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<ul><li><p>7/30/2019 C2166_ch29_Principle of Medical Statistics</p><p> 1/16</p><p>29</p><p>Analysis of Variance</p><p>CONTENTS</p><p>29.1 Conceptual Background</p><p>29.1.1 Clinical Illustration</p><p>29.1.2 Analytic Principles</p><p>29.2 Fishers F Ratio</p><p>29.3 Analysis-of-Variance Table</p><p>29.4 Problems in Performance29.5 Problems of Interpretation</p><p>29.5.1 Quantitative Distinctions</p><p>29.5.2 Stochastic Nonsignificance</p><p>29.5.3 Stochastic Significance</p><p>29.5.4 Substantive Decisions</p><p>29.6 Additional Applications of ANOVA</p><p>29.6.1 Multi-Factor Arrangements</p><p>29.6.2 Nested Analyses</p><p>29.6.3 Analysis of Covariance</p><p>29.6.4 Repeated-Measures Arrangements</p><p>29.7 Non-Parametric Methods of Analysis29.8 Problems in Analysis of Trends</p><p>29.9 Use of ANOVA in Published Literature</p><p>References</p><p>The targeted analytic method called analysis of variance, sometimes cited acronymically as ANOVA,</p><p>was devised (like so many other procedures in statistics) by Sir Ronald A. Fisher. Although often marking</p><p>the conceptual boundary between elementary and advanced statistics, or between amateur fan and</p><p>professional connoisseur, ANOVA is sometimes regarded and taught as elementary enough to be used</p><p>for deriving subsequent simple procedures, such as the t test. Nevertheless, ANOVA is used much less</p><p>often today than formerly, for reasons to be noted in the discussions that follow.</p><p>29.1 Conceptual Background</p><p>The main distinguishing feature of ANOVA is that the independent variable contains polytomous</p><p>categories, which are analyzed simultaneously in relation to a dimensional or ordinal dependent (out-</p><p>come) variable.</p><p>Suppose treatments A, B, and C are tested for effects on blood pressure in a randomized trial. When</p><p>the results are examined, we want to determine whether one of the treatments differs significantly from</p><p>the others. With the statistical methods available thus far, the only way to answer this question would</p><p>be to do multiple comparisons for pairs of groups, contrasting results in group A vs. B, A vs. C, and Bvs. C. If more ambitious, we could compare A vs. the combined results of B and C, or group B vs. the</p><p>combined results of A and C, and so on. We could work out various other arrangements, but in each</p></li><li><p>7/30/2019 C2166_ch29_Principle of Medical Statistics</p><p> 2/16</p><p>instance, the comparison would rely on contrasting two collected groups, because we currently know</p><p>no other strategy.</p><p>The analysis of variance allows a single simultaneous comparison for three or more groups. The result</p><p>becomes a type of screening test that indicates whether at least one group differs significantly from the</p><p>others, but further examination is needed to find the distinctive group(s). Despite this disadvantage, ANOVA</p><p>has been a widely used procedure, particularly by professional statisticians, who often like to apply it even</p><p>when simpler tactics are available. For example, when data are compared for only two groups, a t test or</p><p>Z test is simpler, and, as noted later, produces exactly the same results as ANOVA. Nevertheless, many</p><p>persons will do the two-group comparison (and report the results) with an analysis of variance.</p><p>29.1.1 Clinical Illustration</p><p>Although applicable in experimental trials, ANOVA has been most often used for observational studies.</p><p>A real-world example, shown in Figure 29.1, contains data for the survival times, in months, of a random</p><p>sample of 60 patients with lung cancer,1,2 having one of the four histologic categories of WELL (well-</p><p>differentiated), SMALL (small cell), ANAP (anaplastic), and CYTOL (cytology only). The other variable</p><p>(the five categories of TNM stage) listed in Figure 29.1 will be considered later. The main analytic</p><p>question now is whether histology in any of these groups has significantly different effects on survival.