Statistics, Data Analysis, and Simulation SS 2015

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1 Mainz, June 11, 2015 Statistics, Data Analysis, and Simulation SS Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Dr. Michael O. Distler Statistics, Data Analysis, and Simulation SS / 23

2 Statistical hypothesis testing So far: statistical analysis of a data sample in order to extract unknown parameters. Now we have prior assumptions about the value of those parameters a hypothesis We need to check those hypotheses: the procedure is called statistical test Caveat: a test can never prove a hypothesis to be true. However, one can reject a hypothesis because of observations. The degree of statistical compatibility will be quantified using confidence limits. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

3 The testing process There is an initial research hypothesis of which the truth is unknown. The first step is to state the relevant null and alternative hypotheses. This is important as mis-stating the hypotheses will muddy the rest of the process. The second step is to consider the statistical assumptions being made about the sample in doing the test; for example, assumptions about the statistical independence or about the form of the distributions of the observations. This is equally important as invalid assumptions will mean that the results of the test are invalid. Decide which test is appropriate, and state the relevant test statistic T. Derive the distribution of the test statistic under the null hypothesis from the assumptions. In standard cases this will be a well-known result. For example the test statistic might follow a Student s t distribution or a normal distribution. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

4 The testing process Select a significance level (α), a probability threshold below which the null hypothesis will be rejected. Common values are 5% and 1%. The distribution of the test statistic under the null hypothesis partitions the possible values of T into those for which the null hypothesis is rejected the so-called critical region and those for which it is not. The probability of the critical region is α. Compute from the observations the observed value t obs of the test statistic T. Decide to either reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis H0 if the observed value t obs is in the critical region, and to accept or fail to reject the hypothesis otherwise. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

5 The testing process An alternative process is commonly used: 1 Compute from the observations the observed value t obs of the test statistic T. 2 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. 3 Reject the null hypothesis, in favor of the alternative hypothesis, if and only if the p-value is less than the significance level (the selected probability) threshold. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

6 clairvoyance example Dr. Michael O. Distler Statistics, Data Analysis, and Simulation SS / 23

7 Chi-square distribution If x 1, x 2,..., x n are independend random variables distributed according to the standard Gaussian distribution with mean 0 and variance 1, then the sum u = χ 2 = n i=1 x 2 i ist distributed according to a χ 2 distribution f n (u) = f n (χ 2 ) where n is called the number of degrees of freedom. f n (u) = ( 1 u ) n/ e u/2 Γ(n/2) The χ 2 distribution has a maximum at (n 2). The mean is found to be n and the variance is 2n. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

8 Chi-square distribution pdf(2,x) pdf(3,x) pdf(4,x) pdf(5,x) pdf(6,x) pdf(7,x) pdf(8,x) pdf(9,x) Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

9 Chi-square cumulative distribution function The probability for χ 2 n to take on a value in the interval [0, x] cdf(2,x) cdf(3,x) cdf(4,x) cdf(5,x) cdf(6,x) cdf(7,x) cdf(8,x) cdf(9,x) Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

10 Chi-square distribution with 5 d.o.f % c.l. [ ] Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

11 Student s t-test A t-test is any statistical hypothesis test in which the test statistic follows a Student s t distribution if the null hypothesis is supported. A one-sample location test of whether the mean of a normally distributed population has a value specified in a null hypothesis. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

12 t-verteilung Die t-verteilung tritt auf bei Tests der statistischen Verträglichkeit eines Stichproben-Mittelwertes x mit einem vorgegebenen Mittelwert µ, oder der statistischen Verträglichkeit zweier Stichproben-Mittelwerte. Die Wahrscheinlichkeitsdichte der t-verteilung ist gegeben durch f n (t) = 1 ( ) (n+1)/2 Γ((n + 1)/2) 1 + t2 nπ Γ(n/2) n Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

13 t-verteilung Die Studentschen t-verteilungen f (t) (links) im Vergleich zur standardisierten Gauß-Verteilung (gestrichelt) sowie die integrierten Studentschen t-verteilungen t f (x)dx (rechts). Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

14 t-verteilung Quantile der t-verteilung, P = t f n(x)dx. Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

15 F -Verteilung Gegeben sind n 1 Stichprobenwerte einer Zufallsvariablen x und n 2 Stichprobenwerte derselben Zufallsvariablen. Die beste Schätzung der Varianzen aus beiden Datenkollektionen seien s1 2 und s2 2. Die Zufallszahl F = s2 1 s 2 2 folgt dann einer F-Verteilung mit (n 1, n 2 ) Freiheitsgraden. Es ist Konvention, dass F immer größer als eins ist. Die Wahrscheinlichkeitsdichte von F ist gegeben durch f (F) = ( n1 n 2 ) n1 /2 ( Γ((n 1 + n 2 )/2) Γ(n 1 /2)Γ(n 2 /2) F (n 1 2)/2 1 + n ) (n1 +n 2 )/2 1 F n 2 Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

16 Quantile der F -Verteilung, Konfidenz = 0.68 Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

17 Quantile der F -Verteilung, Konfidenz = 0.90 Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

18 Quantile der F -Verteilung, Konfidenz = 0.95 Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

19 Quantile der F -Verteilung, Konfidenz = 0.99 Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

20 5.3 Kolmogorov-Smirnov-Test Dieser Test reagiert empfindlich auf Unterschiede in der globalen Form oder in Tendenzen von Verteilungen. Die theoretische Wahrscheinlichkeitsdichte f (x) und ihre Verteilungsfunktion F(x) = x f (x )dx sei gegeben. Die x i werden nach ihrer Größe geordnet und die kumulative Größe gebildet: F n = Anzahl der x i-werte x n Die Testgröße ist t = n max F n (x) F (x) Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

21 Kolmogorov-Smirnov-Test Die Wahrscheinlichkeit P, einen Wert t 0 für die Testgröße t zu erhalten, ist P = 1 2 ( 1) k 1 e 2k 2 t0 2 k=1 Werte für den praktischen Gebrauch: P 1% 5% 50% 68% 95% 99% 99.9% t Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

22 Kolmogorov-Smirnov-Test Beispiel: Die Daten 7, -1, 8, 5, 6 sollen einer Normalverteilung mit µ = 5 und σ = 2 entnommen worden sein. Für die Testgröße ergibt sich t = = Verteilungsfunktion F(x) Zufallsvariable x Dr. Michael O. Distler <distler@uni-mainz.de> Statistics, Data Analysis, and Simulation SS / 23

23 Dr. Michael O. Distler Statistics, Data Analysis, and Simulation SS / 23