STATISTICAL CONTROL CHARTS

Variability is a phenomenon innate to every single production process and, regardless of the process design or maintenance, it is unavoidable. The quality level of products is directly related to the process variability. Two kinds of process variability can be discerned: the common cause or expected (innate) variation and the special cause or unexpected variation (Shewhart Reference Shewhart1931). Special causes in livestock production may occur as a result of human error, technical failures, contamination of feed or water, health problems, etc. A production system that shows only expected variation is said to be statistically in control and when a special cause gives rise to an altered process performance (process shift), resulting in inferior product quality, the process is statistically out of control (DeVor et al. Reference DeVor, Chang and Sutherland1992; Wadsworth et al. Reference Wadsworth, Stephens and Godfrey2002; Montgomery Reference Montgomery2009). Hence, the basis of quality control is the understanding of the information contained in process variability (Deming Reference Deming1986). The concepts of SPC have been designed for this purpose.

Essentially, SPC is a collection of tools that aims to detect the occurrence of process shifts as a result of special causes. Control charts are SPC tools that enable visualization and detection of process shifts. The limits of the control charts are formulated based on the common cause of variation of the considered process or quality parameter. When the process is in control, nearly all measurements will fall within the range of the control limits and hence most likely within expected random variability. Any problem which causes unexpected process variation, i.e. the special cause variation, results in control chart recordings outside of its limits. At that moment, investigation is required to identify the cause and corrective measures are required in order to resolve the problem. As such, control charts are useful tools for achieving process stability through a reduction of variability (DeVor et al. Reference DeVor, Chang and Sutherland1992; Wadsworth et al. Reference Wadsworth, Stephens and Godfrey2002; Montgomery Reference Montgomery2009).

Since control charts are related to repeated hypothesis testing and hence probability calculations, two types of signalling errors can be made. A Type I error – false alarm or false positive – occurs when the chart produces a signal without a change of the process. A Type II error – false negative – is made when the chart generates no signal in spite of a process change. The rates of these errors depend on the setting of the control limits. The narrower the control limits, the more sensitive the chart (fewer Type II errors) and yet the more false alarms (Type I error).

Performance measures of control charts are based on the frequency of Type I and II errors, expressed in terms of run lengths, i.e. the number of observations between a certain starting observation and an observation which causes an alarm. The average run length (ARL) is often used for summarizing chart performance. The in-control ARL (ARL₀) measures the in control behaviour and hence indicates the average time between false alarms (Type I errors). The ARL₀ should be relatively long: false alarms are obviously undesirable as they waste time and energy and disrupt operations looking for non-existent special causes (Wieringa Reference Wieringa1999). The out of control ARL (ARL₁) is the average ARL until an alarm is given after a shift of the process. Hence, it indicates the speed of detection and should be as short as possible.

A wide range of control charts has been developed for different types of data and for a desired size of the detectable shift. The first and probably best-known control chart was introduced by Shewhart (Reference Shewhart1931). The Shewhart control chart is typically used to detect large shifts (>3 sigma (σ)). Shewhart charts have been developed for all well-known probability distributions (Table 2). A typical example of a Shewhart control chart is given in Fig. 1. However, the classical Shewhart charts require a specification of the process mean and variance before charting. Quesenberry introduced Q-charts for most statistical measures and probability distributions partly to address these limitations (Quesenberry Reference Quesenberry1991, Reference Quesenberry1995, Reference Quesenberry1997). Basically, Q-charts transform original observations into a normally distributed data series with zero mean and unit variance. These Q-charts are useful for short runs in which process mean and variance are unknown (Quesenberry Reference Quesenberry1991, Reference Quesenberry1995, Reference Quesenberry1997).

Fig. 1.

Example of a typical control chart. T=target (process mean); UCL/LCL=upper/lower control limit=T± 3σ. Figure based on data from Montgomery (Reference Montgomery2009), p. 200.

Table 2.

Overview of the different types of Shewhart charts for different probability distributions

* u=c/n.

