The predictive model
The predictive model for M. graminicola evaluated in this paper is described in detail by te Beest et al. (2009). The model gives an indication of disease risk just before the first fungicide spray is applied. In the UK it is common practice to apply three fungicide sprays, predominantly to control M. graminicola, in correspondence with wheat leaf emergence (Paveley et al., 2009). The first spray is timed when the third leaf (counting down from the top of the plant) is fully emerged, typically at growth stage 31/32 (first/second node detectable, typically midApril); a second spray at growth stage 39 (flag leaf fully emerged, typically end of May); and a third spray at growth stage 59 (emergence of head complete, typically midJune). The prediction is timed at growth stage 31 (1st node detectable).
The model predicts the presence or absence of a ‘damaging epidemic’, defined as 5% or more disease severity at growth stage 75 (medium milk). This 5% threshold has been used before as the economic injury level for M. graminicola on winter wheat in the UK (Gladders et al., 2001, Pietravalle et al., 2003). The model is the result of ‘window pane’ statistical analysis (Coakley & Line, 1982; Pietravalle et al., 2003), with which key weather variables suitable for prediction were identified, resulting in the following discriminant model:
 (1)
A damaging epidemic is predicted if the outcome of Eqn 1, F(Rain,MinT), is larger than 0. The first variable used in the M. graminicola predictive model is daily rain above 3 mm accumulated in the 80day period preceding growth stage 31 (Rain, Eqn 1). This variable works as follows: if, for example, the daily rainfalls over a 4day period were 5 mm, 2 mm, 5 mm and 10 mm, this would contribute 2 mm, 0 mm, 2 mm and 7 mm to the accumulation, which would be 11 mm in total. The second variable is the daily minimum temperature above 0°C accumulated in a window that starts 120 days before growth stage 31 and ends 70 days before growth stage 31 (MinT, Eqn 1).
Data
The disease data used for the evaluation originate from the Cropmonitor project (http://www.cropmonitor.co.uk), which assesses and reports disease severity from experimental sites across the UK (Table 1). The Cropmonitor data are independent of the data used to build the predictive model, and are described in te Beest et al. (2009). These data were measured in 2003–05, while the model was constructed on data from the period 1994 to 2002. At each year–site combination observations of disease severity were made on cultivars differing in their disease resistance. All observations were from plots untreated with fungicides. Averaged over 2003–05, the predictive model predicted a damaging epidemic in 61% of the experiments in these data. Disease severity on leaf 2 at growth stage 75, commonly used as a benchmark (Thomas et al., 1989), was used to summarize the observations and to relate disease severity to yield loss (Table 2). The diseaseseverity observations were divided into two groups according to HGCA Recommended List resistance rating (Anonymous, 2003–05). This rating system ranges from 1 to 9, with 1 being very susceptible and 9 very resistant. The highest resistance rating in the dataset was 7 and the lowest was 3. The resistant group contained cultivars with resistance ratings of 6 and 7 and the susceptible group contained cultivars with ratings of 3, 4 and 5. This split was chosen so that both groups (and subgroups) were large enough for fitting a severity probability distribution. The collection of observations in a group represented the situation without prediction. Two subgroups were constructed in each group, one consisting of the observations for which a damaging epidemic was predicted and one where no damaging epidemic was predicted. To calculate the model predictions, weather data (minimum temperature and rain) were retrieved from meteorological stations near the experimental site (Table 1) from the BBSRC meteorological website (http://www.bits.bbsrc.ac.uk/metweb).
