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distribution. These requirements cannot be satisfied if the relationship is non-rectilinear or if the distribution is “eccentric” for either (or both) of the two variables. [An “eccentric” distribution has an asymmetrical (“skew”) shape, unbalanced outliers, or any other patterns (such as two or more modes) that make the data non-Gaussian.] The non-Gaussian patterns can promptly be recognized when the univariate summary is checked separately for each variable, but detection of a nonlinear relationship will require special examinations such as those discussed in Section 19.6.2.

Because relatively few sets of bivariate data meet the strict rectilinear and bi-Gaussian “eligibility” requirements, most correlations for bi-dimensional data would probably best be expressed with the “nonparametric” Kendall or Spearman coefficients, which do not make draconian demands for the pattern of distribution. Nevertheless, because Pearson’s r is believed to be “robust” (i.e., not greatly affected by violations of the eligibility criteria), it is often applied almost routinely for all bi-dimensional correlations. Because the violations may sometimes distort the results, the non-parametric coefficients may often be “significant” (quantitatively, stochastically, or both) in situations where Pearson’s r was not. In the simple example in Tables 19.1 and 27.2, both the Kendall and Spearman coefficients had higher values than Pearson’s r.

27.3.4Published Examples

In clinical applications, the results of a randomized, double-blind, crossover trial21 of acetazolamide versus placebo for the prevention of acute mountain sickness were expressed with Spearman’s rho and with Wilcoxon’s signed rank test, because the scores for symptoms did “not conform to normal distributions.” Wilcoxon’s test was used to contrast differences in the two treatments, and the rs coefficient expressed the association between symptoms and age.

Another clinical application occurred in a trial22 checking the effect of serum interleukin-6-level on survival in patients with septic shock. The investigators elected not to use the conventional Pearson r coefficient because the interleukin results were expressed in powers of 10. Accordingly, the Spearman rs coefficient was applied to correlate the ranks of the two variables. The results are shown in Figure 27.2. You can decide for yourself whether the data seem to have a correlation as high as the cited “r = .51.”

Other examples of the Spearman rs coefficient appeared in medical publications where the entities being correlated were lipoprotein (a) with hemoglobin Alc in diabetes mellitus,23 collateral blood flow with cardiac wall motion score in recent myocardial infarction,24 mean faculty income at university hospitals vs. percent of women in each subspecialty fellowship,25 and ordinal ratings of “experiences from an imaginary cardiovascular clinical trial.”26

FIGURE 27.2

Correlation between the serum concentrations of IL-6 (ng/mL) measured at study entry and the duration of survival (days) in 33 nonsurvivors (Spearman rank correlation coefficient). [Figure and legend taken from Chapter Reference 22.]

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27.3.5Concerns about “Reduced Efficiency”

When non-parametric analyses were first proposed for dimensional data, many statisticians were worried about a descriptive “loss of information” (as noted in Section 15.9.1). A more substantial concern, however, was the stochastic fear that non-parametric procedures, such as the Wil- coxon–Mann–Whitney U test, were less powerful than the counterpart parametric procedures, such as the t test. The concept was checked with ratios of “power efficiency” or “asymptomatic relative efficiency” that were calculated from the sample sizes needed to give the tests the same “power” when the same null hypothesis and same alternative hypothesis were evaluated at the same level of α . As discussed earlier (Section 23.5), if test A requires nA persons and if test B requires nB persons to have the same power, and if nB is larger than nA, the reduced “power-efficiency” of test B is calculated as nA/nB. With this type of calculation, the relative power-efficiency of non-parametric tests was found to range from about .64 to .96 when compared with the corresponding parametric procedures.

These comparisons, however, were done under conditions that are optimal for the parametric tests, but not when eccentric distributions might handicap the parametric performance. Accordingly, the alleged superiority of the parametric tests may be due to unfair comparisons. Perhaps the best approach is to use a parametric test if it seems appropriate, but don’t let the false specter of “reduced efficiency” keep you from using a non-parametric test if it does what you want or need it to do.

27.4 Trend in r × 2 Tables

Beyond the expressions of trend in bi-ordinal data, probably the most common and useful additional indexes are those that express trend in r × 2 tables. Such tables have two co lumns for a binary variable, which can also be cited as a proportion, and more than two rows for r polytomous categories of a variable that can be nominal or ordinal. (The table can also appear in a counterpart 2 × c arrangement, for 2 rows and c columns. Everything to be discussed here for the r × 2 format will al so pertain to the alternative arrangement.)

