Intervention analysis for low-count time series with applications in public health

Abstract

It is common in many fields to be interested in the evaluation of the impact of an intervention over a particular phenomenon. In the context of classical time series analysis, a possible choice might be intervention analysis, but there is no analogous methodology developed for low-count time series. In this article, we propose a modified INAR model that allows us to quantify the effect of an intervention, and is also capable of taking into account possible trends or seasonal behaviour. Several examples of application in different real and simulated contexts will also be discussed.

Keywords

count data INAR models intervention analysis time series

1 Introduction

This article focuses on the evaluation of the impact of an intervention over the number of occurrences of a particular phenomenon by using discrete time series techniques. Therefore, unlike in many other applications of time series, the main interest is not in forecasting but in the estimation of the effect of the intervention and its further inference. Many models of discrete time series have been considered in the literature (see McKenzie [2003]), although we focus on integer autoregressive (INAR) models, which are a natural extension to the well-known AR models, and are often easily interpretable in practical contexts. It is usual in many contexts such as public health or sociology to design and conduct an intervention to change some phenomenon behaviour. When dealing with continuous valued time series or series with large counts, intervention analysis may be used with this purpose (see Helfenstein [2005] for a comprehensive handbook). When the time point where the potential change occurs is unknown, as studied by some authors, the change-point analysis can be referred (see Csörgö and Horváth [1997] and Horváth and Rice [2014]). However, for count data, there is not a clear analogous methodology, although there have been some recent contributions (Vasileios, 2015; Liboschik et al., 2016) and an application of the change-point techniques to INAR models (Hudecová et al., 2015). Nonetheless, most of these models are focused on real-time monitoring for structural changes in the series while we are focused on transient or definitive changes in the new observed cases after an intervention through a retrospective analysis.

In addition to the potential effect of the intervention, the phenomenon under study may present a higher incidence at certain times of the year or a trend over time, and these behaviours are not covered by the classical $INAR (k)$ time series models.

The proposed model is presented in detail in Section 2, and some approaches to evaluate the goodness of fit are also discussed. In Section 3, three examples of application in different contexts are discussed. The first example is based on a plain simulation, useful to illustrate how the proposed methodology can be applied in a simple framework. The second example discusses the well-known effect of compulsory wearing of seatbelts in Great Britain over the number of killed drivers of light goods vehicles. Finally, the third example deals with a recent public health concern as the effect of the celebration of massive events where unprotected sex is common over the number of cases of sexually transmitted infections (STI).

2 Model definition

$INAR (k)$ models are defined by

X_{t} = p_{1} \circ X_{t - 1} + \dots + p_{k} \circ X_{t - k} + W_{t},

(2.1)

where $p_{1}, \dots, p_{k}$ are fixed parameters, $0 < p_{1}, \dots, p_{k} < 1$ and $W_{t}$ is assumed to follow a Poisson distribution with a fixed mean $λ$ . In addition, $X_{t - 1}$ and $W_{t}$ are assumed to be independent at any time $t$ . The $\circ$ operator in expression (2.1), called binomial thinning, is defined as follows:

p_{j} \circ X_{t - j} = \sum_{i = 1}^{X_{t - j}} Y_{i},

(2.2)

where $Y_{i}$ are independent identically distributed Bernoulli random variables with the probability of success equal to $p_{j}$ . Therefore, if $X_{t - j} = x_{t - j}$ , then $p_{j} \circ x_{t - j}$ is binomially distributed with the number of successes equal to $x_{t - j}$ . An early but clear review of the application of these models to the analysis of disease incidence is Cardinal et al. (1999). A more recent review of INAR models can be found in Scotto et al. (2015). A characterization of count distributions based on these kind of binomial thinning properties can be found in Puig and Valero (2007). These distributions, which include the Poisson, can be used as appropriate distributions for $W_{t}$ .