Table 1. Overview of years and sites used with model predictions and realized outcomes for Mycosphaerella graminicola on winter wheat Site  Year  Prediction^{a}  Resistant^{b}  Susceptible^{c} 


Askham Bryan (Yorkshire)  2004  E  3/0  2/2 
Boxworth (Cambridgeshire)  2003  N   1/0 
2004  E  3/2  1/1 
Drayton (Warkwickshire)  2003  N   1/1 
2004  E  3/1  2/2 
2005  N  2/0  2/0 
Exeter (Devon)  2003  E   1/1 
2004  E  3/1  2/2 
Gleadthorpe (Nottinghamshire)  2003  N   1/1 
2004  E  3/3  2/2 
High Mowthorpe (Yorkshire)  2003  N   0/0 
2004  E  3/3  2/2 
2005  E  2/0  2/2 
Lavenham (Suffolk)  2003  E   1/0 
2004  N  3/1  2/2 
2005  N  2/0  2/0 
Newcastle (Tyne & Wear)  2004  E  3/0  2/2 
2005  E  2/0  2/1 
Rosemaund (Herefordshire)  2003  N   1/1 
2004  E  3/3  2/2 
2005  E  2/2  2/2 
Terrington (Norfolk)  2003  N   1/1 
2005  E  2/0  2/2 
Wye (Kent)  2003  N   1/1 
York (Yorkshire)  2005  E  2/2  2/2 
Table 2. Summary of the data by subgroup for Mycosphaerella graminicola on winter wheat  Severity (mean %)  Observations  Yearsite combinations  α^{a}  Fit^{b} 


All cultivars  9·5  81  25  10·6  0·45 
No epidemic predicted  5·0  20  10  20·2  0·17 
Epidemic predicted  10·9  61  15  9·2  0·66 
Resistant cultivars  5·9  41  16  16·9  0·21 
No epidemic predicted  1·3  7  3  74·6  0·10 
Epidemic predicted  6·8  34  13  14·7  0·35 
Susceptible cultivars  13·1  40  25  7·6  0·41 
No epidemic predicted  6·9  13  10  14·5  0·74 
Epidemic predicted  16·1  27  15  6·2  0·26 
Disease risk
To quantify the economic benefit of using the model predictions we first derived the risk levels needed to define the scenarios. An exponential probability density distribution, α·e^{αS}, in which S is the disease severity and α is the exponential probability distribution parameter, was fitted to each subgroup (Fig. 1). The exponential distribution quantified the probability that, for a given year–site combination, disease severity S develops. A KolmogorovSmirnov goodnessoffit test was performed with genstat (Payne et al., 2004) to test whether the fitted data were significantly different from an exponential distribution. Based on the Pvalues (Table 2) all fits were acceptable and not significantly different from an exponential distribution.
Yield loss increases with disease severity, and the disease severity which causes a certain economic loss (specified later) is termed the ‘critical disease severity’. The probability of a disease severity (S) equal to or greater than this critical disease severity (S_{c}) is equal to the area under the upper tail of the exponential distribution. By integrating over the domain [S_{c},∞] the probability of a disease severity greater than S_{c} can be derived according to:
 (2)
where α is a parameter of the exponential distribution and K is the probability of disease with a severity equal to or greater than the critical disease severity (S_{c}).
Gross margin, M. graminicolarelated costs, and economic optimal dose
To quantify the economic benefit of using the model predictions, we quantified the expected costs related to M. graminicola. The gross margin (Nix, 2005) is defined as the economic output per hectare minus the variable costs per hectare. The economic output per hectare is calculated as the production (P) in tonnes per hectare multiplied by the grain price (PW) per tonne. The variable costs are all costs that vary with crop area, such as purchase of seeds, chemicals and fertilizer. This excludes fixed costs such as building and machinery maintenance or mortgage payments, which are all not directly dependent on crop area and can differ considerably between farms. The variable costs were subdivided into costs that were, and costs that were not, related to M. graminicola. The gross margin was then calculated according to:
 (3)
where VC were the variable costs unrelated to M. graminicola, and MC were the variable costs related to M. graminicola, defined as:
 (4)
The costs related to M. graminicola (MC) were the sum of the costs of fungicides and the costs resulting from M. graminicolarelated yield loss. Costs of fungicides were calculated by multiplying the fungicide price (PD), in £ per dose, by the fungicide dose (D), in dose per hectare. The yield loss was calculated from the expected disease severity without treatment [E(S)], the percentage yield loss for each percent disease severity (L), and the percentage of disease remaining after fungicide dose D was applied (DR). Disease severity remaining (DR) was defined as the percentage of disease severity that occurred at growth stage 75 after using fungicide dose D compared to the disease severity that would have occurred without using fungicides. By multiplying the percentage yield loss with the production of wheat in tonnes per hectare (P) and the price of wheat in £ per tonne (PW) yield loss was quantified into costs in £ per hectare.
The relationship between disease severity and yield loss is typically described with a simple empirically derived equation with a yieldloss ratio describing the percentage yield loss per percent disease (Gaunt, 1995). For M. graminicola such relationships have been previously derived by King et al. (1983), Thomas et al. (1989) and Paveley et al. (1997). There is also evidence for a disease density dependent relationship (Shaw & Royle, 1989), but because of the variability in yield loss, such relationships seldom provide a better estimation. Based on yieldloss data over the years 2003–05 from the UK HGCA Recommend List trials (http://www.hgca.com) a yieldloss ratio (L) was calculated (Table 3).