27.4.1General Association

To illustrate the arrangement, the r × 2 format of Table 27.3 has a row variable with A, B, and C as three polytomous categories. The column variable is a binary outcome event, such as survival. The data are easier to understand and interpret when the two-column frequency counts are rearranged and shown as proportions in Table 27.4.

Three general indexes — lambda, φ 2, and φ — can be used to express association in any r × 2 table, regardless of whether the row categories are nominal or ordinal.

27.4.1.1Lambda Index of Error Reduction — The lambda index discussed in Section

27.2.1 is a simple expression for the statistical “accomplishment” of the categories in Table 27.4. From only the “Y variable” totals, with survival rate 70%, everyone would be estimated as alive. The estimates would be correct in 143 instances, and wrong in 61. When the X variable is used, everyone in Category

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A, with a 35% survival rate, would be estimated as dead. The estimates would be correct in 44 persons and wrong in 24. In categories B and C, where survival rates are 96% and 84%, everyone would be estimated as alive. These estimates would have 2 errors in B and 15 errors in C. The total number of errors in the three groups would be 24 + 2 + 15 = 41.

The lambda value for proportionate reduction in errors will be

errors for total

which is (61 − 41)/61 = .33 in this instance.

27.4.1.2 φ 2 for Variance Reduction — From the customary X2 used in the chi-square test, φ 2 can be calculated as X2/N to form a descriptive index that expresses proportion of group variance reduction in an r × 2 table. The mathematical justification is as follows:

Each row in the table has ni members, of whom ti have the target binary event, such as alive. The proportion in that row will be Pi = ti/ni. For the total table, T = Σ ti , N = Σ ni , and P = T/N, with Q = 1 – P. The original group variance, before any division into categories, is NPQ; and the group variance in each row category is nipiqi. The proportionate reduction in variance achieved by dividing the total group into categories will be

For example, in Table 27.4, NPQ = (204)(143/204)(61/204) = 42.76; Σ nipiqi = [(24)(44)/68] + [(43)(2)/45] + [(76)(15)/91] = 29.97; and NPQ − Σ nipiqi = 42.76 − 29.97 = 12.79. With Formula [27.8], we get (242/68) + (432/45) + (762/91) − (1432/204) = 113.03 − 100.24 = 12.79. The value of φ 2 will be 12.79/42.76 = .299 with Formula [27.7] or 12.79{204/[(143)(61)]} = .299 with Formula [27.9].

These results for φ 2 and the preceding lambda indicate that when the total survival rate of 143/204 (70%) is divided into the three categories shown in Table 27.4, the proportionate reductions are 33% in error rate, and 30% (i.e., φ 2 = .299) in group variance.

Because X2 = Nφ 2, Formula [27.9], when multiplied by N, becomes exactly the same calculation of X2 that was shown earlier for a 2 × 2 table with Formula [14.5]. The extension of the X2 formula to an

Formula [27.10] is particularly convenient because it avoids having to determine all the “expected” values for calculations using Σ (0 − E)2/E. It also avoids the nipiqi calculations of Formula [27.7]. With Formula [27.10], X2 (without Yates correction) can be promptly found for any r × 2 table and used as a stochastic index, interpreted with (r − 1)(c − 1) = (r − 1)(2 − 1) = r − 1 degrees of freedom. In

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Table 27.4, where we already know that φ 2 = .299, multiplication by N = 204 will produce X2, which will have the very high value of (204)(.299) = 61, for which 2P is <.0001 at 2 d.f. If calculated first, the X2 value in the r × 2 table can be divided by N to get φ 2 as a descriptive index of association, representing the proportion of reduced variance.

The φ index is seldom used descriptively, because a correlation coefficient intuitively seems to be a peculiar way of summarizing the results being contrasted in an r × 2 or 2 × 2 table. For larger r × c tables, however, φ may sometimes have the useful “screening” function described later in Section 27.8.1.

