# Median of the p-value Under the Alternative Hypothesis

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Median of the p-value Under the Alternative Hypothesis Bhaskar Bhattacharya Department of Mathematics, Southern Illinois University, Carbondale, IL, USA Desale Habtzghi Department of Statistics, University of Georgia, Athens, GA, USA Abstract Due to absence of an universally acceptable magnitude of the type I error in various fields, p-values are used as a well-recognized tool in decision making in all areas of statistical practice. The distribution of p-values under the null hypothesis is uniform. However, under the alternative hypothesis the distribution of the p-values is skewed. The expected p-value (EPV) has been proposed by authors to be used as a measure of the performance of the test. In this article, we propose the median of the p-values (MPV) which is more appropriate for this purpose. We work out many examples to calculate the MPV s directly and also compare the MPV with the EPV. We consider testing equality of distributions against stochastic ordering in the multinomial case and compare the EPV s and MPV s by simulation. A second simulation study for general continuous data is also considered for two samples with different test statistics for the same hypotheses. In both cases MPV performs better than EPV. KEY WORDS: Comparing test statistics; Expected p-value; Median p-value; Power; Stochastic order. 1

2 1 INTRODUCTION The theory of hypothesis testing depends heavily on the pre specified value of the significance level. To avoid the non uniqueness of the decision of testing the same hypotheses using the same test statistic but different significance levels, it is a popular choice to report the p-value. The p-value is the smallest level of significance at which an experimenter would reject the null hypothesis on the basis of the observed outcome. The user can compare his/her own significance level with the p-value and make his/her own decision. The p-values are particularly useful in cases when the null hypothesis is well defined but the alternative is not (e.g. composite) so that type II error considerations are unclear. In this context to quote Fisher, The actual value of p obtainable from the table by interpolation indicates the strength of evidence against the null hypothesis. We will consider tests of the form Reject H 0 when T c where T is a realvalued test statistic computed from data when testing the null hypothesis H 0 against the alternative H 1. The value c is determined from the pre specified size restrictions such that P H0 (T c) = α. Of course when T is a discrete random variable, one needs to adopt randomization so that all sizes are possible. If t is the observed value of T and the distribution of T under H 0 is given by F 0 ( ), then the p-value is given by 1 F 0 (t) which is the probability of finding the test statistic as extreme as, or more extreme than, the value actually observed. Thus if the p-value is less than the preferred significance level then one rejects H 0. Over the years several authors have attempted for proper explanation of the p-values. Gibbons and Pratt (1975) provided interpretation and methodology of the p-values. Recently, Schervish (1996) treated the p-values as significance probabilities. Discussions on p-values can be found in Blyth and Staudte (1997) and in Dollinger et al. (1996). However these papers do not treat the p-values as random. 2

3 The p-values are based on the test statistics used and hence random. The stochastic nature of the p-values has been investigated by Dempster and Schatzoff (1965) and Schatzoff (1966) who introduced expected significance value. Recently Sackrowitz and Samuel-Cahn (1999) investigated this concept further and renamed it as the expected p-value (EPV). Under the null hypothesis, the p-values have a uniform distribution over (0, 1) for any sample size. Thus, under H 0, EPV is 1/2 always, and there is no way to distinguish p-values derived from large studies and those from small-scale studies. Also it would be impossible under H 0 to differentiate between studies well powered to detect a posited alternative hypothesis and the underpowered to detect the same posited alternative value. In contrast, the distribution of the p-values under the alternative hypothesis is a function of the sample size and the true parameter value in the alternative hypothesis. As the p-values measure evidence against the null hypothesis, it is of interest to investigate the behavior of the p-values under the alternative at various sample sizes. We reject H 0 when p-value is small which is expected when H 1 is true. As noted by Hung et al. (1997), the distribution of the p-values under the alternative is highly skewed. The skewness increases with the sample size and the true parameter value under the alternative reflecting the ability to detect the alternative by increasing power under these situations. Hence it is more appropriate to consider the median of the p-value (MPV) instead of the EPV under the alternative as a measure of the center of its distribution which is the main focus of this article. Applications of the distribution of the p-values under the alternative in the area of meta-analysis of several studies is considered by Hung et al. (1997). Studying the p-value under the alternative is also beneficial over the power of a test and is explained in the next paragraph. When several test procedures are available for the same testing situation one compares them by means of power. However, power calculations depend on the chosen 3

