Modal value

The index of refraction of allophane ranges from below 1.470 to over 1.510, with a modal value about 1.485. The lack of characteristic lines given by crystals in x-ray diffraction patterns and the gradual loss of water during heating confirm the amorphous character of allophane. Allophane has been found most abundantly in soils and altered volcanic ash (101,164,165). It usually occurs in spherical form but has also been observed in fibers. 

Estimates of sales income and other types of forecasts are usually based on the opinions of experts. Experts should be able to estimate maximum, minimum, and most hkely, or modal, values for a quantity. The modal value is not necessarily midway between the minimum and maximum values, since many distributions are skewed. An expert may be asked to estimate the probabilitv of the occurrence of certain values on each side of the mode. Wken experts are questioned separately, the procedure is known as the Delphic method. Strictly speaking, this method requires that the opinion of each expert be assessed by a coordinator, who then feeds the resiilts back to see if the opinions of one expert are modified by those of others. The process is repeated until agreement is reached. In practice, the procedure is too tedious to be repeated more than once. 

With a cost of capital of 10 percent the various cash flows can be discounted and summed. Thus for the base cases Z Af = 2,815,600, Z Ajp/d = 754,716, Z Aofd = 614,457, and Z C c/d = 61,446. With corporate taxes payable at 50 percent the aftertax cash flows of the first three items are (1 — 0.50) of the sums calculated above. The discounted working capital and the fixed-capital outlay are not subject to tax. These most probable values are listed and summed in Table 9-11 and, after adjustment for tax, give the modal value of the (NPV) as 276,224. 

The mean value of each of the distributions is obtained from these high, modal, and low values by the use of Eq. (9-101). If the distribution is skewed, the mean and the mode will not coincide. However, the mean values may be summed to give the mean value of the (NPV) as 161,266. The standard deviation of each of the distributions is calculated by the use of Eq. (9-75). The fact that the (NPV) of the mean or the mode is the sum of the individual mean or modal values implies that Eq. (9-81) is appropriate with all the A s equal to unity. Hence, by Eq. (9-81) the standard deviation of the (NPV) is the root mean square of the individual standard deviations. In the present case s° = 166,840 for the (NPV). 

One further point might be made here. Although the example illustrates the difference between the two types of molecular weight average, the weight average molecular weight in this example cannot be said to be truly representative, an essential requirement of any measure of central tendency. In such circumstances where there is a bimodal, i.e. two-peaked, distribution additional data should be provided such as the modal values (100 and 100000 in this case) of the two peaks. 

The three summary statistics are displayed in Figure 8.3. Clearly, for this data the mean and median are similar, and this is true for any distribution of values that is symmetric, which is the case here. The mode is somewhat removed from both the mean and median. In fact, the mode is not often used as a summary of data because it records only the most frequent value, and this may be far from the centre of the distribution. A second difficulty with the mode is that there can be more than one mode in a sample. For example, had one of the values 3.6 been instead 3.5, there would have been eight distinct modal values 3.3, 3.4, 3.6, 3.8, 4.0,4.1,4.4 and 4.7 mmol/L. 

Moreover, for monoenergetic heavy ions of the same energy per nucleon, the average number of ion pairs increases with a corresponding decrease in the deviation about the mean, so we have a higher modal value with a more sharply defined Bragg peak. 

Within Eq. (7), the selection function S(y) was approximated by Sty) = Kyc with adjustable parameters K and c and the breakage function b(x, y) was simply modeled by a triangular shaped function between a minimal breaking particle size of xq = 25 nm and x with a modal value at... 

Their data indicate that the moles of MgO per 100 g range from approximately 0.4 to 0.7 with a modal value near 0.6. The moles of (Al2 03 + Fe2 O + FeO + Mn) range from approximately 0.0 to 0.15. These data also indicate that eight octahedral sites are filled and in only a few samples is substitution of Mg by trivalent cations enough to account for the filling of only seven octahedral sites. 

Most kaolinites contain appreciable amounts of MgO (range 0.01—1.0% modal value between 0.2—0.3%). Bundy et al. (1965) found that MgO as well as total iron and soluble iron were directly related to the C.E.C. and suggested that the Mg and Fe were present in montmorillonite which they believe is commonly present, in amounts less than 5%, in kaolinites. This may be true in part, but electron probe studies (Weaver,1968) indicate that some of the MgO is related to the Ti02-Fe203 material and some is present in biotite. 

When the modal values are considered, there is relatively little overlap in the composition of the octahedral sheets of the three Fe3+-rich clays whereas, for the Al-rich clays, the illite and mixed-layer illite-montmorillonite fields fall within the montmorillonite-beidellite field. 

The central tendency of a probability distribution typically refers to the mean (arithmetic average) or median (50th percentile) value estimated from the distribution. For some very highly skewed distributions, the mean might not represent central tendency. Some analysts prefer to use the median as a central tendency estimate. For distributions that have only one mode, the modal value is sometimes considered to be a central tendency estimate. 

Consideration (figure 2) of both a Standard Drug File (24082 entries) and a Pharmaprojects dataset (5279 entries) shows that drug-like molecules have a normal distribution, with a modal value of around 2.5, and a LogP of 5.0 is indeed a reasonable upper bound for a candidate drug molecule. 

IV = Modal values (except means average) for the Phosphoria Formation, according to Gulbrandsen (1966). 

FIGURE 28.1 A density plot superimposed on a histogram of modal values of random effect on elimination rate for one subpopulation conditional estimation with e-rj interaction. 

FIGURE 28.4 A density plot of modal values of random effect on weight change asymptote for one subpopulation. The first-order conditional estimation (FOCE) method was used. 

Here, n is the number of subpopulations, p 6)i is the estimated probability that the patient belongs to the fth subpopulation, which may depend on hxed effects parameters, and F(6, rf)i is the prediction for the ith subpopulation. For this computation one may evaluate F(6, if)i at f] = 0 to get the expected prediction or at 7] equal to a post hoc or modal value to get the expected individualized prediction. NONMEM supports the calculation of these expectations and to communicate the need for these expectations requires one to modify the original control stream to include the ABBREViATED record ( abb) and a block of code to compute the expectations. Modify C4. txt to make ci5. txt, and the abb record looks like... 

The scatterplot produced by NONMEM will display 24 points. Each point will correspond to one time point and have as its coordinates the average of the observations and the average of the predictions for that time point. To create a separate scatterplot, one would just plot dv versus com (i) for the first (dummy) patient. In this example we have chosen to set com(1) =f. We also might have set com(1) =y. As it turns out this latter approach produces incorrect output. Here, because we are using method=o, f is equal to pred. If one were using some type of conditional estimation, f would be evaluated at modal values of rj and would not equal pred. One way to trick NONMEM into producing the desired output would be to use the final model estimates from conditional estimation as initial values for a method=o run with maxevals=o. 

FIGURE 28.14 Density of modal values for random effect on Ka (for subpopulation with high Ka) when simulation and estimation models are identical. 

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