Epidemiology Group, Department of Veterinary Science, University of Liverpool, Leahurst, Neston, Wirral, CH64 7TE, UK

Abstract

Background

Colic is an important cause of mortality and morbidity in domesticated horses yet many questions about this condition remain to be answered. One such question is: does season have an effect on the occurrence of colic? Time-series analysis provides a rigorous statistical approach to this question but until now, to our knowledge, it has not been used in this context. Traditional time-series modelling approaches have limited applicability in the case of relatively rare diseases, such as specific types of equine colic. In this paper we present a modelling approach that respects the discrete nature of the count data and, using a regression model with a correlated latent variable and one with a linear trend, we explored the seasonality of specific types of colic occurring at a UK referral hospital between January 1995–December 2004.

Results

Six- and twelve-month cyclical patterns were identified for all colics, all medical colics, epiploic foramen entrapment (EFE), equine grass sickness (EGS), surgically treated and large colon displacement/torsion colic groups. A twelve-month cyclical pattern only was seen in the large colon impaction colic group. There was no evidence of any cyclical pattern in the pedunculated lipoma group. These results were consistent irrespective of whether we were using a model including latent correlation or trend. Problems were encountered in attempting to include both trend and latent serial dependence in models simultaneously; this is likely to be a consequence of a lack of power to separate these two effects in the presence of small counts, yet in reality the underlying physical effect is likely to be a combination of both.

Conclusion

The use of a regression model with either an autocorrelated latent variable or a linear trend has allowed us to establish formally a seasonal component to certain types of colic presented to a UK referral hospital over a 10 year period. These patterns appeared to coincide with either times of managemental change or periods when horses are more likely to be intensively managed. Further studies are required to identify the determinants of the observed seasonality. Importantly, this type of regression model has applications beyond the study of equine colic and it may be useful in the investigation of seasonal patterns in other, relatively rare, conditions in all species.

Background

Analysis of temporal patterns in data (i.e. data that arises over time) constitutes an important area of statistics, with applications in a wide range of fields from economics to engineering

Colic is an important cause of mortality and morbidity in domesticated horses and has a complex, multifactorial nature

Many standard statistical approaches are built upon the assumption that observations are mutually independent. This assumption is likely to be inappropriate in the case of colic since many factors may be interdependent; observations in adjacent months might be more similar than those which occur months apart due to, for example, similarities in feed types and duration of stabling. Time-series methods provide a valid means of investigating seasonal patterns in colic. Traditional approaches, such as the Auto-regressive Integrated Moving Average (ARIMA) of Box and Jenkins

The aim of this study was to determine if there was any evidence of seasonality in horses presented to a UK referral hospital with particular types of colic. Using a Bayesian approach, we fitted a regression model which incorporated autocorrelation as a latent variable, to reflect the fact that, having taken account of seasonality and trend, any remaining serial dependence may operate over a shorter temporal scale and is likely to represent unmeasured influential covariates which themselves vary over time. In addition we fitted a model without latent correlation but with a linear trend. Based on current evidence in the literature, our

Results

Exploratory data analysis

The total numbers of colic cases for each diagnostic category are shown in Table

Colic categories, case definitions and number of cases in each category admitted to the PLEH between January 1^{st }1995 and 31^{st }December 2004

**Colic category**

**Case definition**

**Total number**

All Colics

All confirmed cases of colic admitted to the hospital

2580

All Surgical Colics

Colic cases with surgical lesions confirmed at exploratory laparotomy or post-mortem examination

1612

All Medical Colics

All colic cases that resolved with medical treatment only

968

Pedunculated Lipoma

Obstruction of small intestine by a pedunculated lipoma diagnosed at exploratory laparotomy or post-mortem examination

231

Epiploic Foramen Entrapment

Entrapment of the small intestine in the epiploic foramen diagnosed at exploratory laparotomy or post-mortem examination

92

Equine Grass Sickness

Equine grass sickeness cases confirmed by histological examination of the ileum

109

Large colon displacements or torsions

Displacement or torsion of the large colon diagnosed by rectal examination, clinical signs and response to treatment; treated either surgically or medically or diagnosed at post-mortem examination

435

Large colon impactions

Primary large colon impactions confirmed by rectal examination and response to treatment (medically treated group) or at exploratory laparotomy

214

Boxplots of de-trended (annual average subtracted) colic admissions by month for each colic admitted to a UK referral hospital between January 1995 – December 2004

Boxplots of de-trended (annual average subtracted) colic admissions by month for each colic admitted to a UK referral hospital between January 1995 – December 2004.

