{"id":327,"date":"2018-10-31T09:08:41","date_gmt":"2018-10-31T09:08:41","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=327"},"modified":"2018-10-31T09:30:40","modified_gmt":"2018-10-31T09:30:40","slug":"327","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/327\/","title":{"rendered":"Analysis of variance and Experimental Design: An Introduction to Experimental, Randomized and Block Design"},"content":{"raw":"<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Learning Objectives:<\/strong>\r\n<ul>\r\n \t<li>After the completion of this module the student will understand:<\/li>\r\n \t<li>An Introduction to Experimental Design<\/li>\r\n \t<li>Complete Randomized Design<\/li>\r\n \t<li>Randomization of sample\/population (treatments)<\/li>\r\n \t<li>Merits and Demerits of Complete Randomized Design<\/li>\r\n \t<li>Applications of complete randomization<\/li>\r\n \t<li>Randomized Block Design<\/li>\r\n \t<li>Advantages of the Randomized Block Design<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>1.<\/strong>\u00a0\u00a0\u00a0 <strong>Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Experimental design is a way to carefully plan experiments in advance so that your results are both objective and valid. Ideally your experimental design should:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogenous blocks before being randomly assigned to groups.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Allow you to make inferences about the relationship between independent variables and dependent variables.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Reduce variability, to make it easier for you to find differences in sample\/population (treatment) outcomes.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The choice of experimental design depends on the number and nature of the sample\/population (treatments) under study. It also depends on the object of the\u00a0<span style=\"font-size: 1em;text-align: initial\">experiment. Another consideration for the choice of the design is the question of available resources.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Some considerations under which the different designs are appropriate are as follows:<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-331\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163.png\" alt=\"\" width=\"613\" height=\"485\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>2.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Complete Randomized Design<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">This is the simplest type of design in which the whole experimental material is divided into a number of experimental units depending upon the number of sample\/population (treatments) and the number of replications for each. After that the sample\/population(treatments) are allotted to the units entirely by chance. In case of field experiments, the whole field is divided into a required number of equal plots and then the sample\/population (treatments) are randomized in those plots.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">If there are 5 sample\/population (treatments) A, B, C, D and E and 4 replication to each, the number of plots will be 20 and each sample\/population (treatment) will be allotted to four plots selected at random by means of random numbers.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">3.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Randomization of sample\/population (treatments)<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Here since the number of units is 20, a two digit random number table will be consulted and a series of 20 random numbers will be taken excluding those which are greater than 20. Suppose the random numbers are 4, 18, 2, 14, 3, 7, 13, 1, 6, 10, 17, 20, 8, 15, 11, 5, 9, 12, 16, 19.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">After this the plots will be serially numbered and the sample\/population (treatment) A will be allotted to the plots bearing the serial numbers 4, 18, 2 and 14, sample\/population (treatment) B will be allotted to the plots bearing the serial numbers 3, 7, 13 and 1, and so on for the other sample\/population (treatments) C.D. and E.<\/p>\r\n<strong>\u00a0<\/strong>\r\n\r\nIn a similar way the randomization can be done for any number of sample\/population (treatments).\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>4.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Structure of Analysis of Variance<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">If we suppose the number of sample\/population (treatments) to be n, and the number of replications to be r for every sample\/population (treatment), the total number of experimental units will be nr = N. If the sample\/population (treatments) have varying number of sample\/population (treatments), r1, r2, r3\u2026\u2026..rn then the total number of units (=N) will be given by<\/p>\r\n&nbsp;\r\n\r\nN\u00a0\u00a0 = r1 + r2 + r3 \u2026\u2026+rn Whereas r= number of replication\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In this design, the total number of degrees of freedom will be divided into two parts representing the independent comparisons. These two will be the two independent sources of variation.