4 significance levels, and in discrete cases involves randomization. These steps can be avoided by considering MPV s. Also as the power functions depend on the chosen significance levels, it is difficult to compare them when different power functions use different significance levels. On the other hand, MPV s depend only on the alternative and not on the significance level. The smaller the value of MPV, the stronger the test. The value of an MPV can tell us which alternative an attained p-value best represents for a given sample size. Also, it helps to know the behavior of the MPV s for varying sample sizes. In Section 2, we obtain a general expression for the MPV s. We also derive a computationally favorable form to be used later in the paper. In Section 3, we consider several examples to compute the MPV s directly. In Section 4, we perform a simulation study to compute the MPV s in the case of testing against stochastic ordering for multinomial distributions. We also perform a second simulation study for general distributions when testing the same hypotheses with two samples using three test statistics of t-test, Mann-Whitney-Wilcoxon and Kolmogorov-Smirnov. In Section 5, we make concluding remarks. 2 p-value AS A RANDOM VARIABLE AND ITS MEDIAN For the test statistic T with distribution F 0 ( ) under H 0, let F θ ( ) be its distribution under H 1. Also, let F 1 0 ( ) be the inverse function of F 0 ( ), so that, F 0 (F 1 0 (γ)) = γ, for any 0 < γ < 1. Since the p-value is the probability of observing a more extreme value than the observed test statistic value, as a random variable it can be expressed as X = 1 F 0 (T). 4

5 As F 0 (T) U(0, 1) under H 0, so is X. The power of the test is related to the p-value as β = P θ (X α) = 1 F θ (F 1 0 (1 α)). (1) Note the above is also the distribution function of the p-value under the alternative. As T is stochastically larger under the alternative than under the null hypothesis, it follows that the p-value under the alternative is stochastically smaller than under the null (Lehmann, 1986). This explains why the distribution of the p-value is skewed to the right under the alternative. Hence to estimate the center of the distribution of the p-value, the median is a better choice than the mean. The median is any value of α = α which satisfies P θ (X < α ).5 and P θ (X > α ).5. For continuous distributions, the median is the value of α = α which satisfies P θ (X α ) =.5. It follows from (1) by simple manipulations that α = 1 F 0 (F 1 θ (.5)). (2) Since F θ (t) F 0 (t), t, it is seen that F 0 (F 1 θ (.5)).5 which implies that α.5 and equality holds when H 0 is true. The smaller the MPV the better it is to detect the alternative. Given the stochastic nature of the p-value under the alternative, it is also true that the MPV is smaller than the EPV and hence clearly preferable over the EPV. The MPV being smaller than the EPV produces a smaller indifference region in the sense that higher power is generated closer to the null hypothesis region using MPV than with using EPV. 5

6 It is also possible to express the MPV in another way. Let T F θ ( ) and, independently, T F 0 ( ). If the observed value of T is t, then the p-value is simply g(t) = P(T t T = t). The MPV is med g(t) = MPV(θ) = P(T med T). (3) If H 0 is true then T and T are identically distributed, and hence the above probability is.5 for any continuous distribution. For discrete distributions, although the above expressions still hold, MPV will be slightly higher as P(T = med T) > 0. For an UMP test, the MPV will be uniformly minimal for all θ values in the alternative as compared to the MPV s of any other test of the same H 0 versus H 1. For computational purposes the above form of the MPV in (3) is very useful. 3 EXAMPLES In this section we consider several examples to calculate the MPV s directly. Example 1. Let X 1, X 2,,X n be a random sample of size n from a N(µ, σ 2 ) distribution, and we like to test H 0 : µ µ 0 versus H 1 : µ > µ 0 where σ is known. Using the test statistic T = X and a particular value µ 1 in H 1 it follows from (2) that ( ) n(µ0 µ 1 ) MPV = Φ (4) σ where Φ is the CDF of the standard normal distribution. It is well known that for a size α test to achieve power β at a specified alternative value µ 1 the sample size n satisfies n = (z 1 α + z β ) 2 σ 2 (µ 0 µ 1 ) 2 (5) where z γ is the γth quantile of the standard normal distribution. Using (4) and (5), it follows that MPV = Φ( z 1 α z β ). (6) 6