There are several ways to understand the parameters involved in an $INAR (k)$ model. One of them is to interpret $p_{j}$ as the proportion of observations counted on time $t - j$ that pass to time $t$ . Depending on the context, it can be understood as a kind of survival rate. On the other hand, $W_{t}$ can be interpreted as the innovations or new events produced at time $t$ . In Moriña et al. (2011), for instance, the authors interpreted $p_{j}$ as the proportion of patients with influenza symptoms who went to emergency service at time $t - j$ and had a relapse after $j$ weeks, and $W_{t}$ is understood as the number of new patients who arrived at the emergency service at time $t$ . However, in many situations where $INAR (k)$ models can be applied, this meaningful interpretation for the parameters $p_{j}$ and the term $W_{t}$ is difficult to find. For instance, this is the situation for the example of meningococcal infection analysed in Cardinal et al. (1999). In such cases, the parameters $p_{j}$ could simply represent the impact of the $t - j$ past value of the time series on the value at time $t$ .

For each intervention we are interested in, we define a dummy variable $I_{t}$ , which takes the value 1 if $t$ is within an intervention period or 0 otherwise. The proposed model is a variation of $INAR (k)$ model (2.1):

X_{t} = p_{1} \circ X_{t - 1} + \dots + p_{k} \circ X_{t - k} + W_{t} (λ^{'}),

(2.3)

where

λ^{'} = λ + σ \cdot I_{t}

. The parameters of the model (2.3),

θ = (λ, σ, p_{1}, \dots, p_{k})

, can be estimated by using the method of the conditioned maximum likelihood. The likelihood function conditioned to the first

k

values

x_{1}, \dots, x_{k}

can be written as

\begin{matrix} L (X; θ) \sim P (x_{k + 1} ∣ x_{1}, \dots, x_{k}; θ) \cdot P (x_{k + 2} ∣ x_{2}, \dots, x_{k + 1}; θ) \dots P (x_{n} ∣ x_{n - 1}, \dots, x_{n - k}; θ), \end{matrix}

(2.4)

because we are considering dependences of order $1, \dots, k$ . Here $P (x_{i} ∣ x_{i - 1}, \dots, x_{i - k}; θ)$ indicates the probability function of $X_{i}$ conditioned to the $k$ previous observations, evaluated at $x_{i}$ . Looking at the expression of model (2.3), we realize that this probability distribution corresponds to the sum of $k$ independent binomial variables with parameters $(x_{i - 1}, p_{1}), \dots, (x_{i - k}, p_{k})$ , plus an independent Poisson variable with mean $λ^{'}$ . To compute this probability function, we have used the following result on the basis of Bu et al. (2008):

\begin{matrix} P (x_{i} | x_{i - 1}, \dots, x_{i - k}) = & \sum_{j_{1} = 0}^{\min (x_{i - 1}, x_{i})} (_{j_{1}}^{x_{i - 1}}) \cdot p_{1}^{j_{1}} \cdot {(1 - p_{1})}^{x_{i - 1} - j_{1}} . \\ \cdot \sum_{j_{2} = 0}^{\min (x_{i - 2}, x_{i} - j_{1})} (_{j_{2}}^{x_{i - 2}}) \cdot p_{2}^{j_{2}} \cdot {(1 - p_{2})}^{x_{i - 2} - j_{2}} \dots \end{matrix}

\begin{matrix} \cdot \sum_{j_{k} = 0}^{\min (x_{i - k}, x_{i} - j_{k - 1})} x_{i - k} j_{k} \cdot p_{k}^{j_{k}} \cdot (1 - p_{k})^{x_{i - k} - j_{k}} \cdot \\ \cdot \frac{e^{- (λ + σ \cdot I_{i})} \cdot (λ + σ \cdot I_{i})^{x_{i} - (j_{1} + \dots + j_{k})}}{(x_{i} - (j_{1} + \dots + j_{k}))!} \end{matrix}

(2.5)

In fact, the main interest in our context will be to test the hypothesis

\begin{matrix} H_{0} & : σ = 0 \\ H_{1} & : σ \neq 0, \end{matrix}

(2.6)

The null hypothesis in (2.6) can be tested using the standard error corresponding to

\hat{σ}

, obtained from the inverse of the Hessian matrix. The proposed model is useful to detect changes in the innovations (

W_{t}

), an increase or decrease in the number of new cases following an intervention, more than a structural change in the series, while a methodology for detecting changes in the parameters

p_{i}

i = 1, \dots, k

in (2.1) is proposed in Hudecová et al. (2015), introducing a method for monitoring structural changes in INAR(1) processes. The two approaches will be compared in the proposed examples.