Shewhart charts have no ‘memory’, meaning that they only use the information of the current measurement. Although they are often supplemented with ‘run rules’, they have not proved to be very sensitive for small shifts (<1·5σ); therefore, other types of control charts, which make use of the information in the previous measurements, have been designed to detect these. The cumulative sum (cusum) chart introduced by Page (Reference Page1954) and the exponentially weighted moving average (EWMA) chart introduced by Roberts (Reference Roberts1959) are typical ‘memorizing’ control charts that are more successful in detecting small shifts, which are particularly of interest in biological processes. These charts are also developed for all well-known probability distributions. The EWMA and cusum charts have comparable performance results for detecting small shifts. The cusum chart is usually a little faster in detecting step changes, while the EWMA is less fit for estimation after the step change has occurred. In addition, the cusum control scheme has the possibility to return to zero. The EWMA chart gives an immediate estimate of the current process mean, while the cusum needs some calculation to determine this (Hawkins & Olwell Reference Hawkins and Olwell1998; de Vargas et al. Reference de Vargas, Lopes and Souza2004; Montgomery Reference Montgomery2009). Making a choice between these two charts depends on the process for which control charts have to be designed. Examples of a cusum chart and an EWMA chart are given in Fig. 2b and c. For more detailed information on the different control charts and statistical and technical concepts for control chart construction, the SPC text books by DeVor et al. (Reference DeVor, Chang and Sutherland1992), Wadsworth et al. (Reference Wadsworth, Stephens and Godfrey2002) and especially Montgomery (Reference Montgomery2009) are highly recommended.

Fig. 2.

(a) Example data taken from Montgomery (Reference Montgomery2009), p. 387. (b) Example of a cusum control chart and (c) of an EWMA control chart for a shift in normal mean, based on (a). The first 20 observations are drawn from a normal distribution N(10, 1), the last 10 observations come from a normal distribution N(11, 1), and so a shift of 1σ occurred. The cusum control scheme was designed to detect a shift of 1σ with reference value k=0·5 and control limit H=5. The cusum chart generates an alarm at observation 29 (C⁺>H). The parameters of the EWMA control chart are the weighting factor λ, which defines how much weight is set on previous observations, and the control limit L. The settings for these parameters in this example are, respectively, λ=0·10 and L=2·7. The EWMA chart generates an alarm at observation 29 (z29>UCL). T=target (process mean); UCL/LCL=upper/lower control limit.

The proper use of control charts requires several assumptions about the model underlying the process. The chart calculations are derived based on these assumptions. The first assumption states that the data generated by the in-control state of the production process must be stationary. A time series is said to be stationary when its statistical properties, mean and (co)variance, are not affected in time and hence the probability distribution is the same for all time periods. The second assumption is on the statistical serial independence of consecutive process recordings. In serial dependence, often expressed in terms of autocorrelation, consecutive recordings are strongly correlated, either positively or negatively. The third assumption is that the data follow the statistical distribution function associated with the specific control chart (normal, binomial, chi-square, Poisson, etc.) (Montgomery et al. Reference Montgomery, Jennings and Kulahci2008; Montgomery Reference Montgomery2009). If all assumptions hold exactly for the considered data, the performance of SPC charts can be exactly determined. Unfortunately, this is not usually the case (Montgomery Reference Montgomery2009), especially for livestock process data. Assumption violations in livestock data are the result of the dynamic aspects of these data. Ignoring assumption violations dramatically compromises control chart performance, especially in case of non-stationarity and autocorrelation. The next section discusses the assumption issues of raw livestock production data.

DYNAMIC ASPECTS OF LIVESTOCK PROCESS DATA

The biological character of livestock production processes is reflected in the data collected from these processes. Typically, biological data are seldom stationary and hence have no fixed target value, and they mostly display autocorrelation between successive observations. Furthermore, in many cases biological data series contain missing values as a result of, for instance, technical failure or (either deliberate or unintentional) managerial error. From a statistical point of view, the assumptions usually made about industrial process data are often invalid for biological process data, which complicates the direct application of control charts for these data.

These peculiarities are a result of the fact that many parameters of livestock production processes are subjected to systematic variation caused by, e.g. the age of the animals or seasonal fluctuations (Sard Reference Sard1979). The most dominant driver behind such age-dependent dynamics are of physiological origin and are primarily related to animal growth or to a change of functionality of producing organs, e.g. pig muscle growth in time, chicken ovary development in time and cow milk gland functionality in different lactation phases and numbers.

Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) show that for chickens, all process parameters (except for mortality) of layer production show a dynamic deterministic trend as a function of age (growth to physical and sexual maturity) of the hens. In dairy cows, de Vries et al. (Reference de Vries, Conlin, Marsh and Reneau1997a) and Lukas et al. (Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) consider the trend in daily milk yield as a result of the stage of lactation and the lactation number. Lukas et al. (Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) also indicated that milk electrical conductivity increases as a function of lactation stage and lactation number. Lukas et al. (Reference Lukas, Reneau and Linn2008) talk about the ‘drift’ in the dry matter intake of dairy cows as lactation progresses. In growing pigs, Madsen & Kristensen (Reference Madsen and Kristensen2005) mentioned a daily circadian rhythm and an increasing dynamic trend in water consumption with age. Modelling of day-to-day dynamic trends in livestock process data has been discussed by, amongst others, Lokhorst (Reference Lokhorst1996) for laying hens, Bebbington et al. (Reference Bebbington, Lai and Zitikis2009) and Silvestre et al. (Reference Silvestre, Martins, Santos, Ginja and Colaço2009) for lactation curves of dairy cows, Darmani-Kuhi et al. (Reference Darmani-Kuhi, Kebreab, Lopez and France2003) for broiler growth and Emmans & Kyriazakis (Reference Emmans and Kyriazakis1997) for pig growth.

Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) also discuss the occurrence of missing values in their layer data. While this issue is less important for control chart calculations, it becomes a problem for estimating the coefficients of models, such as time series models, used to handle the other aforementioned data issues. The problem becomes even more complex in the case of multivariate control charting. Since Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) are the only authors discussing this issue, the current review abstains from an in-depth discussion of this problem.

The dynamic aspects of biological production data impede the direct application and use of classical control charts, mainly because of poor chart performance (see section on Synergistic Control). As a result, the use of this methodology in agricultural production and, more specifically, livestock production is rather limited when compared with industrial production. Nevertheless, the approach has gained interest in recent years, and proper pre-processing of the data will account for the dynamic aspects of livestock process data.

In the following sections, the main control chart aspects – data collection, choice of control chart, chart construction and chart performance – are discussed. The publications included were screened for these aspects and reference is made to the appearance of the aforementioned aspects in the cited papers.

DATA COLLECTION

CHOICE OF THE CONTROL CHART

The choice of the control chart should be based on two criteria. The first is the size of the desired shift to detect: larger shifts use Shewhart charts, while smaller shift detections are performed using cusum and EWMA charts. The second criterion is related to the statistical characteristics of the data. For instance, proportions are assumed to be binomially distributed and hence p-charts or binomial cusum or EWMA charts are used.

In some cases it can be questioned as to whether the control chart used is the most suitable for the considered process parameter. For instance, in the case of large sample sizes, the distribution of proportional data approaches the normal distribution. Such issues can be studied through simulation. de Vries & Conlin (Reference de Vries and Conlin2003, Reference de Vries and Conlin2005) investigated both the binomial and the normal distribution for proportional parameters. They suggest that the normal distribution is appropriate for proportions when the size n of the subgroup is large enough (>20) and found that the performance of x-charts and p-charts, and the normal and binomial cusum on the same proportional data, were similar.

The availability of extra information may lead to the addition of an extra chart. The possibility of rational subgrouping provides variability information within samples and hence enables the use of control charts for variability (R-, R _m-, s- and s ²-charts). Differences between samples indicate a shift in the process parameter (control charts for mean), differences within samples indicate inherent variability (control charts for variability).

Some papers discuss how to choose the most suitable control chart, several dealing with the choice between the cusum and EWMA chart. Both charts have comparable performance results on detecting small shifts (de Vargas et al. Reference de Vargas, Lopes and Souza2004), but the cusum chart is usually slightly faster in detecting step changes while the EWMA is less fit for estimation after the step change has occurred (Hawkins & Olwell Reference Hawkins and Olwell1998; Montgomery Reference Montgomery2009). Practical considerations are often mentioned when considering the choice between cusum and EWMA. Engler et al. (Reference Engler, Tölle, Timm, Hohls and Krieter2009) choose the cusum because it is more straightforward and the settings are easier to handle. Krieter et al. (Reference Krieter, Engler, Tölle, Timm and Hohls2009) consider that the EWMA may be easier to understand since actual values are assigned, namely an estimate of the current process mean (Montgomery Reference Montgomery2009), in contrast to the cusum, which works with deviations. But Krieter et al. (Reference Krieter, Engler, Tölle, Timm and Hohls2009) also state that the cusum has the possibility to differentiate between positive and negative deviations. In the framework of his work, Mertens (Reference Mertens2009) chooses the cusum control scheme because it has the possibility to quickly return to zero. Lukas et al. (Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) use a combination of the Shewhart chart and the cusum chart for good detection of both large and small process shifts. This is also advised by DeVor et al. (Reference DeVor, Chang and Sutherland1992), Hawkins & Olwell (Reference Hawkins and Olwell1998) and Montgomery (Reference Montgomery2009).