7 In Table 1, we have provided values of the MPV s in (6) using some commonly used values of α and β. Each MPV value is smaller than the corresponding EPV value of Table 1 of Sackrowitz and Samuel-Cahn (1999). We have also graphed the EPV and the MPV in Figure 1 for various values of µ 1 > 0 when µ 0 = 0 at sample sizes 10 and 50. The MPV s decrease from the.5 value at much faster rate than the EPV s although for both the rate increases with the sample size. In Figure 2, we have graphed the EPV and the MPV for various sample sizes at µ 1 =.3 and at µ 1 =.5. It is observed that the MPV s decrease at a faster rate than the EPV s at smaller sample sizes and when closer to the null hypothesis. ****** Insert Table 1 here ****** ****** Insert Figures 1 and 2 here ****** If the value of σ is unknown, the sample standard deviation S (or any other consistent estimator of σ) may be used to replace it for moderately large n. The formula in (4) is approximately correct in this case, consequently, Table 1 is approximately correct for the one-sample t-test situation. After observing an actual p-value for an approximately normally distributed statistic, the expression in (4) can be used to determine the µ 1 for which the given value would be an MPV. Example 2. Let X 1, X 2,, X n be a random sample of size n from N(µ 1, σ 2 ), and independently, let Y 1, Y 2,,Y m be another random sample of size m from N(µ 2, σ 2 ), and we like to test H 0 : µ 1 = µ 2 versus H 1 : µ 1 > µ 2 where σ is known. Using the test statistic T = X Y and a particular value µ 1 µ 2 in H 1 it follows that ( ) mn µ 2 µ 1 MPV = Φ. (7) m + n σ When m = n, formula (6) is still valid and hence Table 1 is also correct in this case. For unknown σ, a consistent estimator of σ may be used in (7) for moderately large m, n, and the formula in (7) becomes approximately correct in this case. 7

8 Example 3. Suppose X 1, X 2,,X n is a random sample of size n from a N(µ, σ 2 ) distribution, and we like to test H 0 : σ σ 0 versus H 1 : σ > σ 0 when µ is unknown. Using the test statistic (n 1)S 2 /σ 2 where S 2 is the sample variance, it follows from (4) that at the alternative point σ 1, ( ) σ 2 MPV = 1 G 1 G 1 (.5) σ0 2 (8) where G is the CDF of a chi-square distribution with n 1 degrees of freedom. Note we need not use the F-distribution as needed for the computation of the EPV in this problem (Sackrowitz and Samuel-Cahn, 1999). The following two examples are concerned with testing the scale and location parameters of the exponential distribution. Example 4. Suppose T is exponentially distributed with parameter θ (from pdf f(t) = θe tθ for t > 0) denoted by exp(θ) and we like to test H 0 : θ θ 0 versus H 1 : θ < θ 0. It is seen that med T = (1/θ)ln 2. For a particular value of θ 1 < θ 0, if T exp(θ 0 ) and T exp(θ 1 ), the MPV is given by P(T med T) = 2 θ 0/θ 1. For a size α test with power β, since lnα/lnβ = θ 0 /θ 1, it follows that for α =.1 and β =.9, the MPV is The EPV is.0438 in this case (Sackrowitz and Samuel-Cahn, 1999). If a random sample X 1, X 2,, X n is available, the test may be based on T = n i=1 X i. Here T has a gamma distribution with shape parameter n and scale parameter θ, we will denote its CDF by G n,θ ( ). Then it follows that the MPV at alternative point θ 1 is given by P(T G 1 n,θ 1 (.5)) = 1 G n,θ0 (G 1 n,θ 1 (.5)). Example 5. Suppose X 1, X 2,,X n is a random sample of size n from an exponential distribution with parameters µ, θ (with pdf f(x) = (1/θ)e (x µ)/θ for x µ) 8