27.4.2 Problems in General Trend

Table 27.5 shows another 3 × 2 tabl e, but the row variable contains a specifically ordinal array — mild, moderate, severe — rather than the unranked categories of Table 27.4. If the conventional X2, φ 2, and lambda values were calculated, the results in Table 27.5 would be exactly the same as in Table 27.4. In fact, the same φ 2, X2, and lambda values would also be produced for Table 27.6, where the ordinal array sequentially has exactly the same proportions as in Table 27.4.

Despite obvious differences in the results of Tables 27.5 and 27.6, the φ 2, X2, and lambda values are similar because the conventional mathematical strategy and calculational formulas for the association

of whether the rows contain nominal or ordinal categories.

27.4.2.1Inadequate Descriptive Message — Nevertheless, despite the similarities in φ 2, X2, and lambda, Tables 27.5 and 27.6 contain a dramatically different descriptive message. In Table 27.5, the survival rates would be expected to show their downward monotonic gradient as the categories progressively descend from the “better” to “worse” clinical conditions. In Table 27.6, however, the gradient is biologically absurd. The best survival results occurred in the moderate group, and patients with severe disease had a survival rate more than twice that of the mild disease group. On scientific grounds alone, these results would be rejected as bizarre. Something has gone drastically wrong in the concepts or in the data, or perhaps in the statistical index of expression.

27.4.2.2Monotonic Gradient — The scientific message in Tables 27.5 and 27.6 was communicated from the gradient in survival rates as the ordinal categories went from mild to severe. Because the ordinal categories monotonically increase in severity and because severity of clinical condition affects

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survival, we would expect the survival rates to show a corresponding monotonic gradient, going progressively downward.

For nominal categories, a “trend” cannot really be expected, because the categories have no ranks. Thus, if the A, B, and C categories in Table 27.4 represented different treatments or diseases, we could compare results in pairs of categories (such as A vs. B, B vs. C, or A vs. C), but a distinctive trend could not be cited. If we wanted a single descriptive index for the result in the nominal categories, it would have to be the association shown with lambda, φ 2, or φ .

With a monotonic array of ordinal categories, however, a corresponding upward (or downward) trend can be anticipated and checked. The statistical gradients can be noted from simple subtraction of the (survival) proportions in adjacent categories. In Table 27.5, the monotonic descent of gradients is indicated by the two negative values: p2 − p1 = –12% = −.12 and p3 − p2 = −.49. Thus, the results “made sense” in Table 27.5 because it showed the expected gradient in survival. In Table 27.6, where p2 − p1 is +.61 and p3 – p2 = −.12, the results were “absurd” because the gradient had a peculiar reversal of positive and negative values.

Examining gradients in subtracted proportions is an excellent way to “screen” the data, but can be hazardous if some of the categories have small numbers that make the proportions unstable. What we would like, therefore, is a better statistical mechanism to express the gradient for an ordinal array of proportions.

27.5 Linear Trend in an r × 2 Ordinal Array

The overall gradient in proportions of an r × 2 ordinal array can be determined with a common, simple statistical approach: fitting a straight line to the data.

27.5.1Developing a Linear Model

If appropriate dimensional coding digits are assigned to the ordinal grades, we can construct a linear regression model for the data. Despite the apparent mathematical impropriety, the procedure is pragmatically effective. The index of slope will quantitatively describe the rectilinear trend, which can then be evaluated stochastically.

The binary data, coded as 0/1, become the Y variable in the linear regression model; and the ordinal categories become the X variable, coded with arbitrary weights, wi. The arrangement is shown in Table 27.7. The wi codes in this table are assigned to have equal intervals, consistent with dimensional data. The simplest approach is to code the ordinal categories in 1-unit intervals (such as 1, 2, 3, 4, …

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27.5.2 Regression Components

For each of the X categories, the corresponding value of Yi will be ti (1) + (ni − ti)0 = ti; and for the total, Y will be Σ ti/N = T/N. For the regression analysis, the total system variance, ST (which corresponds

to Syy in ordinary regression), will be Σ (Yi − Y )2 = Σ [t1 (1 – Y )2 + (ni − ti )(0 − Y )2], which is algebraically developed to become

At each category, the value of XiYi will be witi(1) + wi(ni − ti)(0) = witi. Thus, for Sx y = Σ Xi Yi – NXY, the result will be Σ witi − [N(Σ niwi/N)(T/N)], and so

