Statistical Methods in Computer Science

1. Hypothesis Testing I: Treatment experiment designs Ido Dagan Statistical Methods in Computer Science

2. Hypothesis Testing: Intro We have looked at setting up experiments Goal: To prove falsifying hypotheses Goal fails => falsifying hypothesis not true (unlikely) => our theory survives Falsifying hypothesis is called null hypothesis, marked H0 We want to show that the likelihood of H0 being true is low.

3. Comparison Hypothesis Testing A very simple design: treatment experiment Also known as a lesion study / ablation test treatment Ind1 & Ex1 & Ex2 & .... & Exn ==> Dep1 control Ex1 & Ex2 & .... & Exn ==> Dep2 Treatment condition: Categorical independent variable What are possible hypotheses?

4. Hypotheses for a Treatment Experiment H1: Treatment has effect H0: Treatment has no effect Any effect is due to chance But how do we measure effect? We know of different ways to characterize data: Moments: Mean, median, mode, .... Dispersion measures (variance, interquartile range, std. dev) Shape (e.g., kurtosis)

5. Hypotheses for a Treatment Experiment H1: Treatment has effect H0: Treatment has no effect Any effect is due to chance Transformed into: H1: Treatment changes mean of population H0: Treatment does not change mean of population Any effect is due to chance

6. Hypotheses for a Treatment Experiment H1: Treatment has effect H0: Treatment has no effect Any effect is due to chance Transformed into: H1: Treatment changes variance of population H0: Treatment does not change variance of population Any effect is due to chance

7. Hypotheses for a Treatment Experiment H1: Treatment has effect H0: Treatment has no effect Any effect is due to chance Transformed into: H1: Treatment changes shape of population H0: Treatment does not change shape of population Any effect is due to chance

8. Chance Results • The problem: • Suppose we sample the treatment and control groups • We find • mean treatment results = 0.7 • mean control = 0.5 • How do we know there is a real difference? • It could be due to chance! In other words: • What is the probability of getting 0.7 given H0 ? • If low, then we can reject H0

9. Testing Errors The decision to reject the null hypothesis H0 may lead to errors Type I error: Rejecting H0 though it is true (false positive) Type II error: Failing to reject H0 though it is false (false negative) Classification perspective of false/true-positive/negative We are worried about the probability of these errors (upper bounds) Normally, alpha is set to 0.05 or 0.01. This is our rejection criteria for H0 (usually the focus of significance tests) 1-beta is the power of the test (its sensitivity)

10. Two designs for treatment experiments One-sample: Compare sample to a known population e.g., compare to specification Two-sample: Compare two samples, establish whether they are produced from the same underlying distribution

11. One sample testing: Basics We begin with a simple case We are given a known control populationP For example: life expectancy for patients (w/o treatment) Known parameters (e.g. known mean) Recall terminology: population vs. sample Now we sample the treatment population Mean = Mt Was the mean Mt drawn by chance from the known control population? To answer this, must know: What is the sampling distribution of the mean of P?

12. Sampling Distributions Suppose given P we repeat the following: Draw N sample points, calculate mean M1 Draw N sample points, calculate mean M2 ..... Draw N sample points, calculate mean Mn The collection of means forms a distribution, too: The sampling distribution of the mean

13. Central Limit Theorem The sampling distribution of the mean of samples of size N, of a population with mean M and std. dev. S: 1. Approaches a normal distribution as N increases, for which: 2. Mean = M 3. Standard Deviation = This is called the standard error of the sample mean Regardless of shape of underlying population

14. So? Why should we care? We can now examine the likelihood of obtaining the observed sample mean for the known population If it is “too unlikely”, then we can reject the null hypothesis e.g., if likelihood that the mean is due to chance is less than 5%. The process: We are given a control population C Mean Mc and standard deviation Sc A sample of the treatment population sample size N, mean Mt and standard deviation St If Mt is sufficiently different than Mc then we can reject the null hypothesis

15. Z-test by example We are given: Control mean Mc = 1, std. dev. = 0.948 Treatment N=25, Mt = 2.8 We compute: Standard error = 0.948/5 = 0.19 Z score of Mt = (2.8-population-mean-given-H0)/0.19 = (2.8-1)/0.19 = 9.47 Now we compute the percentile rank of 9.47 This sets the probability of receiving Mt of 2.8 or higher by chance Under the assumption that the real mean is 1. Notice: the z-score has standard normal distribution Sample mean is normally distributed, and subtracted/divided by constants; Z has Mean=0, stdev=1.

