Fingertips trend markers

Introduction

This section describes the methodology used for the trend markers that are displayed on the Fingertips website. The tests set out below are run by Fingertips on the fly as pages load, including data download pages. The displays are provided to help users identify quickly indicators of interest from the vast array of indicators in Fingertips, but they do not constitute a rigorous analysis of the time series. Anyone interested in a particular indicator should look at the full time series and analyse it more thoroughly.

Criteria for applying the tests

The tests are applied to all indicator series that meet the following criteria:

  • There must be at least five points in the series
  • Time periods must be independent of one another: not overlapping, or rolling time periods
  • Data must be for years: monthly or quarterly data are not tested
  • Indicator type is one of the following: rate (crude, directly standardised or indirectly standardised), proportion (crude or indirectly standardised), ratio (crude, indirectly standardised or rate ratio), life expectancy, slope index of inequality, excess risk, mean, median, gap
  • There must be no breaks in the series (the method described here can be applied to time series with missing values or varying time intervals, but this is not currently implemented in Fingertips)
  • For indicator types other than proportions (crude or indirectly standardised) the indicator values must have confidence intervals. For proportions there must be a numerator and denominator present

Process for testing for ‘recent’ statistical trend

These calculations are derived using the most recent five indicator values in the time series as shown on the Trends display.

There are two different methods: one for proportions (including indirectly standardised proportions) and one for every other indicator type. In each case the gradient of the trend, \(\hat{\beta}\), and the \(\chi^2\) test statistic are calculated – the methods for calculating these are described in the Technical specifications section below.

If \(\chi^2\) > 9.5495 the trend is significant (\(p\) < 0.2%).

The sign of \(\hat{\beta}\) shows the direction of the trend. If \(\hat{\beta}\) < 0 then it is a falling trend; if \(\hat{\beta}\) > 0 then it is a rising trend.

If no significant trend is detected, an amber, horizontal trend marker will be displayed.

If a significant trend is found, the direction of the trend is noted, and the gradient (\(\hat{\beta}\)) calculation is repeated five times, each time excluding one of the five points.

If removing any one of the five points results in a trend in the opposite direction from the trend based on all five points (whether significant or not), then an amber trend marker will be displayed.

If the original trend based on all five points was significant, and the direction of the trend is the same for all six trends, an upward or downward trend marker will be displayed, as appropriate.

The colour of the upward and downward arrows depends on whether the trend is in an unfavourable direction or a favourable one. This is determined by the polarity of the indicator. For example, where ‘high is good’ an upward arrow will be green and a downward arrow red. For indicators without a clear polarity, a blue marker is used to identify significant trends in either direction.

Technical specifications

Proportions – logistic regression

This test requires the numerators (\(r_i\)) and denominators (\(𝑛_i\)) for each of the \(N\) values (\(x_i\)) in the series, and the time values (\(t_i\)). In the Fingertips application, \(N\) is 5.

If only indicator values and numerators are present, calculate the denominators as:

\[n_i = \frac{r_i}{x_i}\]

If only indicator values and denominators are present, calculate the numerators as:

\[r_i =n_ix_i\] If \(t_i\) are not straightforward numeric values (such as years) they must be converted into values – the actual values don’t matter, but the intervals between time values used must be proportional to the real time intervals between the indicator values.

Calculate the gradient of the trend, \(\hat{\beta}\):

\[ \hat{\beta} = \frac{N \sum \left(\ln \dfrac{x_i}{1 - x_i}\right)t_i - \sum \ln \dfrac{x_i}{1 - x_i}\sum t_i}{N \sum {t_i}^2 - (\sum t_i)^2} \]

Calculate the \(\chi^2\) test statistic:

\[ \chi^2 = \frac{\sum n_i(\sum n_i \sum r_i t_i - \sum r_i \sum n_i t_i)^2}{\sum r_i (\sum n_i - \sum r_i)(\sum n_i \sum n_i t_i^2 - (\sum n_i t_i)^2)} \]

All value types except proportions – weighted linear regression

This test requires the lower (\(l_i\)) and upper (\(u_i\)) 95% confidence limits for each of the \(N\) values (\(x_i\)) in the series, and the time values (\(t_i\)). In the Fingertips application, \(N\) is 5.

If \(t_i\) are not straightforward numeric values (such as years) they must be converted into values – the actual values don’t matter, but the intervals between time values used must be proportional to the real time intervals between the indicator values.

The standard errors for each of the values (\(x_i\)) are estimated from the confidence intervals:

\[\sigma_i = \frac{u_i - l_i}{2z_.975}\] where \(z_.975\) is the 97.5th percentile point of the standard normal distribution, usually approximated as 1.9600.

99.8% confidence intervals could be used to derive the standard errors, substituting \(z_.999\) (approximately 3.0902) in the denominator of the equation above.

Calculate the gradient, \(\hat{\beta}\), based on the latest five points in the time series:

\[ \hat{\beta} = \frac{\sum \dfrac{1}{\sigma_i^2} \sum \dfrac{x_it_i}{\sigma_i^2} - \sum \dfrac{t_i}{\sigma_i^2} \sum \dfrac{x_i}{\sigma_i^2}}{\sum \dfrac{1}{\sigma_i^2} \sum \dfrac{t_i^2}{\sigma_i^2} - \left(\sum \dfrac{t_i}{\sigma_i^2}\right)^2} \]

Calculate the variance, \(var (\hat{\beta})\), for the gradient \(\hat{\beta}\):

\[ var(\hat{\beta}) = \frac{\sum \dfrac{1}{\sigma_i^2}}{\sum \dfrac{1}{\sigma_i^2} \sum \dfrac{t_i^2}{\sigma_i^2} - \left(\sum \dfrac{t_i}{\sigma_i^2}\right)^2} \] Calculate the \(\chi^2\) test statistic:

\[ \chi^2 = \frac{\hat{\beta} ^2}{var (\hat{\beta})} \]

Page last updated: August 2025