Trend Analysis

Trend Analysis in Health System Monitoring

Trend analysis estimates how a health system indicator, such as emergency admissions, mortality, waiting times or spending, has moved over a run of years or months, and at what rate. The result is usually a slope, either an absolute change per period or a percentage change per year. In health economics and health policy that slope is the baseline against which performance is judged, the growth assumption inside budget and demand projections, and the counterfactual that interrupted time series and difference-in-differences designs depend on. A rising count of admissions can reflect a larger and older population rather than any change in how care is used, so how the series is built matters as much as the line.

Trend, season and noise in a health series

A time series is a sequence of observations on a population taken repeatedly, normally at equal intervals. Most health series mix a slow movement in the level, a seasonal pattern such as winter pressure on emergency care, and irregular variation from chance and one-off events. Trend analysis isolates the slow movement, and Lopez Bernal and colleagues recommend starting with a scatter plot, which can help to identify the underlying trend, seasonal patterns and outliers. Annual data hide seasonality but give few points, while monthly data give more points but need the seasonal pattern modelled.

Fitting a trend line

The simplest trend model treats time as the only explanatory variable in a linear regression, so the slope is the average absolute change in the indicator per period. It is fitted by ordinary least squares, like most regression analysis.

$$Y_t = \beta_0 + \beta_1 t + \varepsilon_t$$

where $Y_t$ is the value of the indicator in period $t$, $t$ is the period number, $\beta_0$ is the intercept (the fitted value at $t = 0$), $\beta_1$ is the trend, the change in $Y$ per period, and $\varepsilon_t$ is the error in period $t$.

$$\hat{\beta}_1 = \frac{\sum_t (t - \bar{t})(Y_t - \bar{Y})}{\sum_t (t - \bar{t})^2}$$

where $\hat{\beta}_1$ is the least squares estimate of the trend, $\bar{t}$ is the mean period number and $\bar{Y}$ is the mean of the indicator. The intercept estimate is $\bar{Y} - \hat{\beta}_1 \bar{t}$.

Log-linear trends and the annual percentage change

Health spending and disease rates often grow by a roughly constant proportion rather than a constant amount. The Joinpoint help system of the US National Cancer Institute (NCI) describes this case, in which rates change at a constant percentage of the previous year's rate and therefore change linearly on a log scale. The trend is fitted to the natural log of the rate and summarised as an annual percentage change (APC).

$$\ln R_t = b_0 + b_1 t, \qquad \text{APC} = 100 \left( e^{b_1} - 1 \right)$$

where $R_t$ is the rate in year $t$, $b_0$ is the intercept on the log scale, $b_1$ is the slope on the log scale and the APC is the year-on-year percentage change in the rate. The NCI points out that the APC is comparable across scales, so a rare and a common cancer can both be described as changing at 1% a year. The same property lets a commissioner compare growth in a small specialist service and in a large acute budget.

Counts, rates and population change

Many health system indicators are counts, and a count rises when the population grows even if risk is unchanged. A Poisson regression of the count on time with the log of the population as an offset models the rate instead, and Lopez Bernal and colleagues used the age-standardised population as an offset for this purpose. They note that real counts often show over-dispersion, with variance greater than the mean, which gives incorrect standard errors unless corrected, and they apply a scaling adjustment. A negative binomial regression is another common remedy.

Population ageing is the second problem. The Office for National Statistics (ONS) states that age-standardised mortality rates account for population size and age structure and allow more accurate comparison of mortality across time and location, and it standardises them to the 2013 European Standard Population. A trend in a crude mortality rate or in crude admissions per head mixes changing risk with changing age mix, which is why spending studies separate ageing-driven cost growth from other growth.

Joinpoint models for changing trends

A single straight line forces one rate of change across the whole period, which is often implausible over a decade of policy change. Joinpoint regression fits connected line segments and lets the data choose where the slope changes. Kim and colleagues fit the joinpoints by grid search, choose their number by permutation tests with a Bonferroni correction, and extend the tests to rates with Poisson variation and possibly autocorrelated errors.

