= Tutorial 26: Statistics =
'''[TUTORIAL UNDER DEVELOPMENT: NOT READY FOR PUBLIC USE] '''

''Authors: Francois Tadel, Elizabeth Bock, Dimitrios Pantazis, Richard Leahy, Sylvain Baillet''

In this auditory oddball experiment, we would like to test for the significant  differences between the brain response to the deviant beeps and the  standard beeps, time sample by time sample. Until now we have been computing measures of the brain activity in time or time-frequency domain. We were able to see clear effects or slight tendencies, but these observations were always dependent on an arbitrary amplitude threshold and the configuration of the colormap. With appropriate statistical tests, we can go beyond these empirical observations and assess what are the significant effects in a more formal way.

<<TableOfContents(2,2)>>

== Random variables ==
In most cases we are interesting in comparing the brain signals recorded for two populations or two experimental conditions A and B.

'''A''' and '''B''' are two '''random variables''' for which we have a limited number of repeated measures: multiple trials in the case of a single subject study, or multiple subject averages in the case of a group analysis. To start with, we will consider that each time sample and each signal (source or sensor) is independent: a random variable represents the possible measures for one sensor/source at one specific time point.

A random variable can be described with its '''probability distribution''': a function which indicates what are the chances to obtain one specific measure if we run the experiment. By repeating the same experiment many times, we can approximate this function with a discrete histogram of observed measures.

{{attachment:stat_distribution.gif||height="205",width="351"}} {{attachment:stat_histogram.gif||height="222",width="330"}}

== Histograms ==
You can plot histograms like this one in Brainstorm, it may help you understand what you can expect from the statistics functions described in the rest of this tutorial. For instance, seeing a histogram computed with only 4 values would discourage you forever to run any group analysis with 4 subjects...

==== Recordings ====
 * Let's evaluate the recordings we obtained for sensor '''MLP57''', the channel that was showing the highest value at '''160ms''' in the difference of averages computed in the previous tutorial.<<BR>><<BR>> {{attachment:histogram_channel.gif||height="142",width="459"}}
 * We are going to extract only one  value for each trial we have imported in the database, and save these  values in two separate files, one for each condition (standard and  deviant). <<BR>>In order to observe more meaningful effects, we will process together the trials from the two acquisition runs. As explained in the previous tutorials ([[http://neuroimage.usc.edu/brainstorm/Tutorials/Averaging#Averaging_across_runs|#16]]), this is usually not recommended in MEG analysis, but it can be an acceptable approximation if the subject didn't move between runs.
 * In Process1, select all the '''deviant trials''' from both runs. <<BR>>Run process '''Extract > Extract values''': <<BR>>Options: Time='''[160,160]ms''', Sensor="'''MLP57'''", Concatenate''' time''' (dimension 2)<<BR>><<BR>> {{attachment:histogram_extract_process.gif||height="264",width="420"}}
 * Repeat the same operation for all the '''standard trials'''.
 * You obtain two new files in the folder ''Intra-subject''. If you look inside the files, you can observe that the size of the Value matrix matches the number of trials (78 for deviant, 383 for standard). The matrix is [1 x Ntrials] because we asked to concatenate the extracted values in the 2nd dimension. <<BR>><<BR>> {{attachment:histogram_extract_files.gif||height="139",width="623"}}
 * To display the distribution of the values in these two files: <<BR>>select them simultaneously, right-click > '''File > View histograms'''. <<BR>><<BR>> {{attachment:histogram_sensor1.gif||height="217",width="674"}}
 * With the buttons in the toolbar, you can edit the way these  distributions are represented: number of bins in the histogram, total  number of occurances (shows taller bars for ''standard'' because it  has more values) or density of probability (normalized by the total number  of values). <<BR>>You can additionnally plot the corresponding '''normal distribution''' corresponding to the mean μ and standard deviation σ computed from the set of values (using Matlab functions ''mean'' and ''std'').

 * When comparing two sample sets A and B, we try to evaluate if the  distributions of the measures are equal or not. In most of the questions  we explore in EEG/MEG analysis, the distributions are overlapping a  lot. The very sparse sampling of the data (a few tens or hundreds  repeated measures) doesn't help with the task. Some representations will be more convicing than others to estimate the differences between the two conditions. <<BR>><<BR>> {{attachment:histogram_sensor2.gif||height="103",width="330"}}

==== Sources (relative) ====
 * We can repeat the same operation at the source level and extract all  the values for scout '''A1'''.
 * In Process1, select all the '''deviant trials''' from both runs. Select button '''[Process sources]'''. <<BR>>Run process '''Extract > Extract scouts time series''': <<BR>>Options: Time='''[160,160]ms''', Scout="'''A1'''", Flip, Concatenate<<BR>><<BR>> {{attachment:histogram_scout1.gif||height="296",width="449"}}
 * Repeat the same operation for all the '''standard trials'''.
 * Select the two files > Right-click > File > View histogram.<<BR>><<BR>> {{attachment:histogram_scout2.gif||height="254",width="452"}}
 * The distributions still look Gaussian, but the variances are now slightly different. You have to be pay attention to this information when chosing which parametric t-test to run (see below).

