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Appendix D. Tables showing simulation results from synthetic data.

Table D1: Simulation results from synthetic "Mallard data".

Table D2: Simulation results from synthetic "Canvasback data".

Table D3: Standard deviation of process stochasticity () from synthetic data.

Table D1. Simulation results from synthetic Mallard data. Results are shown for both the state-space model as well as the case where sampling errors are ignored (i.e., the AR-model).

a) Average estimated direct density dependence from simulated data. Standard deviation (SD) of the 300 runs are included in parentheses.
		No covariate (c = 0)		Ponds included (c = 0.1)
State-space	True	low noise	high noise	low noise	high noise
	0	-0.06 (0.09)	-0.10 (0.13)	-0.07 (0.07)	-0.08 (0.08)
	-0.2	-0.25 (0.14)	-0.32 (0.21)	-0.27 (0.09)	-0.29 (0.13)
	-0.5	-0.51 (0.17)	-0.63 (0.26)	-0.56 (0.12)	-0.59 (0.17)
AR-model
	0	-0.14 (0.11)	-0.31 (0.20)	-0.14 (0.08)	-0.25 (0.16)
	-0.2	-0.33 (0.14)	-0.56 (0.20)	-0.29 (0.11)	-0.48 (0.17)
	-0.5	-0.61 (0.15)	-0.81 (0.17)	-0.57 (0.16)	-0.74 (0.18)

b) Average uncertainty of estimates given as (i) mean SD of posterior distribution and (ii) average standard error.
		c = 0		c = 0.1
State-space (i)	True	low noise	high noise	low noise	high noise
	0	0.07	0.10	0.06	0.07
	-0.2	0.13	0.20	0.09	0.12
	-0.5	0.18	0.31	0.12	0.17
AR-model (ii)	0	0.08	0.11	0.08	0.10
	-0.2	0.12	0.14	0.11	0.13
	-0.5	0.14	0.15	0.14	0.15

c) The proportion (of 300 series) reflecting number of times the estimated direct density dependence () would be considered significantly lower (using estimate + 1.96*SE)† than the true .
		c = 0		c = 0.1
State-space	True	low noise	high noise	low noise	high noise
	0	0.050	0.026	0.126	0.086
	-0.2	0.026	0.016	0.056	0.036
	-0.5	0.010	0.030	0.050	0.060
AR-model
	0	0.356	0.766	0.270	0.550
	-0.2	0.203	0.696	0.063	0.580
	-0.5	0.097	0.510	0.060	0.296

   Notes: Strength of the direct density dependence (true ) varies between zero, -0.2 and -0.5. Ponds are either included (c = 0.1) or not (c = 0), and there is a low or a high level of sampling error. There are 300 runs of each parameter combination.

† For the state-space method, we use 1.96*SD of posterior distribution (assuming the posterior distribution is normally distributed, this is a good approximation to a 97.5% confidence interval).

Table D2. Simulation results from synthetic Canvasback data. Results are shown for both the state-space model as well as the case where sampling errors are ignored (i.e., the AR-approach).

a) Average estimated direct density dependence from simulated data. Standard deviation of the 300 runs are included in parantheses.
		No covariate (c = 0)		Ponds included (c = 0.1)
State-space	True	low noise	high noise	low noise	high noise
	0	-0.06 (0.09)	-0.08 (0.12)	-0.08 (0.09)	-0.09 (0.10)
	-0.2	-0.25 (0.14)	-0.28 (0.19)	-0.26 (0.11)	-0.27 (0.16)
	-0.5	-0.52 (0.19)	-0.54 (0.23)	-0.55 (0.14)	-0.59 (0.18)
AR-model
	0	-0.15 (0.13)	-0.22 (0.17)	-0.19 (0.12)	-0.25 (0.16)
	-0.2	-0.35 (0.14)	-0.45 (0.18)	-0.33 (0.14)	-0.43 (0.17)
	-0.5	-0.62 (0.15)	-0.70 (0.16)	-0.61 (0.15)	-0.71 (0.17)

b) Average uncertainty of estimates given as (i) mean SD of posterior distribution and (ii) average standard error.
		c = 0		c = 0.1
State-space (i)	True	low noise	high noise	low noise	high noise
	0	0.07	0.08	0.07	0.08
	-0.2	0.13	0.17	0.11	0.13
	-0.5	0.18	0.25	0.15	0.20
AR-model (ii)
	0	0.08	0.10	0.09	0.10
	-0.2	0.12	0.13	0.11	0.13
	-0.5	0.14	0.15	0.14	0.15

c) The proportion (of 300 series) reflecting number of times the estimated direct density dependence () would be considered significantly lower (using estimate + 1.96*SE)† than the true .
		c = 0		c = 0.1
State-space	True	low noise	high noise	low noise	high noise
	0	0.023	0.020	0.096	0.090
	-0.2	0.023	0.023	0.033	0.026
	-0.5	0.023	0.020	0.016	0.040
AR-model
	0	0.406	0.606	0.426	0.590
	-0.2	0.250	0.466	0.176	0.393
	-0.5	0.136	0.250	0.113	0.263

   Notes: Strength of the direct density dependence (true ) varies between zero, -0.2 and -0.5. Ponds are either included (c = 0.1) or not (c = 0), and there is a low or a high level of sampling error. There are 300 runs of each parameter combination.

† For the state-space method, we use 1.96*SD of posterior distribution (assuming the posterior distribution is normally distributed, this is a good approximation to a 97.5% confidence interval).

Table D3. Average standard deviation of process stochasticity () estimated from synthetic Mallard and Canvasback data.

	No ponds (c = 0)				Ponds included (c = 0.1)
State-space	Mallard		Canvasback		Mallard		Canvasback
	low noise	high noise	low noise	high noise	low noise	high noise	low noise	high noise
	0.30	0.31	0.52	0.52	0.32	0.31	0.53	0.52
	0.31	0.31	0.51	0.51	0.32	0.31	0.53	0.52
	0.30	0.28	0.50	0.48	0.33	0.30	0.51	0.51
AR-model
	0.32	0.51	0.43	0.55	0.41	0.56	0.46	0.56
	0.33	0.50	0.47	0.57	0.39	0.54	0.50	0.60
	0.32	0.47	0.47	0.55	0.36	0.49	0.48	0.57

   Notes: There are 300 runs of each parameter combination. Results are shown for both the state-space model as well as the case where sampling errors are ignored (i.e., the AR-approach).