When the date of a change is known in advance, such as a smoking ban, the break is set by design and the model becomes segmented regression, which Wagner and colleagues apply to policy and educational interventions intended to improve medication use or contain costs.

Assumptions behind a trend estimate

Least squares gives an unbiased slope under fairly weak conditions, but the standard errors and confidence intervals that come with it rest on stronger ones. Health series break them often enough that each deserves a check.

Independence is the first. Lopez Bernal and colleagues note that consecutive observations tend to be more similar to one another than those further apart, a phenomenon known as autocorrelation, which in many epidemiological series is largely explained by seasonality. They advise examining residual plots and the partial autocorrelation function, using tests such as the Breusch-Godfrey test where data are normally distributed, and adjusting any remaining autocorrelation with methods such as Prais regression or ARIMA models.

Seasonality is the second. An uneven mix of winter and summer months at either end of a short series can tilt the slope, and the tutorial lists stratifying by calendar month, Fourier terms and splines as ways to control it.

Stable measurement is the third. Lopez Bernal and colleagues cite a new troponin test introduced progressively in Italy, a change in diagnostic procedure that could alter recorded acute coronary events independently of true incidence, and coding changes act the same way on routine health data. The window matters too, since more data points are not always better where historical trends have changed substantially.

Worked example with emergency admissions in an illustrative system

All numbers in this example are illustrative. A health system records emergency admissions, in thousands, over five years, giving 96, 101, 104, 108 and 111 for years 1 to 5. The population is assumed stable, so counts and rates move together.

1. Means. The mean year is 3 and the mean admissions figure is (96 + 101 + 104 + 108 + 111) / 5 = 104 thousand.

2. Slope. The deviations of year from its mean are minus 2, minus 1, 0, 1 and 2, so the sum of squared deviations is 10. The cross-product sum multiplies each year deviation by the admissions deviation (minus 8, minus 3, 0, 4 and 7), giving 16, 3, 0, 4 and 14, which add to 37. The slope is 37 / 10 = 3.7 thousand admissions a year. The intercept is 104 less 3.7 times 3, which is 104 less 11.1, or 92.9.

3. Fit. The fitted values are 96.6, 100.3, 104.0, 107.7 and 111.4, leaving residuals of minus 0.6, 0.7, 0, 0.3 and minus 0.4. The residual sum of squares is 0.36 + 0.49 + 0 + 0.09 + 0.16 = 1.10 and the total sum of squares is 64 + 9 + 0 + 16 + 49 = 138, so the R-squared is one minus 0.008, or 0.992.

4. Precision. The residual standard error is the square root of 1.10 / 3, or 0.606, and the standard error of the slope is 0.606 divided by the square root of 10, or 0.191. With 3 degrees of freedom the t value for a 95% interval is 3.182, so the interval is 3.7 plus or minus 0.609, or 3.09 to 4.31 thousand a year. Five annual points cannot reveal autocorrelation, so the interval rests on an independence assumption that goes unchecked.

5. Relative trend. On the log scale the slope is 0.0357, so the growth factor is e to the power 0.0357, or 1.0364, and the APC is 3.64% a year. The linear slope against the mean, 3.7 / 104, gives a similar 3.56%.

6. Projection. Two years beyond the data, in year 7, the linear model gives 92.9 + 3.7 × 7 = 118.8 thousand and the log-linear model gives about 119.8 thousand. Against the 111 thousand of year 5 the linear projection implies 7.8 thousand extra admissions, and at an illustrative GBP 2,000 per admission that is 7.8 × 2,000 = 15,600 thousand, or GBP 15.6 million.

7. Interpretation. Admissions rose steadily by about 3.7 thousand, or 3.6%, a year, and the two functional forms agree closely just beyond the data. The slope says nothing about why admissions rose, so ageing and coding changes need ruling out before it feeds a budget.

Trend analysis in performance monitoring and economic decisions

In health system performance work the trend is the first question asked of an indicator, because a current value is hard to judge without its direction of travel. Statistical process control and the control chart answer a different question, whether a new point departs from the expected range of variation, and suit fast operational monitoring better than estimating a rate of change. Comparing trends across organisations is a form of benchmarking, and splitting productivity change into efficiency gains and frontier shift is done with the Malmquist index.

