About 16% of those who present to the emergency department for self-harm (self-poisoning or self-injury, irrespective of suicidal intent) ^{Reference Haw, Bergen, Casey and Hawton1} will return within a year. ^{Reference Owens, Horrocks and House2} Self-harm repetition increases the risk of suicide ^{Reference Zahl and Hawton3} and is an outcome of interest in observational ^{Reference Owens, Horrocks and House2} and intervention ^{Reference Evans, Evans, Morgan, Hayward and Gunnell4} studies. However, many such studies ignore multiple repetition. Often, those who present to hospital are followed-up for subsequent self-harm presentations and then categorised and analysed as repeaters or non-repeaters, disregarding the number of repeat presentations. This approach may obscure important clinical and cost differences. For example, an intervention trial aimed at reducing self-poisoning repetition found, after 2 years, no effect on the proportion of repeaters (21.2% v. 22.8% in the intervention and control groups respectively), but when considering the number of repeat presentations, the intervention halved the rate of repetition (rate ratio = 0.49). ^{Reference Carter, Clover, Whyte, Dawson and D'Este5}

Disregarding the number of repeat presentations, despite potential differences, is likely partly attributable to the fact that repetition is not well suited for typical count models (Poisson or negative binomial regression) because of ‘excess zeros’ (e.g. the 84% of the sample that will not repeat within 1 year). Excess zeros is a source of overdispersion, where the observed variance exceeds that expected under the models’ distributional assumptions. ^{Reference Hilbe6}

Here, an alternative method to analyse self-harm repetition is proposed: the hurdle model. ^{Reference Mullahy7} Hurdle models combine a binary (e.g. logit) model with a zero-truncated count (e.g. Poisson) model. For the self-harm repetition example, the first part tests factors associated with any repetition (repeaters v. non-repeaters) and the second part tests factors associated with the number of presentations (among repeaters). Population-based emergency department data are used to illustrate implementing and interpreting hurdle models. Hurdle models are also shown to be more informative than traditional binary analyses, but also adequately fit these data relative to some other count models.

Method

This is a population-based retrospective cohort study of 12- to 17-year-olds presenting to the emergency department for self-harm in Ontario, Canada. Data are from the National Ambulatory Care Reporting System (NACRS), covering a 7-year period (1 April 2002 to 31 March 2009). The data capture every emergency department visit; all legal residents are insured for acute and primary healthcare services and every hospital submitted NACRS emergency department data. The 2006 Ontario population of 12- to 17-year-olds was about 1 million. ⁸ Ethical approval was obtained from St Michael's Hospital.

Self-harm presentations were identified using ICD–10 criteria (intentional self-harm: X60–84). ⁹ Index episodes were identified as an individual's first during the study period. Anonymous identifiers on each record allowed follow-up of subsequent presentations. The exposure was in-patient admission resulting from the index episode, chosen to represent an important, well-defined aspect of clinical management. The outcome was repeat self-harm presentation within 1 year of the index episode, calculated from the emergency department or in-patient discharge date (as applicable). Individuals with less than 1 year of follow-up data (index episodes after March 2008 and those who died), were excluded. The data were analysed in SAS (version 9.1.3). First, two binary models were fitted: logistic regression, categorising the outcome as repeater or non-repeater; and survival analysis (Cox regression), using time to first repeat presentation as the outcome. Next, four count models were fitted: Poisson, negative binomial, Poisson hurdle and negative binomial hurdle. The outcome was the count of repeat self-harm presentations, incorporating random effects for hospital-level clustering. ^{Reference Min and Agresti10} Model fit was compared using Akaike and Bayesian information criteria (AIC and BIC), ^{Reference Lui and Cela11} where smaller values are better.

Results

The cohort included 10 937 individuals (8012 (73.3%) girls and 2925 (26.7%) boys), of whom 3546 (32.4%) were admitted at their index episode. Overall, 1325 (12.1%) made at least one repeat self-harm presentation within 1 year of their index episode (classified as repeaters), and this proportion was almost identical in the two exposure groups (12.2% and 12.1% among admitted and non-admitted respectively).

