Evaluation of four modelling techniques to predict the potential distribution of ticks using indigenous cattle infestations as calibration data

Abstract

Efficient tick and tick-borne disease control is a major goal in the efforts to improve the livestock industry in developing countries. To gain a better understanding of the distribution and abundance of livestock ticks under changing environmental conditions, a country-wide field survey of tick infestations on indigenous cattle was recently carried out in Tanzania. This paper evaluates four models to generate tick predictive maps including areas between the localities that were surveyed. Four techniques were compared: (1) linear discriminant analysis, (2) quadratic discriminant analysis, (3) generalised regression analysis, and (4) the weights-of-evidence method. Inter-model comparison was accomplished with a data-set of adult Rhipicephalus appendiculatus ticks and a set of predictor variables covering monthly mean temperature, relative humidity, rainfall, and the normalised difference vegetation index (NDVI). The data-set of tick records was divided into two equal subsets one of which was utilised for model fitting and the other for evaluation, and vice versa, in two independent experiments. For each locality the probability of tick occurrence was predicted and compared with the proportion of infested animals observed in the field; overall predictive success was measured with mean squared difference (MSD). All models exhibited a relatively good performance in configurations with optimised sets of predictors. The linear discriminant model had the least predictive success (MSD ≥ 0.210), whereas the accuracy increased in the quadratic discriminant (MSD ≥ 0.197) and generalised regression models (MSD ≥ 0.173). The best predictions were gained with the weights-of-evidence model (MSD ≥ 0.141). Theoretical as well as practical aspects of all models were taken into account. In summary, the weights-of-evidence model was considered to be the best option for the purpose of predictive mapping of the risk of infestation of Tanzanian indigenous cattle. A detailed description of the implementation of this model is provided in an annex to this paper.

Annex 1. Implementation of the weights-of-evidence model

For a comprehensive description of this method we refer to Bonham-Carter (1997). Denoting with t and t¯ the respective tick presence and absence and with ${\cal T}\equiv \{{\cal T}_{1}\ldots {\cal T}_{n}\}$ a set of evidential themes (GIS maps) and assuming their values τ≡ {τ₁ ... τ _n } at a place of interest, the logit posterior probability of the tick occurrence given τ was calculated as

$$ {L}\{t\vert {\cal T}=\tau\}=L\{t\}+\sum\limits^{n}_{{i}=1} {W}[\tau_{\rm i}], $$

where L{t} is the logit of prior probability of the tick occurrence (estimated e.g. as a prevalence of infested animals), and W[·] is the ȁ8weight of evidenceȁ9 defined as the natural logarithm of the ratio between (conditional) probabilities that the evidential theme ${\cal T}_{\rm i}$ takes the value τ_i given the tick presence or absence, respectively:

$$ {W}[\tau_{\rm i}]=\log[{P}\{{\cal T}_{\rm i}=\tau_{\rm i} \vert t\}/{P}\{{\cal T}_{\rm i}=\tau_{\rm i}\vert \bar{t}\}]. $$

Making allowance for the spatial uncertainty of the true origin of ticks attached to pastoral livestock, the conditional probabilities were related to a broader neighbourhood around sampling sites conceived as a set of (overlapping) discs of a predefined radius centred on each tick sample setting:

${P}^{\ast}\{{\cal T}_{\rm i}=\tau_{\rm i}\vert \hbox{t}\}$ = proportion of total area of the neighbourhood of tick-presence sites occupied by the pattern where ${\cal T}_{\rm i}=\tau _{\rm i}$, and analogically

${P}^{\ast}\{{\cal T}_{\rm i}=\tau_{\rm i}\vert \bar{t}\}$ = proportion of total area of the neighbourhood of tick-absence sites occupied by the pattern where ${\cal T}_{\rm i}=\tau_{\rm i}$.

This allows for home ranges of the sentinel animals within which each point had the same chance to be the origin of the attached tick(s), and ensures that all potential habitat types in the vicinity of the sampling sites were taken into account. A radius of 12.5 km was selected as a way of minimising the MSD criterion.

Although evidential themes ${\cal T}$ can in principle be of logical (binary) as well as categorical or numerical type, in typical WofE applications both categorical and numerical themes are binary-reclassified prior to estimating the conditional probabilities (Bonham-Carter 1997). In this study we opt for an alternative density function approach to numerical themes (Coolbaugh and Bedell 2006) that is more appropriate in this context. Following Robinson et al. (1997), two probability density functions were considered, one for tick presence (f{x| t}) and one for tick absence (f{x| t¯}), and the weights viewed as the probability density ratio

$$ {W}[x] ={f}\{x\vert t\})/{f}\{x\vert \bar{t}\}). $$

The probability density functions were estimated analogously to simple probabilities shown above from proportions of total neighbourhood areas assigned to appropriate histogram bins of an evidential themeȁ9s data distribution. Each histogram was fitted with a curve using a running medians smoother (Härdle and Steiger 1995) and the area under the curve was normalised. In order to ensure that empty bins would not cause numerical problems, a small constant was added to each bin: as small as a goodness-of-fit test (e.g. Kolmogorov–Smirnov) indicates no significant departure from the initial distribution (generally, an equivalent of 0.01% of total neighbourhoodsȁ9 area was sufficient). For values completely absent from a calibration set, it guarantees the neutral weight 0.

Monte Carlo test was used to assess confidence bands of the probability density functions. Briefly, under the null hypothesis of complete spatial randomness (H0) both tick-presence and tick-absence sites originate in a common spatial process and the division into the two sub-sets is random. Thus, by randomly redistributing the sites between two pseudo-presence/absence sub-sets and by submitting them to the same analysis as the actual data, further estimates of the probability density functions under H0 were obtained. The 2.5 and 97.5 percentiles registered in each bin in a series of such simulations defined 95-% point-wise tolerance intervals under H0 (Kelsall and Diggle 1995) (Fig. 3).

Fig. 3

An illustration of probability density functions employed in the WofE model with the R. appendiculatus even sub-set and mean temperature in November as an example: (a) plots probability density functions associated with the tick presence (thick line) and absence (thin line), (b) shows 95%-confidence limits (dashed) of a random model, (c) plots smoothed probability density functions, and (d) shows the corresponding diagram of weights of evidence. Notice that a preference of the tick for colder habitats manifests itself as a skew of the presence density function towards lower temperatures and that in a range of ca. 20–23°C it reaches the significance limit; this tendency is cleared of statistical noise, and appears as a peak of the weights positively discriminating habitats within this temperature range. Multiple such non-correlated pieces of evidence can put together a picture of suitable habitats, which is the principle on which the weights-of-evidence method works