© 2003 Plant Management Network.
Bayesian Approaches to Plant Disease Forecasting
Jonathan Yuen, Department of Ecology and Crop Production Science, Swedish University of Agricultural Sciences, SE 750 07 Uppsala, Sweden
Yuen, J. 2003. Bayesian approaches to plant disease forecasting. Online. Plant Health Progress doi:10.1094/PHP-2003-1113-06-RV.
Prediction of disease occurrence is a well known historical theme, and has begun to receive new interest due to internet-based prediction systems. The evaluation of these systems in a quantitative manner is an important step if they are to be used in modern agricultural production. Bayes’s theorem is one way in which the performance of such predictors can be studied. In this way, the conditional probability of pest occurrence after a positive or negative prediction can be compared with the unconditional probability of pest occurrence. Both the specificity and the sensitivity of the predictive system are needed, along with the unconditional probability of pest occurrence, in order to make a Bayesian analysis. If there is little information on the prior probability of disease, most predictors will be useful, but for extremely common or extremely rare diseases, a Bayesian analysis indicates that a system predicting disease occurrence or non-occurrence will have limited usefulness.
The prediction of pests and diseases in agricultural crops is a problem that we still strive to solve today. This is an ancient historical theme, and early references to pest prediction in the Bible (Joseph's prediction for the pharaoh) indicate the importance that our forefathers placed on pest prediction, though as modern scientists we may have difficulty justifying such prediction methods and may question the historical accuracy of this account.
Our view of causality with regards to plant disease would probably rule out interpretation of dreams (Joseph's method) as a way to predict occurrence of diseases in crops, but one can ask if modern methods that rely on our knowledge of the biology and ecology of the crops are better than dreams. We would hope so, but one would then want objective methods for evaluating such predictive systems. Such systems have begun to receive increased attention due to Internet-based implementations, but one should keep in mind that Mills’s rules for the prediction of apple scab actually predate modern computer-based methods. Whether a predictive system is a table, a set of printed cards, or elaborate internet-based graphic systems, there is still a basic set of decision rules that should be evaluated in an objective manner.
A key concept to keep in mind is that many of these predictive systems are fallible, and sometimes give incorrect predictions. How often they are incorrect, and how this can affect their usefulness, is the theme of this paper.
Materials and Methods
A number of measures of a predictor can be derived by comparing what the decision rules predict with what actually happens. If we take a simple case, the presence and absence of disease are the two possibilities for both predictions and outcomes, and we thus have a total of four possibilities. More complicated situations (timing of multiple applications during the growing season, for example) are much more complicated and lie outside the scope of this article. This simple example was illustrated quantitatively by Yuen and Hughes (5) with data borrowed from Jones (3) who used the incidence of eyespot (caused by Pseudocercosporella herpotrichoides) at GS 30-31 to predict the need for fungicide treatment (Table 1). In Table 1, the actual requirements are arranged as columns, and the different predictions appear as rows.
Table 1. Actual requirement for fungicide compared with a predictor
One important measure is the sensitivity of the predictor. This is the proportion of correct predictions that the pest will occur among those fields where the pest actually occurred. This is also referred to as the true positive proportion. In the data presented in Table 1, this is 28/41 or about 0.68. Another critical measure is the specificity of the predictor. This is the proportion of fields with the correct prediction that the pest will not occur among those fields where the pest was actually absent. In our data, this is 7/17. Another measure that is used is the false positive rate, which is 1-specificity, or 10/17 with our data. Likewise, a false negative rate can also be calculated (1-sensitivity). Changing the sensitivity of a predictor by varying the decision threshold will affect the specificity, such that increasing one decreases the other. Simultaneous improvement of both sensitivity and specificity requires either a reformulation of the predictive system, additional information, and usually both of these.
An ideal predictive system will have a sensitivity and specificity of 1.0, which would make the analyses proposed in this article superfluous. Our attention is generally focused on the sensitivity of a predictor, since the mistakes associated with poor sensitivity (missing an application of a control measure such as a pesticide) are generally viewed as serious ones by farmers. The mistakes associated with poor specificity (such as applying pesticides when they are not needed) has traditionally not been viewed as serious by plant pathologists, but for environmentalists this may be a valid concern. This quantity is also more difficult to measure, since an untreated portion has to be left in a field trial to see if a control measure truly was justified, whereas imperfections in sensitivity are painfully obvious.