16. One- and two-tailed hypotheses The Z-test computes the percentile rank of the sample mean Assumption: drawn from sampling distribution of control population What kind of null hypotheses are rejected? One-tailed hypothesis testing: H0: Mt = Mc H1: Mt > Mc If we receive Z >= 1.645, reject H0. Z=0 =P50 Z=1.645 =P95 95% of Population

17. One- and two-tailed hypotheses The Z-test computes the percentile rank of the mean Assumption: drawn from sampling distribution of control population What kind of null hypotheses are rejected? Two-tailed hypothesis testing: H0: Mt = Mc H1: Mt != Mc If we receive Z >= 1.96, reject H0. If we receive Z <= -1.96, reject H0. Z=1.96 =P97.5 Z=-1.96 =P2.5 Z=0 =P50 95% of Population

18. Two-sample Z-test Up until now, assumed we have population mean But what about cases where this is unknown? This is called a two-sample case: We have two samples of populations Treatment & control For now, assume we know std of both populations We want to compare estimated (sample) means

19. Two-sample Z-test(assume std known) Compare the differences of two population means When samples are independent (e.g. two patient groups) H0: M1-M2 = d0 H1: M1-M2 != d0 (this is the two-tailed version) var(X-Y) = var(X) + var(Y) for independent variables When we test for equality, d0 = 0

20. Mean comparison when std unknown Up until now, assumed we have population std. But what about cases where std is unknown? => Have to be approximated When N sufficiently large (e.g., N>30) When population std unknown: Use sample std Population std is: Sample std is:

21. The Student's t-test Z-test works well with relatively large N e.g., N>30 But is less accurate when population std unknown In this case, and small N: t-test is used It approaches normal for large N t-test: Performed like z-test with sample std Compared against t-distribution t-score doesn’t distribute normally(denominator is variable) Assumes sample mean is normally distributed Requires use of size of sample N-1 degrees of freedom, a different distribution for each degree t =0 =P50 thicker tails

22. t-test variations Available in excel or statistical software packages Two-sample and one-sample t-test Two-tailed, one-tailed t-test t-test assuming equal and unequal variances Paired t-test Same inputs (e.g. before/after treatment), not independent The t-test is common for testing hypotheses about means

23. Testing variance hypotheses F-test: compares variances of populations Z-test, t-test: compare means of populations Testing procedure is similar H0: H1: OR OR Now calculate f = , where sx is the sample std of X When far from 1, the variances likely different To determine likelihood (how far), compare to F distribution

24. The F distribution F is based on the ratio of population and sample variances According to H0, the two standard deviations are equal F-distribution Two parameters: numerator and denominator degrees-of-freedom Degrees-of-freedom (here): N-1 of sample Assumes both variables are normal

25. Other tests for two-sample testing There exist multiple other tests for two-sample testing Each with its own assumptions and associated power For instance, Kolmogorov-Smirnov (KS) test Non-parametric estimate of the difference between two distributions Turn to your friendly statistics book for help

26. Testing correlation hypotheses • We now examine the significance of r • To do this, we have to examine the sampling distribution of r • What distribution of r values will we get from the different samples? • The sampling distribution of r is not easy to work with • Fisher's r-to-z transform: Where the standard error of the r sampling distribution is:

27. Testing correlation hypotheses We now plug these values and do a Z-test For example: Let the r correlation coefficient for variables x,y = 0.14 Suppose n = 30 H0: r = 0 H1: r != 0 Cannot reject H0

28. Treatment Experiments(single-factor experiments) Allow comparison of multiple treatment conditions treatment1Ind1 & Ex1 & Ex2 & .... & Exn ==> Dep1 treatment2 Ind2 & Ex1 & Ex2 & .... & Exn ==> Dep2 control Ex1 & Ex2 & .... & Exn ==> Dep3 Compare performance of algorithm A to B to C .... Control condition: Optional (e.g., to establish baseline) Cannot use the tests we learned: Why?