Table 3. Overview of the parameters used and their dimensions and values Variable name  Symbol  Value  Dimension 


Wheat yield  P  9·15  t ha 
Wheat price  PW  100  £ t^{−1} 
Fungicide price per dose  PD  25  £ per dose 
Yieldloss ratio  L  0·81   
Doseresponse factor  k  2·8  ha per dose 
Maximum reduction 3 sprays  RD  99·6^{a}  % 
Variable costs excluding fungicides  VC  230  £ ha^{−1} 
Seed cost   40  £ ha^{−1} 
Fertilizer cost   100  £ ha^{−1} 
Chemical cost excl. M. gram. fungicides   90  £ ha^{−1} 
Yield loss is typically described by the maximum attainable yield minus the actual yield (Nutter et al., 1993). An indication of both over the years 2003–05 was obtained from the UK HGCA Recommended list trials (http://www.hgca.com), where the mean yield in fungicidetreated plots was 10 t ha^{−1} (at 85% dry matter) and the mean yield in plots untreated with fungicides was 8·3 t ha^{−1}. Assuming half the damage was caused by M. graminicola gave a yield loss caused by M. graminicola of 0·85 t ha^{−1}. The attainable wheat yield (P) while only considering M. graminicola then became 9·15 t ha^{−1}. The yieldloss ratio was calculated by relating the disease severity over 2003–05 to the yield loss in 2003 to 2005. Data from Tables 1 and 2 were used to quantify M. graminicola disease severities on leaf 2 and the HGCA data were used as an indication of yield loss (disease severity data from leaf 2 are not available from the HGCA trials, and yield loss data were not available for the data from Tables 1 and 2). As both the HGCA data and the data in Tables 1 and 2 were from a wide range of sites and cultivars, they were assumed to represent countrywide averages over the years 2003–05. With an attainable wheat yield (P) of 9·15 tonnes, a yield loss of 0·85 tonne, and a M. graminicola percentage of 11·4%, the yieldloss ratio became 0·81% yield loss per percentage M. graminicola on leaf 2. In Thomas et al. (1989) a yieldloss ratio of 0·42 was calculated and in King et al. (1983) a yieldloss ratio of 0·55 was estimated. The ratio calculated here is higher than both these ratios, but recent work suggests there has been a decrease in tolerance to M. graminicola in wheat (Parker et al., 2004; Foulkes et al., 2006), which suggests that a higher yieldloss ratio may be appropriate.
The grain price (PW) in £ per tonne is marketdependent and variable and was fixed at a value which was representative of a typical value during recent decades based on data from the HGCA (http://www.hgca.com).
The percentage disease severity remaining at growth stage 75 after applying fungicide dose D (DR) depends on the maximum amount of reduction obtainable with fungicides (RD) and the fungicide response factor (k) and was modelled with an exponential doseresponse curve (Paveley et al., 2000, 2001):
 (5)
where S is the disease severity without fungicide treatment, RD is the maximum percentage of reduction in disease severity possible with an infinite fungicide dose, and k describes the curvature of the response curve.
Epoxiconazole was used as an example fungicide, because it is one of most effective and widely used fungicides against M. graminicola. One litre of the commercial product Opus (BASF plc) per hectare is one recommended dose and contains 125 g of the active substance epoxiconazole. It was assumed that all epoxiconazole used without mixture on commercial crops (Garthwaite et al., 2004) was used predominantly against M. graminicola and that other fungicides were used simultaneously to control other diseases. The product label limits the total dose applied in any one season to a maximum of 2 L commercial product ha^{−1}. Values for RD and k were calculated for the period 2003–05. A single dose of Opus had an average k value of 3·55 (k_{s}) over the years 2001–03 and a diseasereduction percentage (RD_{s}) of 84% (Lockley & Clark, 2005). There are typically three fungicide sprays applied per season to wheat in the UK, each at approximately half the recommended dose (as specified on the product label; Finney, 1993). The disease reduction percentage (RD) for triple sprays was calculated as (1−RD_{S})^{3} (Paveley et al., 2003), in which RD_{s} is the diseasereduction percentage of a single spray. The k value was calculated by fitting a single dose response curve, (1−RD + RD·e^{−k·Dose}), over a triple dose response curve, , assuming all three sprays are of equal dose in which k_{s} stands for the k value for a single fungicide spray. Variable costs were based on data from Nix (2005).
