Pólya Regression Fitted to Accident Data

The business question

A health authority records sports accidents each year. Over sixteen years they have grown from about 650 to about 1,400, with large swings from year to year: far more than a Poisson process would give. A Pólya regression describes them: the expected number grows exponentially, exp(β0 + β1·year), and each year's count scatters around it with extra variation h.

The parameters were fitted by maximising the log-likelihood with Excel's Solver. The authority wants the next three years. Projecting with the fitted parameters alone treats the trend as known. With sixteen noisy years it is not, and the model shows how much that widens the forecast.

Who it is for

Analysts forecasting overdispersed counts from a short history: claims, accidents, incidents, demand.

Main outputs

Accidents in each of the next three years at P10 / median / P90, with the fitted parameters and with their uncertainty · the fitted trend and its standard error · a check that the stored fit is the maximum.

How the model works

Four sheets after this one, read from left to right. Each one feeds the next.

1. 1

   Inputs sheet

   β0 (the log of the expected accidents in year 0), β1 (the growth per year on the log scale) and h, as fitted by Solver.

2. 2

   History sheet

   Expected accidents each year, each year's log-likelihood (VosePolyaProb10), and their total, which the fitted parameters maximise. The score (the slope of the log-likelihood) is about 0 at the maximum, which a check confirms.

3. 3

   Forecast sheet

   The information matrix, built from the History sheet's weights, gives the covariance of β0 and β1. With fixed parameters each year is VosePolya(1/h, λh). With parameter uncertainty, β0 and β1 are first drawn from their correlated normal distribution (two VoseNormal draws through a Cholesky factor), and λ follows from them.

4. 4

   Results sheet

   Percentiles of each year's accidents both ways, the fitted trend with its standard error, and the chart of the history and the fit.

ModelRisk techniques

• Pólya regression VosePolya
• Log-likelihood VosePolyaProb10
• Score check of the fit
• Parameter uncertainty in the projection

Find similar models

Product

• ModelRisk

Type

• Simulation
• Fitting and estimation

Application

• Forecasting
• Statistical techniques

Uses

• Normal
• Pólya

Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Units and conventions

Year

Year 0 is the last year of data; years 1 to 3 are forecast.

Pólya

VosePolya(α, β) is a Poisson with a Gamma(α, β) mean: here α = 1/h and β = λh, so the mean is λ and the variance λ + hλ².

Log scale

β1 = 0.029 means about 3% growth a year: exp(β1) − 1.

Key assumptions and limitations

Trend

Exponential in the year, as in the original example.

Independence

Years are independent given the trend.

Parameter uncertainty

Approximately normal around the fit (large-sample theory); h is taken as known, which affects only the year-to-year scatter.

Data

The accident counts and the fitted parameters are those of ModelRisk's earlier Polya Regression example.

How to use it

1. Replace the history on the History sheet with your own counts.
2. Refit: run Solver to maximise the log-likelihood by changing β0, β1 and h on the Inputs sheet. The score check reads OK at the maximum.
3. Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
4. Use the forecast with parameter uncertainty: the history is short.

ModelRisk functions used

VosePolyaProb10

Each year's log-likelihood (base 10), converted to natural logs.

VosePolya

Each forecast year's accidents: a Gamma-mixed Poisson with mean λ.

VoseNormal

The two standard normal draws behind the uncertain β0 and β1.

VoseOutput

The forecast years as output ranges, both ways.

VoseSimPercentile, VoseSimStdev

Statistics of the simulation.

Further reading

• Monte Carlo simulation explained (Vose guide)
• Cameron, A. C. and Trivedi, P. K. (2013). Regression Analysis of Count Data, 2nd edition. Cambridge University Press: the negative binomial (NB2) model.
• Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: parameter uncertainty in fitted models.