argentina | Latvia | Belize | Panama | |
Bangladesh | Benin | Slovenia | Bhutan | Paraguay | Armenia | Lebanon | Bolivia | Peru | |
Chad | Bermuda | Solomon Islands | Bulgaria | Philippines | Australia | Lesotho | Botswana | Romania | |
Guinea-Bissau | Brunei | St Kitts & Nevis | Cambodia | Poland | Austria | Mauritania | Burkina Faso | Singapore | |
East Timor | Burundi | Togo | Central African Republic | Qatar | Bahamas | Mexico | Chile | Slovak Republic | |
Indonesia | Cameroon | Vietnam | Colombia | South Korea | Barbados | Morocco | Croatia | South Africa | |
Iraq | Cayman Islands | Virgin Islands US | Congo | Sri Lanka | Belgium | Netherlands | Cyprus | Suriname | |
Ivory Coast | China | Wallis & Futuna | Corsica France | Swaziland | Bosnia-Herzegovina | Portugal | Denmark | Tajikistan | |
Kyrgyzstan | Comoros | Western Samoa | Cuba | Taiwan | Brazil | Puerto Rico | Ecuador | ||
Maldives | Czech Republic | Djibouti | Trinidad & Tobago | Belarus | Seychelles | El Salvador | |||
Nepal | Czechoslavkia | Egypt | Tunisia | Canada | Spain | Estonia | |||
Nigeria | Equatorial Guinea | Fiji | Uganda | Costa Rica | Sweden | Guadeloupe | |||
Pakistan | French Polynesia | Finland | Uzbekistan | Dominica | Switzerland | Guatemala | |||
Russia | Gambia | French Guiana | Vanuatu | Dominican Republic | Syria | Honduras | |||
Saudi Arabia | Ghana | Gabon | Yemen | East Germany | Tanzania | Iceland | |||
Serbia-Montenegro | Hungary | Gibraltar | Eritrea | Turkey | Japan | ||||
Somalia | Laos | Guinea | Ethiopia | Ukraine | Kazakhstan | ||||
Sudan | Liberia | Guyana | Falkland Islands | United Arab Emirates | Libya | ||||
Thailand | Macao | Haiti | France | United Kingdom | Lithuania | ||||
Zaire | Madagascar | Hong Kong | Georgia | United States | Luxembourg | ||||
Yugoslavia | Malawi | India | Germany | Uruguay | Malaysia | ||||
Mali | Israel | Greece | Venezuela | Malta | |||||
Namibia | Kenya | Grenada | Zambia | Man, Isle of | |||||
North Korea | Macedonia | Iran | Zimbabwe | Martinique | |||||
Papua New Guinea | Mauritius | Ireland | Northern Ireland | Moldova | |||||
Rwanda | Myanmar | Italy | Western Sahara | Mozambique | |||||
New Zealand | Jamaica | North Yemen | New Caledonia | ||||||
Niger | Jordan | South Yemen | Nicaragua | ||||||
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Morris, N.A., Slocum, L.A. Estimating Country-Level Terrorism Trends Using Group-Based Trajectory Analyses: Latent Class Growth Analysis and General Mixture Modeling. J Quant Criminol 28, 103–139 (2012). https://doi.org/10.1007/s10940-011-9158-2
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DOI: https://doi.org/10.1007/s10940-011-9158-2