Marginal sum

Summation row-wise (horizontally) of the elements of a contingency table produces the vector of row-sums with elements Summation column-wise (vertically) yields the vector of column-sums with elements x j. The global sum is denoted by. These marginal sums are defined as follows ... 

Each element ,y of a contingency table X can be thought of as a random variate. Under the assumption that all marginal sums are fixed, we can derive the expected values E(x,y) for each of the random variates [1] ... 

Table 32.5 presents the expected values of the elements in the contingency Table 32.4. Note that the marginal sums in the two tables are the same. There are, however, large discrepancies between the observed and the expected values. Small discrepancies between the tabulated values of our illustrations and their exact values may arise due to rounding of intermediate results.

In the literature we encounter three common transformations of the contingency table. These can be classified according to the type of closure that is involved. By closure we mean the operation of dividing each element in a row or column of a table by its corresponding marginal sum. We reserve the word closure for the specific operation where the elements in a row or column of the table are reduced to unit sum. This way, we distinguish between closure and normalization, as the latter implies an operation which reduces the elements of a table to unit sums of squares. In a strict sense, closure applies only to tables with non-negative elements. 

In this analysis, weight coefficients for rows and for columns have been defined as constants. They could have been made proportional to the marginal sums of Table 32.10, but this would weight down the influence of the earlier years, which we wished to avoid in this application. As with CFA, this analysis yields three latent vectors which contribute respectively 89, 10 and 1% to the interaction in the data. The numerical results of this analysis are very similar to those in Table 32.11 and, therefore, are not reproduced here. The only notable discrepancies are in the precision of the representation of the early years up to 1972, which is less than in the previous application, and in the precision of the representation of the category of women chemists which is better than in the previous analysis by CFA (0.960 vs 0.770). 

Fig. 37.5. Biplot obtained from correspondence factor analysis of the data in Table 37.8 [43], Circles refer to compounds. Squares relate to observations. Areas of circles and squares are proportional to the marginal sums of the rows and columns in the table. The horizontal and vertical components represent 40 and 31 %, respectively, of the interaction in the data.

Absorption, rate-of-return, and marginal pricing have been considered here on the basis of manufacturing cost. Total cost, which is the sum of manufacturing and general costs, can also be considered as the basis. In this case the appropriate profit to consider is the net annual profit rather than the gross annual profit. 

The side depth of the thickener is determined as the sum of the depths needea for the compression zone and for the clear zone. Normally, 1.5 to 2 m of clear liquid depth above the expected pulp level in a thickener will be sufficient for stable, effective operation. When the location of the pulp level cannot be predicted in advance or it is expected to be relatively low, a thickener sidewall depth of 2 to 3 m is usually safe. Greater depth may be used in order to provide better clarity, although in most thickener applications the improvement obtained by this means will be marginal. 

Table 18.4.1 smiinuuizes another inetliod of risk assessment tliat can be applied to an accident system failure. Both probability and consequence have been ranked on a scale of 0 to 1 witli table entries being the sum of probability and consequence. The acceptability of risk is a major decision and can be described by dividing tlie situations presented in Table 18.4.1 into unacceptable, marginally acceptable, and acceptable regions. Figiue 18.4.2 graphically represents tliis risk data. ... 

The average or expected row-profile is obtained by dividing the marginal row in the original table by the global sum. The matrix F of deviations of row-closed profiles from their expected values is defined by ... 

Table of double-closed data Z, with weighted means, weighted sums of squares and weight coefficients added in the margins, from Table 32.4... 

The distortions that patents provoke were reviewed recently by Kremer.14 Monopoly prices create both static and dynamic distortions. On the one hand, some consumers will not be able to pay prices that are fixed above the marginal cost in order to recover the investment in R D. On the other hand, potential investors will not necessarily take the consumer surplus into account when they decide to carry out research projects. The value of a patent - and in this case of a pharmaceutical - may be very different for different consumers, but price discrimination is impossible. The industry may shelve certain projects owing to the lack of a satisfactory return, because of the difficulty of price discrimination. Kremer even claims that the welfare loss due to monopoly prices is in the region of a quarter of the sum of the profits and the consumer surplus. Other authors, such as Giiell and Fischbaum,15 estimate welfare loss as being around 60 per cent of the sales figure. 

To sum up, the factors that enable the supply side to fix prices above the marginal cost are (a) the imperfect agency relationship between the doctor (the agent) and the insurer (the principal) the prescriber may prefer the brand product, about which he or she has acquired knowledge and experience during the patent period (risk aversion), (b) the patient, and sometimes also the doctor, may have imperfect information on the quality of cheaper alternatives, and (c) the lack of incentives to change prescription habits (moral hazard). 

The sum /x (<5+>ri>Si+ Cj) p adds the mean marginal profits for all products, each summand resulting from the demand density 5 , the already received orders r and the available stock s + Xi, where s is the product already in stock and Xi is the additional amount that is produced. The first constraint is obvious, the second requests that the production is feasible within a given time period. 

Detection, identification and quantification of these compounds in aqueous solutions, even in the form of matrix-free standards, present the analyst with considerable challenges. Even today, the standardised analysis of surfactants is not performed by substance-specific methods, but by sum parameter analysis on spectrophotometric and titrimetric bases. These substance-class-specific determination methods are not only very insensitive, but also very unspecific and therefore can be influenced by interference from other compounds of similar structure. Additionally, these determination methods also often fail to provide information regarding primary degradation products, including those with only marginal modifications in the molecule, and strongly modified metabolites. 

The marginal probabilities are obtained by summing over all possible states that we do not care to specify. Thus, the probability of finding the system empty, irrespective of its conformational state, is... 

The dual price of the slack variable sm on this constraint indicates the effect of selling this product at the margin, that is, it indicates the marginal profit on the product. Ifthe constraint is slack, so that the slack variable is positive (basic), the profit at the margin must obviously be zero and this is in line with the zero dual price of all basic variables. Since cost + profit — realization for a product, the sum of the dual prices on its balance and requirement constraints equals its coefficient in the original objective function. 

For each cell inside this matrix, there is an interval, which is the range of possible values for the sum given the ranges of the marginal intervals for A and B, and a probability, which is the product (under independence) of the 2 marginal probabilities. Notice that the elements inside the matrix are another collection of intervals with associated probability masses. Because the probabilities add up to 1, they also specify a p-box. Figure 6.6 shows this p-box reassembled from the 9 interval-mass pairs in the matrix. 

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