Sampling Distribution of the Mean
Marcello Pagano, Kimberlee Gauvreau, Heather Mattie in Principles of Biostatistics, 2022
In the previous chapter we examine theoretical probability distributions, such as the binomial distribution and the normal distribution. In all cases the relevant population parameters are assumed to be known; this allows us to describe the distributions completely and calculate the probabilities associated with various outcomes. In most practical applications, however, we are not given the values of these parameters. Instead, we must attempt to describe or estimate a population parameter – such as the mean of a normally distributed random variable – using the information contained in a sample of observations selected from the population. The process of drawing conclusions about an entire population based on the information contained in a sample is known as statistical inference.
An introduction to statistical tests in SPSS
Perry R. Hinton, Isabella McMurray in Presenting Your Data with SPSS Explained, 2017
Statistical inference tests are used to make the choice between these two possibilities. From the result of the test the researcher makes a judgement of whether the effect (found in the research) is significant – indicating it is (probably) generalisable to the population – or ‘not significant’ – indicating that they should (probably) not rule out chance factors producing the effect in the study. These statistical inference tests go under the name of null hypothesis significance testing (NHST). There are a number of different tests used for different types of data, but they follow the same logic. They are constructed on the assumption that there is no underlying difference between the populations on the outcome measure, so any difference found between the samples is due to chance. This assumption is called the null hypothesis. A statistical test makes further assumptions about the underlying pattern or distribution of the population data, which allows it to calculate a value of an inferential statistic based on the research data. Finally, it then works out how probable such a calculated value of the statistic is, given the assumption of the null hypothesis. Probability ranges from 1 (certainly) down to 0 (certainly not), so a probability of 0.5 is equivalent to a 50% or a 50:50 chance. If the probability is high (when the null hypothesis is assumed) then this indicates that even though a difference was found between the samples, it is not big enough to reject the null hypothesis and we conclude that the difference in samples has probably arisen by chance.
Statistics in medical research
Douglas G. Altman in Practical Statistics for Medical Research, 1990
I have also introduced the two main approaches to statistical inference - estimation and hypothesis testing. The general principles outlined are fundamental to an appreciation of the remaining chapters of this book, and to understanding what statistical analysis and interpretation is all about. Published papers tend to present results in a shorthand way that can be opaque - for example as means and standard errors. It is important to know what can and cannot be inferred from these quantities, especially by constructing confidence intervals. Likewise, most published papers contain P values but the interpretation of them is often faulty. It is important to understand the true meaning of the P value, and to realize that statistical significance and clinical importance are not the same thing.
A bivariate discrete inverse resilience family of distributions with resilience marginals
Published in Journal of Applied Statistics, 2021
Vahid Nekoukhou, Ashkan Khalifeh, Hamid Bidram
Estimation of the unknown parameters is an important problem in any statistical inference. We will see that the maximum likelihood estimations of the proposed model are difficult to obtain. That is, one needs to solve four non-linear equations to compute the maximum likelihood estimates. Hence, a numerical method is used to compute the maximum likelihood estimates of the parameters. A simulation experiment is performed to consider the effectiveness of the proposed numerical method. In addition, two real bivariate data sets are analyzed to illustrate the capacity of the NBDGE distribution in practice. It is observed that the performances of the models work quite satisfactory. The NBDGE distribution has four parameters and due to the presence of its parameters, can be a very flexible bivariate discrete distribution.
Burnout among high school students is linked to their telomere length and relatedness with peers
Published in Stress, 2023
Frances Hoferichter, Armin Jentsch, Lou Maas, Geja Hageman
In this paper, statistical inference relies on Bayesian estimation techniques (Gelman et al., 2013) in the sense that unknown parameters are considered randomly distributed (i.e., in contrast to the fixed value of a point estimate). This means that we must explicitly state our prior knowledge by formulating the corresponding distributions for the unknown parameters. In light of the data, this prior knowledge is updated and yields a posterior distribution which contains all of the necessary information on the parameters. Because there is still limited evidence on the association between TL and school burnout in adolescents, we apply vague priors that are implemented in Mplus as a default (e.g. broad normal priors centered at zero for regression coefficients and inverse Gamma or Wishart distributions for variance and covariance parameters, respectively). However, recent critiques have stated that default priors for variance parameters should be used with care, particularly in a multilevel context (e.g. Zitzmann et al., 2020). Therefore, we conducted a small sensitivity analysis in which we also applied more informative priors (for details, see the electronic supplemental material, Appendix A). Because the results across different distributions were similar and, in our opinion, vague priors better reflect the state of research, we report the latter.
Significance test for linear regression: how to test without P-values?
Published in Journal of Applied Statistics, 2021
Paravee Maneejuk, Woraphon Yamaka
To further illustrate, we consider an experiment to make comparisons directly among p-value, Bayes factor and plausibility approaches under the linear regression context. We start with the following data generating process, p-value, we use the conventional statistical inference, in which the p-value is equal to or lower than thresholds namely 0.10, 0.05, and 0.01, to make a decision about the null hypothesis. Likewise, the plausibility-based belief function is interpreted in the same way as the p-value. On the other hand, in the case of the minimum Bayes factor approach, the interpretation is different from the first two methods as we make the decision upon the MBF following the Held and Ott [9] labeled intervals as presented in Table 2. Our interest is to see whether these methods will reveal any non-significant outcome when the null is false and reveal the significant outcome when the null is true. The results of the method comparison are provided in the following Figures and Tables.
Related Knowledge Centers
- Bayesian Inference
- Biostatistics
- Cluster Analysis
- Proportional Hazards Model
- Statistical Hypothesis Testing
- Sampling
- Hodges–Lehmann Estimator
- Empirical Distribution Function
- Sample Mean & Covariance
- Hellinger Distance