Segmented regression analysis

What questions will the segmented regression help to answer?

Segmented regression analysis is one approach to comparing pre- and post–intervention series; this method has increasingly been used to estimate the effects of health services and policy interventions, but is still rarely used to evaluate the impact of national public health policy.^203-207

The segmented regression approach can be used to assess how much an intervention, such as the introduction of smokefree legislation,

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changed an outcome of interest, such as the national prevalence of smoking, immediately and over time. Segmented regression analysis estimates the trend in a time series before an intervention, and step changes in the level and changes in the trend of the series at the time point immediately following an intervention, taking into account any seasonal patterns in the data.²⁰³

Data requirements

Segmented regression analysis requires data to be in regular, equally-spaced time intervals.²⁰³ Segmented regression can be undertaken with relatively short time series, although recommendations as to the minimum length of the time series vary. Cochrane‘s Effective Practice and

Organisation of Care Group recommend that interrupted time series models (but not ARIMA models, for which the recommendations are longer) have at least three observations prior to the intervention and at least three in the post-intervention period.²⁰⁸ However, a more appropriate recommendation when using monthly data is for at least 12 data points prior to and after the intervention.²⁰³ This allows seasonal variation to be taken into account. This recommendation is not based on power, however, and the power of the model is more likely to depend on the complexity of the autocorrelation.²⁰⁹ Power calculations for this type of analysis are in their infancy.²⁰⁹

Fitting a linear segmented regression model

Wagner et al. describe in detail how to fit a segmented regression model.²⁰³ Simple segmented regression models can be fitted as linear regression models that include terms to describe the trend in the outcome in the pre-intervention period, any immediate step change in the level of outcome following the intervention, and any change in the trend in the

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outcome in the post-intervention compared to pre-intervention data, as shown in equation (1).²¹⁰

Yt = 0 + 1*timet + 2*interventiont + 3*time since intervention + t (1)

In equation (1), 1 is the trend prior to the intervention, 2 is the change in level immediately after the intervention, and 3 is the change in the trend following the intervention. Thus ( 1+ 3) is the post-intervention slope. A parsimonious model is built by eliminating non-significant terms from the model. Table 4-2 shows an example of how one would structure a dataset to investigate a change in prescribing following an intervention

introduced at time point 16.

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Table 4-2. Structure of data for segmented regression analysis

Time

111 Interpreting the results

Running a segmented regression model generates estimates of coefficients, confidence intervals and p-values for the pre-intervention trend, immediate change in level and change in trend of the time series.

The values of the outcome variable predicted by the model can also be obtained and plotted against the original time series to demonstrate, graphically, any changes in the trend and/or level of the time series following the intervention.

Dealing with autocorrelation in segmented regression models: General Additive Models series, we risk drawing incorrect conclusions about the impact of a policy.

Alternative models can be used to take account of more complex autocorrelation structures.

When autocorrelation structures in data are found to be more complex than AR(1), more flexible types of model can be used:

Generalised Additive Models (GAMs) and Generalised Additive Mixed Models (GAMMs). GAMs enable the fitting of seasonal smoothing terms, and therefore allow seasonal autocorrelation to be taken into account.²¹² GAMMs can also fit AR and MA autocorrelation terms, thus ensuring that all autocorrelation is taken into account.²¹⁰

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GAMs and GAMMs can both be fitted in the statistical package R by first running a linear regression model as described above, and then using stepwise elimination to obtain a parsimonious model. If there is evidence of seasonal autocorrelation in the residuals of the model, a smoothing term is included to model any regular seasonal pattern in the outcome. The ACFs and PACFs of this model are then used to establish which autocorrelation terms are required.

These methods have recently been used to evaluate the impact of smokefree legislation in England on hospital admissions for myocardial infarction.¹⁴⁵ The code required to conduct a segmented regression analysis using GAMs and GAMMs in R is presented in Appendix 9.5.

Strengths and limitations of segmented regression analysis

The key strength of segmented regression analysis is that it can control for underlying trends, which is important for ensuring that correct conclusions are drawn. By using GAMs and GAMMs, seasonal effects and autocorrelation can also be taken into account. Further to this, segmented regression can be used to estimate two key parameters – immediate changes in the mean of a series and changes in the trend following an intervention, which gives a useful indication of the immediate and longer term impact of an intervention.

The main strength of segmented regression is that it can be used with relatively short time series. This is particularly important in public health research where long series with regularly-spaced data are not always at hand for the variables of interest.

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Segmented regression also has several limitations. A fundamental limitation of all analysis that investigates the effect of a policy introduced at one point in time is that it is not possible to reliably separate the effects of different policies that are introduced at the same time or in proximate time. For tobacco control policies this is often the case, and care must therefore be taken in interpreting the results.

A further limitation of segmented regression analysis is that it may fail to capture effects that occur prior to the intervention (for example, people trying to quit smoking in the run up to smokefree legislation). It may also fail to detect any transient effects on the trend of a time series.

This may be overcome by shortening the time series, so that a shorter time period following the introduction of a policy is analysed, but this will reduce the power of the model. Autoregressive Integrated Moving Average (ARIMA) models provide a more flexible method for investigating the effects of policy changes which does not have these limitations.

Finally, because segmented regression involves fitting linear regression models, it can only be used when the trends before and after the policy change are linear. ARIMA models can be used when there are non-linear trends in the time series.

4.3.2 Autoregressive Integrated Moving Average (ARIMA)