==== Sources (absolute) ====
 * In Process1, select the extracted scouts value. Run process "'''Pre-process > Absolute values'''".
 * Display the histograms of the the two rectified files. <<BR>><<BR>> {{attachment:histogram_scout3.gif||height="251",width="487"}}
 * The rectified source values are definitely '''not''' following a normal distribution, the shape of the histogram has nothing to do with the corresponding Gaussian curves. As a consequence, if you are using rectified source maps, you will not be able to run independent parametric t-tests.<<BR>>Additionally, you may have issues with the detection of some effects (see tutorial [[Tutorials/Difference|Difference]]).

==== Sources (norm) ====
 * When processing '''unconstrained '''sources, you are most of the time interested in comparing the norm of the current vector between conditions, rather than the three orientations separately.
 * To get these values, you can run the process "'''Sources > Unconstrained to flat maps'''" on all the trials of both conditions, then to run again the scout values extraction for '''A1/160ms'''. This is the distribution you would get if you use the option "Average of absolute values: mean(abs(x))" in the t-test of averaging processes. <<BR>><<BR>> {{attachment:histogram_scout4.gif||height="138",width="300"}}

==== Time-frequency ====
 * Time-frequency power for recordings: MLP57 / 55ms / 48Hz. <<BR>> {{attachment:histogram_tf1.gif||height="138",width="300"}}
 * Time-frequency power for sources: A1 / 55ms / 48Hz. <<BR>> {{attachment:histogram_tf2.gif||height="138",width="300"}}
 * Time-frequency power for  sources, Z-score normalized: A1 / 55ms / 48Hz. <<BR>> {{attachment:histogram_tf3.gif||height="138",width="300"}}
 * None of these sample sets are distributed following normal distributions. Parametric t-tests in the case of time-frequency power will never be good candidates.

== Statistical inference ==
==== Hypothesis testing ====
To show that there is a difference between A and B, we can use a statistical hypothesis test. We start by assuming that the two sets are identical then reject this hypothesis. For all the tests we will use here, the logic is similar:

 * Define a '''null hypothesis''' (H<<HTML(<sub>)>>0<<HTML(</sub>)>>:"A=B") and alternative hypotheses (eg. H<<HTML(<sub>)>>1<<HTML(</sub>)>>:"A<B").
 * Make some assumptions on the samples we have (eg. A and B are independent, A and B follow normal distributions, A and B have equal variances).
 * Decide which test is appropriate, and state the relevant '''test statistic''' T (eg. Student t-test).
 * Compute from the measures (A<<HTML(<sub>)>>obs<<HTML(</sub>)>>, B<<HTML(<sub>)>>obs<<HTML(</sub>)>>) the observed value of the test statistic (t<<HTML(<sub>)>>obs<<HTML(</sub>)>>).
 * Calculate the '''p-value'''. This is the probability, under the null hypothesis, of sampling a test statistic at least as extreme as that which was observed. A value of (p<0.05) for the null hypothesis has to be interpreted as follows: "If the null hypothesis is true, the chance that we find a test statistic as extreme or more extreme than the one observed is less than 5%".
 * Reject the null hypothesis if and only if the p-value is less than the significance level threshold (<<HTML(&alpha;)>>).<<BR>><<BR>> {{attachment:stat_tails.gif||height="322",width="598"}}

==== Evaluation of a test ====
The quality of test can be evaluated based on two criteria:

 * '''Sensitivity''': True positive rate = power = ability to correctly reject the null hyopthesis and control for the false negative rate (type II error rate). A very sensitive test detects a lot of significant effects, but with a lot of false positive.
 * '''Specificity''': True negative rate = ability to correctly accept the null hypothesis and control for the false positive rate (type I error rate). A very specific test detects only the effects that are clearly non-ambiguous, but can be too conservative and miss a lot of the effects of interest. <<BR>><<BR>> {{attachment:stat_errors.gif||height="167",width="455"}}

==== Different categories of tests ====
Two families of tests can be helpful in our case: parametric and non-parametric tests.

 * '''Parametric tests''' need some strong assumptions on the probability distributions of A and B then use some well-known properties of these distributions to compare them, based on a few simple parameters (typically the mean and variance). The estimation of these parameters is highly optimized and require very little memory. The examples which will be described here are the Student's t-tests.
 * '''Non-parametric tests''' do not  require any assumption on the distribution of the data. They are threrefore more realiable and more generic. On  the other hand they are a lot more complicated to process: they require a lot more memory  because all the tested data has to be loaded at once, and a lot more  computation time because the same test is repeated many times.

