Does Filtering Preclude Us from Studying ERP Time-Courses?

Filtering can distort signals (Lyons, 2004), a problem well documented for ERP data (see, e.g., Luck, 2005; Kappenman and Luck, 2010; May and Tiitinen, 2010). It is thus recommended to filter ERPs as little as possible (Luck, 2005). Recently, VanRullen (2011) provided a healthy reminder of filtering dangers. Using simulated data, VanRullen demonstrated that an effect occurring randomly between 150 and 180 ms post-stimulus can be smeared back in time by a 30-Hz low-pass filter, and appears to start at 100 ms. From this result, VanRullen concluded that if researchers filter their data, they cannot interpret the onsets of ERP effects and should limit their conclusions to peak amplitudes and latencies, without interpreting precise ERP time-courses. However, as we are going to see, we can study ERP onsets by using causal filters.

Non-Causal Filters Distort Onsets

For his demonstration, VanRullen used a zero-phase FIR filter. Zero-phase is achieved by applying the filter in the forward direction and then in the reverse direction – a non-causal filter. Causal filtering results from applying the filter in the forward direction only. Here is the crucial point: non-causal filters smear effects back in time; causal filters do not.

To illustrate, let us consider three sorts of 30 Hz low-pass filters (Figure 1; Appendix). The Butterworth filter has the lowest order, with a slow cut-off in its magnitude response. The FIR and elliptic filters have faster cut-offs but ripples in the pass-band and the stop-band. The FIR filter has a linear phase response over the pass-band, which means that every frequency is delayed by the same amount. On the contrary, the elliptic and Butterworth filters have non-linear phase responses, which can lead to phase distortions.

Figure 1

Magnitude, phase, impulse, and step responses of three 30 Hz low-pass filters. The FIR filter had 48 points and a transition bandwidth of 5 Hz. The elliptic filter had order 11 and a 1-Hz transition bandwidth. The Butterworth filter had order 4.

The performance of these filters is best appreciated by looking at their impulse and step responses. When these filters are applied in the forward direction only, they show only a response at or after time zero. Because of differences in magnitude response, these filters attenuate the peak of the response differently: maximum attenuation is observed for the high-order elliptic filter, followed by the FIR filter and then the Butterworth filter. In addition, faster cut-offs lead to larger side-lobes in the impulse and step responses. The peak of the response is also delayed differently by the filters, as predicted from their phase response: the FIR filter produces more delay than the elliptic filter, which is turn produces more delay than the Butterworth filter. Applying the filters in both direction leads to symmetric impulse responses. This is a necessary property of non-causal filters, but it comes at the cost of introducing side-lobes before signal onset. Thus, if we want to study ERP onsets, causal filtering is the appropriate tool.

Causal and Non-Causal Filter Distortion of Real Data

To evaluate the consequences of causal and non-causal filtering on real ERP data, I used data from a previous study (Rousselet et al., 2010) and repeated the analyses with different filters, with the unfiltered condition serving as benchmark (Appendix). The continuous raw data were filtered using the same EEGLAB FIR filter function used by VanRullen for his demonstration (Delorme et al., 2011). Similar results were obtained with a Butterworth filter. ERPs from two conditions were compared using t-tests and onsets were determined using cluster statistics (Pernet et al., 2011a).

In keeping with previous reports, high-pass filtering above 1 Hz had dramatic effects on ERP shapes. Moderate distortions were observed at 0.5 Hz and below. Compared to the raw data, ERP onsets were much shorter at 0.4 Hz and above. Filtering at 0.2 or 0.3 Hz seemed to have negligible effects. The effect of high-pass filtering was also much weaker on the median of individual onsets compared to the group onsets, which speaks in favor of single-subject analyses (Pernet et al., 2011b; Rousselet and Pernet, 2011). In contrast with VanRullen's simulations, low-pass filtering had very little effects on the onset of ERP differences, except in few subjects. Most affected subjects actually showed delayed onsets, not shorter ones.

In sharp contrast to the distortions introduced by non-causal filters, much more drastic causal high-pass filters at 2–5 Hz can be used to remove DC drifts without affecting onsets (Appendix).

Finally, the distortions caused by non-causal filters are only problematic if researchers actually use similar filter settings. A non-exhaustive survey of the literature suggests that it is the case. Across a sample of 158 ERP studies, only one study used a high-pass causal filter. Modes and medians were both 0.1 Hz for high-pass and 30 Hz for low-pass; 21 studies used high-pass filters at 1 Hz, 1.5 Hz, or even 2 Hz; 38 studies used low-pass filters at 20 Hz or lower, and as low as 4 Hz. I am guilty of having used non-causal high-pass filters at 0.5 and 1 Hz.

In conclusion, causal high-pass filtering provides an excellent solution to study ERP onsets. Non-causal filtering might nevertheless be safe if the cut-off frequencies are far from the frequencies of the effects. Importantly, it seems that low-pass filtering has negligible effects on the time-course of the sort of large ERP effects I used here. The discrepancy with VanRullen's results could be due to his use of a step function, which is particularly prone to ringing in the Fourier domain. However, it remains possible that low-pass filter distortions could be more detrimental at lower signal-to-noise ratios. Indeed, the results described here might not generalize to other contexts – you will need to check your own data.

Table A1

Median onsets (ms) of face-noise ERP differences before and after application of a non-causal FIR filter.

	Raw data	High-pass filter cut-offs						Low-pass filter cut-offs
		0.2	0.3	0.4	0.5	1	2	20	30	40	50
Onset	104	105	105	100	98	−2	−9	107	107	106	106
CI	92	97	94	87	84	−37	−26	101	100	100	100
	110	110	110	108	107	39	0	112	110	110	111
Group	114	114	114	66	−298	−298	−242	112	112	110	112

Medians were estimated using a Harrell–Davis estimator of the 0.5 quantile. Percentile bootstrap 95% confidence intervals (CI) are reported in square brackets. Onsets from group statistics are provided in the last row for comparison.

Table A2

Median of onset differences (ms) between unfiltered and filtered data.

	High-pass filter cut-offs						Low-pass filter cut-offs
	0.2	0.3	0.4	0.5	1	2	20	30	40	50
Onset	0	−0.1	−0.1	−0.1	88.4	106.3	1.7	2.0	2.5	1.5
CI	−0.2	−1.3	−1.3	−1.5	51.0	92.9	0.3	0.4	1.7	0.3
	0.1	0.2	1.0	3.3	138.6	130.1	2.3	3.5	4.7	2.3

Data were iltered using a non-causal FIR filter. A positive difference means that an earlier onset was obtained in the filtered condition compared to the raw data.

Table A3

Median onsets (ms) of face-noise ERP differences before and after application of a causal elliptic filter.

	Raw data	High-pass filter cut-offs
		0.5	1	2	3	4	5
Onset	104	108	109	106	104	105	106
CI	92	100	103	100	98	101	103
	110	113	114	110	109	109	111
Group	114	108	110	110	68	104	98

Medians were estimated using a Harrell–Davis estimator of the 0.5 quantile. Percentile bootstrap 95% CI are reported in square brackets. Onsets from group statistics are provided in the last row for comparison.

Table A4

Median of onset differences between unfiltered and filtered data.

	High-pass filter cut-offs
	0.5	1	2	3	4	5
Onset	−0.3	−0.9	0	0	0.3	−2.6
CI	−1.8	−4.3	−1.8	−1.7	−2.7	−12.3
	0.0	0.5	1.7	1.9	2.6	2.3

Data were filtered using a causal elliptic filter. A positive difference means that an earlier onset was obtained in the filtered condition compared to the DC condition.