The yield loss [YL(D)] caused by M. graminicola as a function of fungicide dose D can be calculated from Eqns 4 and 5 as:
 (6)
To calculate the expected yield loss caused by M. graminicola,Eqn 6 is multiplied by the probability density function of disease severities (Eqn 2) and integrated over the possible range of disease severities from 0 to infinity:
 (7)
This can also be written as:
 (8)
In this calculation, the integral is the equivalent of the mean expected severity [E(S)]. By combining Eqns 4–8 the expected costs at a dose D can be calculated as:
 (9)
If the fungicide dose is increased, the costs of fungicide will increase and the costs of M. graminicola yield loss will decrease. For any given disease severity which would have occurred without treatment, there is a dose where the combination of these two costs is minimal. This dose will be called the economic optimal dose as defined by Paveley et al. (2001). To calculate this economic optimal dose, Eqn 9 is differentiated with respect to dose D, the derivative set to 0, and the derivative is rewritten as a function of the economic optimal dose (D_{E}), resulting:
 (10)
Costs with risk
Although Eqn 10 defines the economic optimal dose that minimizes the costs and maximizes the gross margin, as argued in the introduction, the actual aim of growers in a spray decision may not necessarily be to find the average economic optimal dose, but to reduce the risk of a severe economic loss.
The risk of a certain economic loss, measured with the costs or the (reduction in) gross margin, was earlier defined as being equal to the risk of a ‘critical disease severity’ as risk level K (Eqn 2). What level of disease severity is critical depends on what costs are acceptable. To calculate the critical disease severity, S_{c}, Eqn 9 can be rewritten as:
 (11)
The probability of certain costs (MC) can then be calculated by substituting Eqn 11 (S_{c}) into Eqn 2, resulting in:
 (12)
From equation 12 the costs with risk (MC_{K}) can be calculated according to Eqn 13, where MC_{k} are those costs where the actual costs in a certain year can be expected to be higher with probability of K (risk), and lower with probability 1−K:
 (13)
The dose at which the costs at risk level K are minimal (D_{K}) can then be calculated by differentiating Eqn 13 with respect to fungicide dose D and equating the derivative to zero:
 (14)
Equation 13 is similar to the equation for expected costs (Eqn 9) with the expected severity E(S) replaced by ln(K)/α. Since for the exponential distribution E(S) = 1/α both equations have the same outcome for K = e^{−1 }=^{ }0·37.
All parameter values used in the calculations were derived from literature (Table 3). In correspondence with the disease data, all parameters were, where possible, estimated for the period 2003–05 for the UK.
Value of prediction
The value of prediction can be quantified in fungicide dose or costs. The value of prediction measured by fungicide dose is the dose that would be saved if predictions were followed. The value of prediction measured by costs can be calculated in expected costs, which quantifies the gain in income that can be expected if predictions are followed. The value of prediction is calculated as the difference between scenarios with and without model predictions. Both are calculated with the same risk level to allow comparison. In general, the value of prediction (VP) is calculated as:
 (15)
where Sce(WP) is the measure (fungicide dose or costs) resulting from the scenario without predictions, Sce(E) is the measure if a damaging epidemic is predicted, and Sce(A) is the measure if no epidemic is predicted. The scenario with model predictions is calculated by combining Sce(A) and Sce(E) according to a weighted average based on the proportion of cases in which a damaging epidemic is predicted (θ). On average the risk prediction model predicts a damaging epidemic in 61% (Table 1) of the cases on the dataset (θ is 0·61). This means that in this paper the result of predicting a damaging epidemic has greater weight in calculating the value of prediction.
In the scenario without model predictions, the optimum fungicide dose is calculated using all data in a particular resistance rating group (Fig. 1a, d, g) and the appropriate α value (Table 2). In the scenario with model predictions, the optimum fungicide dose is calculated separately for the subgroups of data where a damaging epidemic is predicted (Fig. 1c, e, i) and where no damaging epidemic is predicted (Fig. 1b, e, h), also with the appropriate α value (Table 2). To calculate the value of prediction measured by fungicide dose, Eqn 14 is substituted for Sce() in Eqn 15 according to Eqn 16 (see appendix). For the value of prediction measured by expected costs the optimum dose at a certain risk level, Eqn 14 is substituted into Eqn 9, which is then substituted for Sce() in Eqn 15 according to Eqn 18 (see appendix).