</p><p>29.1.1.1 Direct Examination The best thing to do with these data, before any formal statis -tical analyses begin, is to examine the results directly. In this instance, we can readily determine the group</p><p>sizes, means, and standard deviations for each of the four histologic categories and for the total. The results,</p><p>shown inTable 29.1, immediately suggest that the data do not have Gaussian distributions, because the</p><p>standard deviations are almost all larger than the means. Nevertheless, to allow the illustration to proceed,</p><p>the results can be further appraised. They show that the well-differentiated and small-cell groups, as</p><p>expected clinically, have the highest and lowest mean survival times, respectively. Because of relatively</p><p>small group sizes and non-Gaussian distributions, however, the distinctions may not be stochasticallysignificant.</p><p>Again before applying any advanced statistics, we can check these results stochastically by using simple</p><p>t tests. For the most obvious comparison of WELL vs. SMALL, we can use the components of Formula</p><p>[13.7] to calculate sp = = 21.96; (1/nA) + (1/nB) = (1/22) + (1/11) =.369; and = 24.43 4.45 = 19.98. These data could then be entered into Formula [13.7] toproduce t = 9.98/[(21.96)(.369)] = 2.47. At 31 d.f., the associated 2P value is about .02. From thisdistinction, we might also expect that all the other paired comparisons will not be stochastically</p><p>significant. (If you check the calculations, you will find that the appropriate 2P values are all >.05.)</p><p>29.1.1.2 Holistic and Multiple-Comparison Problems The foregoing comparisonindicates a significant difference in mean survival between the WELL and SMALL groups, but does</p><p>not answer the holistically phrased analytic question, which asked whether histology has significant</p><p>effects in any of the four groups in the entire collection. Besides, an argument could be made, using</p><p>TABLE 29.1</p><p>Summary of Survival Times in Four Histologic Groups</p><p>of Patients with Lung Cancer inFigure 29.1</p><p>Histologic</p><p>Category</p><p>Group</p><p>Size</p><p>Mean</p><p>Survival</p><p>Standard</p><p>Deviation</p><p>WELL 22 24.43 26.56</p><p>SMALL 11 4.45 3.77</p><p>ANAP 18 10.87 23.39</p><p>CYTOL 9 11.54 13.47</p><p>Total 60 14.77 22.29</p><p>21 26.56( )2 10 3.77( )2+[ ]/ 21 10+( )XA XB</p></li><li><p>7/30/2019 C2166_ch29_Principle of Medical Statistics</p><p> 3/16</p><p>distinctions discussed in Section 25.2.1.1, that the contrast of WELL vs. SMALL was only one of the</p><p>six (4 3/2) possible paired comparisons for the four histologic categories. With the Bonferroni correc-tion, the working level of for each of the six comparisons would be .05/6 = .008. With the lattercriterion, the 2P value of about .02 for WELL vs. SMALL would no longer be stochastically significant.</p><p>We therefore need a new method to answer the original question. Instead of examining six pairs of</p><p>contrasted means, we can use a holistic approach by finding the grand mean of the data, determining</p><p>the deviations of each group of data from that mean, and analyzing those deviations appropriately.</p><p>OBS ID HISTOL TNMSTAGE SURVIVE</p><p>1 62 WELL I 82.32 107 WELL II 5.33 110 WELL IIIA 29.64 157 WELL I 20.35 163 WELL I 54.96 246 SMALL I 10.37 271 WELL IIIB 1.68 282 ANAP IIIA 7.69 302 WELL I 28.0</p><p>10 337 CYTOL I 12.811 344 WELL II 4.012 352 ANAP IIIA 1.313 371 WELL IIIB 14.114 387 SMALL IIIA 0.215 428 SMALL II 6.816 466 ANAP IIIB 1.417 513 ANAP I 0.118 548 ANAP IV 1.819 581 ANAP IV 6.020 605 CYTOL IV 1.021 609 CYTOL IV 6.222 628 SMALL IV 4.423 671 SMALL IV 5.524 764 SMALL IV 0.325 784 ANAP IV 1.626 804 WELL I 12.2</p><p>27 806 ANAP IIIB 6.528 815 WELL I 39.929 852 WELL IIIB 4.530 855 WELL II 1.631 891 CYTOL IIIB 8.132 892 WELL IIIB 62.033 931 CYTOL IIIB 8.834 998 WELL IIIB 0.235 1039 SMALL IV 0.636 1044 ANAP II 19.337 1054 WELL IIIB 