Trend estimates are also inputs to economic decisions. The ISPOR good practice report on budget impact analysis recommends data specific to the decision maker's population and, where eligible population size or treatment patterns change, a model that accounts for people entering and leaving the eligible population over time, which in practice draws on local trends. Demand forecasting for services and staff starts from past utilisation in the same way, as does work on health spending as a share of GDP.

In evaluation, the pre-intervention trend is the counterfactual. Lopez Bernal and colleagues describe the interrupted time series design as using the series to establish an underlying trend that an intervention interrupts at a known point. A difference-in-differences analysis instead assumes that treated and comparison groups would have followed parallel trends, a claim that pre-period data can probe but not prove. In both designs, and in any projection forward, the fitted line is an assumption, and the further it runs beyond the data the more it depends on the chosen functional form, as with extrapolation in decision models.

Misreadings of health system trends

Most errors come from reading more into a fitted slope than the data allow. Performance reports and budget papers repeat the same few.

1. Treating counts as rates. A rise in admissions in a growing, ageing population may show no change in age-specific risk.
2. Straight-line projection far beyond the data. Linear and exponential trends that agree over five years can diverge sharply over fifteen.
3. Overstating precision. Positively autocorrelated monthly data make ordinary least squares intervals too narrow.
4. Ignoring recording changes. A new code, test or counting rule can create a step that a single line reads as trend.
5. Choosing the start year after seeing the data. An unusually high or low first year changes the slope, and picking organisations for an extreme recent value invites regression to the mean.
6. Confusing absolute and relative change. A constant absolute rise is a falling percentage rise, and the two can rank services differently.

Excel functions for trend estimation

Excel has least squares trend functions that suit routine monitoring. The formulas assume the worked example's years in column A and admissions in column B, rows 2 to 6.

Microsoft documents that TREND and LINEST fit a straight line by least squares and that LOGEST fits an exponential curve. None of these functions tests for autocorrelation or seasonality.

Sources

• Lopez Bernal J, Cummins S, Gasparrini A. Interrupted time series regression for the evaluation of public health interventions: a tutorial. International Journal of Epidemiology. 2017;46(1):348-355. https://doi.org/10.1093/ije/dyw098
• Kim HJ, Fay MP, Feuer EJ, Midthune DN. Permutation tests for joinpoint regression with applications to cancer rates. Statistics in Medicine. 2000;19(3):335-351. https://pubmed.ncbi.nlm.nih.gov/10649300/
• Microsoft. LINEST function, LOGEST function, SLOPE function and TREND function. Microsoft Support, accessed 28 October 2026. https://support.microsoft.com/en-us/office/linest-function-84d7d0d9-6e50-4101-977a-fa7abf772b6d
• National Cancer Institute, Surveillance Research Program. Annual Percent Change (APC) and Confidence Interval. Joinpoint Help System, accessed 28 October 2026. https://surveillance.cancer.gov/help/joinpoint/setting-parameters/method-and-parameters-tab/apc-aapc-tau-confidence-intervals/estimate-average-percent-change-apc-and-confidence-interval
• Office for National Statistics. Deaths registered in England and Wales: 2024. Statistical bulletin, released 9 October 2025. https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/deaths/bulletins/deathsregistrationsummarytables/2024
• Sullivan SD, Mauskopf JA, Augustovski F, Jaime Caro J, Lee KM, Minchin M, et al. Budget impact analysis: principles of good practice. Report of the ISPOR 2012 Budget Impact Analysis Good Practice II Task Force. Value in Health. 2014;17(1):5-14. https://doi.org/10.1016/j.jval.2013.08.2291
• Wagner AK, Soumerai SB, Zhang F, Ross-Degnan D. Segmented regression analysis of interrupted time series studies in medication use research. Journal of Clinical Pharmacy and Therapeutics. 2002;27(4):299-309. https://doi.org/10.1046/j.1365-2710.2002.00430.x