<\/p>\r\n&nbsp;\r\n\r\n(a)\u00a0\u00a0\u00a0 Between sample\/population (treatments)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(b)\u00a0\u00a0 Within sample\/population (treatments) (Within sample\/population\/treatment components provides a basis for the estimation of error)<\/p>\r\n&nbsp;\r\n\r\nThus the structure of analysis of variance will be as follows:\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-332\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164.png\" alt=\"\" width=\"570\" height=\"174\" \/>\r\n\r\n&nbsp;\r\n\r\nWhereas n= number of sample\/population (treatments)\r\n\r\nN= Total number of observations\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Here, since the total number of observations is N, the total degrees of freedom will be (N-1). Similarly, population\/sample (treatment) degrees of freedom will be (n-1), one less than the number of population\/sample (treatments), and the remaining (N-n) degrees of freedom will be for \u2018Within sample\/population (Treatments)\u2019 or \u2018Error\u2019.<\/p>\r\n&nbsp;\r\n\r\n<strong>5.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Standard Errors<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">The standard error of the difference between the two population\/sample (treatment) means based on r1 and r2 replications is estimated by the following relation:<\/p>\r\n1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\r\n\r\n( . \u00a0. )\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 . = \u221a\u00a0\u00a0 ( 1 + 2)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Where <strong>V<\/strong><strong>E<\/strong> <strong>is the pooled error variance or error mean square.<\/strong> If r1 = r2= r the formula reduces to \u221a2VrE\u00a0 . The degree of freedom for t-test are the error degrees of freedom.<\/p>\r\n&nbsp;\r\n\r\nWhere r is the number of replication.\r\n\r\n&nbsp;\r\n\r\n<strong>6.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Merits and Demerits of Complete Randomized Design Merits<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">(i)\u00a0 In this design any number of sample\/population (treatments) and replicates may be used. The number of replicates may be used. The number of replicates can also be varies at will form population to population (treatment to treatment).<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">(ii) The statistical analysis of the data is very easy and it remains easy even if the numbers of replicates are not the same for all the sample\/population (treatments).<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">(iii) The method of analysis remains simple when the results from some units are missing or rejected<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">(iv) The relative loss of information due to missing data is smaller than that with any other design.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v) This design is especially useful in small experiments where the supply of experimental material is scarce and homogeneous, as the whole of the material is utilized in the experiment.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(vi) The design provides maximum number of degrees of freedom for the estimation of error as compared with other design, for a given number of sample\/population\/treatments and a given number of experimental units.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Demerits<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">There is one and the main objection in this design and that is on the grounds of accuracy. Since there is no restriction on the randomization of the population\/sample (treatments), we cannot be sure about the fact that the units receiving one population\/sample (treatment) are similar to those receiving the other population\/sample (treatment) and, therefore, the whole of the variation among the units enters into the experimental error.<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>7.\u00a0<\/strong><strong>Applications of Complete Randomization<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\nThis design is especially advantageous and appropriate under the following circumstances:\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n(i)\u00a0 Where the experimental material is limited in quantity and homogeneous.\r\n\r\n&nbsp;\r\n\r\n(ii) Where it is expected that some of the units will be destroyed or will fail to respond.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(iii) In small experiments where the increased accuracy from the alternative design is not sufficient to exceed in importance the loss of error degrees of freedom.<\/p>\r\n&nbsp;\r\n\r\n<strong>8.