2.1 Model properties

The number of cases at time $t$ can be estimated by the expectation conditioned to the last known information, that is, $E (X_{t} | X_{t - 1} = x_{t - 1}, X_{t - 2} = x_{t - 2}, \dots, X_{t - k} = x_{t - k}) = p_{1} x_{t - 1} + p_{2} x_{t - 2} + \dots + p_{k} x_{t - k} + {λ^{'}}_{t}$ . It leads to the following estimator or predicted value of $X_{t}$ :

X_{t} = {\hat{p}}_{1} x_{t - 1} + {\hat{p}}_{2} x_{t - 2} + \dots + {\hat{p}}_{k} x_{t - k} + {\hat{λ^{'}}}_{t} .

(2.7)

This can be used to validate how the model fits the real data, calculating the variance of ${\hat{X}}_{t}$ from expression (2.7):

\begin{matrix} V ({\hat{X}}_{t}) = V ({\hat{p}}_{1}) x_{t - 1}^{2} + V ({\hat{p}}_{2}) x_{t - 2}^{2} + \dots + V ({\hat{p}}_{k}) x_{t - k}^{2} + V ({\hat{λ^{'}}}_{t}) \\ + \sum_{j = 1}^{k - 1} \sum_{i = 2}^{k} 2 \cdot x_{t - j} x_{t - i} C O V ({\hat{p}}_{j}, {\hat{p}}_{i}) + 2 x_{t - 1} C O V ({\hat{p}}_{1}, {\hat{λ}}^{'}_{t}) \\ + 2 x_{t - 2} C O V ({\hat{p}}_{2}, {\hat{λ^{'}}}_{t}) + \dots + 2 x_{t - k} C O V ({\hat{p}}_{k}, {\hat{λ^{'}}}_{t}) . \end{matrix}

(2.8)

In order to estimate

V ({\hat{X}}_{t})

, the variances and covariances needed in expression (2.8) can be replaced by their corresponding estimates. From here,

100 (1 - α) %

approximate confidence intervals can be constructed in a standard way by means of

X_{t} = {\hat{X}}_{t} \pm z_{α / 2} \sqrt{\hat{V} ({\hat{X}}_{t})}

As will be discussed in the examples in the following section, model selection can be based on statistical significance of the parameters and any usual information criterion as the Akaike information criterion (AIC).

2.2 Goodness of fit

The goodness of fit of the selected model can be assessed through a discretised version of the Cox–Snell residuals (Cox and Snell, 1968), computed from the estimated conditional distribution:

\hat{P} (Y_{n} = y_{n} ∣ (Y_{1}, \dots, Y_{n - 1}, Y_{n + 1}, \dots, Y_{T})) = \frac{\hat{P} (Y_{n} = y_{n}, Y_{1}, \dots, Y_{n - 1}, Y_{n + 1}, \dots, Y_{T})}{\hat{P} (Y_{1}, \dots, Y_{n - 1}, Y_{n + 1}, \dots, Y_{T})},

(2.9)

where $T$ is the length of the series. The normal pseudo-residual segment $[z_{n}^{-}, z_{n}^{+}]$ can be obtained as

z_{n}^{-} = Φ^{- 1} (\hat{P} (Y_{n} < y_{n} ∣ (Y_{1}, \dots, Y_{n - 1}, Y_{n + 1}, \dots, Y_{T}))) = Φ^{- 1} (u_{n}^{-})

(2.10)

z_{n}^{+} = Φ^{- 1} (\hat{P} (Y_{n} \leq y_{n} ∣ (Y_{1}, \dots, Y_{n - 1}, Y_{n + 1}, \dots, Y_{T}))) = Φ^{- 1} (u_{n}^{+}),

(2.11)

where

Φ

is the standard normal distribution function. As pointed out in Fernández-Fontelo et al. (2016), when

y_{n} = 0

the probability

u_{n}^{-}

in (2.10) is 0 and

z_{n}^{-}

is not defined. The solution proposed by the authors is to take a standard normal quantile corresponding to a probability close to 0 as

z_{n}^{-} = - 4

. The mid-pseudo-residuals, defined by

z_{n}^{m} = Φ^{- 1} (\frac{u_{n}^{-} + u_{n}^{+}}{2})

(2.12)

are commonly used in practice to check the validity of a fitted model, as a white noise behaviour is expected. Other approaches might be used for assessing the goodness of fit in this context, as the adapted version of the probability integral transform described in Czado et al. (2009). A comprehensive description of some of these methods is available in Zucchini and MacDonald.