The binary models, logistic regression and survival analysis found no statistically significant association between admission and repetition (odds ratio (OR) 1.01, P = 0.8309; hazard ratio 1.01, P = 0.8614). The count models' AIC and BIC (Table 1) suggest substantial improvement in model fit from selecting the negative binomial, Poisson hurdle and negative binomial hurdle models over the Poisson model. Both fit indices favour the negative binomial hurdle model, demonstrating their flexibility in accounting for overdispersion from excess zeroes as well as other sources. ^{Reference Rose, Martin, Wannemuehler and Plikaytis12} Interpreting the negative binomial hurdle model, similar to the binary analyses, the logit portion shows admission subsequent to the index episode was not associated with repetition (OR = 1.02, P = 0.7269). However, the negative binomial portion shows that, among repeaters, the estimated number of repeat presentations is lower among those admitted (P = 0.0179).

Table 1 Count model results for the association between in-patient admission and repeat self-harm presentation(s) within 1 year for 12- to 17-year-olds in Ontario, Canada

Count models	Coefficient (standard error)	P	AIC	BIC
Poisson	-0.0537 (0.0495)	0.2791	12317	12327
Negative binomial	-0.0606 (0.0697)	0.3854	10601	10613
Poisson hurdle			10940	10962
Logit	0.0345 (0.0637)	0.5889
Poisson	-0.2527 (0.0832)	0.0027
Negative binomial hurdle			10580	10605
Logit	0.0223 (0.0637)	0.7269
Negative binomial	-0.2854 (0.1194)	0.0179

Discussion

These results highlight the importance of considering the number of repeat presentations when studying self-harm. Others have already acknowledged the tendency for self-harm repetition studies to ignore multiple repetition and proposed alternative analyses, including recurrent event survival analysis ^{Reference Lilley, Owens, Horrocks, House, Noble and Bergen13,Reference Reith, Whyte and Carter14} and multinomial logistic regression. ^{Reference Haw, Bergen, Casey and Hawton1} Here, we have shown that hurdle models are also an appropriate and useful statistical method. They are more informative than binary analyses because the investigator retains the ‘repeaters v. non-repeaters’ analysis while gaining the second part of the model (the number of subsequent self-harm presentations among repeaters). The hurdle model's two-part results have a similar substantive advantage over conventional count models, as well as the statistical advantage of accounting for overdispersion from excess zeros. Aside from hurdle models, zero-inflated models might also be considered for these data. Both are two-part models, address overdispersion from excess zeros, and tend to produce similar fits to the data, so the decision between them depends on the study's design and purpose. ^{Reference Rose, Martin, Wannemuehler and Plikaytis12} Although hurdle models assume all those in the study sample are at risk of events, the zero-inflated models assume some will not experience any events because they are never at risk. Given that this sample was assembled from individuals who presented to the emergency department for self-harm, the assumption of the former seems more appropriate. The main weakness of hurdle models, however, is that unlike survival analysis, they require uniform follow-up. This drawback was not problematic here; each cohort member had 1 year of follow-up after their index episode and loss to follow-up was considered minimal. Yet, in other studies, accommodating variable follow-up times (e.g. from attrition, timing of analysis, or including fatal events as outcomes) may be an important consideration.

In this example, admission subsequent to a self-harm presentation did not influence the odds of repetition within 1 year, but it was associated with fewer repeat presentations (among repeaters). If this association is causal, it has important clinical and cost implications. It suggests that hospitalisation may be beneficial in reducing future self-harm episodes. Potential explanations involve service access, either during admission or as follow-up. However, without accounting for potential biases, any such substantive interpretations remain speculative. Further work is needed to assess this finding, accounting for severity, self-harm method and service use.

Ultimately, these results demonstrate that when studying self-harm repetition, incorporating the number of repeat presentations can be of value for policy, research and clinical practice. Hurdle models are one way of assessing these patterns.