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Randomized Block Design<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">In this design the whole experimental material is divided into homogeneous groups, each of which constitutes a single replication. Each of these groups is further divided into a number of experimental units which are equal in all respects. The sample\/population (treatments) are applied to these units by any random process. In case of field experiments, if it is observed that the fertility gradient of the field is in one direction, the whole field may be divided into a number of equal plots. The number of plots in each\u00a0<span style=\"font-size: 1em;text-align: initial\">block is equal to the number of sample\/population (treatments), so that each block is a replicate of each sample\/population (treatment).<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nThe following important points are to be kept in mind for this design:\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(1) In this design the number of blocks must be equal to the number of replications fixed for each sample\/population (treatment).<\/p>\r\n&nbsp;\r\n\r\n(2)\u00a0\u00a0 The number of plots in each block should be equal to the number of sample\/population (treatments).\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(3)\u00a0\u00a0 An important and essential point, on which the attention is kept, is that the experimental errors within each block are to be kept as small as practically possible and the variation from block to block as great as possible. In this way all the Population\/sample (treatments) which are assigned to one block, experience the same type of environmental effects, and are, therefore, comparable.<\/p>\r\n&nbsp;\r\n\r\n(4)\u00a0\u00a0 Randomization of population\/sample (treatments) in each block should be afresh.\r\n\r\n&nbsp;\r\n\r\n<strong>Experiments other than the field experiments<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In other types of experiments the replicates can be identified with the sources of variation corresponding with position, time, classification of the experimental units, etc. For instance, if the experimental material is cows, they can be divided into groups according to breed, age, weight, location number etc. If in a herd the other sources of variation except the \u2018lactation number\u2019 are constant the cows can be grouped according to \u2018lactation number\u2019. One group will consist of cows of one \u2018lactation number\u2019, the other group will consist of cows of the 2nd \u2018lactation number\u2019, and so on. Here the \u2018lactation number\u2019 will be the replicates and a cow will be the experimental unit.<\/p>\r\n&nbsp;\r\n\r\n<strong>9.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Randomization of Population\/Sample (Treatments)<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">The sample\/population (treatments) are assigned to the units (plots) within each group (block) entirely at random with the help of random numbers. It is important to note that for every group the randomization should be afresh. The same set of random numbers should not be used for all the groups.<\/p>\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>10.\u00a0 <\/strong><strong>Structure of Analysis of Variance<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">Taking the case of agricultural experiment, if we suppose the number of sample\/population (treatments) to be n and the number of replications to be r, the total\u00a0<span style=\"font-size: 1em;text-align: initial\">number of degrees of freedom will be divided into three parts representing the independent comparisons:<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n(a)\u00a0\u00a0\u00a0 Between blocks\r\n\r\n(b)\u00a0\u00a0 Between sample\/population (treatments)\r\n\r\n(c)\u00a0\u00a0\u00a0 Random variation which provides a basis for the estimation of error. Thus the structure of analysis will\r\n\r\nbe as follows:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-333\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165.png\" alt=\"\" width=\"590\" height=\"209\" \/>\r\n\r\n(r-1)= Degree of freedom between blocks\r\n\r\n&nbsp;\r\n\r\nn-1= Degree of freedom between sample\/population (treatment)\r\n\r\n&nbsp;\r\n\r\n(n-1)(r-1)=Degree of freedom due to Error\r\n\r\n&nbsp;\r\n\r\nnr-1= Total degree of freedom\r\n\r\n&nbsp;\r\n\r\nVB= Mean Sum of Square (Blocks)\r\n\r\n&nbsp;\r\n\r\nVT= Mean Sum of Square (Sample\/Population\/Treatment)\r\n\r\n&nbsp;\r\n\r\nVE= Mean Sum of Square (Error)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Here, since the total number of observations is nr the total degrees of freedom will be (nr-1). Similarly, since the blocks and the sample\/population (treatments) are respectively r and n in number, their corresponding degrees of freedom will be (r-1) and (n-1).<\/p>\r\n&nbsp;\r\n\r\n<strong>11.