0.638 1057 ANAP I 10.939 1155 ANAP I 0.240 1192 SMALL IV 11.241 1223 ANAP IV 0.942 1228 ANAP II 27.943 1303 ANAP IIIB 2.9</p><p>44 1309 ANAP II 99.945 1317 ANAP IV 4.746 1355 CYTOL IIIB 1.847 1361 WELL IV 1.048 1380 CYTOL IV 10.649 1405 SMALL IV 3.750 1444 WELL II 55.951 1509 SMALL IV 3.452 1515 WELL I 79.753 1521 ANAP IV 1.954 1556 ANAP IIIB 0.855 1567 SMALL IV 2.556 1608 CYTOL I 8.657 1612 WELL IIIA 13.358 1666 CYTOL IV 46.059 1702 WELL II 23.960 1738 WELL II 2.6</p><p>FIGURE 29.1Printout of data on histologic type, TNM Stage, and months of survival in a random sample of 60 patients with primary cancer</p><p>of the lung. [OBS = observation number in sample; ID = original indentification number; HISTOL = histology type; TNMSTAGE= one of five ordinal anatomic TNM stages for lung cancer; SURVIVE = survival time (mos.); WELL = well-differentiated;SMALL = small cell; ANAP = anaplastic; CYTOL = cytology only.]</p></li><li><p>7/30/2019 C2166_ch29_Principle of Medical Statistics</p><p> 4/16</p><p>Many different symbols have been used to indicate the entities that are involved. In the illustration</p><p>here, Yij will represent the target variable (survival time) for person i in group j. For example, if WELL</p><p>is the first group inFigure 29.1, the eighth person in the group has Y8,1= 4.0. The mean of the valuesin group j will be = Yij/nj, where nj is the number of members in the group. Thus, for the last</p><p>group (cytology) inTable 29.1, n4= 9, Yi,4= 103.9, and = 103.9/9 = 11.54. The grand mean, ,will be (nj )/N, where N = nj= size of the total group under analysis. From the data inTable 29.1,G = [(22 24.43) + (11 4.45) + (18 10.87) + (9 11.54)]/60 = 885.93/60 = 14.77.</p><p>We can now determine the distance, , between each groups mean and the grand mean. Forthe ANAP group, the distance is 10.87 14.77 =3.90. For the other three groups, the distances are3.23 for CYTOL, 10.32 for SMALL, and +9.66 for WELL. This inspection confirms that the meansof the SMALL and WELL groups are most different from the grand mean, but the results contain no</p><p>attention to stochastic variation in the data.</p><p>29.1.2 Analytic Principles</p><p>To solve the stochastic challenge, we can use ANOVA, which like many other classical statisticalstrategies, expresses real world phenomena with mathematical models. We have already used such models</p><p>both implicitly and explicitly. In univariate statistics, the mean, , was an implicit model for fitting</p><p>a group of data from only the values in the single set of data. The measured deviations from that model,</p><p>Yi , were then converted to the groups basic variance, .In bivariate statistics for the associations in Chapters 18 and 19, we used an explicit model based on</p><p>an additional variable, expressed algebraically as = a + bXi. We then compared variances for threesets of deviations: Yi , between the items of data and the explicit model; Yi , between the itemsof data and the implicit model; and , between the explicit and implicit models. The group variances</p><p>or sums of squares associated with these deviations were called residual (or error) for ,</p><p>basic for , and model for ( )2.</p><p>29.1.2.1 Distinctions in Nomenclature The foregoing symbols and nomenclature havebeen simplified for the sake of clarity. In strict statistical reasoning, any set of observed data is regarded</p><p>as a sample from an unobserved population whose parameters are being estimated from the data. If</p><p>modeled with a straight line, the parametric population would be cited as Y = + X. When theresults for the observed data are expressed as = a + bXi, the coefficients a and b are estimates ofthe corresponding and parameters.