Some papers use the V-mask type of cusum, a graphical alternative to the more frequently used algorithmic cusum mentioned earlier. For more information concerning this V-mask cusum, refer to Montgomery (Reference Montgomery2009), who strongly advises against the use of the V-mask procedure, mainly because of statistical issues in the choice of the false alarm rate. Sard (Reference Sard1979), Wilson et al. (Reference Wilson, McMillan and Swaminathan1980) and Wrathall & Hebert (Reference Wrathall and Hebert1982) use a V-mask cusum for snout-deformity scores in pigs, percent return to service and number of piglets born alive. Madsen & Kristensen (Reference Madsen and Kristensen2005) and Cornou et al. (Reference Cornou, Vinther and Kristensen2008) use a V-mask cusum for detecting estrus and health problems in sows. Madsen & Kristensen (Reference Madsen and Kristensen2005) suggest the use of the V-mask as a tool for discriminating between the age-dependent trend in the data as a result of the pigs growing and changes due to sudden problems such as disease outbreaks. The long-term, age-dependent cusum drift can indeed be taken into account by the V-mask; nevertheless, it is an arbitrary setting of tolerated age-related dynamics. Madsen & Kristensen (Reference Madsen and Kristensen2005) partly try to overcome this issue by standardizing the data used.

The actual investigation of compliance with the model assumptions underlying the control charts is easily overlooked by most authors. Some papers mention the probability distribution assumption. de Vries et al. (Reference de Vries, Conlin, Reneau, Kinsel and Marsh1997b) (bulk tank somatic cell count) and Engler et al. (Reference Engler, Tölle, Timm, Hohls, Krieter and Cox2005) and Krieter et al. (Reference Krieter, Engler, Tölle, Timm and Hohls2009) (total piglets born) refer to other papers that provide proof of the normal distribution for their parameters. Only Mertens (Reference Mertens2009) and Mertens et al. (Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) graphically analyse their raw data for assumption violations prior to control chart design, and found violations of the stationarity and independence assumptions in layer data. de Vries et al. (Reference de Vries, Conlin, Marsh and Reneau1997a) acknowledge the issue of non-stationarity in daily milk weight. Krieter et al. (Reference Krieter, Kirchner, Engler and Tölle2005, Reference Krieter, Engler, Tölle, Timm and Hohls2009) mention the problem of serial correlation, indicating that it has to be removed before the application of control charts. Lukas et al. (Reference Lukas, Reneau and Linn2008, Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) indicated the presence of non-stationarity and autocorrelation in their data. Madsen & Kristensen (Reference Madsen and Kristensen2005) and Cornou et al. (Reference Cornou, Vinther and Kristensen2008) acknowledge that the systematic variation on top of the random variation complicates the application of SPC techniques in their hourly water consumption data.

Most of the aforementioned authors provide some solution to solve the issue of assumption violations. A detailed discussion on this is provided in the section on Synergistic Control.

Control charts are especially sensitive to the first two assumptions (stationarity and independence). Their degree of (in)sensitivity is referred to as ‘chart robustness’. This robustness can be expressed in terms of the performance measures, ARL₀ and ARL₁. The performance (e.g. ARL₀) of a non-robust control chart in response to violating data will be a strong deviation from the expected performance (more false alarms, smaller ARL₀). Borror et al. (Reference Borror, Montgomery and Runger1999) state that the Shewhart chart is very sensitive to non-normality. However, Woodall (Reference Woodall2000) questions the importance of distribution robustness on practical applications. Indeed, Montgomery (Reference Montgomery2009) states that the EWMA is very insensitive to the normality assumption since it is a weighted average of all past and current observations, and that the cusum is less robust than EWMA for violations of the normality assumption. However, Hawkins & Olwell (Reference Hawkins and Olwell1998) state that the impact of non-normality on the performance of a cusum chart depends on size of the shift for which the chart is tuned. They found that the effect of non-normality is greatest when the cusum is tuned to detect large shifts (large k and h). Wieringa (Reference Wieringa1999) shows that the serial autocorrelation (often also resulting in non-stationarity) has a strong influence on performance of the Shewhart, cusum and EWMA chart, which was confirmed by del Hawkins & Olwell (Reference Hawkins and Olwell1998), del Castillo (Reference del Castillo2002) and Montgomery (Reference Montgomery2009).