3 Examples

The maximization of (2.4) for the examples discussed in this section has been done with a program developed in R (R Core Team, 2016) using the nonlinear minimization procedure nlm, which is available as a supplementary material. The standard errors of the estimates have been calculated from the inverse of the corresponding Hessian matrix.

3.1 Example 1: Simulation study

Two INAR(1) processes were simulated, with parameters $p_{1} = 0.7$ and $λ = 5$ for $t = 1, \dots, 400$ and $p_{1} = 0.7$ and $λ = 1$ for $t = 401, \dots, 500$ . The simulated series is shown in Figure 1(a).

Figure 1:

Simulated INAR(1) process with an intervention at time $t = 400$ (a) and no intervention (b)

Therefore, a change is expected to be detected for $t > 400$ , and the fitted model is

X_{t} = p_{1} \circ X_{t - 1} + W_{t} (λ^{'}),

where

λ^{'} = λ

for

t \leq 400

and

λ^{'} = λ + σ

for

t > 400

. The estimate for

σ

-

5.89 with 95% confidence interval (

-

6.82,

-

4.96), meaning that the intervention at time

t = 400

had a significant impact on the series.

Similarly, an INAR(1) process consisting of 500 observations was simulated with parameters $p_{1} = 0.7$ and $λ = 5$ . The simulated process can be seen in Figure 1(b). In that case, as expected, an estimate of $\hat{σ} = 0.39 (- 0.25, 1.03)$ is obtained, which can be interpreted as no effect of the hypothetical intervention at time $t = 400$ .

Another approach would be to test for structural changes in the series, following, for example, the methodology described in Hudecová et al. (2015). In this case, as the change in series (a) is structural and the series does not return to the pre-intervention values after the intervention, this methodology is able to detect a change at $t = 396$ . No change is observed for series (b), as expected. It is important to notice that our approach is retrospective and the potential intervention time is known, while the real-time monitoring alternatives are able to estimate when the change occurred.

3.2 Example 2: Road casualties in Great Britain

The series consists in the monthly number of killed drivers of light goods vehicles in Great Britain from January 1969 to December 1984, including a period after compulsory wearing of seatbelts (February 1983December 1984 1983/ 02 –1984/12), which is the intervention to be evaluated. These data are a subset of a time series discussed in Harvey and Durbin (1986), including all casualties for drivers and passengers of cars, resulting in numbers large enough to be analysed through methods for continuous valued data. Later, the same dataset we are considering here was analysed in Liboschik et al. (2016) finding a significant seasonal effect. This monthly seasonality can be included in (2.3) in a similar way to that of Moriña et al. (2011):

X_{t} = p_{1} \circ X_{t - 1} + \dots + p_{k} \circ X_{t - k} + W_{t} (λ^{'}),

(3.1)

where

{λ^{'}}_{t} = λ_{i} + σ \cdot I_{t}

i = 1, \dots, 12

. To fit a model comparable to the example reported in Liboschik et al. (2016), the petrol price was also taken into account, together with the linear trend and the annual cycle. Several alternative models were tried, but the best according to residual exploration and AIC was

X_{t} = p_{1} \circ X_{t - 1} + p_{12} \circ X_{t - 12} + W_{t} (λ^{'}),

(3.2)

where

{λ^{'}}_{t} = λ_{i} \cdot P e t r o l P r i c e_{t} + σ \cdot I_{t} + β \cdot \frac{t}{12}

i = 1, \dots, 12

. Table 1 shows the obtained estimates.

The AIC for an equivalent model using the R package tscount as in Liboschik et al. (2016) is 966.3269, while using this methodology, the AIC is reduced to 923.2088. Using the seasonal INAR(2) model introduced in Moriña et al. (2011) (and omitting the intervention effect), the AIC is 931.1917. From a methodological point of view, the best model can be chosen on the basis of the statistical significance of the parameters or AIC and in this case both criteria point to the model (2.3). The code used to fit all these models is available as a supplementary material.