Substituting NW for Σ niwi, Formula [27.14] can be algebraically simplified to

27.5.3Slope of “Trend”

With these regression concepts and symbols, the categorical data can be fitted with a straight line, and the slope of the line will indicate the average trend in the array of proportions. Using b = Sxy/Sxx for the slope of the regression line, the values of N will cancel in the two denominators of [27.14] and [27.11], and the algebraic calculation will become

Using the simplifications of Formulas [27.12] and [27.15], the formula for slope would become

27.5.3.1 Illustration for Two Categories — To illustrate the calculation, suppose we compare just two proportions t1/n1 vs. t2/n2, and suppose (for this example) we let w1 = 0 and w2 = 1. The components of Formula [27.16] become NΣ witi = Nt2; TΣ niwi = Tn2; NΣ ni wi2 = Nn2; and (Σ niwi)2 = n22 . When all the algebra is then carried out, these components for Formula [27.16] produce b = p2 − p1. Thus, with two X categories coded as 0/1, the slope is simply the increment in proportions, p2 − p1.

If the codes had been w1 = −1 and w2 = +1, the numerator of [27.16] would be N(−t1 + t2) – T(−n1 + n2), which becomes 2(n1t2 − n2t1), which is twice the previous value. The denominator would be N(n1 + n2) − (n2 − n1)2 = 4n1n2. The slope would be 2(n1t2 − n2t1)/4n1n2 which reduces to (p2 – p1 )/2,

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which is half the previous value. The halving would be expected since the −1/+1 coding made the slope span 2 units in the X-variable, rather than 1.

27.5.3.2Standardized Slope in 2 × 2 Table — The standardized slope for a 2 × 2 table

would be the correlation coefficient, r, which is bsx/sy, or b Sx x ⁄Sy y. With the 0,1 coding for w1 and

w2, the value of Sxx in Formula [27.11] becomes [Nn2 − (n1)2 ]/N, which is n1n2/N. The value that

Z2/N = X2/N = φ 2. Thus, in a 2 × 2 table, the values of r and φ are identical. (The similarity is seldom true for larger tables.)

27.5.3.3 Illustration for Three Ordinal Categories — For the three ordinal categories inspected in Tables 27.4 through 27.6, the simplest codes (for “hand” electronic calculation) would be w1 = −1, w2 = 0, and w3 = +1. With these codes, Formula [27.16] becomes

For Table 27.5, the numerator of [27.18] will be 204(24 − 43) − 143(68 − 45) = 204(−19) − 143(23) = −7165. The denominator of [27.18] will be 204(45 + 68) (68 − 45)2 = 204(113) − (23)2 = 22523. The slope will be −7165/22523 = –.32.

Note that this slope is just what would have been expected from a judgmental evaluation of Table 27.5. Because the gradient drops 12% (= 96% − 84%) from the first to the second category, and then drops 49% (= 84% − 35%), the “average” drop would be −(12% + 49%)/2 = −30.5% or −.305, which is consistent with the linear calculation of −.32. Because the trend is monotonic, another judgmental approach could have been used to determine the total gradient as 96% − 35% = 61%, going downward from the first to third categories. Because the drop of 61% is spread over two zones of change, the “average” gradient would be –61%/2 = −.305.

For Table 27.6, however, judgmental examination would argue against calculating an average gradient. The gradient rises by 61% (= 96% − 35%) from the first to second category, but then falls by 12% (= 84% − 96%) from the second category to the third. With this reversal in trend, a single expression of directional gradient would be misleading. Nevertheless, if such an expression is desired, the judgmental “average” rise over the two categorical intervals would be .49/2 = .245. When the statistical “average” for these data is calculated with Formula [27.16], the numerator is [204(76 − 24)] − [143(91 − 68)] = 7319. The denominator is [204(91 + 68)] − [(91 − 68)2] = 31907. The slope would be 7319/31907 =

.23, which is close to the judgmentally approximated value of .245.

