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Theoretical Probability Distributions
Published in Marcello Pagano, Kimberlee Gauvreau, Heather Mattie, Principles of Biostatistics, 2022
Marcello Pagano, Kimberlee Gauvreau, Heather Mattie
The value of the normal distribution will become more apparent when we begin to work with the sampling distribution of the mean. For now, however, it is important to note that many random variables of interest – including blood pressure, serum cholesterol level, height, weight, and body temperature – are approximately normally distributed. The normal curve may therefore be used to estimate probabilities associated with these variables. For example, in a population in which serum cholesterol level is normally distributed with mean μ and standard deviation σ, we might wish to find the probability that a randomly chosen individual has a serum cholesterol level greater than 250 mg/100 ml to help us to plan for future cardiac services. Since the total area beneath the normal curve is equal to 1, we can estimate this probability by determining the proportion of the area under the curve that lies to the right of the point , or P. This may be done using a computer program, or a table of areas calculated for the normal curve.
Clinical Trial Designs
Published in Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson, Bayesian Thinking in Biostatistics, 2021
Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson
The previous example assumed a discrete set of outcomes at each of a finite set of decision times, with a discrete set of actions from which to choose at each time. As the set of possible outcomes or actions increases, the calculations become more difficult, growing exponentially. The problem becomes even more complex if one is dealing with a continuous outcome space, as when one is designing a clinical trial with a continuous primary endpoint, such as change in blood pressure. One can apply numerical methods to find the optimal solution or, at least, a close approximation to the set of optimal decisions. A particularly useful approach is based on generating random trajectories of the clinical study based on the assumed sampling distribution and some prior probability distribution for the parameters in the sampling model. If the outcome space is continuous, the computational burden is still enormous. When there is a sufficient or nearly sufficient summary statistic that one can use, over which one forms a grid, then one can get a reasonably good finite and discrete approximation to the full outcome space [53, 245]. One can then apply backward induction to the finite grid to find an approximate set of optimal decisions.
Measurement Uncertainty in Ultrasonic Exposimetry
Published in Marvin C. Ziskin, Peter A. Lewin, Ultrasonic Exposimetry, 2020
Suppose we consider taking samples of size 10 from a parent population of 1000 individuals. There are 1000!/(10!·990!) = 2.63 × 1023 possible samples, each with its own sample mean, . The population of sample means has a frequency distribution, called the sampling distribution, which possesses a mean, , and a standard deviation, . is usually called the standard error of the mean, or sometimes simply the standard error. Sampling theory shows that the sample mean, , is equal to the mean of the parent population, μ, and that
The Effect of Self-Stigma on the Hope of Chinese with Mental Illness: The Mediating Role of Family Function
Published in Psychiatry, 2023
Collected data were organized, coded, and entered into SPSS version 22.0 (IBM Corp., Armonk, NY, USA). Responses with missing data were excluded. The demographic variables, including age, sex, diagnostic category, MI duration, education level, and household income, were analyzed using descriptive statistics and controlled in the analyses to test their possible effects on the outcomes. Pearson’s correlation analysis was performed to examine the relationships between self-stigma, family function, and hope. The indirect effect of family function in the relationship between self-stigma and hope was further estimated using the serial mediation Model 4 test (designed for testing a single or parallel mediation model) based on Andrew Hayes’ PROCESS macro for SPSS, an observed variable ordinary least squares regression analysis and logistic regression path analysis modeling tool. To form the confidence interval, bias-corrected accelerated confidence intervals (CI) were estimated using 5,000 re-samples. This is the most commonly used default method in the SPSS PROCESS by Hayes (2017) that corrects for bias and skew in the distribution of bootstrap estimates (Banjanovic & Osborne, 2016). A 95% CI was calculated by finding the exact values at the 2.5th and 97.5th percentiles on the bootstrap sampling distribution. The indirect effect of family function was considered significant when the 95% CI did not contain zero.
Inferences for multiple interval type-I censoring scheme
Published in Journal of Applied Statistics, 2023
Shubham Agnihotri, Sanjay Kumar Singh, Umesh Singh
Next, the ML estimators γ and λ, respectively, can be obtained from the simultaneous solution of the normal Equations (4) and (5) given as: 4) and (5). The contour plot technique can be effectively used to set the initial guess of the parameters γ and λ. Since no explicit form of ML estimators is obtained, one may not be able to find the estimators' sampling distribution; for this reason, exact confidence intervals for the parameters are hard to construct. However, one can use the asymptotic distribution of ML estimators and develop the 6), (7) and (8) have the following meaning; γ and λ are
Fidelity of a Traffic Safety Education Intervention for Combat Veterans
Published in Occupational Therapy In Health Care, 2021
Sandra M. Winter, Katelyn R. Caldwell, Babette A. Brumback, Mary E. Jeghers, Sherrilene Classen
Next, bootstrap analysis was used to indicate the sampling distribution of the resulting measure. The bootstrap estimates the sampling distribution of the statistic, i.e., the proportion of concordant ranks, commonly represented as p̂, using a resampling approach (Efron & Tibshirani, 1993). The six raters of the research team belong to a population of all hypothetical raters. The sampling distribution is the distribution of the statistic had sampling from the population of raters many different times been possible, each time computing a value of the statistic. Therefore, bootstrapping allows the research team to make assumptions about the data with greater confidence. Because the proportion of ranked pairs is assessed, the measure of concordance provides a value between 0.00 and 1.00, with 0.00 indicating no agreement and 1.00 indicating total agreement (Healey, 2012).