Table 1:

Maximum likelihood estimates

Parameter	$λ_{1}$	$λ_{2}$	$λ_{3}$	$λ_{4}$	$λ_{5}$	$λ_{6}$	$λ_{7}$	$λ_{8}$	$λ_{9}$	$λ_{10}$	$λ_{11}$	$λ_{12}$	$p_{1}$	$p_{2}$	$σ$	$β$
Estimate	99.10	76.22	88.00	83.66	87.38	95.65	86.49	90.06	91.21	105.71	99.01	101.02	0.18	0.13	$-$ 1.30	$-$ 0.38
St. Error	15.84	13.74	13.57	13.26	13.46	14.21	14.19	13.77	14.27	15.10	15.47	15.54	0.07	0.07	0.68	0.08

The estimate for $σ$ is $-$ 1.30 with the 95% confidence interval $(- 2.63; 0.02)$ , which can be interpreted as a relevant reduction on the number of killed drivers of light goods vehicles in Great Britain after the introduction of mandatory wearing of seatbelts but non-significant at this level of confidence.

The observed number of road casualties in the considered period and the model estimates given by (2.7) are shown in Figure 2, together with the 95% approximate confidence interval built using (2.8). It can be seen that most of the observed values are inside this interval (85.9%). The vertical line represents the moment when the use of the seatbelt became mandatory (February 1983).

Figure 2:

Observed and estimated values with the 95% confidence interval

Using (2.8) again to build the 95% approximate confidence interval but with estimates based on the seasonal INAR(2) model introduced in Moriña et al. (2011), only 69.3% of the observed values were within the limits.

In Figure 2, it can be seen that the observed counts are rather persistently below the model estimated mean. To check whether it is due to a poor fit of the time series model or it is just a consequence of the fact that the one-step ahead predictive distributions are skewed, so that we should indeed expect to see a majority of the observed counts below the model's mean, the mid-pseudo-residuals approach described in Section 2.2 can be used. The results of this approach are shown in Figure 3, showing that the latter explanation is correct and thus supporting the suitability of this model regarding our purpose of detecting the impact of the introduction of mandatory wearing of seatbelts on the number of road causalties.

Figure 3:

Autocorrelation function (ACF) and partial ACF (PACF) of the mid-pseudo-residuals of the model for the number of road casualties in the United Kingdom (1969–1984)

3.3 Example 3: LGV and massive events in Barcelona

Venereal lymphogranuloma (LGV) is an STI caused by the bacteria chlamydia trachomatis. Due to the popularity that the so-called circuit parties have reached recently, especially among gay and bisexual men, where unprotected sex encounters are common, the impact of these massive events over the number of cases of this and other STI is a public health concern (Cheung et al., 2015; Zenilman and Avery, 2015). The analysed data correspond to the number of LGV cases registered in the Barcelona area from January 2007 to December 2014. The time evolution of these data is shown in Figure 4, and no trend or seasonal behaviour is observed.

Figure 4:

Observed number of LGV cases in Barcelona (2007–2014)

According to the ACF and PACF of the process, a model of order 1 seems to be appropriate, so the model

X_{t} = p_{1} \circ X_{t - 1} + W_{t} (λ^{'}),

(3.3)

where

λ^{'} = λ + σ \cdot I (t)

is proposed. In this case, the indicator variable takes the value 1 for all periods of time after the celebration of any circuit festival in Barcelona within the LGV incubation time (between 3 and 30 days; Ceovic and Gulin, 2015). For instance, the first festival took place in early July 2008, so the indicator takes the value 1 in August 2008. In 2009, there were two circuit parties in Barcelona, the first in early June and the second again in July, so the indicator takes the value 1 in July and August 2009.

The estimates of the parameters are shown in Table 2, and all of them are again different from zero. In particular, $\hat{σ} = 1.51 (0.55; 2.47)$ , so a significant effect of the celebration of circuit parties over the number of new LGV cases is detected. This result can be interpreted as a relevant increase in the new LGV cases due to the celebration of the circuit parties.