\u00a0 <\/strong><strong>Standard Errors and Critical Difference<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\nThe standard error of the difference between the sample\/population (treatment) means based on r replications is estimated by the relation\r\n\r\n2\r\n\r\n( .\u00a0 . )\u00a0 \u00a0 = \u221a\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">Where VE is the pooled Error Mean Square.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Critical differences at 5% level of significance<\/span><span style=\"text-align: initial;font-size: 1em\">= ( .\u00a0 . )\u00a0\u00a0\u00a0\u00a0 \u00d7 5%<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">If some sample\/population (treatments) receive extra replications, the general formula for the standard error is<\/p>\r\n1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\r\n\r\n( .\u00a0 . )\u00a0 \u00a0 \u00a0 = \u221a\u00a0\u00a0 ( 1 + 2)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Where, r1 and r2 are the numbers of replications of the sample\/population (treatments) to be compared.<\/p>\r\n&nbsp;\r\n\r\n<strong>12.\u00a0 <\/strong><strong>Advantages of the Randomized Block Design<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">(1)\u00a0\u00a0 When the material is heterogeneous, the residual variance can be reduced by choosing blocks or plots such that the plots within any block are fairly similar.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">(2)\u00a0\u00a0 This design allows of any number of sample\/population (treatments) and any number of replications, but when the number of sample\/population (treatments) is very large (approximately 20 or more) the efficiency of error control decreases.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">(3)\u00a0\u00a0 Although a reduction in the number of replications leads to a larger standard error, yet it furnishes a result of some value at least.<\/p>\r\n&nbsp;\r\n\r\n(4)\u00a0\u00a0 By means of grouping more accurate results are usually obtained than with the completely randomized design.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(5)\u00a0\u00a0 If we find that the experimental error variance is larger for some sample\/population (treatments) than for others, we can obtain an unbiased error for testing any specific combination of the sample\/population (treatment) means.<\/p>\r\n&nbsp;\r\n\r\n<strong>13.\u00a0 <\/strong><strong>Self-Check Exercise with solutions<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">Q.1. The following table gives the yields in pounds per plot, of five varieties of wheat after being applied to each of 4 plots, completely randomized.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-334\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166.png\" alt=\"\" width=\"613\" height=\"222\" \/>\r\n\r\nAnalyse the data and state your conclusions.\r\n\r\n&nbsp;\r\n\r\n<strong>Analysis<\/strong>\r\n\r\n224)2\r\n\r\n= = 2508.8\r\n\r\n. . = (82 + 82 + \u2026 \u2026 \u2026 . +92 + 82) \u2212 . . = 207.2\r\n\r\n. . = (322+ 442+ 642+482+362) \u2212 . . = 155.2\r\n\r\n4\r\n\r\n. . = . . \u2212 . .\r\n\r\n= 207.2 \u2212 155.2 = 52.\r\n\r\n&nbsp;\r\n\r\n<strong>Table for Analysis of Variance<\/strong>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-335\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167.png\" alt=\"\" width=\"657\" height=\"131\" \/>\r\n\r\n&nbsp;\r\n\r\nHere, F-test indicates that there are significant differences between the sample\/population (treatment) means, since the observed value of the variance ratio is highly significant at 0.1% level of significance. Now we wish to know as to which variety is the best and also which varieties show the significant differences among themselves. This can be done with the help of critical difference and confidence interval.\r\n\r\n&nbsp;\r\n\r\nStandard error of the difference between two sample\/population (treatment) means\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">= \u221a 2 = \u221a 2 \u00d7 3.47 = 1.317<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">= ( .\u00a0 . )\u00a0 \u00d7 5% .\u00a0 . = 15<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">= 1.317 \u00d7 2.131 = 2.81<\/span>\r\n\r\n= ( . . ) \u00d7 5% . . = 15\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Here, F-test in the analysis of variance indicates significant differences between the varieties and therefore we are justified in comparing the individual varieties with the help of critical difference.<\/p>\r\n&nbsp;\r\n\r\n<strong>Summary of results<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In agricultural experiments it is advisable to express the results in the commercial units of measurement like pounds per acre, or quintals per hectare, etc. This can be done by calculating the appropriate conversion factor, depending upon the area of each plot and unit of measurement and multiplying each sample\/population (treatment) mean by this conversion factor.