To get standardized slopes for the three categories, we need to determine r = b Sxx ⁄Syy . The value that corresponds to Syy is NPQ = T(N−T)/N, which is (143)(204 − 143)/204 = 42.76 in both tables. The value of Sxx is 1/N times the denominator of Equation [27.16]. It was 22523 for Table 27.5 and 31907 for Table 27.6. Thus, for Table 27.5,

r = (.32) 22523/[(204 )(42.76 )] = .51

and for Table 27.6,

r = (.23) 31907 ⁄[(204 )(42.76)] = .44

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27.5.4Precautions and Caveats

The linear trends calculated for Tables 27.5 and 27.6 should serve as an important warning whenever ranked data — expressed in either ordinal or dimensional values — are fitted with a straight-line regression model. The mathematical model will fit an average constant rectilinear slope to the data, regardless of whether the data do or do not have the constant monotonic trend denoted by the slope. Because the value of the constant slope may be used for conclusions such as “You live 2 years longer for every 5-point drop in Substance X,” the conclusion can be misleading or grossly distorted if the data do not conform to the linear model.

For example, in Table 27.5, the slope of −.32 would denote an average drop of 32% in survival from one category to the next. This result would be misleading because the gradient is not constant. It drops 12% from the first to second category, and then drops 49% from the second to the third. In Table 27.6, the slope of .23 denotes a constant rise of 23% from one category to the next. This result is a gross distortion of the sharp rise followed by a fall in gradient for the actual data. In both instances, the statistically calculated slopes are correct as “average” values, but wrong in different zones of the data. Similarly, the standardized slopes of .51 and .44 in the two tables seem moderately impressive, although the impression is not strictly accurate in Table 27.5 and egregiously misleading in Table 27.6.

The point to remember is that fitting a linear slope is an excellent “screening” mechanism, but it cannot replace a direct examination of the data, and it should not be used for final conclusions until confirmation that the data actually conform to the linear mathematical pattern.

27.5.5X2L Test of Linear Trend

The descriptive index for trend in the linear slope is regularly checked stochastically with a chi-square test for linear trend. In the conventional simple regression discussed in Chapter 19, the stochastic procedure was a t test on either b or the correlation coefficient, r. For an ordinal r × 2 table, the stochastic approach is more complex because it involves using a partition of X2.

27.5.5.1 Strategy for X2 Partition — The X2 partition is derived from the principles of partitioning variance that were introduced in Section 19.2.1.2.

The diverse concepts and symbols are shown in Table 27.8. In the ordinary arrangement that is used for chi-square, a group of binary data, {Yi}, having N members with mean (or summary proportion) Y = P, is divided into k categories, each having pi as its mean. The total system variance, ST = NPQ, becomes partitioned as a sum of the within-category group variance, SW, and the between-category group variance, SB. The latter value is the numerator of the ordinary chi-square test, when X2 is formed as

SB/PQ = (NPQ − Σ nipiqi)/PQ.

When a regression model is imposed on the same set of data, the group variance becomes partitioned into ST = Sr + SM. (The latter value is used for the calculation of r2 as SM/ST.) The linear accomplishment of the model can be stochastically examined in two ways. In the first way, SM represents the effect of a linear model for the entire set of unpartitioned data. This procedure, as discussed in Chapter 19, uses a t test on the correlation coefficient r. In the second way, the goal is to check what the linear model does for the categorical partition of data. For the latter examination, the group variance between categories, SB, is divided, as noted by Armitage and Berry,27 into

In the symbols of Table 27.8, the foregoing expression is

SB = SU + SM

When all terms in this expression are divided by PQ, the result is a partitioning of X2 as

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TABLE 27.8

Arrangement of Deviations and Group Variances in Tests of Ordinary X2 and X2L for Linear Trend

regression estimate

In Equation [27.20], the X2L component, derived from SM, represents the linearity of the model’s fit, and the X2R component represents nonlinearity in the residual categories. The degrees of freedom for interpreting the partitioned chi-square values in results for the k categories are k − l for X2, 1 for X2L , and k − 2 for X2R .