Few papers mention chart robustness. Based on simulations, de Vries & Conlin (Reference de Vries and Conlin2003, Reference de Vries and Conlin2005) found that both Shewhart and cusum charts are rather dependent on the goodness of fit of the assumed probability distribution model. Sard (Reference Sard1979) says that his cusum chart is fairly robust to the normality assumption, and yet does not present proof.

CHART CONSTRUCTION

The construction of control charts consists of two phases. Phase I could be considered as the calibration phase. The process is explored and recursively purified of out-of-control observations to obtain in-control process data. Next, the parameters required for setting up the control chart target (process mean) and limits, used for monitoring in Phase II, are estimated.

Phase I has to be long enough, because a higher difference between the true values of statistical measures (μ, σ) and their estimates (, s) could have a large effect on control chart performance (Quesenberry Reference Quesenberry1997). However, this depends on the chart type: for instance, cusum charts are more sensitive to such errors than Shewhart charts (Hawkins & Olwell Reference Hawkins and Olwell1998). It is recommended that at least 25–30 subgroups of a minimum of five observations are available to obtain reliable estimates of the in control process. However, procedures for short-run processes and processes where quick monitoring is necessary, which is in particular of interest for livestock processes, have been proposed (Quesenberry Reference Quesenberry1997; Hawkins & Olwell Reference Hawkins and Olwell1998; Montgomery Reference Montgomery2009).

Only a few papers mention the application of the two phases explicitly when designing their charts. Pastell & Madsen (Reference Pastell and Madsen2008), Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) describe the procedure of control chart construction in the two phases. Pastell & Madsen (Reference Pastell and Madsen2008) use c. 40 observations (10–20 days) as Phase I to estimate in control variation, while Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) take 40 and 21 days for Phase I. de Vries & Conlin (Reference de Vries and Conlin2003, Reference de Vries and Conlin2005) describe the difference between Phase I and Phase II in their study. Galli et al. (Reference Galli, Signori and Balduzzi1998) mention Phase I and Phase II, but it is not clear whether they actually performed the steps. Cornou et al. (Reference Cornou, Vinther and Kristensen2008) talk about a reference period for chart estimation, and yet Phase I is not mentioned and no purification sequence is performed. Lukas et al. (Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) use historical data to construct a head-start control chart.

CONTROL CHART PERFORMANCE

Ultimately, it is good performance of the control chart – detecting as many problems as possible with as few false alarms as possible – in combination with the actual identification and resolving of a root cause in case of an alarm that will lift SPC in livestock production from an attractive theory up to an integrated management support tool.

As mentioned earlier, the performance of control charts is expressed in terms of ARL₀ and ARL₁. In livestock production, often data are pooled over periods, e.g. data for 1 week are collected for charting. In that case it is useful to express the performance in terms of the time related to these periods. The term used then is the average time to signal (ATS) with ATS=ARL.k and with k the period length (Hawkins & Olwell Reference Hawkins and Olwell1998; Montgomery Reference Montgomery2009). Derivation of these performance measures is obtained analytically from known mathematical probability distributions (e.g. normal or binomial) or Monte Carlo simulations are used when analytical solution is not feasible (Hawkins & Olwell Reference Hawkins and Olwell1998).

When performance is evaluated in terms of correctness rather than in terms of errors, the sensitivity and the specificity of control charts are used. The sensitivity of a control chart measures the proportion of correct alarms, while the specificity measures the proportion of correct in controls. These two measures should be as high as possible (Kutner et al. Reference Kutner, Nachtsheim, Neter and Li2005).