== Parametric Student's t-test ==
==== Assumptions ====
The [[https://en.wikipedia.org/wiki/Student's_t-test|Student's t-test]] is a widely-used parametric test to evaluate the difference between the means of two random variables (two-sample test), or between the mean of one variable and one known value (one-sample test). If the assumptions are correct, the t-statistic follows a [[https://en.wikipedia.org/wiki/Student's_t-distribution|Student's t-distribution]].

 . {{attachment:stat_tvalue.gif||height="127",width="490"}}

The main assumption for using a t-test is that the random variables involved follow a [[https://en.wikipedia.org/wiki/Normal_distribution|normal distribution]] (mean:  μ, standard deviation: σ). The figure below shows a few example of normal distributions.

 . {{attachment:stat_distrib_normal.gif||height="190",width="298"}} <<latex($$ f(x \; | \; \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi} } \; e^{ -\frac{(x-\mu)^2}{2\sigma^2} } $$)>>

==== t-statistic ====
Depending on the type of data we are testing, we can have different variants for this test:

 * '''One-sample t-test''' (testing against a known mean μ<<HTML(<sub>)>>0<<HTML(</sub>)>>): <<BR>><<latex($$ t = \frac{\overline{x} - \mu_0}{\sigma/\sqrt{n}} \hspace{40pt} \mathrm{d.f.} = n - 1 $$)>> <<BR>>where <<latex($$\overline{x}$$)>> is the sample mean, σ is the sample standard deviation and ''n'' is the sample size.
 * '''Dependent t-test''' for paired samples (eg.when testing two conditions across subject). Equivalent to testing the difference of the pairs of samples against zero with a one-sample t-test:<<BR>><<latex($$ t = \frac{\overline{X}_D}{\sigma_D/\sqrt{n}} \hspace{40pt} \mathrm{d.f.} = n - 1 $$)>>  <<BR>>where D=A-B, <<latex($$\overline{X}_D$$)>> is the average of D and σ'',,D,,'' its standard deviation.
 * '''Independent two-sample test, equal variance''' (equal or unequal sample sizes):<<BR>><<latex($$ t = \frac{\bar{X}_A - \bar{X}_B}{s_{X_AX_B} \cdot \sqrt{\frac{1}{n_A}+\frac{1}{n_B}}} \hspace{40pt} s_{X_AX_B} = \sqrt{\frac{(n_A-1)s_{X_A}^2+(n_B-1)s_{X_B}^2}{n_A+n_B-2}} \hspace{40pt} \mathrm{d.f.} = n_A + n_B - 2 $$)>> <<BR>>where <<latex($$s_{X_A}^2$$)>> and <<latex($$s_{X_B}^2$$)>> are the unbiased estimators of the variances of the two samples.
 * '''Independent two-sample test, u''''''nequal variance''' (Welch's t-test):<<BR>> <<latex($$t = {\overline{X}_A - \overline{X}_B \over s_{\overline{X}_A - \overline{X}_B}} \hspace{40pt}s_{\overline{X}_A - \overline{X}_B} = \sqrt{{s_A^2 \over n_A} + {s_B^2  \over n_B}} \hspace{40pt} \mathrm{d.f.} = \frac{(s_A^2/n_A + s_B^2/n_B)^2}{(s_A^2/n_A)^2/(n_A-1) + (s_B^2/n_B)^2/(n_B-1)} $$)>> <<BR>> where <<latex($$n_A$$)>> and <<latex($$n_B$$)>> are the sample sizes of A and B.

==== p-value ====
Once the t-value is computed (t<<HTML(<sub>)>>obs<<HTML(</sub>)>> in the previous section), we can convert it to a p-value based on the known distributions of the t-statistic. This conversion depends on two factors: the number of degrees of freedom and the tails of the distribution we want to consider. For a two-tailed t-test, the two following commands in Matlab are equivalent and can convert the t-values in p-values. <<BR>>To obtain the p-values for a '''one-tailed t-test''', you can simply '''divide the p-values by 2'''.

<<HTML(<BLOCKQUOTE>)>>

{{{
p = betainc(df./(df + t.^2), df/2, 0.5);    % Without the Statistics toolbox
p = 2*(1-tcdf(abs(t),df));                  % With the Statistics toolbox
}}}
<<HTML(</BLOCKQUOTE>)>>

The distribution of this function for different numbers of degrees of freedom:

 . {{attachment:ttest_tcdf.gif||height="177",width="366"}}

== Example 1: Parametric t-test on recordings ==
Parametric t-tests require the values tested to follow a normal distribution. The recordings evaluated in the histograms sections above (MLP57/160ms) show distributions that are matching relatively well this assumption: the shape of the histogram follows the trace of the corresponding normal function, and with very similar variances. It looks reasonable to use a parametric t-test in this case.