Information about sensitivity and specificity of a predictor is usually not what is required by a decision maker. A farmer, for example, would most likely want to know whether or not to apply control measures, and this is most likely available from the probability of pest occurrence, which is a formalized way of referring to the experience of the farmer. Most farmers can say whether they have had a problem before, or if this is something new. Using this experience to make inferences about the future is of course dependent on assumptions about unchanging conditions, cultivars, farming practices, pathogen genotypes, etc. Thus, without any additional information, a farmer (or any decision maker) would act as though the future would be similar to the past.
An alternate measure can be obtained by examining the rows of Table 1 instead of the columns. One calculates the proportion of correct decisions of the 38 positive predictions. In this case, the measure, known as the positive predictive value (PPV), is 28/38. A similar measure is also available for the negative predictions (7/20) and is known as the negative predictive value (NPV). These measures, which are closer to the information needed by a decision maker, are unfortunately dependent on the prevalence of the disease (i.e., how common the disease is). As prevalence decreases PPV also decreases but NPV increases. Conversely, as prevalence increases, PPV increases, but NPV decreases. Due to their dependence on prevalence, these measures have limited utility.
In statistical terms, this can be defined in terms of probabilities. If disease occurrence is referred to as A, the probability of disease occurrence can be written Pr(A). In the absence of any predictive system, this is the prevalence of the disease, and is the probability of disease occurrence in the absence of any additional information. In Bayesian terms this is often referred to as a prior probability, since this is the probability of disease before we use the predictor. Note the connection to the experience of the farmer. What we would like is the probability of pest occurrence after using a positive prediction. These are called conditional probabilities, and are written as Pr(A|B), where B represents a positive prediction. Likewise, there is the probability of pest occurrence after a negative prediction. This is denoted Pr(A|B), where B represents a negative prediction.
We can calculate the conditional probabilities by combining the sensitivity and specificity of a predictor with information on the prior probability of disease using Bayes’s theorem. The usual form of Bayes’s theorem is presented as Equation 1, and one can see that it contains the both the sensitivity and prevalence in the denominator and the first part of the numerator, and even the false positive rate appears in the second portion of the numerator.
Thus, the probability of pest presence if the predictor is positive (Pr(A|B)) equals the sensitivity (Pr(B|A)) times the prior probability of the pest being present (Pr(A)) divided by the sum of sensitivity (Pr(B|A)) times probability of the pest being present (Pr(A)) and false positive rate (Pr(B|A) times probability of the pest not being present (Pr(A)). The probability of the pest being absent is written as Pr(A); thus Pr(A) + Pr(A) = 1.0.
This form of Bayes’s theorem is difficult to use, and can be simplified by the use of likelihood ratios (LR) and conversion of probabilities to odds. The LR for a positive prediction (LR+) is sensitivity/(1-specificity) and the LR for a negative test (LR-) is (1-sensitivity)/specificity. Odds can be calculated from probabilities.
Using this form yields a much simpler form of Bayes’s theorem:
OddsPosterior = OddsPrior LR 
Thus the odds after using the predictor equals the odds before using the predictor times the likelihood ratio.
Using the data from the eyespot example (with sensitivity 0.68 and specificity 0.41), the LR+ is 0.68/(1-0.41) or 1.15 and the LR- is (1-0.68)/0.41 or 0.78. Thus, if we were to use the predictive system proposed by Jones (3), a positive prediction would increase the odds of disease by 15% and a negative prediction would decrease it by 22%. The prior probabilities would vary from field to field, but in the absence of any other information, one might use the overall probability of eyespot. If we borrow information from Cook et al. (1) we might start with an odds of 0.19 (5). With the corresponding likelihood ratios and a prior odds, we can now use Equation 3. Thus after a positive prediction the odds of eyespot would increase from 0.19 to yield a posterior odds of 0.19 × 1.15 or 0.23. Likewise a negative prediction would yield posterior odds of 0.19 × 0.78 or 0.15. These odds can be converted to probabilities using the inverse of Equation 2, which would be 0.19 after a positive prediction for eyespot, and 0.13 after a negative prediction for eyespot.
The critical value for eyespot incidence in that example was 20% of the tillers (a fungicide application was justified if it was greater than or equal to 20%). This is an example of what is referred to as a decision threshold (2), and it could of course vary, with resulting changes in both sensitivity and specificity (Yuen et al., 1996). A plot of these (usually the true positive rate as a function of the false positive rate) is referred to as an ROC curve. An example of a ROC for the prediction of plant disease, borrowed from Yuen et. al., 1996, is presented in Fig. 1. Many of the predictive systems for disease prediction have a continuous or semi-continuous point scale, with a decision threshold, and these can easily be presented along with an ROC curve.