Regression model with seasonal components, trend and an autocorrelated latent variable

The posterior distribution summaries for each colic type are presented in Table

Parameter estimates from the regression models for each colic type.

**Colic type**

**Parameter**

**Posterior Mean**

**Posterior Standard Deviation**

**95% Credible Interval**

All Colics

Intercept

2.849

0.966

1.059, 4.737

S_{12}

0.082

0.043

-0.002, 0.167

C_{12}

0.029

0.043

-0.055, 0.113

S_{6}

-0.132

0.033

-0.196, -0.067

C_{6}

-0.007

0.033

-0.071, 0.058

α

0.005

0.012

-0.018, 0.029

All Surgical

Intercept

2.159

1.089

-0.017, 4.156

S_{12}

0.065

0.054

-0.042, 0.173

C_{12}

0.034

0.055

-0.073, 0.142

S_{6}

-0.114

0.042

-0.196, -0.032

C_{6}

-0.037

0.041

-0.119, 0.044

α

0.007

0.015

-0.024, 0.037

All Medical

Intercept

2.218

1.035

0.271, 4.225

S_{12}

0.117

0.061

-0.001, 0.237

C_{12}

0.021

0.059

-0.095, 0.136

S_{6}

-0.167

0.051

-0.267, -0.067

C_{6}

0.044

0.049

-0.054, 0.140

α

0.004

0.014

-0.023, 0.031

Equine Grass Sickness

Intercept

-1.430

1.278

-3.750, 1.244

S_{12}

-0.275

0.190

-0.655, 0.093

C_{12}

-1.060

0.206

-1.481, -0.673

S_{6}

-0.638

0.172

-0.980, -0.306

C_{6}

0.041

0.163

-0.277, 0.357

α

0.006

0.024

-0.042, 0.054

Epiploic Foramen Entrapment

Intercept

-0.698

1.029

-2.710, 1.456

S_{12}

0.396

0.199

0.013, 0.794

C_{12}

0.590

0.168

0.271, 0.929

S_{6}

0.028

0.167

-0.302, 0.356

C_{6}

0.404

0.169

0.077, 0.736

α

0.002

0.020

-0.038, 0.041

Pedunculated Lipoma

Intercept

-0.253

1.123

-2.489, 1.872

α

0.010

0.019

-0.028, 0.049

Large Colon Impaction

Intercept

0.057

0.957

-1.643, 1.999

S_{12}

0.265

0.118

0.033, 0.497

C_{12}

0.389

0.118

0.162, 0.622

α

0.005

0.021

-0.038, 0.046

Large Colon Displacement/Torsion

Intercept

-0.275

1.112

-2.388, 2.065

S_{12}

0.116

0.101

-0.084, 0.315

C_{12}

0.166

0.110

-0.049, 0.383

S_{6}

-0.234

0.090

-0.410, -0.058

C_{6}

-0.256

0.090

-0.433, -0.080

α

0.005

0.022

-0.039, 0.049

For compactness, S_{12} = _{12} = _{6} = _{6} =

Estimate of model's seasonal component for each colic type

**Estimate of model's seasonal component for each colic type**. For each colic type an estimate of the model's seasonal component was extracted using the posterior mean of the parameter associated with each of the sine and cosine terms based on the frequencies detected for each group in Table 2. With the exception of the large colon impaction group (12 month cycles only) all models incorporated 12- and 6-monthly cycles.

The inclusion of trend and serial correlation together in models of this nature where the number of cases observed at a particular time point is small is potentially problematic, as it may prove difficult to separate positive serial dependence and trend. Indeed, if positive trend exists and there may be positive serial correlation, parameters in the model are potentially highly correlated and the MCMC algorithm struggles in the presence of low counts. As expected there were problems with convergence for many of the models including both terms; we therefore do not include the DICs from models incorporating latent serial correlation together with a linear trend in Table

Deviance information criteria (DICs) for models with a latent autocorrelation structure.

Model

Total

Total surgical

Total medical

EFE

Grass sickness

Large colon impaction

Large colon displacement

Lipoma

No seasonality, no trend

756.83

718.59

607.42

282.14

339.02

414.80

483.27

**419.58**

12-month seasonality, no trend

754.20

720.39

604.86

261.33

280.39

**391.20**

481.77

421.54

12- and 6-month seasonality, no trend

**732.29**

**708.46**

**592.65**

**258.93**

**267.09**

394.47

**459.30**

425.44

A lower DIC statistic can be considered to represent a better model

Models either without trend/with latent serial correlation or with trend/without latent serial correlation, provided better convergence of the MCMC algorithm. For the same data set we find situations where a model with latent serial correlation and 12- and 6-month cycles but no trend term is selected as optimal by DIC comparison (Table

Deviance information criteria (DICs) for models without a latent autocorrelation structure but with trend (Poisson GLMs).

