Comparison of two means

In Section 3.3.3 we introduced the general structure for a significance test with the comparison of two means in a parallel group trial. This resulted in a procedure which goes under the general heading of the two-sample (or unpaired) t-test. This test was developed for continuous data, although it is applicable more widely and, in particular, is frequently used for score and count data. 

Calculate t for comparison of two means, where S is estimated for each from the data and 1 does not differ significantly from S2. 

The standard /-value for the comparison of two means may be calculated as follows ... 

Depending on the circumstances at hand, several different types of mean comparisons can be made. In this section we review the method for comparison of two means with independent samples. Other applications, such as a comparison of means with matched samples, can be found in statistical texts. Suppose, for example, we have two methods for the determination of lead (Pb) in orchard leaves. The first method is based on the electrochemical method of potentiometric stripping analysis [1], and the second is based on the method of atomic absorption spectroscopy [2], We perform replicate analyses of homogeneous aliquots prepared by dissolving the orchard leaves into one homogeneous solution and obtain the data listed in Table 3.1. 

Comparison of two means yx and y2 of different measurement series with ni resp. n2 measurements having the standard deviations Si resp. s2 is done by the t-test ... 

Figure 2.1 la illustrates the relationship between the error of the first kind, also called a error, and the error of the second kind P error) for the comparison of two means. An error of the first kind is that the means are taken to be different, although they deviate from each other randomly. The error of the second kind is that it is wrongly stated that the two means are comparable. 

Table 1.7. Comparison of two mean value notation systems...

A systematic comparison of two sets of data requires a numerical evaluation of their likeliness. TOF-SARS and SARIS produce one- and two-dhnensional data plots, respectively. Comparison of sunulated and experimental data is accomplished by calculating a one- or two-dimensional reliability (R) factor [33], respectively, based on the R-factors developed for FEED [34]. The R-factor between tire experimental and simulated data is minimized by means of a multiparameter simplex method [33]. 

The analysis of cigarette smoke for 16 different polyaromatic hydrocarbons is described in this experiment. Separations are carried out using a polymeric bonded silica column with a mobile phase of 50% v/v water, 40% v/v acetonitrile, and 10% v/v tetrahydrofuran. A notable feature of this experiment is the evaluation of two means of detection. The ability to improve sensitivity by selecting the optimum excitation and emission wavelengths when using a fluorescence detector is demonstrated. A comparison of fluorescence detection with absorbance detection shows that better detection limits are obtained when using fluorescence. 

A comparison of two or more means can be made with a one-way analysis of variance. This tool compares... 

Another frequent question asked considers whether the mean of a value measured by two separate methods differs significantly. For example, does the mean pKB value of an antagonist measured in a binding experiment differ significantly from its affinity as an antagonist of agonist function The value of t for the comparison of the mean values xml and xm2 can be calculated with the following equation ... 

The most widely used test is that for detecting a deviation of a test object from a standard by comparison of the means, the so-called t-test. Note that before a f-test is decided upon, the confidence level must be declared and a decision made about whether a one- or a two-sided test is to be performed. For details, see shortly. Three levels of complexity, a, b, and c, and subcases are distinguishable. (The necessary equations are assembled in Table 1.10 and are all included in program TTEST.)... 

Because no direct way exists to test simultaneously more than two means for significant differences, the comparison of several means, J1,J2, >JV traced back to the comparison of variances. 

When a comparison of two separate replicate sets of data is required, the first stage is normally to compare their respective precisions by means of the F-test. This test uses the ratio of the variances of the two sets to establish any statistically significant difference in precision. F is calculated from... 

Having established that the standard deviations of two sets of data agree at a reasonable confidence level it is possible to proceed to a comparison of the mean results derived from the two sets, using the t-test in one of its forms. As in the previous case, the factor is calculated from the experimental set of results and compared with the table of critical values (Table 2.3). If /jX ) exceeds the critical value for the appropriate number of degrees of freedom, the difference between the means is said to be significant. When there is an accepted value for the result based on extensive previous analysis t is computed from equation (2.9)... 

Various model-dependent methods for the comparison of two cumulative dissolution data sets have been proposed (21). Usually, these methods involve prior characterization of both profiles by one to three parameters per profile. In some models, these parameters can be interpreted in terms of the kinetics, the shape, and/or the plateau, but in other instances, they have no physical meaning. One issue that requires some attention is that, in cases where more than one parameter is estimated, a multi-variate procedure for the comparison of the parameters must be applied (9,21). 

While the value for coefficient of variation is a general statement about the imprecision of a method, only the value for standard deviation can be used in any statistical comparison of two methods. The use of coefficient of variation assumes a constant relationship between standard deviation and the mean value and this is not always true (Table 1.5). 

Table 1.1. Comparison of the mean sizes and distribution widths at the two intersecting points A and B (Fig. 1.22) of shear thinning and Newtonian rheogramms...

More generally the test statistic is constructed as the signal/noise (signal-to-noise) ratio or something akin to this. We will develop this methodology in relation the comparison of two independent means for a between-patient design. The resulting test is known as the unpaired t-test or the two-sample t-test. 

Explosive Performance, Comparison of Two Methods for Its Evaluation. Until, the middle of 1950 s, the choice of expls for blasting a particular rock was made on the basis of the following methods a) Trial and error gained from previous experience in similar rocks b) Calculation of meaningful performance parameters of expls from their chemical compns by means of laws of thermochemistry or thermodynamics or c) Laboratory determination of expl characteristics... 

Figure 4. Comparisons of two IMPROVEd DRUM samplers (12 LI min, three size cuts below 2.5 xm, and slit orifices) running side by side at Sacramento, California. The size range shown is for a particle diameter of 0.34 to 2.5 xm. Iron is used as a tracer of soil particles. The mean r2 for all stages less than 2.5 xm was 0.87 with a slope of 1.03.

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