1. Introduction to resampling in MATLAB

2. So you've done an experiment... Two independent datasets: control experimental Have n numbers in each dataset, representing independent measurements Question: Did the experimental manipulation have an effect, i.e. did it make a difference?

3. The question restated Is there a significant (i.e. “real”) numerical difference between the control and experimental datasets? What is the probability that the two datasets are subsets of two different/independent distributions? What is the probability that the two datasets are NOT subsets of the same underlying distribution?

4. Answering the question What do we need? A number(s) to represent/summarize each dataset → data descriptors A test for comparing these numbers → test statistic A way to assess the significance of the test statistic → distribution of test statistics generated from datasets that ARE from the same underlying distribution

5. Descriptors and tests Know your data! Visualize it Does it have a central tendency? (i.e. is histogram vaguely mound shaped?)  If yes, use the mean as descriptor  If no, median may be better Typical test: compute difference (subtraction) between descriptors

6. We'll discuss how to assess the significance of your test statistic soon, that's the resampling part... But first let's visualize some data

7. Simulate and inspect data ctrl = 10.*(5.75+randn(25,1)); exp = 10.*(5+randn(25,1)); boxplot([ctrl exp])groups = {'control' 'experimental'} boxplot([ctrl exp],'labels',groups)

8. Central tendency? hist(ctrl) hist(exp) title('Control group') title('Experimental group')

9. Compute descriptors and test stat Our data looks pretty mound shaped, so let's use the mean as data descriptor Take the absolute value of the difference as our test stat ctrl_d = mean(ctrl); exp_d = mean(exp); test_stat = abs(ctrl_d - exp_d);

10. Assessing the test statistic What we would really like to test: The probability that the two datasets are subsets of two different/independent distributions Problem: We don't know what those hypothetical independent distributions are, if we did we wouldn't have to go through all of this!

11. Assessing the test statistic Solution: Compute the probability that the two datasets are NOT subsets of the same underlying distribution How to do this? Start by assuming datasets are subsets of same distribution → null hypothesis See what test statistic looks like when null hypothesis is true

12. Assessing the test statistic Need to generate a distribution of test statistic generated when datasets really ARE from the same underlying distribution → distribution of test stat under null hypothesis Then we can quantify how (a)typical the value of our test stat is under null hypothesis if typical, our datasets are likely from same distribution → no effect of experiment if atypical, there is a good chance datasets are from different distributions → experiment had an effect

13. Generate the null distribution using bootstrap Basic procedure: Combine datasets Randomly sample (w/ replacement) from combined dataset to create two pseudo datasets w/ same n as real datasets Compute descriptors and test statistic for pseudo datasets Repeat 10,000 times, keeping track of pseudo test stat

14. Combine datasets and resample box = [ctrl; exp]; % combine datasets into 'box' % could use concat() pseudo_stat_dist = zeros(1,10000); % create vector 'pseudo_stat_dist' % to store results of resampling % --- start resampling --- for trials = 1:10000 % resample 10000 times pseudo_ctrl =... % create pseudo groups to be sample(length(ctrl), box); % same size as actual groups pseudo_exp =... sample(length(exp), box); pseudo_ctrl_d =... % compute pseudo group mean(pseudo_ctrl); % descriptors pseudo_exp_d =... mean(pseudo_exp); pseudo_stat =... % compute pseudo test stat abs(pseudo_ctrl_d - pseudo_exp_d); pseudo_stat_dist(trials) =... % store pseudo test stat to pseudo_stat; % build null distribution end

15. Now what? Compute how many values of pseudo test stat are greater than our actual test stat, then divide by the total number of data points in the null distribution This numerically approximates the likelihood of getting our actual test stat (i.e. the likelihood of seeing a difference as big as we see) if our two datasets were truly from the same underlying distribution → p-value

16. Compute p-val and visualize null distribution bigger = count(pseudo_stat_dist > test_stat); % count pseudo test stats % bigger than actual stat pval = bigger / length(pseudo_stat_dist) % divide by total number % of pseudo test stats to % compute p-value % could use proportion() hist(pseudo_stat_dist) % show histogram of pseudo title('Pseudo test stat distribution') % test stats xlabel('Pseudo test stat') ylabel('Frequency')

17. How to interpret the p-value? Assuming that the null hypothesis is true, the p- value is the likelihood that we would see a value of the test statistic that is greater than the value of our actual test statistic Restated → Assuming both datasets really are from the same underlying distribution, the p-value is the likelihood that we would see a difference as big as we do

18. function [] = bootstrap2groupsMEANS (ctrl, exp) % % function [] = bootstrap2groupsMEANS (ctrl, exp) % % 'ctrl' and 'exp' are assumed to be column vectors, this is % necessary for creation of 'box' % % This function will use bootstrap to compute the probability % that data in 'ctrl' and 'exp' are from the same distribution, % using the group means as descriptors, and the absolute value % of the difference between means as the test statistic % (similar to Student's t-test) % --- prepare for resampling --- ctrl_d = mean(ctrl); % compute descriptors exp_d = mean(exp); % 'ctrl_d' and 'exp_d' test_stat = abs(ctrl_d - exp_d) % compute test statistic 'test_stat' box = [ctrl; exp]; % combine datasets into 'box' % could use concat() pseudo_stat_dist = zeros(1,10000); % create vector 'pseudo_stat_dist' % to store results of resampling % --- start resampling --- for trials = 1:10000 % resample 10000 times pseudo_ctrl =... % create pseudo groups to be sample(length(ctrl), box); % same size as actual groups pseudo_exp =... sample(length(exp), box); pseudo_ctrl_d =... % compute pseudo group mean(pseudo_ctrl); % descriptors pseudo_exp_d =... mean(pseudo_exp); pseudo_stat =... % compute pseudo test stat abs(pseudo_ctrl_d - pseudo_exp_d); pseudo_stat_dist(trials) =... % store pseudo test stat to pseudo_stat; % build null distribution end % --- end resampling --- bigger = count(pseudo_stat_dist > test_stat); % count pseudo test stats % bigger than actual stat pval = bigger / length(pseudo_stat_dist) % divide by total number % of pseudo test stats to % compute p-value % could use proportion() hist(pseudo_stat_dist) % show histogram of pseudo title('Pseudo test stat distribution') % test stats xlabel('Pseudo test stat') ylabel('Frequency')