Difference between two means

Tsong Y, Hammerstrom T, Sathe P, Shah VP. Statistical assessment of mean differences between two dissolution data sets. Drug Inf J 1996 30 1105-1112. 

In general, bioequivalence is demonstrated if the mean difference between two products is within 20% at the 95% confidence level. This is a statistical requirement, which may require a large number of samples (e.g. volunteers), if the drug exhibits variable absorption and disposition pharmacokinetics. For drugs for which there is a small therapeutic window or low therapeutic index, the 20% limit may be reduced. The preferred test method is an in vivo crossover study and, since this occurs in the development phase, necessitates the emplo)unent of volim-teers. These studies are, therefore, expensive and animal experiments may be substituted, or in vitro experiments if they have been correlated with in vivo studies. 

Comments Bias-corrected standard error measurements allow the characterization of the variance attributable to random unexplained error. The bias value is calculated as the mean difference between two columns of data, most commonly actual minus NIR predicted values. 

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

Confidence Interval for the Difference in Two Population Means The confidence intei val for a mean can be extended to include the difference between two population means. This intei val is based on the assumption that the respective populations have the same variance <7 ... 

It is easy to prove that there are differences between two given means using the t-test (Case bi or c). 

Welch BL (1937) The significance of the difference between two means when the population variances are unequal. Biometrika 29 350... 

Student f-test A statistical test to establish if there is a significant difference between two mean values, taking account of the uncertainties associated with both values. 

The process described above is irreversible. Irreversibility means that, given two states A and B of an adiabatically enclosed system of constant composition, either of the processes A - B or B —> A may be driven mechanically or electromagnetically, but not both. By such a procedure the energy difference between two states AU — Ub - Da can always be measured. 

The same experiments were carried out with and without an oscillation of the QCM during the transfer process. The difference between two cases was within experimental error, which means an oscillation of a QCM has no effect on the detachment of LB films. 

As discussed in section 2.4.4 the coordinating ability of a solvent will often affect the rate of nucleation and crystal growth differently between two polymorphs. This can be used as an effective means of process control and information on solvent effects can often be obtained from polymorph screening experiments. There are no theoretical methods available at the present time which accurately predict the effect of solvents on nucleation rates in the industrial environment. 

Blanco ° proposed the use of the mean square difference between two consecutive spectra plotted against the blending time in order to identify the time that mixture homogeneity was reached. 

Another issue is how to interpret a clinical trial with equivocal results. While Schor and Karten established the probability of less than 1 in 20 P < 0.05) that a difference between two groups was due to chance as meaning that it was due to the drug, they did not establish criteria for how to properly interpret studies that failed to find this big a difference. Can this lack of evidence of effect be considered as evidence of lack of effect People have settled on the convention that a clinical trial must include enough patients to have at least an 80% chance of finding an effect if an effect really exists. Failure to find an effect in this large a trial is considered evidence of true lack of effect. This has been named the power of the study. How can we handle studies that do not have this power ... 

Example 2.3 Standard error for the difference between two means... 

We have seen in the previous chapter that it is not possible to make a precise statement about the exact value of a population parameter, based on sample data, and that this is a consequence of the inherent sampling variation in the sampling process. The confidence interval provides us with a compromise rather than trying to pin down precisely the value of the mean p or the difference between two means — p2> for example, we give a range of values, within which we are fairly certain that the true value lies. 

At the end of the previous chapter we saw how to extend the idea of a standard error for a single mean to a standard error for the difference between two means. The extension of the confidence interval is similarly straightforward. Consider the placebo controlled trial in cholesterol lowering described in Example 2.3 in Chapter 2. We had an observed difference in the sample means 3cj — 3c2 of 1.4 mmol/1 and a standard error of 0.29. The formula for the 95 per cent confidence interval for the difference between two means — p.2) i -... 

This expression is essentially the same as that for a single mean statistic (constant x se). The rules for obtaining the multiplying constant however are slightly different. For the difference between two means we use Table 3.1 as before, but now we go into that table at the row + U2 — 2, where rii and 2 e the sample sizes for treatment groups 1 and 2 respectively. 

In general, the calculation of the confidence interval for any statistic, be it a single mean, the difference between two means, a median, a proportion, the difference between two proportions and so on, always has the same structure ... 

In the case of an electrode-solution interface, the thermodynamic equilibrium between the media in contact is attained by means of electron-ion exchange processes. Since the process of attaining the equilibrium is governed by charged particles, the equilibrium state is characterized by a certain potential difference between the phases. If they are in direct contact, we deal with the potential difference between two points in different media, for example, inside a semiconductor electrode and inside a solution. This potential difference is called the Galvani potential. 

Equation (9) is an important result since it describes the relationship among Rs, v, r/, and Ap, the density difference. Any one of these quantities may be evaluated by Equation (9) when the other three are known. Thus, Equation (9) can be used to determine the density difference between two phases or to determine the viscosity of a liquid. In this chapter, however, our interest is in the characterization of colloidal particles by means of observations of their sedimentation behavior. Therefore, we are primarily concerned with Equations (11) and (12), which are specifically directed toward this objective. 

Once equilibrium has been reached, the height difference between the two liquid surfaces is all that remains to be measured. The primary factor to note here is that capillaries are used to minimize the dilution effects. This means that corrections for capillary rise must be taken into account unless the apparatus allows the difference between two carefully matched capillaries to be measured. We discuss capillary rise in Chapter 6, Sections 6.2 and 6.4. Finally, there is an extremely important practical reason, in addition to the theoretical requirement of isothermal conditions, for good thermostating in osmometry experiments. The apparatus consists of a large liquid volume attached to a capillary and therefore has the characteristics of a liquid thermometer The location of the meniscus is quite sensitive to temperature fluctuations. 

We use a t test to compare one set of measurements with another to decide whether or not they are the same. Statisticians say we are testing the null hypothesis, which states that the mean values from two sets of measurements are not different. Because of inevitable random errors, we do not expect the mean values to be exactly the same, even if we are measuring the same physical quantity. Statistics gives us a probability that the observed difference between two means can arise from purely random measurement error. We customarily reject the null hypothesis if there is less than a 5% chance that the observed difference arises from random... 

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