9 denoted by exp(µ, θ) and we like to test H 0 : µ µ 0 versus H 1 : µ > µ 0 where θ is known. The test statistic is T = min(x 1, X 2,, X n ) whose distribution is exp(µ, θ/n). It is seen that med T = µ + (θ/n)ln 2. For a particular alternative µ 1 > µ 0, the MPV is given by.5e n(µ 1 µ 0 )/θ. 4 SIMULATION STUDIES We perform two simulation studies to calculate and compare the EPV s and MPV s. For the discrete case we consider the binomial distribution, with m trials and probability of success p, given by m p j = p j (1 p) m j, j = 0,, m (9) j which is symmetric when p =.5. When p >.5, the binomial distribution is skewed to the left, that is, it becomes stochastically larger than the p =.5 case. Let q = (q 0,, q m ) and p = (p 0,,p m ) be the vectors of binomial probabilities obtained from (9) with p =.5 and p >.5, respectively. We consider testing H 0 : p = q (i.e., p i = q i, i) against H 1 : p is stochastically larger than q (i.e., m i=j p i m i=j q i, j = 1,, m, and m i=0 p i = m i=0 q i = 1). The likelihood ratio test statistic is given by m T = 2n ˆp i ln (p i /q i ) i=0 where p i is the ith coordinate of p = ˆpE ˆp (q/ˆp A), E ˆp (q/ˆp A) is the isotonic regression of q/ˆp (all multiplications and divisions of vectors are done coordinatewise) onto the non increasing cone A = {x = (x 0, x 1,, x m ) : x 0 x 1 x m } with weights ˆp. It is well established (Robertson et. al., 1988) that under H 0, asymptotically, the statistic T has a chi-bar squared distribution. We create a random sample T 1,,T n distributed like T, and independently, another random sample T 1,, T n distributed like T. An unbiased estimator of 9

10 EPV is given by A E = 1 n n i=1 and an unbiased estimator of MPV is given by A M = 1 n n i=1 I(T i T i ) I(T i med T i ). The variance of A E is EPV(1 EPV)/n and that of A M is MPV(1 MPV)/n We consider m = 3, 6 with sample sizes n = 50, 100 and replications 10,000. The results are given in Table 2. When p =.5, both of the EPV and MPV start slightly higher than.5 as expected for discrete distributions. For p >.5, the MPV s are consistently smaller than the EPV s, both being very close to zero when p >.65. The effects are more pronounced for larger n. Note the exact value of the EPV or the MPV is difficult to calculate in this case. ****** Insert Table 2 here ****** We consider a second simulation study to compare the performance of several tests using the MPV s for a general continuous case. We use the same set up as Sackrowitz and Samuel-Cahn (1999) but calculate the MPV s instead. Thus we consider the twosample problem of testing H 0 : F = G versus H 1 : F is stochastically larger than G using two independent random samples from F and G respectively. The test statistics considered are the two-sample t-test, the Mann-Whitney-Wilcoxon (MWW) test and the Kolmogorov-Smirnov (KS) test. The comparison is made for shift alternatives so that G(x) = F(x + ) where F is chosen as various distributions. We consider F as normal (0,1), exponential, chi-square with degrees of freedom 10, uniform (0,1) and double exponential. We chose sample sizes 10, 20, 50 and 0 = 0, 1 = 2.546σ/ 50, 2 = 2.546σ/ 20, 3 = 2.546σ/ 10, where σ is the actual standard deviation of the underlying distribution F. This choice is made so that when the 10

11 underlying distribution is normal, a test based on the normal two-sample statistic and n observations would have size α =.10 and power β =.7 (as opposed to β =.9 of Sackrowitz and Samuel-Cahn, 1999 in their table 4) for = (2.546σ/ n) and thus from Table 1 we have MPV= Note that we have used the same i values for all n. We consider 10,000 replications. ****** Insert Table 3 here ****** The MPV values in Table 3 have similar magnitude as the EPV values of Sackrowitz and Samuel-Cahn (1999, Table 4) even at β =.7. They also have similar pattern as the corresponding EPV values. So their conclusions are also valid in our case. However our KS values of MPV perform worse than the corresponding EPV values of Sackrowitz and Samuel-Cahn (1999) in all cases considered. 5 CONCLUSION The distribution of the p-values under the alternative is a skewed distribution to the right, and hence the median of this distribution is advocated as a more appropriate tool than its mean for determination of the strength of a test for a particular alternative. The alternatives closer to H 0 are detected easily with MPV than with EPV. The MPV is easily computed in most cases and does not depend on the specified significance level of a test. Thus it may be used as a single number which can help to choose among different test statistics when testing the same hypotheses. For approximately normally distributed statistics, Table 1 can be consulted to relate the MPV value to the usual significance level and power combinations. 11