 * In the Process2 tab, select the following files, from '''both runs '''(approximation discussed previously):
  * Files A: All the '''deviant '''trials, with the [Process recordings] button selected.
  * Files B: All the '''standard '''trials, with the [Process recordings] button selected.
 * Run the process "'''Test > Parametric t-test'''", Equal variance, Arithmetic average.<<BR>><<BR>> {{attachment:stat_ttest1_process.gif||height="376",width="610"}}
 * Double-click on the new file for displaying it. With the new tab "Stat"  you can control the p-value threshold and the correction you want to  apply for multiple comparisons (see next section). <<BR>><<BR>> {{attachment:stat_ttest1_file.gif||height="136",width="674"}}

== Correction for multiple comparisons ==
==== Multiple comparison problem ====
The approach  described in this first example performs many tests simulateneously. We  test independently each MEG sensor and each time sample across the  trials, so we run a total of 274*361 = 98914 t-tests.

If  we select a critical value of 0.05 (p<0.05), it means that we want  to see what is significantly different between the conditions while  accepting the risk to observe a false positive in 5% of the cases. If we  run around 100,000 times the test, we can expect to observe around  5,000 false positives. We need to control better for false positives  (type I errors) when dealing with multiple tests.

==== Bonferroni correction ====
The probability to observe at least one false positive, or '''familywise error rate (FWER)''' is almost 1: <<BR>> '''FWER''' = 1 - prob(no significant results) = 1 - (1 - 0.05)^100000 '''~ 1'''

A classical way to control the familywise error rate is to replace the p-value threshold with a corrected value, to enforce the expected FWER. The Bonferroni correction sets the significance cut-off at <<latex($$\alpha$$)>>/Ntest. If we set (p <<HTML(&#8804;)>> <<latex($$\alpha$$)>>/Ntest), then we have (FWER <<HTML(&#8804;)>> <<latex($$\alpha$$)>>). Following the previous example:<<BR>>'''FWER''' = 1 - prob(no significant results) = 1 - (1 - 0.05/100000)^100000 '''~ 0.0488 < 0.05'''

This works well in a context where all the tests are strictly independent. However, in the case of MEG/EEG recordings, the tests have an important level of dependence: two adjacent sensors or time samples often have similar values. In the case of highly correlated tests, the Bonferroni correction tends too be conservative, leading to a high rate of false negatives.

==== FDR correction ====
The '''false discovery rate (FDR)''' is another way of representing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. It is designed to control the expected proportion of false positives, while the Bonferroni correction controls the probability to have at least one false positive. FDR-controlling procedures have greater power, at the cost of increased rates of Type I errors ([[[https://en.wikipedia.org/wiki/False_discovery_rate|Wikipedia]])

In Brainstorm, we implement the Benjamini–Hochberg step-up procedure for FDR correction:

 * Sort the p-values p(''k'') obtained across all the multiple tests (''k''=1..Ntest).
 * Find the largest ''k'' such as p(''k'') < ''k'' / Ntest * <<latex($$\alpha$$)>>.
 * Reject the null hypotheses corresponding to the first ''k'' smallest p-values.

==== In the interface ====
<<HTML(<div   style="padding: 0px; float:  right;">)>> {{attachment:stat_mcp_gui.gif}} <<HTML(</A></div>)>>

You can select interactively the type of correction to apply for multiple comparison while reviewing your statistical results. The checkboxes "Control over dims" allow you to select which are the dimensions that you consider as multiple comparisons (1=sensor, 2=time, 3=not applicable here). If you select all the dimensions, all the values available in the file are considered as the same repeated test, and only one corrected p-threshold is computed for all the time samples and all the sensors.

If you select only the first dimension, only the values recorded at the same time sample are considered as repeated tests, the different time points are corrected independently, with one different corrected p-threshold for each time point.

When changing these options, a message is displayed in the Matlab command window, showing the number of repeated tests that are considered, and the corrected p-value threshold (or the average if there are multiple corrected p-thresholds, when not all the dimensions are selected):

<<HTML(<BLOCKQUOTE>)>>

{{{
BST> Average corrected p-threshold: 5.0549e-07  (Bonferroni, Ntests=98914)
BST> Average corrected p-threshold: 0.00440939  (FDR, Ntests=98914)
}}}
<<HTML(</BLOCKQUOTE>)>>

== Nonparametric permutation tests ==
==== Principle ====
A '''permutation test''' (or randomization test) is a type of test in which the distribution of the test statistic under the null hypothesis is obtained by calculating all possible values of the test statistic under rearrangements of the labels on the observed data points. ([[https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests|Wikipedia]])

If the null hypothesis is true, the two sets of tested values A and B follow the same distribution. Therefore the values are '''exchangeable '''between the two sets: we can move one value from set A to set B, and one value from B to A, and we expect to obtain the same value for the test statistic T.

 . {{attachment:stat_exchange.gif||height="37",width="302"}}

By taking all the possible permutations between sets A and B and computing the statistic for each of them, we can build a histogram that approximates the '''permutation distribution'''.

