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2.3 Extending Cluster-Mass Test using the Slope of Signals

2.3.4 Simulation Study

The simulation study shows the advantages and limits of the use of the slopes in the cluster-mass test. We keep the design simple in order to highlight the difference between the classical cluster-mass test and its extension described previously. We simulate signals using a design with 2 groups, with n= 22 participants and 600 time-points2. At statistic is used to test the hypothesis. The slopes are computed using smoothing splines with a smoothing parameter such that the roughness of the original signals match the one of their slopes. The FWER of both methods is, as expected, close to the nominal level (for the classical cluster-mass test, ˆq = .053 with 95% CI [.046; .060], and for its extension using the slope, ˆq = .050 with 95% CI [.044; .058]). Moreover, we simulate data with a true effects similar to a ”wave” shape, with 3 spikes (positive, negative and then positive as shown in the top panel of Figure 2.11). In Figure 2.11, we see the average power of the test for each time-point using 3 different effect sizes. When the effect size on the original signals is small (Figure 2.11, second panel from the top), the average power of

2The error are simulated using an exponential autocorrelation functionρ(τ) =−3(τ /30)2 (Abraham-sen,1997) andσ= 1.5

Simulated Effect

βdβ/ds

Average Power per time

Index

0.00.20.4 Effect size: .25

Index

0.00.20.40.60.81.0

Cluster−mass Extended with slope True effect Effect size: .75

Index

0.00.20.40.60.81.0

0 100 200 300 400 500 600

Effect size: 1

Figure 2.11: Average power of the classical cluster-mass test and its extension using slopes.

The top panel represents the shape of the true effect and its slope. The bottom three panels show the average power for effect size (β’s) multiplied by .25, .75 and 1. The average power is higher between spikes when using the slopes.

2.4. Conclusion 51

Table 2.1: The False Positive Rate corresponds to the average rate of discovery for the time-points under the null hypothesis (1ms to 100ms and 419ms to 600ms, or white areas in Figure 2.11). The False Positive Rate for the 10 time-points near the true effect (from 96ms to 100ms and 419ms to 423ms) is more influenced when using the slope. Finally, the true discovery rate corresponds to the average rate of discovery for the time-points under the alternative (between 101ms and 418ms or gray area in Figure 2.11). Confidence intervals are computed using Agresti and Coull (1998).

Method Effect .25 Effect .75 Effect 1.0

False Positive Rate

Cluster-mass test .0027 [.0014;.0049] .0022 [.0011;.0042] .0020 [.0010;.0040]

Extension using slope .0039 [.0023;.0064] .0042 [.0026;.0068] .0043 [.0026;.0069]

False Positive Rate (10 nearest)

Cluster-mass test .0041 [.0025;.0067] .0038 [.0022;.0063] .0040 [.0024;.0065]

Extension using slope .0052 [.0033;.0080] .0118 [.0088;.0156] .0162 [.0127;.0206]

True Discovery Rate

Cluster-mass test .0390 [.0335;.0455] .5080 [.4928;.5237] .6364 [.6217;.6515]

Extension using slope .0375 [.0321;.0439] .6499 [.6353;.6649] .7904 [.7779;.8031]

both methods is small and using the slopes does not results in an increase of the average power. However, when the effect size becomes larger (two bottom panel of Figure 2.11), the transition between positive and negative spike is more often declared significant when using the slopes. However, due to the smoothing of the slopes, more false positive tests are measured at the edge of the true effects (see Table 2.1, section ”False Positive Rate (10 nearest)”). This may be a negative counterpart of the cluster-mass test using the slopes as it increases the number of false positive. It results in clusters that may be larger than the true effect. In our simulation settings, the average false positive rate is still low even for the tests near the true effect (for the 10 nearest tests and high effect size: F P R= 0.0162) but it may depend on the shape of the true effect. However, from a practical perspective, neuroscientists may not be interested by the precise position (in time) of the edge of the clusters and this uncertainty may be a reasonable trade-off for an increase in power.

2.4 Conclusion

In Section 2.1, we explain that permutation methods have a geometrical representation. It helps to understand links between the permutation methods. However, the permutation methods which modify the design (like dekker) are not well adapted to this graphical representation. Indeed, in this case, the whole plan [D X] rotates for each permutation which is more complicated both to represent graphically and also to understand using a 3D graphic. However, a further exploration of this graphical representation may come from the interpretation of the F statistic as a function of an angle. Using this interpretation, permuting the response variables modify only one vector of this angle while permuting the design modifies the other one. A better understanding of the effect of permutations on this angle may leads to a clever graphical representation of the methods permuting the design, especially for the dekker method.

In Section 2.2, we describe a real data analysis of a full scalp EEG experiment

us-ing the cluster-mass test. We explain some challenges encounter from implementus-ing the cluster-mass test to the graphical representation of the results. We plan to implement the functions used for this data analysis in the next release of the permuco package.

In Section 2.3, we present a cluster-mass test using the slopes of the signals to increase the power of the test. This method has still to be investigated in details to be useful for real data applications. Indeed, it has the drawback to increase the false positive rate.

Moreover, the choice of the smoothing parameter when using splines or local polynomial is actually not made on an optimality criterion. A better understanding of the effect of the smoothing parameter on the false positive rate could lead to a clever choice of the smoothing parameters.

Chapter 3

Finite Sample and Asymptotic Properties of the Conditional Distribution by Permutations

3.1 Introduction

In this Chapter, we introduce theoretical findings on permutation methods for regression and factorial designs. We propose a formalization of the distribution by permutation as a conditional distribution given the observation of the response variable. We then show that the expectation and variance of the conditional distribution by permutation can be computed analytically. These findings are first applied to investigate the conditional distribution by permutation of the F statistic for finite sample size. This approach is general and is applied using several permutation methods including the one introduced by Manly (1991), Kennedy (1995), Freedman and Lane (1983) or ter Braak (1992). It allows to produce a correction of the permutation distribution of the kennedy method.

Similarly to Pauly et al. (2015) which found the asymptotic distribution for a Wald type statistic in the Behrens–Fisher problem in a factorial design, we then derive the asymptotic distribution by permutation for the F statistic in a similar setting. Finally, we show the validity of several permutation methods as we give the asymptotic distribution of the F statistic not only for the manly permutation method but also for the kennedy, freedman lane and terBraakpermutation methods.