</p><p>Also in strict reasoning, variance is an attribute of the parametric population. Terms such as</p><p>or , which are used to estimate the parametric variances, should be called sums</p><p>of squares, not group variances. The linguistic propriety has been violated here for two reasons: (1) the</p><p>distinctions are more easily understood when called variance, and (2) the violations constantly appear</p><p>in both published literature and computer print-outs. The usage here, although a departure from strictformalism, is probably better than in many discussions elsewhere where the sums of squares are called</p><p>variances instead ofgroup variances.</p><p>Another issue in nomenclature is syntactical rather than mathematical. In most English prose,</p><p>between is used for a distinction of two objects, and among for more than two. Nevertheless, in the</p><p>original description of the analysis of variance, R. A. Fisher used the preposition between rather than</p><p>among when more than two groups or classes were involved. The term between groups has been</p><p>perpetuated by subsequent writers, much to the delight of English-prose pedants who may denounce</p><p>the absence of literacy in mathematical technocracy. Nevertheless, Fisher and his successors have been</p><p>quite correct in maintaining between. Its use for the cited purpose is approved by diverse high-echelon</p><p>authorities, including the Oxford English Dictionary, which states that between has been, from its</p><p>earliest appearance, extended to more than two.3 [As one of the potential pedants, I was ready to useamong in this text until I checked the dictionary and became enlightened.]</p><p>29.1.2.2 Partition of Group Variance The same type of partitioning that was used forgroup variance in linear regression is also applied in ANOVA. Conceptually, however, the models are</p><p>Yj</p><p>Y4 GYj</p><p>Yj G</p><p>Y</p><p>Y Yi Y( )2</p><p>Yi</p><p>Yi Y</p><p>Yi Y</p><p> Yi Yi( )2</p><p> Yi Y( )2</p><p>Yi Y</p><p>Yi</p><p> Yi Y( )2</p><p> Yi Y( )2</p></li><li><p>7/30/2019 C2166_ch29_Principle of Medical Statistics</p><p> 5/16</p><p>expressed differently. Symbolically, each observation can be labelled Y ij, with j representing the group</p><p>and i, the person (or other observed entity) within the group. The grand mean, , is used for the implicit</p><p>model when the basic group or system variance, , is summed for the individual values of</p><p>Yi in all of the groups. The individual group means, , become the explicit models when the total</p><p>system is partitioned into groups. The residual group variance is the sum of the values of (Yi )2</p><p>within each of the groups. [In more accurate symbolism, the two cited group variances would be written</p><p>with double subscripts and summations as (Yij )2 and (Yij )2.] The model group variance,summed for each group of njmembers with group mean , is nj( )2. These results for data inthe four groups ofFigure 29.1 andTable 29.1are shown inTable 29.2.</p><p>Except for minor differences due to rounding, the components ofTable 29.2have the same structure</p><p>noted earlier for simple linear regression in Section 19.2.2. The structure is as follows:</p><p>{Basic Group Variance} = {Model Variance between Groups} + { Residual Variance within Groups}</p><p>or Syy = SM + SR.</p><p>The structure is similar to that of the deviations</p><p>Total Deviation = Model Deviation + Residual Deviation</p><p>which arises when each individual deviation is expressed in the algebraic identity</p><p>If is moved to the first part of the right side, the equation becomes</p><p>and is consistent with a parametric algebraic model that has the form</p><p>Yij=+j+ij</p><p>In this model, each persons value of Yij consists of three contributions: (1) from the grand parametric</p><p>mean, (which is estimated by ); (2) from the parametric increment, j (estimated by ),between the grand mean and group mean; and (3) from an error term, ij (estimated by Yij ), forthe increment between the observed value of Yij and the group mean.</p><p>For stochastic appraisal of results, the null hypothesis assumption is that the m groups have the same</p><p>parametric mean, i.e., 1=2= = j= =m.</p><p>TABLE 29.2</p><p>Group-Variance Partitions of Sums of Squares for the Four Histologic Gro...</p></li></ul>