Statistical Evaluation of the Amount of Recordings

When conducting the event-related MEG experiment, we usually repeat the experiment many times and retrieve tens or hundreds of recordings to go on synchronized-averaging or single-trial analysis. If the amount of recordings is enough, we can get the more accurate solution to the inverse problem. But how to understand whether the amount of recordings is ”enough” is still an issue. There is still no criteria for the experimentalist to follow.

Therefore, the number of recordings is usually manually determined by the experimentalist.

Besides, there are many variations during experiments, including the condition of the subject, the various recordings between different subjects, different experimental paradigms, and interferences from the environment. As existing such many variations, it is difficult for the experimentalist to understand whether the amount of recordings is enough. We use statistical hypothesis test to reveal the discrimination between the selected active and con-trol states that the experimentalist can have some reference indicators about the amount of recordings.

2.1.1 Statistical Hypothesis Test

One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. The hy-pothesis can be divide into two categories, research hyhy-pothesis and statistical hyhy-pothesis.

Research hypothesis is also called scientific hypothesis that the scientists raise their ratio-nal hypothesis to their problems, according to theorems or documentations. The research hypothesis is represented by a statement.

When we express the research hypothesis with statistical terms or symbols, it becomes the statistical hypothesis. The statistical hypothesis includes null hypothesis and alternative hypothesis. We call a false hypothesis as null hypothesis and the alternative hypothesis is the opposite.

The statistical hypothesis must be stated in statistical terms and is able to calculate the probability of possible samples assuming the hypothesis is correct. Applying the statis-tical hypothesis, we often decide a null hypothesis which is opposite to our expectation.

Then test the abnormality with this null hypothesis by some statistical methods and decide whether this null hypothesis is hold. If this null hypothesis is rejected, our expectation is proved; otherwise, rejected. Therefore, how to state the null hypothesis is very important.

However, no matter rejection or acceptance of the null hypothesis, false decision still exists. If the null hypothesis is true, we still decide to accept it from the result of statistical test, and this is called Type I error. Type II error is just the opposite. The probability of type I error is often denoted as α and is called level of significance. The probability of type II error is often denoted as β. We often set the value of α as 0.05 or 0.01. In real world, Type I error is more serious than Type II error. As a result, we pay more attention to α than β.

Following the concept and purpose of statistical hypothesis test, we want to transform our problem about the amount of recordings into hypothesis for evaluation.

2.1.2 Evaluation of the Amount of Recordings with Statistical Hy-pothesis Test

Assuming that the control state and the active state of brain signal are both random pro-cesses. Furthermore, the baseline state is weakly stationary and ergodic. We take a period of pre-stimulus recordings as the baseline state and a period of post-stimulus recordings as the active state, and take advantage of some statistical hypothesis test to reveal whether there is a significant difference between these two states.

We can follow the below procedure to construct a correct and reliable hypothesis test [28].

1. From the defined problem, identify the parameter of interest.

2. State the null hypothesis, H₀ and specify an appropriate alternative hypothesis, H₁. 3. Specify an appropriate test statistic.

4. Choose a significance level α and state the rejection region for the statistic.

5. Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value.

6. Decide whether or not H₀should be rejected and report that in the defined problem and retrieve the results.

Applying our problem into the above procedure, we can get the below statements.

1. The parameter of interest is the difference in mean between two different states, µ_cand µ_a. For each MEG channel i, the mean µciis calculated by averaging the recordings m_i during the selected control period, t = t_cs. . . t_ce. The mean µ_ai is calculated by averaging the recordings during the selected control period, t = tas. . . tae.

For each channel i,

and N is the number of MEG sensors.

2. H₀:µ_c = µ_a, H₁:µ_c 6= µ_a

3. There are many kinds of statistical hypothesis tests (Table 2.1), we need to examine the data to be tested carefully to find the most appropriate hypothesis test. From 1., we want to test the difference in mean between two different states where the states are selected from each channel of recordings. We want to investigate whether there is a meaningful one-to-one correspondence between the data points in one state and those in the other. In the other words, we compare the data points of one channel in one state to those of the same channel in the other state by the statistical hypothesis test. Because of this special relationship, we take paired t-test as our hypothesis test.

4. α=0.05

5. Compute t- and p-value from the control and active states under this significant level.

6. From 5, we can get whether the null hypothesis is rejected or accepted to get the solution to our problem.

We apply the statistical hypothesis to the selected control and active states from the recordings and examine how the discrimination between two states variate as the number of recordings increase. From this variation, we can generate some reference indicators to help the experimentalist understand the amount of recordings.

Table 2.1: This Table lists some major statistical hypothesis tests, its assumption and independent observations and σ₁ = σ₂ and σ₁and σ₂unknown independent observations and σ1 6= σ₂ and σ₁and σ₂unknown

Each data point in one group corresponds to a matching data point in the other group