Department Medical Statistics, London School of Hygiene and Tropical Medicine, Keppel Street, WC1E 7HT, London, UK

Department of Epidemiology, Lazio Regional Health Service, Via di Santa Costanza 53, 00198 Rome, Italy

Abstract

Background

Measures of attributable risk are an integral part of epidemiological analyses, particularly when aimed at the planning and evaluation of public health interventions. However, the current definition of such measures does not consider any temporal relationships between exposure and risk. In this contribution, we propose extended definitions of attributable risk within the framework of distributed lag non-linear models, an approach recently proposed for modelling delayed associations in either linear or non-linear exposure-response associations.

Methods

We classify versions of attributable number and fraction expressed using either a forward or backward perspective. The former specifies the future burden due to a given exposure event, while the latter summarizes the current burden due to the set of exposure events experienced in the past. In addition, we illustrate how the components related to sub-ranges of the exposure can be separated.

Results

We apply these methods for estimating the mortality risk attributable to outdoor temperature in two cities, London and Rome, using time series data for the periods 1993–2006 and 1992–2010, respectively. The analysis provides estimates of the overall mortality burden attributable to temperature, and then computes the components attributable to cold and heat and then mild and extreme temperatures.

Conclusions

These extended definitions of attributable risk account for the additional temporal dimension which characterizes exposure-response associations, providing more appropriate attributable measures in the presence of dependencies characterized by potentially complex temporal patterns.

Background

Epidemiological studies usually rely on effect summaries based on

Problems in the definition of these measures may arise in the presence of

In this contribution, we extend this research and attempt to formalize the definition of attributable risk measures within the DLNM modelling framework. In particular, we illustrate how complex non-linear and temporal patterns can be accounted for in the computation of the attributable risk. Also, we show how attributable components related to different exposure ranges can be separated. We propose an example on the estimation of the health burden attributable to temperature with time series data, a problem which has received quite a lot of interest recently due to climate change predictions. However, the approach can be easily applied to other exposure-lag-response associations. The method is implemented in simple functions developed within the

Methods

Attributable risk measures

A general definition of the attributable fraction AF_{
x
} and number AN_{
x
} for a given exposure

with _{
x
} used in Eq. (1a) represents the risk associated with the exposure, and it usually corresponds to the logarithm of a ratio measure such as relative risk, relative rate or odds ratio. It is generally obtained from regression models while adjusting for potential confounders. The general definition of _{
x
} used here refers to the association with a specific exposure intensity _{0}. For linear exposure-response relationships, the association can also be reported as _{
x
}, which is easily applicable to non-linear exposure-response relationships, throughout the manuscript.

The theoretical nature of these effect measures is based on a _{0}. Typically, such a reference is represented by the absence of association, meaning _{0}=0 and _{
x
} can be simply re-parameterized as

Eq. (1a) can be extended to define the risk attributable to multiple exposures _{1},…,_{
p
}:

with

A review of the DLNM modelling framework

The basic idea underpinning the development of DLNMs is that the risk at time _{0},…,_{0} and **
η
**), written in terms of parameters

The function **w**
_{
x,t
} linearly combined with the parameters **
η
**. Simpler DLMs are defined by Eq. (3) by assuming

The complex parameterization of exposure-lag-response associations provided by Eq. (3) can be more easily interpreted by computing effect summaries from the original parameters **
η
**. Specifically, the bi-dimensional exposure-lag-response risk surface modelled through

This _{
x,ℓ
} from exposures

Forward and backward perspectives

The term _{
x,ℓ
} for each intensity _{
x,ℓ
} are the contributions from the exposure _{
t
} occurring at time _{0},…,_{
x,ℓ
} are the contributions to the risk at time _{0},…,

Conceptual model for the interpretation of exposure-lag-response associations: forward (left panel) and backward (right panel) perspectives

**Conceptual model for the interpretation of exposure-lag-response associations: forward (left panel) and backward (right panel) perspectives.**

Attributable risk from DLNMs

The effect summaries provided above can be used for defining attributable risk measures within the DLNM framework. The idea is to treat the associations with exposures at different lags as independent contributions to the risk. A neat definition can be developed using a backward perspective, assuming the risk at time _{
x,t
} and number b-AN_{
x,t
} at time

with _{
t
} as the number of cases at time _{
x,t
} and b-AF_{
x,t
} are interpreted as the number of cases and the related fraction at time _{0},…,_{0} throughout the same period.