 . {{attachment:stat_perm_distrib.gif||height="184",width="400"}}

Then we compare the observed statistic with the permutation distribution. From the histogram, we calculate the proportion of permutations that resulted in a larger test statistic than the observed one. This proportion is called the '''p-value'''. If the p-value is smaller than the critical value (typically  0.05), we conclude that the data in the two experimental conditions  are significantly different.

 . {{attachment:stat_perm_pvalue.gif||height="183",width="400"}}

The number of possible permutations between the two sets of data is usually too large to compute an exhaustive permutation test in a reasonable amount of time. The permutation distribution of the statistic of interest is approximated using a '''Monte-Carlo approach'''. A relatively small number of random selection of possible permutations can give us a reasonably good idea of the distribution.

Permutation tests can be used for '''any test statistic''', regardless  of whether or not its distribution is known. The hypothesis is about the  data itself, not about a specific parameter. In the examples below, we  use the t-statistic, but we could use any other function.

==== Practical limitations ====
'''Computation time''': If you increase the number of random permutations you use for estimating the distribution, the computation time will increase linearly. You need to find a good balance between the total computation time and the accuracy of the result.

'''Memory (RAM)''': The implementations of the permutation tests available in Brainstorm require all the data to be loaded in memory before starting the evaluation of the permutation statistics. Running this function on large datasets or on source data could quickly crash your computer. For example, loading the data for the non-parametric equivalent to the parametric t-test we ran previously would require: <<BR>> 276(sensors) * 461(trials) * 361(time) * 8(bytes) / 1024^2(Gb) = '''0.3 Gb of memory'''.

This is acceptable on most recent computers. But to perform the same at the source level, you need: <<BR>>45000*461*361*8/1024^3 = '''58 Gb of memory''' just to load the  data. This is impossible on most computers, we have to give up at least  one  dimension and run the test only for one region  of  interest or one time sample (or average over a short time window).

== FieldTrip implementation ==
==== FieldTrip functions in Brainstorm ====
We have the possibility to call some of the FieldTrip toolbox functions from the Brainstorm environment. For this, you need first to [[http://www.fieldtriptoolbox.org/download|install the FieldTrip toolbox]] on your computer and [[http://www.fieldtriptoolbox.org/faq/should_i_add_fieldtrip_with_all_subdirectories_to_my_matlab_path|add it to your Matlab path]].

==== Cluster-based correction ====
One interesting method that has been promoted by the FieldTrip developers is the cluster-based approach for non-parametric tests. In the type of data we manipulate in MEG/EEG analysis, the neighbouring channels, time points or frequency bins are expected to show similar behavior. We can group these neighbours into '''clusters '''to "accumulate the evidence". A cluster can have a multi-dimensional extent in space/time/frequency.

In the context of a non-parametric test, the test statistic computed at each permutation is the extent of the '''largest cluster'''. To reject or accept the null hypothesis, we compare the largest observed cluster with the randomization distribution of the largest clusters.

This approach '''solves the multiple comparisons problem''', because the test statistic that is used (the maximum cluster size) is computed using all the values at the same time, along all the dimensions (time, sensors, frequency). There is only one test that is performed, so there is no multiple comparisons problem.

The result is simpler to report but also '''a lot less informative''' than other non-parametric tests, FDR-corrected . We had only one null hypothesis, "the two sets of data follow the same probability distribution", and the outcome of the test is to accept or reject this hypothesis. Therefore we '''cannot report the spatial or temporal extent''' of the most significant clusters. Make sure you read [[http://www.fieldtriptoolbox.org/faq/how_not_to_interpret_results_from_a_cluster-based_permutation_test|this recommendation]] before reporting cluster-based results in your publications.

==== Reference documentation ====
For a complete description of non-parametric cluster-based statistics in FieldTrip, read the following:

 * Article:  [[http://www.sciencedirect.com/science/article/pii/S0165027007001707|Maris & Oostendveld (2007)]]
 * Video: [[https://www.youtube.com/watch?v=x0hR-VsHZj8|Statistics using non-parametric randomization techniques]] (E Maris)
 * Video: [[https://www.youtube.com/watch?v=vOSfabsDUNg|Non-parametric cluster-based statistical testing of MEG/EEG data]] (R Oostenveld)

 * Tutorial: [[http://www.fieldtriptoolbox.org/tutorial/eventrelatedstatistics|Parametric and non-parametric statistics on event-related fields]]

 * Tutorial: [[http://www.fieldtriptoolbox.org/tutorial/cluster_permutation_timelock|Cluster-based permutation tests on event related fields]]
 * Tutorial: [[http://www.fieldtriptoolbox.org/tutorial/cluster_permutation_freq|Cluster-based permutation tests on time-frequency data]]
 * Tutorial: [[http://www.fieldtriptoolbox.org/faq/how_not_to_interpret_results_from_a_cluster-based_permutation_test|How NOT to interpret results from a cluster-based permutation test]]

 * Functions references: [[http://www.fieldtriptoolbox.org/reference/timelockstatistics|ft_timelockstatistics]], [[http://www.fieldtriptoolbox.org/reference/sourcestatistics|ft_sourcestatistics]], [[http://www.fieldtriptoolbox.org/reference/freqstatistics|ft_freqstatistics]]

==== Process options ====
There are three seprate processes in Brainstorm, to call the three FieldTrip functions.