27.5.5.2 Illustration of Calculations — If you do the calculations yourself, the easiest approach is to note that X2L is determined as SM/ PQ, and that SM was established when a regression line was fitted to the data in Sections 27.5.1 and 27.5.2. From Chapter 19, we can recall that SM = (r2)(total system variance) = r2(NPQ) = bSxy. We can then recognize that both b and Sxy have already been found. The value of b is the slope previously obtained in [27.16] (or for a three-category coding in [27.18]); and the value of NSxy, as noted in [27.14], is the numerator in the formulas for b. The value of X2L can then be calculated as

For example, in Tables 27.5 and 27.6, NPQ = (143)(61)/204 = 42.76. For Table 27.5, the slope was calculated as −7165/22523, which will be the value of b in Formula [27.21]. From the numerator of the previous slope calculation, we get NSxy = 7165, and so Sxy = 7165/204. Since NPQ = 42.76, PQ = 42.76/204. The value of Sxy/PQ will be 7165/42.76 = 167.56. The value of X2L in Formula [27.21] will be (7165)(167.56)/(22523) = 53.3. For Table 27.6, the slope was calculated as 7319/31907. Substituting NSxy/NPQ for Sxy/PQ, the corresponding value of X2L will be (7319)(7319)/(31907)(42.76) = 39.3. At 1 d.f., both these values of X2L are “highly significant” with 2P < .001. The “high significance” found for the linear trend in X2L values for both of these two tables should serve to warn against drawing conclusions by inspecting P values alone.

27.5.5.3 Relationship of r2 and X2L — An even simpler approach is to realize that X2L = SM/PQ and that when a regression line is fitted to the data, SM = r2(NPQ). Therefore,

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A linear regression computer program applied to the original results will produce a value of r2, from which X2L can promptly be found after multiplication by N.

Conversely, if a calculation has produced X2L , but not the value of r2, it can be easily found as

Thus, r2 = 53.3/204 = .26 and r = .51 for Table 27.5, whereas r2 = 39.9/204 = .19 and r = .44 for Table 27.6. Table 27.6 thus continues to show a modestly high linear correlation coefficient despite the reversed gradient in the last category.

The main point conceptually is that X2 reflects the variation that is “explained” by imposing categories on the total data, and X2L reflects the further “explanation” produced when a linear trend is imposed on the categories. Because φ 2 = X2/N and r2 = X2L ⁄N, the relationship can be expressed as X2L /X2 = r2/φ 2. In a 2 × 2 table, as noted in Section 27.5.3.2, the difference in the two proportions (p 1 − p2) always forms a straight line, and so φ 2 = r2. Because the total system variance is NPQ and the variance within the categories is Σ nipiqi, the value of NPQ − Σ nipiqi becomes the numerator of X2, which can be expressed as Σ (Yi – Y)2 , as shown in Table 27.8. The latter value becomes partitioned into the non-linear and linear components shown in Equation [27.19]. The reason for the easy calculation of r2 is that the SM term is used for both the ordinary linear regression partition of variance and the subsequent partition of chi-square.

27.5.5.4 Check of Residual Nonlinear Variance (X2R ) — With the ordinal linear regression arrangement, we can check the residual variance of the pi category values around the regression line to see whether a stochastically “significant” amount of “nonlinear variance” still remains. For this purpose, the value of X2L is subtracted from X2, and the result is interpreted (using k for the number of categories) with k − 2 degrees of freedom. Thus, X2R = X2 – X2L = 61.0 – 53.3 = 7.7 at 2 d.f. for Table 27.5, and the corresponding value is 61.0 − 39.3 = 21.7 for Table 27.6. Both of these values are highly stochastically significant, indicating that a great deal of variance has not been explained by imposition of the linear model alone. The particularly high value of X2R for Table 27.6 could serve as warning of linear inadequacy in that table.

27.5.6Applications of Linear Trend Test

The linear trend in a set of ordinal proportions can serve as an alternative to the Wilcoxon–Mann–Whitney U test. It can be used to appraise ordinal staging systems, to demonstrate “significance” for ordinal partitions, or to screen for a “double gradient” before the conjunctive consolidation procedure of targeted multivariable analyses.15

27.5.6.1 Alternative to Wilcoxon–Mann–Whitney U Test — A two-group rank test, such as the Wilcoxon–Mann–Whitney U procedure, is customarily done for a 2 × c structure, such as the earlier data in Table 15.5. Such data can also be regarded, however, as an r × 2 structure, and then evaluated for linear trend.

For example, the data in Table 15.5 could be rearranged as follows:

The average crude gradient of decline in the binary proportions is (80% − 34%)/3 = 15.33%. With a more formal calculation, assigning codes of 1, 2, 3, 4 to the ordinal categories, and using Formula

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