Twengström et al. (4) developed a predictor for Sclerotinia stem rot based on weather, cropping history, and other variables. They presented their results in part as an ROC curve, and suggested decision thresholds with varying specificity and sensitivity. Lower thresholds will increase the sensitivity of a predictive system but will also increase the false positive rate (decrease specificity). Higher thresholds can reduce the false positive rate (increase specificity) at the expense of decreased sensitivity. This leads to varying sensitivity and specificity, and the likelihood ratios for the different decision thresholds are presented in Table 2.
Table 2. Likelihood ratios for positive and negative prediction of Sclerotinia stem rot based on varying decision thresholds.
a Sensitivity is the conditional probability of prediction disease occurrence given that disease actually occurred.
b Specificity is the conditional probability of predicting the absence of disease given that disease actually was absent.
c LR+ is the likelihood ratio of a positive prediction, calculated by sensitivity/(1-specificity).
d LR- is the likelihood ratio of a negative prediction, calculated by (1-sensitivity)/specificity.
Although prior probabilities will vary from field to field (or from farmer to farmer), a rough estimate of them can be obtained by examining the average need for fungicide sprays from two regions or the single worst year (5). These are shown in Table 3.
Table 3. Prior probabilities and odds for Sclerotinia stem rot based on twenty-year averages or single worst year from two provinces in Sweden.
Use of Bayes’s theorem gives both increases (Table 4) and decreases (Table 5) in the odds of disease occurrence following positive and negative predictions.
Table 4. Increases in the odds or probability of Sclerotinia occurrence after a positive prediction based on varying decision thresholds. Values in body of table are odds, probabilities in parentheses, and are calculated using the prior odds from Table 3 (5).
Table 5. Decreases in the odds or probability of Sclerotinia occurrence after a negative prediction based on varying decision thresholds. Values in body of table are odds, probabilities in parentheses, and are calculated using the prior odds from Table 3 (5).
Sensitivity and specificity can be used to characterize disease prediction rules, and can be summarized as likelihood ratios (LR) for positive and negative predictions. Good predictors have high LR’s for positive predictions and low LR’s for negative predictions. Bayes’s theorem can be used to examine how the probability of disease occurrence changes after using the predictor.
It can be difficult to perform these calculations for many predictive systems used for plant pests. This is in part because the sensitivity and specificity are often not accurately quantified. The fact that the use of a predictive system does not result in more disease than the common practice (or routinely used control measures) is related to the sensitivity, but is not a direct measure of this quantity. Likewise, we can make inferences about the specificity of a predictor by comparing the frequency of pesticide use while following its recommendation with the "usual" frequency, but, as with sensitivity, we have not accurately measured this quantity either.
For extremely common or extremely rare diseases, it may be difficult to develop predictors of pest occurrence that have sufficient sensitivity or specificity such that they would be able to change the behavior of the decision maker. For example, if a decision threshold is selected from Table 2 that equally weighs selectivity and sensitivity (40 points), this gives a value for LR+ of 4.8. If the prior odds is 0.05, a positive prediction will increase the odds to only 0.24. A prior odds of 0.01 would only increase to 0.048 after a positive prediction. A similar argument can be made for extremely common diseases and negative predictions. Thus, these predictive systems will have their greatest usefulness when the prior probabilities lie near 50%, i.e. for diseases that are neither very common nor very rare.
For many pests, the goal of the predictive system is to schedule multiple control measures (such as the frequency of repeated fungicide applications). Thus the problem is not so much if the pest will occur (which is the focus of this article) but when the pest will occur, and the validation of these methods lie outside of this discussion. Many of these systems assume that the pest will occur (i.e., the prior assumption is that the pest will occur) and then try to refine the control measures accordingly. Thus, the fact that such systems do not try to predict the occurrence of the pest is entirely consistent with the Bayesian analyses presented here. This does agree with the concepts presented here, however, since the focus of the problem is not whether the pest will occur during the growing season, but when the pest will occur.
Even for such diseases that are neither very common nor very rare, with unconditional probabilities that lie around 50%, the use of a Bayesian analysis will indicate the magnitude with which disease probabilities will change when the predictor is used. If the sensitivity and/or specificity is poor (with resulting values for the LR’s close to 1.0), only small changes in disease probability will occur when the predictors are used.
Thus, knowledge of sensitivity and specificity of prediction rules would permit targeting of systems where the prior probabilities would allow success. If the prior probability of disease is too large or small, and the performance of the predictive system is poor, one would not expect adoption of the system due to small changes in the disease probabilities.
Even in the absence of predictive systems, a Bayesian analysis would be useful. Knowing the prior probabilities of disease occurrence would determine minimum performance criteria necessary for success.
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