The majority of the reviewed publications are based on experimental data. Hence, performance measures, if presented, are experience or perception based (de Vries Reference de Vries2001). Another, rather pragmatic, approach for measuring chart performance presented by several papers is related to comparing the control chart detection speed with that of an existing system. Quimby et al. (Reference Quimby, Sowell, Bowman, Branine, Hubbert and Sherwood2001), who detected morbidity in steers, provide a specificity of 87% and a sensitivity of 90%. Compared with human morbidity detection, their cusum chart was on average 4·1 days earlier. Grennstam (Reference Grennstam2005) provides a comparison on specificity and sensitivity with an existing (SPC-free) algorithm. The derived cusum-based algorithm had a specificity of 85·4 v. 77·0% for the existing algorithm, and a sensitivity of 45·2 v. 51·7%. This was probably due to assumption violations (no assumption testing presented) and few Phase I in-control data. Madsen & Kristensen (Reference Madsen and Kristensen2005) indicated that their control chart could signal a change 1 day before clinical disease signs appeared. Cornou et al. (Reference Cornou, Vinther and Kristensen2008) report a higher sensitivity than detection via current sow praxis, and yet too many false alarms due to low specificity. This is probably due to their very short Phase I. Lukas et al. (Reference Lukas, Reneau and Linn2008) mention an alarm from the control chart 2 days before a mastitis treatment was applied. Lukas et al. (Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) compared their disease detection control charts with an existing software tool; their charts detected the onset of different diseases in more cases (more true positives) and earlier. Analysis of weekly pig weaning related farm data using control charts by Engler et al. (Reference Engler, Tölle, Timm, Hohls and Krieter2009) resulted in a detection of a process shift 2 and 4 weeks before the diagnosis of porcine reproductive and respiratory syndrome (PRRS) was made. For fertility data, a change was detected 3·5 months and 6 weeks before the diagnosis was made. The control chart of Lukas et al. (Reference Lukas, Hawkins, Kinsel and Reneau2005) detected a change in bulk tank somatic cell count (BTSCC) (as a result of subclinical mastitis) earlier than the official control authority that tested on a monthly basis. Finally, Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) detected unnoticed problems using average egg weight and hen-day egg production, and they detected a drop in egg weight 5 days earlier than caretakers noticed there was a problem. Pastell & Madsen (Reference Pastell and Madsen2008) did not present such performance measures, but they showed an estimation of drift and they used this to calculate when the problem was most likely initiated.

Although simulation studies can be very useful to investigate the actual performance of control charts (de Vries Reference de Vries2001), only de Vries (Reference de Vries2001), de Vries & Conlin (Reference de Vries and Conlin2003, Reference de Vries and Conlin2005), Krieter et al. (Reference Krieter, Kirchner, Engler and Tölle2005, Reference Krieter, Engler, Tölle, Timm and Hohls2009) and Engler et al. (Reference Engler, Tölle, Timm, Hohls, Krieter and Cox2005) provide control chart performance analysis based on simulations.

SYNERGISTIC CONTROL

As mentioned in the section ‘Statistical Control Charts’, the proper use of control charts requires three assumptions to hold, namely that the data have to be stationary, independent and that they have to follow the distribution function associated with the control chart used. This section goes into more detail regarding violations in assumptions with raw livestock process data mentioned earlier. It introduces a possible and promising approach to deal with the dynamic aspects of most livestock process data. This issue might as well prove to be the most important hurdle to tackle in the application of control charts on livestock production. Several publications have indicated and clearly recognized the complexities of real-life process data, like livestock process data, causing violations of control chart assumptions (DeVor et al. Reference DeVor, Chang and Sutherland1992; Hawkins & Olwell Reference Hawkins and Olwell1998; del Castillo Reference del Castillo2002; Ostyn et al. Reference Ostyn, De Ketelaere, Mertens, Anthonis, Zoons, De Baerdemaeker, Kataoka, Noguchi and Murase2006; Box & Paniagua-Quiñones Reference Box and Paniagua-Quiñones2007; Montgomery Reference Montgomery2009).

The assumptions of stationarity and autocorrelation are critical for an even basic performance of the control chart (Montgomery Reference Montgomery2009). Montgomery (Reference Montgomery2009) indicated that with an autocorrelation between two successive observations of at least 0·25, the false alarm rate of a control chart is increased significantly. In a negatively autocorrelated process the common cause variation is overestimated, resulting in limits that are too wide and a decreased detection rate. In a positively autocorrelated process the common cause variation is underestimated, resulting in limits that are too narrow and an increased false alarm rate. Furthermore, the risk of misinterpretation of the information present in the data is high (Wieringa Reference Wieringa1999; del Castillo Reference del Castillo2002). According to Alwan & Roberts (Reference Alwan and Roberts1995) in practice the fundamental assumptions for standard implementation of control charts are violated in over 0.85 data sets.