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-336\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168.png\" alt=\"\" width=\"669\" height=\"126\" \/>\r\n<p style=\"text-align: justify\">The varieties which do not differ significantly have been underlined by a bar. This method of underlying the sample\/population (treatments) which do not differ significantly is the concise way of indicating the significance and non significance of individual comparisons.<\/p>\r\n&nbsp;\r\n\r\nQ.2. The yields of 6 varieties of crop in lbs., along-with the plan of the experiment, are given bellow. The number of blocks is 5, plot size is 1\/20 acre and the varieties have been represented by A, B, C, D, E and F.\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-337\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169.png\" alt=\"\" width=\"428\" height=\"363\" \/>\r\n\r\nTest the significance of the variation due to strains.\r\n\r\n&nbsp;\r\n\r\n<strong>Analysis<\/strong>\r\n\r\n&nbsp;\r\n\r\nTabulation:\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>Varieties<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>Blocks<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>Variety<\/td>\r\n<td>Variety<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>I<\/td>\r\n<td>II<\/td>\r\n<td>III<\/td>\r\n<td>IV<\/td>\r\n<td>V<\/td>\r\n<td>Totals<\/td>\r\n<td>means<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A<\/td>\r\n<td>20<\/td>\r\n<td>26<\/td>\r\n<td>30<\/td>\r\n<td>28<\/td>\r\n<td>23<\/td>\r\n<td>127<\/td>\r\n<td>25.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B<\/td>\r\n<td>9<\/td>\r\n<td>12<\/td>\r\n<td>10<\/td>\r\n<td>9<\/td>\r\n<td>7<\/td>\r\n<td>47<\/td>\r\n<td>9.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C<\/td>\r\n<td>12<\/td>\r\n<td>15<\/td>\r\n<td>16<\/td>\r\n<td>14<\/td>\r\n<td>14<\/td>\r\n<td>71<\/td>\r\n<td>14.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<\/td>\r\n<td>17<\/td>\r\n<td>10<\/td>\r\n<td>20<\/td>\r\n<td>23<\/td>\r\n<td>20<\/td>\r\n<td>90<\/td>\r\n<td>18.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>E<\/td>\r\n<td>28<\/td>\r\n<td>26<\/td>\r\n<td>23<\/td>\r\n<td>35<\/td>\r\n<td>30<\/td>\r\n<td>142<\/td>\r\n<td>28.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>70<\/td>\r\n<td>62<\/td>\r\n<td>56<\/td>\r\n<td>64<\/td>\r\n<td>75<\/td>\r\n<td>327<\/td>\r\n<td>65.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Totals<\/td>\r\n<td>156<\/td>\r\n<td>151<\/td>\r\n<td>155<\/td>\r\n<td>173<\/td>\r\n<td>169<\/td>\r\n<td>804<\/td>\r\n<td>G.M.=26.8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<strong>Sum of Squares for different sources of variation<\/strong>\r\n\r\n8042\r\n\r\n=\u00a0 30\u00a0 = 21547.2\r\n\r\n.\u00a0 . = (202 + 92 + \u2026 \u2026 \u2026 + 302 + 752) \u2212\u00a0 .\u00a0 . = 10646.8\r\n\r\n.\u00a0 .= (1562 + \u22ef + 1692) \u2212\u00a0 .\u00a0 . = 61.5 6\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">.\u00a0 .= (1272 + \u2026 \u2026 + 3272) \u2212\u00a0 .\u00a0 . = 10167.2 5<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">.\u00a0 .= (\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 .\u00a0 . ) \u2212\u00a0 .\u00a0 . (\u00a0 . ) \u2212\u00a0 .\u00a0 . (\u00a0 )<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">= 10646.8 \u2212 61.5 \u2212 10167.2 = 418.1<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Table of analysis of variance<\/strong>\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>Sources<\/td>\r\n<td>of<\/td>\r\n<td>D.F.<\/td>\r\n<td>S.S.<\/td>\r\n<td>M.S.<\/td>\r\n<td>F (Cal.)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>variation<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>5%<\/td>\r\n<td>1%<\/td>\r\n<td>0.1%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Blocks<\/td>\r\n<td><\/td>\r\n<td>4<\/td>\r\n<td>61.5<\/td>\r\n<td>15.38<\/td>\r\n<td>97.25***<\/td>\r\n<td>2.71<\/td>\r\n<td>4.10<\/td>\r\n<td>6.46<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Varieties<\/td>\r\n<td><\/td>\r\n<td>5<\/td>\r\n<td>10167.2<\/td>\r\n<td>2033.44<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Error<\/td>\r\n<td><\/td>\r\n<td>20<\/td>\r\n<td>418.1<\/td>\r\n<td>20.91<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td><\/td>\r\n<td>29<\/td>\r\n<td>10646.8<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n***Significant at 0.1% level of significance.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">It is clear from the table that this high observed value of F is significant at 0.1% level of significance which proves that there are significant differences between the variety means. Now, we have to test the significance of the difference between the individual varieties, and that will be done with the help of critical difference.<\/p>\r\n&nbsp;\r\n\r\n<strong>Summary<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Randomization, replication and error control are the three main principles of design of experiments.