Day-of-week effects — how three effects combine to shape reported data

The same weekly rhythm, three different causes

Each one belongs in a different part of the model

Real reporting rarely follows a single fixed delay. One of the most common departures is a day-of-week pattern, and it helps to keep three distinct causes apart because each acts on a different thing and each belongs in a different module of the model.

Reference-date effect on onsets. Events genuinely happen at different rates by day of week — for example fewer onsets are recorded at weekends. This changes when events occur, and belongs in the expectation model.
Reference-date effect on the delay. The reporting delay distribution itself differs by the day an event occurred — for example weekend onsets take longer to be reported. This reshapes how long reporting takes, and belongs in the reference (delay) model.
Report-date effect. The reporting system processes fewer reports at weekends regardless of when the event occurred. This acts on the day the report lands, and belongs in the report-date model.

The figure below mirrors the simulation in the session: it starts from a smooth underlying onset curve and applies the three effects with the same weekly weights used there.

Switch each effect on and watch the reported series build up

Grey line: the smooth underlying onsets we would like to recover. Bars: what we actually observe, reported by report day.

underlying onsets (smooth truth) observed reports by report day weekend (Sat/Sun)

The three weekly ingredients behind the series above. The onset weekday effect scales how many onsets occur on each weekday, the mean delay by weekday reshapes how long weekend onsets take to report, and the report weekday effect scales reports on the day they land. Each panel is full colour when its toggle is on and greyed when off; weekends (Sat/Sun) are highlighted.

Reference-date effect on onsets fewer onsets on Fri–Sun (weights 0.95, 0.7, 0.7) Reference-date effect on the delay weekend onsets reported more slowly (meanlog 1 → 1.4) Report-date effect fewer reports processed on Fri–Sun (weights 0.9, 0.4, 0.5)

effects on: 0 of 3 weekend report dip: 0% weekly variation in reports: 0%

With no effects the reports are just a smoothed, delayed copy of the onset curve. Turn on the report-date effect alone and a clean weekly dip appears on the report axis. Add the two reference-date effects and the pattern gets richer: onsets themselves dip at weekends, and weekend onsets arrive later, smearing their weekly signature into the following week.

All three act on different things, so they are separable and each is a one-line addition to the relevant module. That is exactly why a single fixed delay cannot capture them and why the report-date effect had no place in the hand-rolled p_t,d, which was a function of the delay alone.

Things worth remembering

Weekly wiggle is not one thing. A day-of-week pattern in the data can come from real variation in events, from a delay that depends on the onset day, or from the reporting system itself — often all three at once.
Each cause lives in a different module. The reference-date onset effect belongs in the expectation model, the reference-date delay effect in the delay model, and the report-date effect in the report model.
They are separable. One shifts the level of onsets, one reshapes their delay, and one scales reports on the calendar day they land, so fitting them together lets the model absorb the weekly reporting dip without distorting the nowcast of the underlying onsets.