To deal with these characteristics in the framework of SPC, the control chart limits can be modified (Wieringa Reference Wieringa1999), but most textbooks advise a strategy based on predictions with a model capturing the systems dynamics (del Castillo Reference del Castillo2002; Montgomery et al. Reference Montgomery, Jennings and Kulahci2008; Montgomery Reference Montgomery2009). This kind of strategy is called engineering process control (EPC). Essentially, EPC encompasses all the activities that focus on the mathematical modelling of (production) systems (del Castillo Reference del Castillo2002). Such models can be used to perform a sort of pre-processing of the raw data. The EPC strategy describes the dynamic aspects of the process data and as such accounts for sources of variability which cannot be dealt with by SPC. Next, the proper use of the SPC chart on the model residuals allows detecting assignable causes in the process and leads to permanent process improvement or reduction in variability (Montgomery Reference Montgomery2009).

The mutual objective of both process control procedures, SPC and EPC, is the reduction of variability (Montgomery Reference Montgomery2009). The cited literature states the necessity of combining the adjustment power of EPC and the monitoring power of SPC in order to achieve control of a process and improve product quality. Box & Paniagua-Quiñones (Reference Box and Paniagua-Quiñones2007) defined this combined strategy as synergistic control (SGC). The general philosophy of this approach is that after the model-based correction of the data by means of a model, the resulting residuals are normally and independently distributed with mean zero and constant variance, N(0, σ ²). Hence, control charts can then be applied to the corrected time series of residuals. The disadvantage of this approach is that model estimation errors are introduced into the resulting monitoring scheme (del Castillo Reference del Castillo2002). Nevertheless, this approach has proved to be very useful (DeVor et al. Reference DeVor, Chang and Sutherland1992; Hawkins & Olwell Reference Hawkins and Olwell1998; del Castillo Reference del Castillo2002; Box & Paniagua-Quiñones Reference Box and Paniagua-Quiñones2007; Montgomery Reference Montgomery2009).

The recognition of this systematic variation in livestock process data is not new. It was already mentioned by Sard (Reference Sard1979), who suggested that data series should be filtered for the removal of such variation. Publications by Roush et al. (Reference Roush, Tomiyama, Garnaoui, D'Alfonso and Cravener1992), de Mol et al. (Reference de Mol, Keen, Kroeze and Achten1999) and Van Bebber et al. (Reference Van Bebber, Reinsch, Junge and Kalm1999) introduced an approach for modelling the dynamic behaviour of different dairy and poultry data. They did this in consideration of signalling changes, and yet did not use SPC. Roush et al. (Reference Roush, Tomiyama, Garnaoui, D'Alfonso and Cravener1992) present a Kalman filter to detect significant features in poultry feed consumption data. Van Bebber et al. (Reference Van Bebber, Reinsch, Junge and Kalm1999) also used a Kalman filter for the prediction of milk yield in a recursive estimation pattern. They generated a signal if the model prediction error exceeds ±2 times its standard deviation. de Mol et al. (Reference de Mol, Keen, Kroeze and Achten1999) use time series models combined with a Kalman filter for dairy cow production (milk yield, milk temperature, electrical conductivity of quarter milk and the cows activity, and a probability distribution for concentrates leftover). The problem with using a Kalman filter as a means of detection is that it could be considered as a moving average displaying a backlog to the actual course of the observed time series, thus making it less sensitive for quick detection purposes on a daily basis (Brockwell & Davis Reference Brockwell and Davis1991). The reviewed papers that acknowledged the issue of the systematic variation on top of the random variation for direct application of SPC techniques have suggested several SGC solutions. Lukas et al. (Reference Lukas, Reneau and Linn2008) use first-order autoregressive (AR(1)) models for water and dry matter intake of cows and heifers. For both cows and heifers they make a model for the 21 first days and one for the following 129 days, and hence end up with four models. Doing so, they aim to divide the trend in the data into two approximately linear intervals. Their overall model R ² ranged from 0·61 to 0·71. The problem with this approach is that they arbitrarily divide the course of the parameters into two pieces while there is a large variability between cows in the course of these parameters. Mertens (Reference Mertens2009) showed that the autocorrelation structure in his layer data was more complex than an AR(1) model. Lukas et al. (Reference Lukas, Reneau, Wallace, Hawkins and Munoz-Zanzi2009) used a fourth-order autoregressive (AR(4)) model extended with cow parity for individual milk yield and milk conductivity data.