Model

Total

Total surgical

Total medical

EFE

Grass sickness

Large colon impaction

Large colon displacement

Lipoma

No seasonality, no trend

796.18

728.37

645.77

280.79

329.39

429.29

513.58

422.95

12-month seasonality, no trend

793.62

729.59

645.22

258.15

289.65

411.25

513.64

425.78

12- and 6-month seasonality, no trend

773.29

721.87

633.73

**255.35**

277.67

414.21

500.42

429.93

No seasonality, trend

740.80

704.58

613.87

282.11

324.70

422.58

480.01

**413.20**

12-month seasonality, trend

735.36

704.83

611.45

259.15

284.20

**403.98**

478.60

415.88

12- and 6-month seasonality, trend

**717.63**

**698.06**

**601.9**

256.60

**272.58**

407.02

**466.76**

419.74

A lower DIC statistic can be considered to represent a better model.

In the model incorporating latent serial correlation but no trend, it is interesting that although the parameter which controls the dependence (α) does not have a marked effect on the model (as judged by the fact that the credible interval contains 0) the posterior mean for α in all cases, though small, is positive. Whilst we must be cautious concerning over-interpretation of this finding in the presence of large uncertainty, a small but positive effect may represent positive serial correlation, or it could in part be measuring the increasing trend which we were unable to include simultaneously for statistical reasons. (Note that, whilst comparisons within Tables are valid, comparisons between DICs presented in Table

For our purposes, given that our primary interest concerns seasonality, whether we included latent serial correlation or trend, the estimates of the seasonal components were broadly similar across models and this renders our findings regarding seasonality robust in the presence of these largely statistical effects.

Discussion

The aim of the present study was to investigate the seasonality of different types of colic presented at a UK equine referral hospital. Cohen

Two studies in the UK have described an apparent peak in cases of colic of any cause in spring and autumn months

This modelling approach confirmed our hypothesis that EGS would exhibit seasonality, as demonstrated by other workers using different approaches. Although EGS may occur at any time of the year, the peak incidence of this condition in the UK is reported in the months of spring and summer, and the month of May in particular

Use of this model also confirmed our hypothesis that EFE would exhibit seasonality. Using data arising over a 10 year period at the same hospital (1991–2001), multivariable modelling confirmed that EFE was consistently more prevalent in the months of December, January and February

The large colon impaction colic group exhibited 12 month cyclicity, with an increasing number of cases identified in the autumn and winter months (peak December/January) decreasing over the spring months with the lowest incidence over the months of July and August. A slightly different cyclical pattern was identified in the large colon displacement/torsion colic group with peak incidence in the months of Spring and Autumn, similar to that seen in the all colic and all medically or surgically treated colic groups. Hillyer et al.

Obstruction of intestine by pedunculated lipomas in theory should be a random event, and this model confirmed our

We have alluded to the difficulties in detecting serial dependence in the presence of trend when samples are small. With larger samples it might be possible to separate more conclusively trend and latent serial dependence and further research using larger samples sizes is warranted.

Considering first the possible interpretation of latent serial correlation in the context of colic, we take EGS as an example. The role of

Considering now the interpretation of a positive linear trend which was evident in all models excepting that for EFE not including latent correlation, knowledge of continued improvements in the medical and surgical management of colic and resultant increased success rates following treatment

Weather-related factors have not been shown to be statistically significant in relation to colic using traditional methods of analysis, despite many anecdotal reports to the contrary

A number of approaches may be used to investigate temporal patterns in data and, when choosing the most suitable method, it is important to recognise that different types of dependence which are context-specific may occur. First, the number of events in month

An important issue in Markov Chain Monte Carlo (MCMC) based analysis is that of convergence of the Markov Chains and whether the samples being generated are from the true posterior distribution under the model framework. In order to test this, we ran two chains simultaneously using differing starting values, and found that in each case the posterior summaries obtained were analogous. In addition, we examined the

A further issue in Bayesian analysis concerns the sensitivity of the resultant posterior distribution to the choice of prior distribution. Given that, for all parameters, we have selected vague priors we do not believe this to be an issue here; in addition, although the counts at each time point were relatively small, the length of each series was large (n = 120 in all but one case where n = 119) so we would expect the data to dominate.