12 REFERENCES Blyth, C. R. and Staudte, R. G. (1997), Hypothesis estimation and acceptability profiles for 2 2 contingency tables, Journal of the American Statistical Association, 92, Dempster, A. P. and Schatzoff, M. (1965), Expected significance level as a sensibility index for test statistics, Journal of the American Statistical Association, 60, Dollinger, M. B., Kulinskaya, E., Staudte, R. G. (1996), When is a p value a good measure of evidence, Robust statistics, Data Analysis and Computer intensive methods, ed. H. Reider, New York, Springer-Verlag, Gibbons, J. D. and Pratt, J. W. (1975) P-values: Interpretation and methodology, The American Statistician, 29, Hung, H.M.J., O Neill, R. T., Bauer, P. and Kohne, K (1997), The behavior of the p-value when the alternative hypothesis is true, Biometrics, 53, Lehmann, E. L. (1986) Testing Statistical Hypotheses, (2nd edition), John Wiley & Sons, New York. Robertson, T, Wright. F. T. and Dykstra, R. L. (1988) Order Restricted Statistical Inference, John Wiley & Sons, New York. Sackrowitz, H. and Samuel-Cahn, E.(1999), P-values as random variables-expected p-values, The American Statistician, 53, Schatzoff, M. (1966), Sensitivity comparisons among tests of the general linear hypotheses, Journal of the American Statistical Association, 61,

13 Schervish, M. J. (1996), P-values: What they are and what they are not, The American Statistician, 50,

14 . 14

15 . 15

16 Table 1: MPV s in the testing for the normal mean as a function of the significance level α and power β α/β Table 2: EPV s and MPV s for testing H 0 : p = q against H 1 : p is stochastically larger than q for different combinations of m, n, p m = 3 m = 6 EPV MPV EPV MPV p n = 50 n = 100 n = 50 n = 100 n = 50 n = 100 n = 50 n =

17 Table 3: MPV s for the one sided, two-sample t, Mann-Whitney-Wilcoxon (MWW) and Kolmogorov-Smirnov (KS) tests for various sample sizes, shift parameters and underlying distributions n = 10 n = 20 n = 50 t MWW KS t MWW KS t MWW KS N ormal(0, 1) 0 = = = = Exponential 0 = = = = Chi Square(10) 0 = = = = U niform(0, 1) 0 = = = = DoubleExponential 0 = = = =

### MAT X Hypothesis Testing - Part I

MAT 2379 3X Hypothesis Testing - Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating

### Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. 2-Sample

### Chapter 9 Testing Hypothesis and Assesing Goodness of Fit

Chapter 9 Testing Hypothesis and Assesing Goodness of Fit 9.1-9.3 The Neyman-Pearson Paradigm and Neyman Pearson Lemma We begin by reviewing some basic terms in hypothesis testing. Let H 0 denote the null

### Test of Hypotheses. Since the Neyman-Pearson approach involves two statistical hypotheses, one has to decide which one

Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Imagine that you play a repeated Bernoulli game: you win \$1 if head and lose \$1 if tail. After 10 plays, you lost \$2 in net (4 heads

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Sampling Distribution of a Normal Variable

Ismor Fischer, 5/9/01 5.-1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,

### Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

### Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling

Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling Instructor: Sargur N. University at Buffalo The State University of New York srihari@cedar.buffalo.edu Topics 1. Hypothesis Testing 1.

### Classical Hypothesis Testing and GWAS

Department of Computer Science Brown University, Providence sorin@cs.brown.edu September 15, 2014 Outline General Principles Classical statistical hypothesis testing involves the test of a null hypothesis

### 3.6: General Hypothesis Tests

3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

### 0.1 Solutions to CAS Exam ST, Fall 2015

0.1 Solutions to CAS Exam ST, Fall 2015 The questions can be found at http://www.casact.org/admissions/studytools/examst/f15-st.pdf. 1. The variance equals the parameter λ of the Poisson distribution for

### The Modified Kolmogorov-Smirnov One-Sample Test Statistic

Thailand Statistician July 00; 8() : 43-55 http://statassoc.or.th Contributed paper The Modified Kolmogorov-Smirnov One-Sample Test Statistic Jantra Sukkasem epartment of Mathematics, Faculty of Science,

### . (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.

Chapter 3 Kolmogorov-Smirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models

On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models Hisashi Tanizaki Graduate School of Economics Kobe University (tanizaki@kobe-u.ac.p) ABSTRACT In this paper we