 * '''ft_timelockstatistics''': Compare imported trials (recordings).
 * '''ft_sourcestatistics''': Compare source maps or time-frequency files include the entire cortex.
 * '''ft_freqstatistics''': Compare time-frequency decompositions of sensor recordings or scouts signals.
 * See below the correspondance between the options in the interface and the FieldTrip functions.

Test statistic options (process options in bold):

 * cfg.numrandomization = "'''Number of randomizations'''"
 * cfg.statistic = "'''Independent t-test'''" ('indepsamplesT') or "'''Paire t-test'''" ('depsamplesT)
 * cfg.tail         = '''One-tailed''' (-1), '''Two-tailed''' (0), '''One-tailed''' (+1)
 * cfg.correctm = "'''Type of correction'''" ('no', 'cluster', 'bonferroni', 'fdr', 'max', 'holm', 'hochberg')
 * cfg.method = 'montecarlo'<<HTML(<div   style="padding: 0px; float:  right;">)>> {{attachment:ft_process_options.gif||height="318",width="296"}} <<HTML(</div>)>>
 * cfg.correcttail = 'prob'

Cluster-based correction:

 * cfg.clusteralpha = "'''Cluster Alpha'''"
 * cfg.minnbchan = "'''Min number of neighours'''"
 * cfg.clustertail = cfg.tail (in not, FieldTrip crashes)
 * cfg.clusterstatistic = 'maxsum'

Input options: All the data selection is done before, in the process code and functions out_fieldtrip_*.m.

 * cfg.channel      = 'all';
 * cfg.latency      = 'all';
 * cfg.frequency          = 'all';
 * cfg.avgovertime  = 'no';
 * cfg.avgchan      = 'no';
 * cfg.avgoverfreq        = 'no';

== Example 2: Uncorrected non-parametric t-test (FieldTrip) ==
Let's run the non-parametric equivalent to the test we ran in the first example.

 * In the Process2 tab, select the following files, from '''both runs '''(approximation discussed previously):
  * Files A: All the '''deviant '''trials, with the [Process recordings] button selected.
  * Files B: All the '''standard '''trials, with the [Process recordings] button selected.
 * Run the process "'''Test > FieldTrip: ft_timelockstatistics'''", set the options as shown below <<BR>>(expect this to run for at least 15min - check the Matlab command window for progress report): <<BR>><<BR>> {{attachment:ft_nocorrection_options.gif||height="431",width="576"}}
 * Open the parametric and non-parametric results side by side, the results should be very similar. <<BR>><<BR>> {{attachment:ft_nocorrection_display.gif||height="241",width="407"}}
 * In this case, the distributions of the values for each sensor and each time point (non-rectified MEG recordings) are very close to '''normal distributions'''. This was illustrated at the top of this page, in the section "Histograms". Therefore the assumptions behind the parametric t-test are verified, the results of the parametric tests are correct and very similar to the nonparametric ones.
 * If you get different results with the parametric and nonparametric approaches: trust the nonparametric one. If you want to increase the precision of a nonparametric test, increase the number of random permutations.

== Example 3: Cluster-based test (FieldTrip) ==
Run again the same test, but this time select the cluster correction.

 * Keep the same files selected in Process2.
 * Run the process "'''Test > FieldTrip: ft_timelockstatistics'''", Type of correction = '''cluster'''<<BR>>(expect this to run for at least 20min - check the Matlab command window for progress report): <<BR>><<BR>> {{attachment:ft_cluster_process.gif||height="149",width="264"}}
 * Double-click on the new file to display it, add a 2D topography to is (CTRL+T).<<BR>>Note that in the Stat tab, the options for multiple comparisons corrections are disabled, because the values saved in the file are already corrected, you can only change the significance threshold.
 * Alternatively, you get a list of significant clusters, which you can display separately if needed. The colored dots on the topography represent the clusters, blue for negative clusters and blue for positive clusters. You can change these colors in the Stat tab. Note that the clusters have a spatio-temporal extent: at one time point they can be represented as two separate blobs in the 2D topography, but these blobs are connected at other time points.<<BR>><<BR>> {{attachment:ft_cluster_display1.gif}}
 * Additional options are available for exploring the clusters, try them all: <<BR>><<BR>>
 * '''Don't spend too much time exploring the clusters''': While in the previous case the all the tests at each sensor and each data point were computed independently, we could report individually that each of them was significant or not. The cluster-based approach just allows us to report that the two conditions are different, not where or when, which makes the visual exploration of clusters relatively useless. Make sure you read [[http://www.fieldtriptoolbox.org/faq/how_not_to_interpret_results_from_a_cluster-based_permutation_test|this recommendation]] before reporting cluster-based results in your publications.