The issue of determining a suitable autocorrelation structure for the error term in these models is also important. There exists only a single series of data, in contrast with a longitudinal data set for which we can gain knowledge about the autocorrelation structure by exploiting the replication in the data

The exact gastrointestinal dysfunction or lesion is unknown in many cases of colic that occur within the general equine population

The models produced in this paper are biologically plausible and provide useful information on the temporal patterns of different colic types. This work demonstrates in principle how standard and non-standard Poisson regression-based approaches can be used in other veterinary applications where disease incidence is relatively rare. These results also provide an insight into the aetiology of different colic types admitted to a UK referral hospital. There is a suggestion of increased admissions of certain colic types at times of managemental change (surgically and/or medically treated colics, large colon displacements/torsions and EGS) and during periods of intensive management (months of the year when horses are more likely to be stabled or stabled for longer periods of time) e.g. EFE and large colon impaction. These results are based on the findings from a single UK referral equine hospital; further studies are required to determine the relationship between season and colic incidence in other geographical locations using hospital and non-hospital based populations.

Conclusion

We have used a regression model which has the flexibility to incorporate latent serial correlation to explore the seasonal prevalence of different colic types presented at a UK equine referral hospital. This is a novel statistical approach in the field of equine colic research and it has enabled us to confirm a seasonal pattern for equine grass sickness, as demonstrated by other workers using different methods of analysis, and to formally establish the existence of a marked seasonal effect in cases of epiploic foramen entrapment. In addition, a seasonal pattern was evident to admissions of all colic types, all surgical and medical colics and in cases of large colon impaction and large colon displacement/volvulus. Use of this model confirmed that intestinal obstruction by pedunculated lipomas showed no seasonal effect. Knowledge of the seasonal associations with certain types of colic is consistent with an aetiological role for managemental change and periods of intense management such as prolonged stabling. Further studies are required to identify the determinants of the observed seasonality. This type of regression model has applications beyond the study of equine colic and it may be useful in the investigation of seasonal patterns in other, relatively rare, conditions in all species.

Methods

Colic data

All cases of colic admitted to the Philip Leverhulme Equine Hospital, University of Liverpool between 1^{st }January 1995 and 31^{st }December 2004 were reviewed retrospectively. The numbers of colic cases occurring in each of the 120 months under investigation were recorded and aggregated as counts per month in the groups defined in Table

Exploratory data analysis

For each colic type, the effect of increasing yearly case numbers was removed (de-trended) by subtracting an annual average to create a residual

Regression model

Our chosen model for incorporating latent correlation was similar to the generalised linear model with Poisson response and logarithmic link function, which is commonly used to model independent count data

The most general model incorporating cycles at both 6 and 12-month frequencies is as follows: Let _{t }be the number of admissions in month t, and

_{t }~ Poisson(μ_{t})

log(μ_{t}) = β_{0 }+ β_{1 }_{2 }_{3 }_{4 }_{5}_{t}

_{t }~ _{t},σ_{τ}^{2}

μ_{t }= α + _{t-1}

The model detailed above treats the unobserved variables as a latent, temporally varying process (here autoregressive of order 1 so that the latent variable in the current month is allowed to depend via a Normal distribution on the equivalent latent variable in the previous month; in principle in its most general form the structure could be of order

The model was fitted within a Bayesian framework as described in ^{th }observation to reduce correlation between samples in the posterior distribution. Vague prior distributions were adopted for each of the β parameters (reflecting a lack of prior belief concerning parameter values), and the prior distribution for α was Uniform on [-1, 1] (although we believe

Within each selected "best" model for each colic, the posterior mean, posterior standard deviation and 95% credible interval for each parameter are given in Table

For each colic type, an estimate of the model's seasonal component was calculated by exponentiating from the chosen "best" model the sum of the posterior means of the seasonal components on the log scale, thus representing a multiplicative term in a model for the original observations. This enabled us to produce a graphical representation of the cyclical patterns in each group in relation to months of the year (Figure

Authors' contributions

DA designed the study, acquired the data and drafted the manuscript. HC participated in design of the study, performed the statistical modelling and drafted the manuscript. GP conceived the study, participated in its design and critically reviewed the manuscript. CP compiled the data, participated in the design of the study and critically reviewed the manuscript. All authors have read and approved the final manuscript.

Acknowledgements

DA's research training scholarship is funded by the Horserace Betting Levy Board. We are grateful to the Home of Rest for Horses and Petplan Charitable Trust for funding the colic survival study from which some of the data was obtained. We also thank Patrick Brown for helpful discussions regarding modelling.