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Non Parametric Inference

Maura Department of Economics and Finance Università Tor Vergata Outline 1 2 3 Inverse distribution function Theorem: Let U be a uniform random variable on (0, 1). Let X be a continuous random variable

### Lecture 1: t tests and CLT

Lecture 1: t tests and CLT http://www.stats.ox.ac.uk/ winkel/phs.html Dr Matthias Winkel 1 Outline I. z test for unknown population mean - review II. Limitations of the z test III. t test for unknown population

### The t Test. April 3-8, exp (2πσ 2 ) 2σ 2. µ ln L(µ, σ2 x) = 1 σ 2. i=1. 2σ (σ 2 ) 2. ˆσ 2 0 = 1 n. (x i µ 0 ) 2 = i=1

The t Test April 3-8, 008 Again, we begin with independent normal observations X, X n with unknown mean µ and unknown variance σ The likelihood function Lµ, σ x) exp πσ ) n/ σ x i µ) Thus, ˆµ x ln Lµ,

### Testing: is my coin fair?

Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

Math 1 MPS Homework 6 Answers to additional problems 1. What are the two types of hypotheses used in a hypothesis test? How are they related? Ho: Null Hypotheses A statement about a population parameter

### MATH4427 Notebook 3 Spring MATH4427 Notebook Hypotheses: Definitions and Examples... 3

MATH4427 Notebook 3 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MATH4427 Notebook 3 3 3.1 Hypotheses: Definitions and Examples............................

### Comparing Two Populations OPRE 6301

Comparing Two Populations OPRE 6301 Introduction... In many applications, we are interested in hypotheses concerning differences between the means of two populations. For example, we may wish to decide

### Statistics 2014 Scoring Guidelines

AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### Hypothesis testing. Null hypothesis H 0 and alternative hypothesis H 1.

Hypothesis testing Null hypothesis H 0 and alternative hypothesis H 1. Hypothesis testing Null hypothesis H 0 and alternative hypothesis H 1. Simple and compound hypotheses. Simple : the probabilistic

### Null Hypothesis Significance Testing Signifcance Level, Power, t-tests. 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is

### One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots

1 / 27 One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis

### FALL 2005 EXAM C SOLUTIONS

FALL 005 EXAM C SOLUTIONS Question #1 Key: D S ˆ(300) = 3/10 (there are three observations greater than 300) H ˆ (300) = ln[ S ˆ (300)] = ln(0.3) = 1.0. Question # EX ( λ) = VarX ( λ) = λ µ = v = E( λ)

### Continuous Random Variables

Probability 2 - Notes 7 Continuous Random Variables Definition. A random variable X is said to be a continuous random variable if there is a function f X (x) (the probability density function or p.d.f.)

### Regression analysis in the Assistant fits a model with one continuous predictor and one continuous response and can fit two types of models:

This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. The simple regression procedure in the

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### Variance estimation in clinical studies with interim sample size re-estimation

Variance estimation in clinical studies with interim sample size re-estimation 1 Variance estimation in clinical studies with interim sample size re-estimation Frank Miller AstraZeneca, Clinical Science,

### Lecture 10: Other Continuous Distributions and Probability Plots

Lecture 10: Other Continuous Distributions and Probability Plots Devore: Section 4.4-4.6 Page 1 Gamma Distribution Gamma function is a natural extension of the factorial For any α > 0, Γ(α) = 0 x α 1 e

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part II)

Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part II) Florian Pelgrin HEC September-December 2010 Florian Pelgrin (HEC) Constrained estimators September-December

### Lecture 13: Kolmogorov Smirnov Test & Power of Tests

Lecture 13: Kolmogorov Smirnov Test & Power of Tests S. Massa, Department of Statistics, University of Oxford 2 February 2016 An example Suppose you are given the following 100 observations. -0.16-0.68-0.32-0.85

### 2 Testing under Normality Assumption

1 Hypothesis testing A statistical test is a method of making a decision about one hypothesis (the null hypothesis in comparison with another one (the alternative using a sample of observations of known

### Inference in Regression Analysis. Dr. Frank Wood

Inference in Regression Analysis Dr. Frank Wood Inference in the Normal Error Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters

### Generalized Linear Model Theory

Appendix B Generalized Linear Model Theory We describe the generalized linear model as formulated by Nelder and Wedderburn (1972), and discuss estimation of the parameters and tests of hypotheses. B.1

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Pooling and Meta-analysis. Tony O Hagan

Pooling and Meta-analysis Tony O Hagan Pooling Synthesising prior information from several experts 2 Multiple experts The case of multiple experts is important When elicitation is used to provide expert

### Math 62 Statistics Sample Exam Questions

Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected

### Test Code MS (Short answer type) 2004 Syllabus for Mathematics. Syllabus for Statistics

Test Code MS (Short answer type) 24 Syllabus for Mathematics Permutations and Combinations. Binomials and multinomial theorem. Theory of equations. Inequalities. Determinants, matrices, solution of linear

### Use of Pearson s Chi-Square for Testing Equality of Percentile Profiles across Multiple Populations

Open Journal of Statistics, 2015, 5, 412-420 Published Online August 2015 in SciRes. http://www.scirp.org/journal/ojs http://dx.doi.org/10.4236/ojs.2015.55043 Use of Pearson s Chi-Square for Testing Equality

### Chi-Square Distribution. is distributed according to the chi-square distribution. This is usually written

Chi-Square Distribution If X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chi-square distribution. This

### Asymptotic Joint Distribution of Sample Mean and a Sample Quantile. Thomas S. Ferguson UCLA

Asymptotic Joint Distribution of Sample Mean a Sample Quantile Thomas S. Ferguson UCA. Introduction. The joint asymptotic distribution of the sample mean the sample median was found by aplace almost 2

### Chi-Square & F Distributions

Chi-Square & F Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Chi-Square & F Distributions p. 1/55 Chi-Square & F Distributions... and Inferences about Variances The Chi-square Distribution Definition,

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### Power and Sample Size Determination

Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

### tests whether there is an association between the outcome variable and a predictor variable. In the Assistant, you can perform a Chi-Square Test for

This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. In practice, quality professionals sometimes

### Sufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution

### p Values and Alternative Boundaries

p Values and Alternative Boundaries for CUSUM Tests Achim Zeileis Working Paper No. 78 December 2000 December 2000 SFB Adaptive Information Systems and Modelling in Economics and Management Science Vienna

### Factorial Analysis of Variance

Chapter 560 Factorial Analysis of Variance Introduction A common task in research is to compare the average response across levels of one or more factor variables. Examples of factor variables are income

### 2. The mean birthweights from this hospital is lower than 120 oz.

Probability February 13, 2013 Debdeep Pati Hypothesis testing Two sample inference Obstetrics: Mothers with low socioeconomic status (SES) deliver babies with lower than normal birthweights. The mean birthweights

### The purpose of this section is to derive the asymptotic distribution of the Pearson chi-square statistic. k (n j np j ) 2. np j.

Chapter 7 Pearson s chi-square test 7. Null hypothesis asymptotics Let X, X 2, be independent from a multinomial(, p) distribution, where p is a k-vector with nonnegative entries that sum to one. That

### t-statistics for Weighted Means with Application to Risk Factor Models

t-statistics for Weighted Means with Application to Risk Factor Models Lisa R. Goldberg Barra, Inc. 00 Milvia St. Berkeley, CA 94704 Alec N. Kercheval Dept. of Mathematics Florida State University Tallahassee,

### Statistics 641 - EXAM II - 1999 through 2003

Statistics 641 - EXAM II - 1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the

### DISTRIBUTION FITTING 1

1 DISTRIBUTION FITTING What Is Distribution Fitting? Distribution fitting is the procedure of selecting a statistical distribution that best fits to a data set generated by some random process. In other

### Standard Deviation Calculator

CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

### AP Statistics 2007 Scoring Guidelines

AP Statistics 2007 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### Sample size and power calculations using the noncentral t-distribution

The Stata Journal (004) 4, Number, pp. 14 153 Sample size and power calculations using the noncentral t-distribution David A. Harrison ICNARC, London, UK david@icnarc.org Anthony R. Brady ICNARC, London,

### The Chi-Square Distributions

MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

### Applications of the Central Limit Theorem

Applications of the Central Limit Theorem October 23, 2008 Take home message. I expect you to know all the material in this note. We will get to the Maximum Liklihood Estimate material very soon! 1 Introduction

### Lecture Topic 6: Chapter 9 Hypothesis Testing

Lecture Topic 6: Chapter 9 Hypothesis Testing 9.1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should

### Continuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.

UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,

### Statistiek (WISB361)

Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

### The written Master s Examination

The written Master s Examination Option Statistics and Probability Fall Full points may be obtained for correct answers to 8 questions. Each numbered question (which may have several parts) is worth the

### AP STATISTICS 2009 SCORING GUIDELINES (Form B)

AP STATISTICS 2009 SCORING GUIDELINES (Form B) Question 5 Intent of Question The primary goals of this question were to assess students ability to (1) state the appropriate hypotheses, (2) identify and

### Non-Inferiority Tests for Two Proportions

Chapter 0 Non-Inferiority Tests for Two Proportions Introduction This module provides power analysis and sample size calculation for non-inferiority and superiority tests in twosample designs in which

### Exact Nonparametric Tests for Comparing Means - A Personal Summary

Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola

### I. Basics of Hypothesis Testing

Introduction to Hypothesis Testing This deals with an issue highly similar to what we did in the previous chapter. In that chapter we used sample information to make inferences about the range of possibilities

### The Delta Method and Applications

Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435)

Confidence Intervals for Poisson data For an observation from a Poisson distribution, we have σ 2 = λ. If we observe r events, then our estimate ˆλ = r : N(λ, λ) If r is bigger than 20, we can use this

### Linear and Piecewise Linear Regressions

Tarigan Statistical Consulting & Coaching statistical-coaching.ch Doctoral Program in Computer Science of the Universities of Fribourg, Geneva, Lausanne, Neuchâtel, Bern and the EPFL Hands-on Data Analysis

### Chapter 9: Hypothesis Testing Sections

Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.6 Comparing the

### For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random

Lecture.9 Test of significance Basic concepts null hypothesis alternative hypothesis level of significance Standard error and its importance steps in testing Test of Significance Objective To familiarize

### Statistical foundations of machine learning

Machine learning p. 1/45 Statistical foundations of machine learning INFO-F-422 Gianluca Bontempi Département d Informatique Boulevard de Triomphe - CP 212 http://www.ulb.ac.be/di Machine learning p. 2/45

### Fisher Information and Cramér-Rao Bound. 1 Fisher Information. Math 541: Statistical Theory II. Instructor: Songfeng Zheng

Math 54: Statistical Theory II Fisher Information Cramér-Rao Bound Instructor: Songfeng Zheng In the parameter estimation problems, we obtain information about the parameter from a sample of data coming

### 1 Confidence intervals

Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.

### 9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

### SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Measuring the Power of a Test

Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection

### Chapter 4: Statistical Hypothesis Testing

Chapter 4: Statistical Hypothesis Testing Christophe Hurlin November 20, 2015 Christophe Hurlin () Advanced Econometrics - Master ESA November 20, 2015 1 / 225 Section 1 Introduction Christophe Hurlin

### Robust nonparametric tests for the two-sample location problem SFB 823. Discussion Paper. Roland Fried, Herold Dehling

SFB 823 Robust nonparametric tests for the two-sample location problem Discussion Paper Roland Fried, Herold Dehling Nr. 19/2011 Robust nonparametric tests for the two-sample location problem Roland Fried

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### SAMPLE SIZE CALCULATION FOR COMPARING PROPORTIONS

SAMPLE SIZE CALCULATION FOR COMPARING PROPORTIONS INTRODUCTION HANSHENG WANG Guanghua School of Management, Peking University Department of Business Statistics &Econometrics Beijing, P. R. China SHEIN-CHUNG

### Chapter G08 Nonparametric Statistics

G08 Nonparametric Statistics Chapter G08 Nonparametric Statistics Contents 1 Scope of the Chapter 2 2 Background to the Problems 2 2.1 Parametric and Nonparametric Hypothesis Testing......................

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### Statistical Significance and Bivariate Tests

Statistical Significance and Bivariate Tests BUS 735: Business Decision Making and Research 1 1.1 Goals Goals Specific goals: Re-familiarize ourselves with basic statistics ideas: sampling distributions,

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

### OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance

Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance (if the Gauss-Markov

### AP Statistics: Syllabus 3

AP Statistics: Syllabus 3 Scoring Components SC1 The course provides instruction in exploring data. 4 SC2 The course provides instruction in sampling. 5 SC3 The course provides instruction in experimentation.

### Wording of Final Conclusion. Slide 1

Wording of Final Conclusion Slide 1 8.3: Assumptions for Testing Slide 2 Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation σ is known