== === STOP READING HERE === ==
== Example 2: t-test on sources ==
Parametric  method require the values tested to follow a Gaussian distribution.  Some  sets of data could be considered as Gaussian (example: screen  capture of the histogram of the values recorded by one sensor at one  time point), some are not (example: histogram ABS(values) at one   point).

'''WARNING''': This parametric t-test is valid only for '''constrained '''sources.  In the case of unconstrained sources, the norm of the three  orientations is used, which breaks the hypothesis of the tested values  following a Gaussian distribution. This test is now illustrated on  unconstrained sources, but this will change when we figure out the  correct parametric test to apply on unconstrained sources.

 * In the Process2 tab, select the following files:
  * Files A: All the deviant trials, with the '''[Process sources]''' button selected.
  * Files B: All the standard trials, with the '''[Process sources]''' button selected.
 * Run the process "'''Test > Student's t-test'''", Equal variance, '''Absolute value of average'''.<<BR>>This  option will convert the unconstrained source maps (three dipoles at  each cortex  location) into a flat cortical map by taking the norm of  the three  dipole orientations before computing the difference. <<BR>><<BR>> {{attachment:ttest_process.gif||height="516",width="477"}}
 * Double-click  on the new file for displaying it. With the new tab "Stat" you can  control the p-value threshold and the correction you want to apply for  multiple comparisons (see next section). <<BR>><<BR>> {{attachment:ttest_file.gif||height="243",width="492"}}

 * From  the Scout tab, you can also plot the scouts time series and get in this  way a summary of what is happening in your regions of interest.  Positive peaks indicate the latencies when '''at least one vertex''' of  the scout has a value that is significantly higher in the deviant  condition. The values that are shown are the averaged t-values in the  scout. <<BR>><<BR>> {{attachment:ttest_scouts.gif||height="169",width="620"}}

== Time-frequency files ==

== One-sided or two-sided ==
two-tailed test on either (A-B) or (|A|-|B|)

The  statistic, after thresholding should be displayed as a bipolar signal  (red – positive, blue – negative) since this does affect interpretation

Constrained   (A-B), null hypothesis  H0: A=B, We compute a z-score through  normalization using a pre-stim period and apply FDR-controlled  thresholding to identify significant differences with a two tailed test.

Constraine (|A|-|B|), null hypothesis  H0:  |A|=|B|, We compute a z-score through normalization using a pre-stim  period and apply FDR-controlled thresholding to identify significant  differences with a two tailed test.

Unconstrained RMS(X-Y), H0: RMS(X-Y)=0, Since the statistic is non-negative a one-tailed test is used.

Unconstrained  (RMS(X) –RMS(Y)), H0: RMS(X)-RMS(Y)=0, we use a two-tailed test and use  a bipolar display to differentiate increased from decreased amplitude.

== Workflow: Single subject ==
You  want to test for one unique subject what is significantly different  between two experimental conditions. You are going to compare the single  trials corresponding to each condition directly. <<BR>>This  is the case for this tutorial dataset, we want to know what is  different between the brain responses to the deviant beep and the  standard beep.

'''Sensor recordings''': Not advised for MEG with multiple runs, correct for EEG.

 * '''Parametric t-test''': Correct
  * Select the option "Arithmetic average: mean(x)".
  * No restriction on the number of trials.
 * '''Non-parametric t-test''': Correct
  * Do not select the option "Test absolute values".
  * No restriction on the number of trials.
 * '''Difference of means''':
  * Select the option "Arithmetic average: mean(x)".
  * Use the same number of trials for both conditions.

'''Constrained source maps''' (one value per vertex):

 * Use the non-normalized minimum norm maps for all the trials (no Z-score).
 * '''Parametric t-test''': Incorrect
  * Select the option "Absolue value of average: abs(mean(x))".
  * No restriction on the number of trials.
 * '''Non-parametric t-test''': Correct
  * Select the option "Test absolute values".
  * No restriction on the number of trials.

 * '''Difference of means''':
  * Select the option "Absolue value of average: abs(mean(x))".
  * Use the same number of trials for both conditions.

'''Unconstrainted source maps''' (three values per vertex):

 * '''Parametric t-test''': Incorrect
  * ?
 * '''Non-parametric t-test''': Correct
  * ?
 * '''Difference of means''':
  * ?

'''Time-frequency maps''':

 * Test the non-normalized time-frequency maps for all the trials (no Z-score or ERS/ERD).
 * '''Parametric t-test''': ?
  * No restriction on the number of trials.
 * '''Non-parametric t-test''': Correct
  * No restriction on the number of trials.

 * '''Difference of means''':
  * Use the same number of trials for both conditions.

== Workflow: Group analysis ==
You have the same two conditions recorded for multiple subjects.

'''Sensor recordings''': Strongly discouraged for MEG, correct for EEG.

