Confidence limits for the mean

An approach for analyzing data of a quantitative attribute that is expected to change with time is to determine the time at which the 95% one-sided confidence limit for the mean curve intersects the acceptance criterion. If analysis shows that the batch-to-batch variability is small, it is advantageous to combine the data into one overall estimate by applying appropriate statistical tests (e.g., p-values for level of significance of rejection of more than 0.25) to the slopes of the regression lines and zero-time intercepts for individual batches. If it is inappropriate to combine data from several batches, the overall shelf life should be based on the minimum time a batch can be expected to remain within the acceptance criteria. 

The shelf life for a single batch is usually computed based on regression techniques. An appropriate approach to shelf life estimation when using regression analysis is by calculating the earliest time at which the 95% confidence limit for the mean intersects the proposed acceptance criterion [8]. A detailed description of shelf life calculations is provided in Sections 7.2.3 and 7.2.4. 

The confidence limits for the means corresponding to the four colorimetric methods are ... 

An empirical, but often practical way of sampling has been presented by EXNER et al. [1985]. They divided an area into suitable clean-up units of practical size to solve a problem of hazardous waste. Inside these strips, samples were taken along transects and some composite samples were formed from different combinations of the individual samples. After homogenization of the composite samples randomly selected aliquots of soil were taken and analyzed. The estimated arithmetic mean and its standard deviation enabled computation of an upper confidence limit for the mean concentration of the clean-up units. 

Which is more important confidence limits for the mean or confidence limits for the distribution It depends on the application. The vendor of the standardized solutions above should report confidence limits for the distribution to its customers, who will use one bottle at a time (for example, in preparing a saline solution to be injected into a patient). In other cases, however, the error is in the measurement process itself. We believe that all electrons have the same mass, but 1000 measurements of electron mass will likely all give slightly different answers. Then we want to know confidence limits for the mean. In addition, 95% confidence limits for the mean are used by pollsters to predict the results of an election. The fact that individual preferences vary is not interesting what is interesting is whether, on average, more than 50% of the voters prefer one specific candidate. 

One-tailed or two-tailed confidence limits for the mean content, activity, purity, or residual moisture may also be calculated. For each batch produced, the 99.9999% confidence limits for the mean should be found to be within specifications. For a well-controlled process, 99.9999% one-tailed limits may typically lie in the range of 50-75% of the upper specification limit. 

A volumetric calcium analysis on triplicate samples of the blood serum of a patient believed to be suffering from a hyperparathyroid condition produced the following data meq Ca/L = 3.15, 3.25, 3.26. What is the 95% confidence limit for the mean of the data, assuming... 

An acceptable approach for analyzing an attribute that is expected to decrease with time is to determine the time at which the 95% one-sided lower confidence limit, also known as the 95% lower confidence bound, for the estimated curve intersects the acceptable lower specification limit. Where the estimated curve is assumed to be linear based on 24 months of real-time data and the lower specification limit is assumed to be 90% of label claim, an expiration dating period of 24 months could be granted. When analyzing an attribute that is expected to increase with time, the 95% one-sided upper confidence limit for the mean is recommended. 

We may likewise calculate the confidence interval for predicted value for solubility at the centre point being the 95% confidence limits for the mean of two... 

Figure 1. Calculated mean pH values for California. For each site the top value is calculated by the charge balance. The middle row is the 68.3% confidence limits for the mean pH assuming a 10% coefficient of variation in the known mean major ion concentrations the bottom row, the 95% confidence limits assuming a 30% coefficient of variation(5).

The confidence limits for the mean of a small sample where x is normally distributed with mean and variance (with x and s as the sample estimates based on a r.s.s. n) can be determined using the Student s distribution. Let... 

When a is not known from previous experience, it becomes necessary to calculate the standard deviation, S, for a small set of measurements. Thus, S is a less precise value than a, and the confidence limit for the mean will be larger when S is used. W. S. Cosset, writing under the pseudonym of Student , worked out a t distribution that is used for calculating confidence Hmits of the mean when S, instead of a, is known. 

The essential complication in trying to estimate confidence limits for the mean is the presence of strong correlation in the raw data. An expression for the variance of means for a stationary time series, which includes the effects of correlation, has been given by Davenport. However, to evaluate this expression requires knowledge of the autocorrelation function of the data. This function is not known a priori, but has to be computed from the data that we want to analyze. From an examination of a large amount of MD data on metallic sodium in solid and liquid phases, we find that calculation of the autocorrelation function requires an enormous amount of data, and hence we reach the following conclusion. 

Observation. For a given amount of data X(t), from a particular MD calculation, trying to calculate the autocorrelation function from this data, and then calculating the variance of means from the autocorrelation function, is an inefficient method for estimating confidence limits for the mean. 

Statistical analysis of the data demonstrated that the three generic lots were not only non-bioequivalent to the innovator, they were also significantly different from each other P <0.01). They failed to meet the 80-120% requirement for the 90% confidence limits for the mean AUC and Cmax with respect to the originally launched drug form. 

According to ICH QIE, an appropriate approach to retest period estimation is to analyze a quantitative attribute (e.g., assay, degradation products) by determining the earliest time at which the 95% confidence limit for the mean intersects the proposed acceptance criterion. For an attribute known to decrease with time, the lower one-sided 95% confidence limit should be compared to the acceptance criterion. For an attribute known to increase with time, the upper one-sided 95% confidence limit should be compared to the acceptance criterion. For an attribute that can either increase or decrease, or whose direction of change is not known, two-sided 95% confidence limits should be calculated and compared to the upper and lower acceptance criteria. If the data show that the batch-to-batch variability is small, it may be worthwhile to combine the data into the overall estimate. The appropriate statistical modeling is used to analyze the data. ... 

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