de Vries et al. (Reference de Vries, Conlin, Marsh and Reneau1997a) use an autoregressive integrated moving average (ARIMA) model to describe the deterministic trend in daily milk weight. The approach using ARIMA models is recommended by several papers on industrial processes (DeVor et al. Reference DeVor, Chang and Sutherland1992; Hawkins & Olwell Reference Hawkins and Olwell1998; del Castillo Reference del Castillo2002; Ostyn et al. Reference Ostyn, De Ketelaere, Mertens, Anthonis, Zoons, De Baerdemaeker, Kataoka, Noguchi and Murase2006; Box & Paniagua-Quiñones Reference Box and Paniagua-Quiñones2007; Montgomery Reference Montgomery2009). An ARIMA model pre-processes the data as follows: the non-stationarity is treated by the integrating part of the model (I), while the autocorrelation issue is dealt with by the ARMA part. This approach is useful when the non-stationary trend can be described by a model linear in its coefficients. For most livestock production processes, the age-dependent trend in process parameters displays a non-linear relation with the age (e.g. Lokhorst (Reference Lokhorst1996) for layers, Bebbington et al. (Reference Bebbington, Lai and Zitikis2009) and Silvestre et al. (Reference Silvestre, Martins, Santos, Ginja and Colaço2009) for dairy). Applying first- or second-order differencing to such data series would often not produce a stationary time series. Moreover, there is the risk of over-differencing and introducing an artificial error into the data (del Castillo Reference del Castillo2002).

To overcome this problem, Mertens et al. (Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) presented a sequential model approach. To deal with the non-stationarity in layer process data, they used non-linear trend models to obtain stationary model residuals. Next, they use ARMA models to correct for the autocorrelation in the residuals.

The papers by Mertens et al. (Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) dealt with data with a maximum sampling frequency of one day. When this sampling frequency increases and approaches real-time data logging, possible diurnal fluctuations are introduced into the data series. For instance, Madsen & Kristensen (Reference Madsen and Kristensen2005) had to deal with diurnal patterns in their hourly water consumption data. They tried to model them with a combination of a State-space model and a Kalman filter. The detection rate of their cusum was acceptable, and yet the same remark as before can be made concerning the Kalman filter. The same approach was used by Cornou et al. (Reference Cornou, Vinther and Kristensen2008). From these results it can be stated that an increased sampling frequency will lead to more complex models.

REAL-TIME LIVESTOCK MONITORING

From the aforementioned section, several suggestions for dealing with the dynamic aspects are available. Nevertheless, there is a necessity for a real-time approach when developing such EPC algorithms for pre-processing raw livestock process data. None of the reviewed papers presented an on-farm real-time validation of their presented control charts. They all provided a retrospective post-hoc analysis, i.e. after all data were collected. Since the aim of introducing control charts as livestock management support tools is to quickly detect process aberrations, the EPC algorithms need to perform the model developments in an ad-hoc or online way. Mertens et al. (Reference Mertens, De Ketelaere, Vaesen, Löffel, Ostyn, Kemps, Kamers, Bamelis, Zoons, Darius, Decuypere and De Baerdemaeker2008, Reference Mertens, Vaesen, Löffel, Kemps, Kamers, Zoons, Darius, Decuypere, De Baerdemaeker and De Ketelaere2009) and Mertens (Reference Mertens2009) included such an approach. Their models are updated by using the information from newly acquired observations. As such, the principle of recursive model estimation is applied: each time a new measurement is performed, the model coefficients are re-estimated. Furthermore, they use the information of the cusum chart on the current observation (in-control or out-of-control) as feedback input for the next model estimation. In this way out-of-control estimations are excluded from the model development.

CONCLUSION

The concepts of SPC, and more specifically control charts, form a natural basis for fully exploiting the increasing amounts of data available on both the input and output parameters of livestock production processes and hence for generating fact-based information to improve the quality of management decisions. The high number of research papers in the last decade indicates the increasing interest. One of the most important hurdles to tackle is the statistical characteristics of the raw livestock production data which are mostly not in agreement with the assumptions for control charts. Also, the operational aspects of control charts and the economic consequences of decision making based on control charts should be further developed for successful implementation in (precision) livestock farming. An integrated approach, such as synergistic control, appears to be promising in handling this issue. The availability of real-time on-farm validation of proposed systems will be crucial for lifting them from the potential level to direct practical relevance.