An alternative version can be obtained using a forward perspective. Among other possible definitions, forward attributable number f-AN_{
x,t
} and fraction f-AF_{
x,t
} can be defined as:

This alternative version has some advantages if compared to the backward definition. First, the counterfactual condition is simpler: f-AF_{
x,t
} and f-AN_{
x,t
} are interpreted as the fraction and number of future cases in the period _{0},…,_{0}. Moreover, the overall cumulative risk _{
t
} in (6a) is available also when the bi-dimensional exposure-lag-response is reduced to uni-dimensional exposure-response relationship, a step often needed in multi-site studies

However, the forward version also has an important limitation, related to the fact that the contributions are associated to risks measured at different times. The attributable number f-AN_{
x,t
} in (6b) is computed by averaging the total counts experienced in the next _{0},…,

Separating attributable components

The definitions provided in Eq. (5)–(6) can be extended to separate the attributable components related to specific exposures or exposure ranges. This will be used later in the example to single out the contributions from cold and heat in temperature-health associations. Let’s define a range

simply selecting the risk contributions from past exposures included in the range _{
ℓ
}=_{0} for the whole lag period _{0},…,

The forward version has the additional advantage that for two non-overlapping ranges _{1} and _{2} the sum of the components is equal to the overall attributable risk, namely

Total attributable risk

The attributable risk measures provided above can be computed for each of the _{
t
o
t
} and fraction AF_{
t
o
t
} is provided by:

The equations above can be applied either to forward or backward attributable risk and to separate components, simply substituting the related attributable numbers in Eq. (8a).

Computing uncertainty intervals

Analytical formulae for confidence intervals of attributable risk measures are not easily produced **
η
**

Results

The methods illustrated in the previous section are applied to estimate the all-cause mortality risk attributable to temperature, using daily time series from two cities, London and Rome, in the periods 1993-2006 and 1992-2010 respectively. R scripts and data implementing the method and partly replicating the results are provided as Additional files

**R script implementing the function to compute attributable risk measures.**

Click here for file

**Documentation of the attrdl function.**

Click here for file

**R script for fitting the regression models in the example.**

Click here for file

**R script for computing attributable risk in the example.**

Click here for file

**R script for producing the graphs in the example.**

Click here for file

**File with data used in the example.**

Click here for file

Modelling strategy

We fitted a standard time series Poisson model allowing for overdispersion, controlling for seasonal and long term trends and day of the week, using a 10 df/year spline and indicator variables, respectively. Model selection is still an issue of current research within the DLNM framework, although simulation studies indicate a good performance of methods based on the Akaike Information criterion (AIC)

In the specific case of temperature where a null exposure condition cannot be defined, a reasonable choice is to center the cross-basis in Eq. (3) to the temperature of minimum risk, as suggested in previous publications _{0} for the computation of the attributable risk measures. These are obtained for the whole temperature range, and then for cold and heat contributions by separating the associations with temperatures lower or higher than _{0}. In addition, the attributable components are separated further in mild and extreme cold and heat by selecting as cut-off values the 1^{st} and 99^{th} percentiles of city-specific distributions, corresponding to 0.4°C and 23.7°C in London and 2.6°C and 28.6°C in Rome.

We derived empirical confidence intervals for backward total attributable numbers and fractions, computed overall and for separated components, by simulating 5,000 samples from the assumed distribution of

Risk attributable to temperature

The estimated associations between temperature and all-cause mortality in the two cities are illustrated in Figure

Association between temperature and all-cause mortality

**Association between temperature and all-cause mortality.**

Table _{
t
o
t
} and f-AF_{
t
o
t
}, with 95%eCI. The backward version indicates that overall 13.59% and 12.58% deaths are attributable to temperature in London and Rome within the study periods, respectively. As expected, the corresponding estimates computed forward are affected by a certain degree of negative bias associated to the averaging of future mortality within the lag period, as described above. Nonetheless, the difference is not substantial in this case.

**Deaths**

**Overall**

**Cold**

**Hot**

London 1993–2006 and Rome 1992–2010.

London

845,215

b-AF_{
t
o
t
}

13.59 (10.04–17.09)

12.95 (9.32–16.38)

0.66 (0.52–0.80)

f-AF_{
t
o
t
}

13.41 (9.72–16.87)

12.84 (9.38–16.33)

0.57 (0.45–0.68)

Rome

395,691

b-AF_{
t
o
t
}

12.58 (9.30–15.64)

10.84 (7.37–14.23)

1.74 (1.12–2.37)

f-AF_{
t
o
t
}

12.27 (8.94–15.41)

10.72 (7.19–14.00)

1.55 (0.95–2.13)

Cold, heat and extreme components

The total backward attributable risk is then separated into components due to cold and hot temperatures, defined as those below and above the optimal temperature, respectively. The estimates, computed using Eq. (7), are reported in Table _{
t
o
t
} equal to 12.95% and 10.84%, compared to 0.66% and 1.74% for heat, in the two cities. Estimates of forward attributable risk are very similar, and as expected their sum is equal to the overall burden, differently than for the backward version.

The analysis is extended by further separating the attributable components into contributions from mild and extreme temperatures, with results summarized in Table

**Extreme cold**

**Mild cold**

**Mild hot**

**Extreme hot**

London 1993–2006 and Rome 1992–2010.