Table 2:

Maximum likelihood estimates

Parameter	$λ$	$p_{1}$	$σ$
Estimate	1.21	0.50	1.51
St. Error	0.22	0.07	0.49

The AIC of this model is 706.0132, while the AIC corresponding to the standard INAR(1) model is 716.7405. Therefore, the proposed model is preferred. To check the goodness of fit of the model (3.3), the mid-pseudo-residuals approach described in Section 2.2 was used, and the results are shown in Figure 5, supporting the suitability of the model (3.3).

Figure 5:

ACF and PACF of the mid-pseudo-residuals of the model for the number of LGV cases in Barcelona (2007–2013)

Following the approach described in Hudecová et al. (2015), using values of $γ = 0, 0.25$ and $0.45$ , no significant change in the parameters can be detected at a 95% confidence level. This fact highlights the difference between the two approaches, as the focus here is on detecting changes in the innovations due to an intervention or several interventions but then possibly returning to the original pre-intervention stage. Thus, the proposed methodology is able to detect non-structural changes.

4 Discussion

Measuring the impact of planned actions or unexpected events over a time series is an issue that often arises in many fields, including public health, sociology, economics or politics. When dealing with continuous time series, intervention analysis has been widely used in the literature. For instance, in Gilmour et al. (2006), the authors propose a modification of classical intervention analysis to handle the change point with unknown date analysis (a situation where the exact time of an intervention is uncertain). In Huitema et al. (2014), intervention analysis is used to analyse the effect of the introduction of pedestrian countdown timers over the number of accidents involving pedestrians. In this work, we propose a flexible model based on INAR discrete time series models, which is able to suit a wide range of situations focused on the evaluation of the impact of a planned intervention, an unexpected event or the combination of one or more actions or circumstances, so it might be useful to researchers in many areas. In addition to estimating the impact of the intervention, the proposed methodology is capable of providing forecasts, allowing for the midterm estimation of the intervention impact on the counts, improving the performance of the usual INAR models. Regarding the last example, it is shown that this methodology might be used in the public health context to evaluate the impact of certain events that attract a huge (and increasing) number of participants over the incidence of ITS, drug consumption or other problems, although further research is needed to overcome some difficulties as the long incubation periods and the considerable proportion of attendants coming from abroad and therefore unnoticed cases for the event place health authorities.

Supplementary materials

Supplementary materials for this article including the R code and data used in all the examples (a similar simulated dataset was generated for Example 3 as original data were not publicly procurable) are available from http://www.statmod.org/.

Acknowledgments

We would like to thank Professor Pedro Puig, Professor Tilmann Gneiting and the reviewers for their valuable suggestions that helped to improve the article remarkably. We also thank the Agència de Salut Pública de Barcelona, specially Professor Joan Caylà and Patricia García de Olalla for providing us with the LGV data discussed in the last example.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The research leading to these results has received funding from Recer-Caixa (2015ACUP00129) and was partially supported by grants from the Instituto de Salud Carlos III-ISCIII (Spanish Government) co-funded by FEDER funds/European Regional Development Fund (ERDF)—a way to build Europe (References: RD12/0036/0056, PI11/02090), from the Agència de Gestió dAjuts Universitaris i de Recerca (2014SGR 756, 2014SGR 1307) and the Spanish MINECO (FIS2015-71851-P, MTM2015-69493-R). David Moriña acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445) and from Fundación Santander Universidades.

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Intervention analysis for low-count time series with applications in public health

Abstract

Keywords

1 Introduction

2 Model definition

3.1 Example 1: Simulation study

Figure 1:

Simulated INAR(1) process with an intervention at time t = 400 (a) and no intervention (b)

Maximum likelihood estimates

Observed and estimated values with the 95% confidence interval

Autocorrelation function (ACF) and partial ACF (PACF) of the mid-pseudo-residuals of the model for the number of road casualties in the United Kingdom (1969–1984)

Figure 4:

Observed number of LGV cases in Barcelona (2007–2014)

Maximum likelihood estimates

ACF and PACF of the mid-pseudo-residuals of the model for the number of LGV cases in Barcelona (2007–2013)

Supplementary materials

Acknowledgments

Declaration of conflicting interests

Funding

References

Simulated INAR(1) process with an intervention at time $t = 400$ (a) and no intervention (b)