Automated public health records provide the necessary data for rapid outbreak detection. An adaptive exponentially weighted moving average (EWMA) plan is developed for signalling unusually high incidence when monitoring a time series of nonhomogeneous daily disease counts. A Poisson transitional regression model is used to fit background/expected trend in counts and provides “one-day-ahead” forecasts of the next day's count. Departures of counts from their forecasts are monitored. The paper outlines an approach for improving early outbreak data signals by dynamically adjusting the exponential weights to be efficient at signalling local persistent high side changes. We emphasise outbreak signals in steady-state situations; that is, changes that occur after the EWMA statistic had run through several in-control counts.
Early detection of disease outbreaks is essential for the efficient control of acute public health risks. For example, identifying the source of, and restricting exposure to, contaminated food products can only proceed once there is enough information for an outbreak data signal. Automated detection systems that accumulate sufficient information for earlier identification of an outbreak data signal are common now days. Identifying unusual disease incidence is normally carried out by monitoring data streams such as daily, weekly, or monthly counts or rates.
Statistical process control (SPC) methods, including control charts, are increasingly being applied to public health surveillance [
Exponentially weighted moving average (EWMA) control charts have been useful for monitoring industrial production processes (e.g., see Lucas and Saccucci [
Grigg and Spiegelhalter [ the annual seasonal influence, day of the week influences, usual transitional (day-to-day) carry over influences of the lag counts.
EWMA plans are used to accumulate memory of local trends. The benefit of accumulating memory is having sufficient power to signal a change. The exponential weights
Monitoring performance is measured in terms of out-of-control average time to signal. We used the average number of days from the day of onset of outbreak to the day the outbreak signal to define performance. The average time to signal for a step change of
Most of the work on control chart efficiency in the literature on surveillance has been carried out for EWMA plans in their initial state where the out-of-control behaviour is generated starting the EWMA at its in-control value. In practice, process changes seldom start when EWMA statistics equals the in-control value. Therefore, all
Section
Let the daily count on day
Suppose outbreak data cause the mean to increase from
For Poisson counts with mean
Threshold values
1 | 2 | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 | 6.5 | 7 | |
0.05 | 1.232 | 2.336 | 3.410 | 3.939 | 4.466 | 4.991 | 5.515 | 6.038 | 6.562 | 7.085 | 7.608 |
0.07 | 1.319 | 2.450 | 3.547 | 4.088 | 4.625 | 5.159 | 5.692 | 6.224 | 6.756 | 7.287 | 7.818 |
0.10 | 1.435 | 2.602 | 3.732 | 4.288 | 4.839 | 5.387 | 5.932 | 6.476 | 7.018 | 7.560 | 8.100 |
0.15 | 1.604 | 2.827 | 4.005 | 4.583 | 5.154 | 5.720 | 6.283 | 6.843 | 7.401 | 7.957 | 8.511 |
0.20 | 1.758 | 3.033 | 4.254 | 4.849 | 5.438 | 6.020 | 6.597 | 7.171 | 7.741 | 8.309 | 8.875 |
0.25 | 1.905 | 3.229 | 4.489 | 5.101 | 5.705 | 6.301 | 6.891 | 7.477 | 8.058 | 8.637 | 9.213 |
0.30 | 2.039 | 3.412 | 4.710 | 5.339 | 5.957 | 6.566 | 7.168 | 7.765 | 8.357 | 8.946 | 9.532 |
0.35 | 2.142 | 3.565 | 4.904 | 5.549 | 6.182 | 6.805 | 7.419 | 8.027 | 8.631 | 9.230 | 9.827 |
The
Sparks [
For disease counts that are Poisson distributed, a model for interpolating “optimal’’ exponential weights (denoted by
“Optimal’’
