Statistics interval estimation

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

The estimation of a parameter alone is not sufficient since a single estimate tells us nothing about how accurate the estimate is. The main purpose of confidence intervals is to indicate the precision, or imprecision, of the estimated statistic as representing the population values. The confidence interval will give us a range of values within which we can have a chosen confidence of it containing the population value. The degree of confidence usually presented is 95%. 

Obviously, we need tests and estimates on the variability of our experimental data. We can develop procedures that parallel the tests and estimates on the mean as presented in the previous section. We might test to determine whether the sample was drawn from a population of a given variance or we might establish point or interval estimates of the variance. We may wish to compare two variances to determine whether they are equal. Before we proceed with these tests and estimates, we must consider two new probability distributions. Statistical procedures for interval estimates of a variance are based on chi-square and F-distributions. To be more precise, the interval estimate of a a2 variance is based on x -distribution while the estimate and testing of two variances is part of a F-distribution. 

By comparing absolute values of regression coefficients with their interval estimates, with 95% confidence, these regression coefficients are statistically significant b0, b3, b44 and bss. Following this the regression model (2.97) becomes ... 

Veldhuis JD, Evans WS, Johnson ML. Complicating effects of highly correlated model variables on nonlinear least-squares estimates of unique parameter values and their statistical confidence intervals Estimating basal secretion and neurohormone half-life by deconvolution analysis. Methods Neurosci 1995 28 130-8. 

The statistical evaluation of bioequivalence studies should be based on confidence interval estimation rather than hypothesis testing (Metzler, 1988, 1989 Westlake, 1988). The 90% confidence interval approach, using 1 —2a (where a = 0.05), should be applied to the individual parameters of interest (i.e. the pharmacokinetic terms that estimate the rate and extent of drug absorption) (Martinez Berson, 1998). Graphical presentation of the plasma concentrationtime curves for averaged data (test vs. reference product) can be misleading, as the curves may appear to be similar even for drug products that are not bioequivalent. 

Ideally, the group of reference individuals should be a random sample of all the individuals fulfilling the defined inclusion criteria in the parent population. Statistical estimation of distribution parameters (and their confidence intervals) and statistical hypothesis testing require this assumption. 

With these numbers calculated, all that is left to compute the three confidence intervals are the reliability factors associated with each. For the 90% confidence interval, the value of the reliability factor will be the value of t that cuts off the upper 5% of the area (half the size of a) under the t distribution with 99 df. This value is 1.66 and can be verified from a table of values or from statistical software. Note that the t value of -1.66 is the value of t that cuts off the lower 5% of the area (half of the size of a) under the t distribution with 99 df. The reliability factors listed previously for the two-sided 95% and 99% confidence intervals can also be used to compute the following interval estimates ... 

To summarize, the computational aspects of confidence intervals involve a point estimate of the population parameter, some error attributed to sampling, and the amount of confidence (or reliability) required for interpretation. We have illustrated the general framework of the computation of confidence intervals using the case of the population mean. It is important to emphasize that interval estimates for other parameters of interest will require different reliability factors because these depend on the sampling distribution of the estimator itself and different calculations of standard errors. The calculated confidence interval has a statistical interpretation based on a probability statement. 

This chapter started with an introduction to the concepts of probability and random variable distributions. The role of probability is to assist in our ability to make statistical inferences. Test statistics are the numeric results of an experiment or study. The yardstick by which a test statistic is measured is how extreme it is. The term "extreme" in Statistics is used in relation to a value that would have been expected if there was no effect, that is, the value that would be expected by random chance alone. Confidence intervals provide an interval estimate for a population parameter of interest. Confidence intervals of (1 — a)% can also be used to test hypotheses, as seen in Chapter 8. 

In Chapter 6 we described the basic components of hypothesis testing and interval estimation (that is, confidence intervals). One of the basic components of interval estimation is the standard error of the estimator, which quantifies how much the sample estimate would vary from sample to sample if (totally implausibly) we were to conduct the same clinical study over and over again. The larger the sample size in the trial, the smaller the standard error. Another component of an interval estimate is the reliability factor, which acts as a multiplier for the standard error. The more confidence that we require, the larger the reliability factor (multiplier). The reliability factor is determined by the shape of the sampling distribution of the statistic of interest and is the value that defines an area under the curve of (1 - a). In the case of a two-sided interval the reliability factor defines lower and upper tail areas of size a/2. 

Statistics can effectively be used to provide a best estimate of the value of a repeatedly measured variable, establish the reliability of such an estimate (confidence interval), estimate parameter values of a model from experimental data, help to discriminate between rival models on the basis of goodness of fit, and guard against acceptance of a model whose superior fit may well be due to chance. It can also help to design experimental data gathering to be most efficient [48], On the other hand, statistics alone cannot be relied upon to identify or verify reaction pathways or mechanisms. 

Estimation of Cp and Cpk indices, confidence intervals, summary statistics... 

The first objective is dealt with using statistical hypothesis testing, while the second one gives rise to confidence interval estimation. 

We are aware that the examined control networks have different shapes, and also different number of redundant observations. The last parameter has essential influence not only on the results of accuracy analysis, but first of all on reliability of these results, as well as on reliability of computed coordinates. So in order to draw final conclusions from the analysis of control networks with widely diversified numbers of redundant observations, one has to perform statistical analysis of results, based, for example, on the interval estimation. Confidence intervals for true values of the vector of unknowns X can be determined with the known formula... 

The mean of the difference was calculated by using the statistical hypothesis test and confidence interval estimation on the 20 participants data sample. The normality of the distribution of the sample data was tested by the Ryan—Joiner test at 5% significant level. The significance of the results with over 80% confidence level is reported in the following section. 

When the difference data approximation obeys normal distribution, we can use Matlab statistics function parameters for the forecast error, such as average value, variance and significant level d of confidence interval estimation. Meanwhile, it is examined whether the difference between unknown average error parameter is equal to the estimation of mean value. 

Confidence interval A statistical interval estimate which is used to indicate the reliability of an estimate. 

The overall goal of Bayesian inference is knowing the posterior. The fundamental idea behind nearly all statistical methods is that as the sample size increases, the distribution of a random sample from a population approaches the distribution of the population. Thus, the distribution of the random sample from the posterior will approach the true posterior distribution. Other inferences such as point and interval estimates of the parameters can be constructed from the posterior sample. For example, if we had a random sample from the posterior, any parameter could be estimated by the corresponding statistic calculated from that random sample. We could achieve any required level of accuracy for our estimates by making sure our random sample from the posterior is large enough. Existing exploratory data analysis (EDA) techniques can be used on the sample from the posterior to explore the relationships between parameters in the posterior. 

Interval estimates and hypothesis tests for the parameters can be performed, in principle, by considering the as)miptotic normal distribution of the maximum likelihood estimates and the asymptotic chi-squared distribution of the likelihood ratio statistics, respectively (Lawless, 1982). 

Limits to the mean and standard deviation have been discussed in the previous sections based upon Student s t function and the Chi-squared function. While the theory of confidence intervals for these two quantities is well developed, such is not the case for the general nonlinear fitting of parameters to a data set. This will be discussed in the next chapter on general parameter estimation. For such cases, about the only approach to confidence intervals is through the Monte Carlo simulation of a number of test data sets. This approach is also applicable to limits on the mean and standard deviation and will be discussed here partly as background for the next chapter and as another approach to obtaining confidence intervals on statistical quantities. 

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