Measurement error in time-series analysis: a simulation study comparing modelled and monitored data

Table 2 Summarising the analysis of 1000 simulated data sets: urban background pollution concentrations with additive error

Description of simulated data	No. of grids per region containing a monitor	Ozone β × 10 = 0.00399		log_e(Nitrogen Dioxide) β = 0.0419
		$\hat{β} \times 10$	Coverage probability;	$\hat{β}$	Coverage probability;
		$SE (\hat{β}) \times 10$	Power	$SE (\hat{β})$	Power
*Monitor data:* regional average used for each 5 km × 5km grid within region (instrument and monitor-site location error included)	1	0.00375 (0.00209)	94%; 43%	0.0297 (0.0104)	78%; 81%
	2	0.00388 (0.00214)	95%; 44%	0.0348 (0.0113)	91%; 86%
	3	0.00393 (0.00215)	95%; 45%	0.0369 (0.0117)	93%; 89%
	5	0.00396 (0.00216)	96%; 45%	0.0387 (0.0120)	94%; 90%
	10	0.00400 (0.00217)	95%; 45%	0.0403 (0.0122)	95%; 90%
*“true” data:* grid-specific monitor data (no instrument or monitor-site location error)	25	0.00394 (0.00217)	95%; 45%	0.0417 (0.0124)	95%; 93%
*Model data:* grid-specific model data	-	0.00325 (0.00226)	94%; 29%	0.0193 (0.0076)	15%; 72%

The table presents estimated regression coefficients $(\hat{β})$ , standard errors $(SE (\hat{β}))$ , coverage probabilities and power, each based on the analysis of 1000 sets of simulated time-series data. The “true” value of the regression coefficient β for ozone (i.e. β × 10 = 0.00399) equates to a 0.4% increase in mortality per 10 μg/m³ increase in ozone and the “true” value of the regression coefficient for log_e(NO₂) (i.e. β = 0.0419) equates to a 0.4% increase in mortality per 10% increase in NO₂.