1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 | 6.5 | 7 | |
1.3 | 0.05 | 0.06 | 0.07 | 0.07 | 0.08 | 0.08 | 0.09 | 0.09 | 0.10 | 0.10 | |||
1.4 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 | 0.11 | 0.12 | 0.13 | 0.14 | 0.15 | 0.16 | 0.17 | 0.18 |
1.5 | 0.06 | 0.08 | 0.10 | 0.11 | 0.13 | 0.15 | 0.16 | 0.18 | 0.19 | 0.21 | 0.23 | 0.24 | 0.26 |
1.6 | 0.07 | 0.10 | 0.12 | 0.14 | 0.16 | 0.19 | 0.21 | 0.23 | 0.25 | 0.27 | 0.30 | 0.32 | 0.34 |
1.7 | 0.08 | 0.11 | 0.14 | 0.17 | 0.20 | 0.23 | 0.25 | 0.28 | 0.31 | 0.34 | |||
1.8 | 0.10 | 0.13 | 0.16 | 0.20 | 0.23 | 0.27 | 0.30 | 0.33 | |||||
2.0 | 0.12 | 0.16 | 0.21 | 0.26 | 0.30 | 0.35 | |||||||
2.5 | 0.18 | 0.25 | 0.32 | ||||||||||
3.0 | 0.23 | 0.34 |
For nonhomogeneous counts
Outbreak data signals involving a step change in means (
Model number | |||||||||
2 | 3 | 4 | 2 | 3 | 4 | 2 | 3 | 4 | |
Poisson adaptive EWMA | Poisson adaptive EWMA | Adaptive CUSUM | |||||||
of pseudo residuals | |||||||||
0.075 | 0.1 | 0.1 | |||||||
1.0075 | 1.0076 | 1.0144 | 1.0055 | 1.0050 | 1.0200 | 2.104 | 1.65 | 1.372 | |
ATS (Average time to signal/detection in days) | |||||||||
0.0 | 102.38 | 100.96 | 97.23 | 100.21 | 104.70 | 99.50 | 102.90 | 100.09 | 100.11 |
0.5 | 25.22 | 29.21 | 35.11 | 24.75 | 29.89 | 35.32 | 44.31 | 47.89 | 52.95 |
1.0 | 12.59 | 15.34 | 19.85 | 12.44 | 15.40 | 20.49 | 26.33 | 28.99 | 33.71 |
1.5 | 7.79 | 9.72 | 13.21 | 7.73 | 9.73 | 13.41 | 17.71 | 19.71 | 24.01 |
2.0 | 5.50 | 6.99 | 9.83 | 5.40 | 6.90 | 9.35 | 12.87 | 14.19 | 17.83 |
2.5 | 4.25 | 5.46 | 6.98 | 4.20 | 5.34 | 6.69 | 9.85 | 11.11 | 13.66 |
3.0 | 3.46 | 4.40 | 5.26 | 3.37 | 4.31 | 5.16 | 7.80 | 8.61 | 10.95 |
3.5 | 2.88 | 3.65 | 4.38 | 2.84 | 3.58 | 4.19 | 6.61 | 7.21 | 9.07 |
4.0 | 2.49 | 3.06 | 3.73 | 2.47 | 3.05 | 3.60 | 5.70 | 6.15 | 7.57 |
5.0 | 2.03 | 2.46 | 2.89 | 2.01 | 2.45 | 2.68 | 4.46 | 4.72 | 5.60 |
Seasonal outbreak data involving a parabolic change in means (
Model number | ||||||
1 | 2 | 3 | 1 | 2 | 3 | |
Dynamic Poisson adaptive EWMA | Poisson adaptive EWMA | |||||
0.075 | 0.1 | 0.1 | ||||
1.0076 | 1.0075 | 1.0076 | 1.0056 | 1.0055 | 1.0050 | |
Overall average number of extra victims in the outbreak data | ||||||
42 | 42 | 84 | 42 | 42 | 84 | |
ATS (Pr(no outbreak data signal)) | ||||||
(0,0) | 102.38 (0.00) | 100.96 (0.00) | 97.23 (0.00) | 99.50 (0.00) | 100.19 (0.00) | 103.03 (0.00) |
22.05 (0.16) | 14.73 (0.03) | 16.11 (0.07) | 18.56 (0.14) | 15.31 (0.03) | 17.00 (0.07) | |
18.93 (0.14) | 12.36 (0.02) | 13.57 (0.06) | 15.72 (0.14) | 12.59 (0.02) | 14.17 (0.06) | |
14.62 (0.12) | 9.66 (0.01) | 10.88 (0.03) | 12.85 (0.13) | 10.37 (0.01) | 11.61 (0.03) | |
11.20 (0.11) | 7.54 (0.00) | 8.43 (0.03) | 9.81 (0.11) | 7.60 (0.00) | 8.65 (0.03) | |
7.44 (0.10) | 4.75 (0.00) | 5.03 (0.02) | 6.66 (0.08) | 4.93 (0.00) | 5.69 (0.02) | |
3.50 (0.04) | 2.40 (0.00) | 2.81 (0.00) | 3.33 (0.05) | 2.43 (0.00) | 2.83 (0.00) |
A disadvantage with
If future shifts are known in advance for homogeneous counts then the
An adaptive EWMA plan using Estimate Use the estimates in step Calculate Signal an unusual epidemic footprint whenever
Let the one-day-ahead model forecast for mean made at time
The one-day-ahead forecast is used to remove the natural variation that comes from day of the week, holidays, seasonal and transitional influences, that is,
Suppose the aim of surveillance is identifying the start of a seasonal epidemic as early as possible, for example, the start of the annual influenza season. An epidemic footprint is expected each year, but the start time varies from season to season and varies in magnitude. Assume we have prior information that the seasonal epidemic event is rare in a certain period of the year (e.g., summer for influenza). Let the average (nearly homogenous) mean during the nonepidemic period be denoted by