London

b-AF_{
t
o
t
}

0.55 (0.45–0.64)

12.48 (8.86–15.88)

0.31 (0.23–0.38)

0.36 (0.29–0.43)

f-AF_{
t
o
t
}

0.47 (0.40–0.53)

12.38 (8.98–15.78)

0.29 (0.22–0.35)

0.28 (0.23–0.33)

Rome

b-AF_{
t
o
t
}

0.59 (0.47–0.70)

10.37 (6.88–13.63)

1.45 (0.89–2.01)

0.33 (0.25–0.40)

f-AF_{
t
o
t
}

0.47 (0.39–0.54)

10.27 (6.69–13.50)

1.32 (0.75–1.85)

0.25 (0.19–0.30)

In contrast, the comparison between the two cities is rather different for the components attributable to mild and extreme hot temperatures. In spite of the stronger risk in London, the attributable fraction is similar for extreme heat and even higher in Rome for mild heat (1.32%–1.45% versus 0.25%–0.33%). This apparent contradiction is explained by the different temperature distribution, and in particular the percentile corresponding to the optimal temperature, corresponding to 93.6^{th} and 72.5^{th} in London and Rome. This result suggests the hypothesis that the population in Rome is more adapted to the range of temperatures corresponding to extreme hot if compared to London, where the population experienced only a few days of unusually high temperatures.

The harvesting paradox

Accounting for the additional lag dimension in exposure-lag-response associations involves further complexities in the interpretation of attributable risk measures. We now focus our attention to the association with hot temperature in Rome. The left panel of Figure

Lag structure and harvesting paradox

**Lag structure and harvesting paradox.** Left panel: lag-response associations between various temperatures and all-cause mortality. Rome 1992–2010. Right panel: daily number of deaths attributable to heat, computed forward (green circles) and backward (yellow squares), and temperature trend. Rome July-Sept 1992.

This phenomenon has interesting implications. An example is offered by the right panel of Figure _{
x,t
} and f-AN_{
x,t
} attributable to heat, computed backward and forward for the first summer in the time series for Rome, with related temperature trend. As expected, a substantial number of deaths are attributable to temperatures above the optimal value, represented by the horizontal dotted line, in the period mid-July to mid-August. The trend of forward attributable deaths f-AN_{
x,t
} closely follows the daily temperatures, consistently with the definition of number of deaths attributable to the temperature in day _{
x,t
} decreases to zero and even becomes negative in late summer days, although the overall cumulative exposure-response in Figure

This paradox is explained by the counterfactual condition associated with the backward perspective. Specifically, each b-AN_{
x,t
} compares the association with the observed temperatures in the past _{0}. In the presence of harvesting, the observed population becomes ‘healthier’ than the counterfactual population after a series of heat days, due to the depletion of the susceptible pool. This explains the negative attributable numbers for specific combinations of lagged exposures. This fact emphasises that harvesting should not be interpreted as a true protective association at longer lags, but rather as an artefact due to a change in the underlying population following a stress, which affects the counterfactual condition. This issue is relevant when using backward attributable risk measures b-AN_{
x,t
} and b-AF_{
x,t
} to assess the contribution of specific days. However, similarly to the net overall cumulative risk, the total attributable number b-AN_{
t
o
t
} and fraction b-AN_{
t
o
t
}, produced by Eq. (8) and reported in Tables

Discussion and conclusions

In this contribution we illustrate an extended definition of attributable risk measures based on the DLNM framework. Consistently with this class of models, such a definition accounts for the complex pattern of potentially non-linear and delayed associations described through exposure-lag-response associations.

Two alternative definitions of attributable risk are proposed, assuming backward or forward perspectives. The former provides more consistent estimators which naturally arise from the structure of the regression model, where distributed lag terms at times

Strictly speaking, the definition given in Eq. 1a is interpreted as the attributable fraction among the sub-population of exposed subjects. In the setting of time series analysis for environmental stressors, the whole population is usually considered as exposed, and this definition can be more generally interpreted as the population attributable fraction. If only a subset is instead exposed, Eq. (5)–(8) can be easily extended using the equations proposed by Steenland and Armstrong

Previous papers suggested approaches for producing attributable risk from distributed lag models when applied to heat-mortality associations. Baccini and colleagues applied DLMs and computed attributable risk measures, specifically addressing the issue of harvesting

An advantage of the proposed method is the provision of estimates for separate components of the attributable risk, associated with different exposure ranges. In the specific case of temperature-health associations, this allows the separation of attributable risks from cold and heat, and further from mild and extreme temperatures. The estimates reported in the example highlights how the simple analysis of exposure-response curves can be misleading in the attribution of risk, and that most of the mortality in the two cities is in fact attributable to mild cold temperatures, in spite of the relatively low RR.

The availability of attributable risk measures, complementary to estimates of exposure-response associations, is essential for the identification and planning of public health interventions. Their extension to exposure-lag-response associations allows the computation of such measures from dependencies showing potentially complex non-linear and temporal patterns.

Abbreviations

AF: Attributable fraction; AN: Attributable number of cases; RR: Relative risk; DLM: Distributed lag models; DLNM: Distributed lag non-linear models; eCI: empirical confidence interval.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AG conceived the idea of attributable risk measures for DLNMs and worked out their algebraic definitions. AG and ML developed the final version of the measures, planned the example and carried out the analysis, drafted the final version of the manuscript. AG provided the software implementation through the R scripts. Both authors read and approved the final manuscript.

Acknowledgements

AG is supported through a Methodology Research fellowship awarded by Medical Research Council-UK (grant ID G1002296).

Pre-publication history

The pre-publication history for this paper can be accessed here: