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Statistical Inference II
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
Whenever a decision is based on the result of a hypothesis test, there is a chance that it is incorrect. Consider Table 2.1. In this classical Neyman–Pearson methodology, the sample space is partitioned into two regions. If the observed data reflected through the test statistic falls into the rejection or critical region, the null hypothesis is rejected. If the test statistic falls into the acceptance region, the null hypothesis cannot be rejected. When the null hypothesis is true, there is α percent chance of rejecting it (Type I error). When the null hypothesis is false, there is still a β percent chance of accepting it (Type II error). The probability of Type I error is the size of the test. It is conventionally denoted by α and called the significance level. The power of a test is the probability that it will correctly lead to rejection of a false null hypothesis, and is given as 1 − β.
Detection of Changes
Published in Zbigniew W. Kundzewicz, Changes in Flood Risk in Europe, 2019
Sheng Yue, Zbigniew W. Kundzewicz, Linghui Wang
In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The amount of evidence required to accept that an event is unlikely to have arisen by chance is known as the significance level, i.e. the measure of the probability of rejecting H0 when it is true. The significance level can also be called the Type I error. In contrast, the Type II error is the probability of accepting the null hypothesis when it is false. The power of a test is the probability of correctly rejecting the null hypothesis when it is false. Table 2 indicates the relationship between significance level, Type I and Type II errors and the power of a test. Yue et al. (2002a) and Yue & Pilon (2004) indicated that the power of a test for detecting trend depends on the pre-assigned significance level, magnitude of trend, sample size, distribution type, variation and skewness of the tested time series.
Experiments
Published in Patrick F. Dunn, Fundamentals of Sensors for Engineering and Science, 2019
Hypothesis testing [9] incorporates the tools of statistics into a decision-making process. The subject of statistics is covered in Chapter 12. In the terminology of statistics, a null hypothesis is indicated by H0 and an alternative hypothesis by H1. The alternative hypothesis is considered to be the complement of the null hypothesis. There is the possibility that H0 could be rejected by considering it false when it is actually true. This is called a Type I error. Conversely, H0 could be accepted by considering it true when it is actually false. This is termed a Type II error. Type II errors are of particular concern in engineering. Sound engineering decisions should be based upon the assurance that Type II error is minimized. For example, if H0 states that a structure will not fail when its load is less than a particular safety-limit load, then it is important to assess the probability that the structure can fail below the safety-limit load. This can be quantified by the power of the test, where the power is defined as 1 – probability of Type II error. For a fixed level of significance (see Section 12.7), the power increases as the sample size increases. Large values of power signify better precision. Null hypothesis decisions are summarized in Table 2.1.
On scoping a test that addresses the wrong objective
Published in Quality Engineering, 2019
Thomas H. Johnson, Rebecca M. Medlin, Laura J. Freeman, James R. Simpson
When performing the hypothesis test on the coefficients, we are formally testing if the model coefficients are significantly different from some constant, t. The power of the test is the probability that the test correctly rejects the null hypothesis, when the alternative hypothesis is true. The test statistic for analysis of variance is an F-statistic which depends on the collected data y, the model matrix X, and the coefficients estimated from the collected data , and is calculated as
Countervailing Risk Management Through Knowledge Transfer
Published in Engineering Management Journal, 2020
Jeffery Temple, Rafael E. Landaeta
The power of a test is calculated through the use of beta and provides the probability of correctly rejecting the null hypothesis. Exhibit 6 provides results from the regression analysis and details the standardized coefficients of beta. Knowledge transfer has a strong positive impact on both the identification and handling of countervailing risk variables. The R Square values from the regression show data associated with both independent variables education and role have a low predictive capability on the dependent variables. Although they may not show as predictive, they may still have a marginal impact.
Mixture of Birnbaum-Saunders Distributions: Identifiability, Estimation and Testing Homogeneity with Randomly Censored Data
Published in American Journal of Mathematical and Management Sciences, 2021
Walaa A. El-Sharkawy, Moshira A. Ismail
The size of the test or significance level is the rejection rate of a true null hypothesis and is also known as type I error rate. It is an important term in hypothesis testing where the critical value and the power are affected by its choice and therefore it influences the inferential result. The power of the test is the probability of rejecting the null hypothesis when it is false. An important part of the research study is to make conclusions about the factors that affect achieving the desired power level for the test. Since the EM test has a nonstandard limiting distribution, the size and the power of the test can computed by applying Monte Carlo simulations. First, the empirical distribution of the EM test statistic under the null hypothesis is constructed to obtain estimates of the critical values. This is obtained by generating M random samples of size n according to the model under H0 and computing the EM test statistic for each of the samples. For a given nominal significance level α, the critical value is obtained as the quantile at th position in ordered EM test statistic values. Then, to compute the size and power of the test, the empirical distribution of EM test statistic under the null and alternative hypotheses are also required. To achieve this, two random samples of size n are generated, M times according to the model under H0 and Ha, respectively and the EM test statistic is computed for each of the samples. For a given the size and power are computed as follows and where is an indicator function.