A simulation study is used to evaluate the practical efficiency of the plans. Tables
For Models
Two outbreak data used for the results in Tables A step change of A parabolic change in mean simulated to mimic say seasonal increases in say influenza counts. Here the mean increases of
The model in ( if the conventional CUSUM is applied to the pseudoforecast errors (without adjustment), the plan will deliver too many false alarms, the threshold needs an adjustment for each change in the mean distribution.
On the other hand, the Poisson adaptive EWMA plans need very little adjustments for the design control limit of one (
In summary, the dynamic adaptive Poisson EWMA plan signals, on average, earlier than the adaptive Poisson EWMA plans with
The application involves influenza A daily counts in New South Wales (NSW), Australia ranging from the end of 2003 to the end of 2005. Temporal trends in the daily disease counts are found in the first plot of Figures a moving window of 730 days of data (2 years), discounted weights
EWMA plans optimised for detecting seasonal epidemic footprints applied to influenza A daily counts in NSW.
EWMA plans optimised for detecting unexpectedly high influenza A counts in NSW.
The autocorrelations of standardised forecast errors for influenza were insignificant (see Figure
Autocorrelation function of the model standardised residuals for influenza A daily counts in NSW.
The forecasts, together with the smoothed forecast errors, are used to optimise the adaptive EWMA plan as outlined previously. After each signal, EWMA plans reset the EWMA statistic to its initial value, and therefore follow-up signals indicate that either the epidemic footprint persisted, or that the counts continued to be higher than expected. The “optimal’’ adaptive CUSUM plans for the one-day-ahead pseudoforecast errors
The first plot in Figure
The second plot in Figure
Two epidemic footprints were detected in each of years 2004 and 2005 (see Figure
The second plot in Figure
A laboratory diagnosing patients as having influenza A was found to produce many false positives in early 2005. Poisson adaptive EWMA plan signalled these whereas the “optimal’’ adaptive CUSUM plan fails to signal (see southern hemisphere summer of 2004/2005 in Figure
Further analysis (not reported) demonstrated that the Poisson adaptive EWMA plan for
In summary, we offer a dynamic Poisson adaptive EWMA plan for early detection of epidemic footprints when disease counts follow a Poisson regression model. Results demonstrate that in the case of step changes in means, the dynamic Poisson adaptive EWMA detects outbreaks earlier than the CUSUM plan of pseudoforecast errors. Results for parabolic changes demonstrate the superiority of the dynamic Poisson adaptive EWMA over adaptive EWMA with
If the in-control mean changes dramatically from season to season, then the adaptive CUSUM of pseudo residuals plan threshold adjust would need revising from season to season. This aspect makes the implementation of the adaptive CUSUM of pseudo residuals plan more difficult.
Most of the effort has been placed on sequential hypothesis testing, but the performance of plans is also influenced by recursive estimation of model parameters. More effort in modelling would improve the performance of the Poisson adaptive EWMA plan. There is also value in monitoring model lack of fit for each moving window of data (see Zeileis [
The authors are grateful to Dr. Petra Graham for writing R programs for fitting the models in the application. In addition, the authors thank the referees for their valuable comments that have improved the presentation of this paper.