 * For each subject: Compute the average of the trials for each condition individually.
  * Do not apply an absolute value.
  * Use the same number of trials for all the subjects.
 * Test the subject averages.
  * Do no apply an absolute value.

'''Source maps (template anatomy)''':

 * For each subject: Average the non-normalized minimumn norm maps for the trials of each condition.
  * Use the same number of trials for all the subjects.
  * If you can't use the same number of trials, you have to correct the Z-score values ['''TODO'''].
 * Normalize the averages using a Z-score wrt with a baseline.
  * Select the option: "Use absolue values of source activations"
  * Select the option: "Dynamic" (does not make any difference but could be faster).
 * Test the normalized averages across subjects.
  * Do not apply an absolute value.

'''Source maps (individual anatomy)''':

 * For each subject: Average the non-normalized minimumn norm maps for the trials of each condition.
  * Use the same number of trials for all the subjects.
  * If you can't use the same number of trials, you have to correct the Z-score values ['''TODO'''].
 * Normalize the averages using a Z-score wrt with a baseline.
  * Select the option: "Use absolue values of source activations"
  * Select the option: "Dynamic" (does not make any difference but could be faster).
 * Project the normalized maps on the template anatomy.
 * Option: Smooth spatially the source maps.
 * Test the normalized averages across subjects.
  * Do not apply an absolute value.

'''Time-frequency maps''':

 * For  each subject: Compute the average time-frequency decompositions of all  the trials for each condition (using the advanced options of the "Morlet  wavelets" and "Hilbert transform" processes).
  * Use the same number of trials for all the subjects.
 * Normalize the averages wrt with a baseline (Z-score or ERS/ERD).
 * Test the normalized averages across subjects.
  * Do no apply an absolute value.

== Convert statistic results to regular files ==
 * Process: Extract > Apply statistc threshold
 * Process: Simulate > Simulate recordings from scout [TODO]

== Export to SPM ==
An alternative to running the statical tests in Brainstorm is to export all the data and compute the tests with an external program (R, Matlab, SPM, etc). Multiple menus exist to export files to external file formats (right-click on a file > File > Export to file).

Two tutorials explain to export data specifically to SPM:

 * [[http://neuroimage.usc.edu/brainstorm/ExportSpm8|Export source maps to SPM8]] (volume)
 * [[http://neuroimage.usc.edu/brainstorm/ExportSpm12|Export source maps to SPM12]] (surface)

== On the hard drive [TODO] ==
Right  click one of the first TF file we computed > File > '''View file  contents'''.

==== Useful functions ====
 * process_ttest:
 * process_ttest_paired:
 * process_ttest_zero:
 * process_ttest_baseline:
 * bst_stat_thresh:
 * out_fieldtrip_data:
 * out_fieldtrip_results:
 * out_fieldtrip_timefreq:
 * out_fieldtrip_matrix:

== References ==
 * Bennett CM, Wolford GL, Miller MB, [[http://scan.oxfordjournals.org/content/4/4/417.full|The principled control of false positives in neuroimaging]] <<BR>> Soc Cogn Affect Neurosci (2009), 4(4):417-422.
 * Maris E, Oostendveld R, [[http://www.sciencedirect.com/science/article/pii/S0165027007001707|Nonparametric statistical testing of EEG- and MEG-data]] <<BR>>J Neurosci Methods (2007), 164(1):177-90.
 * Maris E, [[http://www.ncbi.nlm.nih.gov/pubmed/22176204|Statistical testing in electrophysiological studies]]<<BR>>Psychophysiology (2012), 49(4):549-65
 * Bennett CM, Wolford GL, Miller MB, <<BR>>[[http://scan.oxfordjournals.org/content/4/4/417.full|The principled control of false positives in neuroimaging]]<<BR>>Soc Cogn Affect Neurosci (2009), 4(4):417-422.

 * Pantazis D, Nichols TE, Baillet S, Leahy RM. [[http://www.sciencedirect.com/science/article/pii/S1053811904005671|A comparison of random field theory and permutation methods for the statistical analysis of MEG data]], Neuroimage (2005), 25(2):383-94.
 * FieldTrip video: Non-parametric cluster-based statistical testing of MEG/EEG data: <<BR>> https://www.youtube.com/watch?v=vOSfabsDUNg

== Additional documentation ==
 * Forum: Multiple comparisons: http://neuroimage.usc.edu/forums/showthread.php?1297
 * Forum: Cluster neighborhoods: [[http://neuroimage.usc.edu/forums/showthread.php?2132-Fieldtrip-statistics|http://neuroimage.usc.edu/forums/showthread.php?2132]]
 * Forum: Differences FieldTrip-Brainstorm: http://neuroimage.usc.edu/forums/showthread.php?2164

== Delete all your experiments ==
Before moving to the next tutorial, '''delete '''all  the statistic results you computed in this tutorial. It will make the database structure less confusing for the following tutorials.

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