\u00a0\u00a0\u00a0 Randomization\u00a0\u00a0\u00a0 which\u00a0\u00a0\u00a0 defines\u00a0\u00a0 the\u00a0\u00a0\u00a0 manner\u00a0\u00a0\u00a0 of\u00a0\u00a0\u00a0 allocation\u00a0\u00a0\u00a0 of\u00a0\u00a0\u00a0 the sample\/population (treatments) to the experimental units, replication which specifies the number of units to be provided for each of the sample\/population (treatments) and error control which increases the precision by choosing appropriate type of experimental units and also their grouping. We\u00a0 have\u00a0 seen\u00a0 that\u00a0 in\u00a0 a\u00a0 completely\u00a0 randomized\u00a0 design\u00a0 no\u00a0 local\u00a0 control\u00a0 is\u00a0 adopted excepting that the experimental units should be homogenous. Usually when experiments require a large number of experimental units, completely randomized designs cannot ensure\u00a0\u00a0 precision\u00a0\u00a0\u00a0 of\u00a0\u00a0 the\u00a0\u00a0 estimates\u00a0\u00a0 of\u00a0\u00a0 sample\/population\u00a0\u00a0\u00a0 (treatment)\u00a0\u00a0 effects.\u00a0\u00a0 An improvement\u00a0 of completely\u00a0 randomized\u00a0 designs\u00a0 can\u00a0 be\u00a0 obtained\u00a0 by providing\u00a0 error control measures as randomized block design. The error control measures in this design\u00a0<span style=\"text-align: initial;font-size: 1em\">consist of making the units in each of these groups homogeneous. These groups are commonly known as blocks and the experimental units in the blocks are known as plots. This type of homogeneous grouping of the experimental units and the random allocation of sample\/population (treatments) separately in each block are the two main characteristic features of randomized block design.<\/span><\/p>\r\n\r\n<\/div>\r\n<p style=\"text-align: center\"><strong>Learn More:<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li>http:\/\/www.statisticshowto.com\/experimental-design\/<\/li>\r\n \t<li>Chandel, S.R.S. (2006). In: A Handbook of Agricultural Statistics, Anchal Prakashan mandir, Kanpur.<\/li>\r\n \t<li>Sharma, J. K. (2014). In: Business Statistics, II eds., S. Chand &amp; Company, N Delhi.<\/li>\r\n \t<li>http:\/\/stattrek.com\/experiments\/experimental-design.aspx?Tutorial=AP<\/li>\r\n \t<li>Das, M.N. and Giri, N.C. (1991). In: Design and Analysis of Experiments. Wiley Eastern Limited, Second Eds., New Delhi.<\/li>\r\n<\/ol>","rendered":"<div>\n<p>&nbsp;<\/p>\n<p><strong>Learning Objectives:<\/strong><\/p>\n<ul>\n<li>After the completion of this module the student will understand:<\/li>\n<li>An Introduction to Experimental Design<\/li>\n<li>Complete Randomized Design<\/li>\n<li>Randomization of sample\/population (treatments)<\/li>\n<li>Merits and Demerits of Complete Randomized Design<\/li>\n<li>Applications of complete randomization<\/li>\n<li>Randomized Block Design<\/li>\n<li>Advantages of the Randomized Block Design<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>1.<\/strong>\u00a0\u00a0\u00a0 <strong>Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Experimental design is a way to carefully plan experiments in advance so that your results are both objective and valid. Ideally your experimental design should:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogenous blocks before being randomly assigned to groups.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Allow you to make inferences about the relationship between independent variables and dependent variables.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Reduce variability, to make it easier for you to find differences in sample\/population (treatment) outcomes.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The choice of experimental design depends on the number and nature of the sample\/population (treatments) under study. It also depends on the object of the\u00a0<span style=\"font-size: 1em;text-align: initial\">experiment. Another consideration for the choice of the design is the question of available resources.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Some considerations under which the different designs are appropriate are as follows:<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-331\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163.png\" alt=\"\" width=\"613\" height=\"485\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163.png 613w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163-300x237.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163-65x51.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163-225x178.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-163-350x277.png 350w\" sizes=\"auto, (max-width: 613px) 100vw, 613px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Complete Randomized Design<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">This is the simplest type of design in which the whole experimental material is divided into a number of experimental units depending upon the number of sample\/population (treatments) and the number of replications for each. After that the sample\/population(treatments) are allotted to the units entirely by chance. In case of field experiments, the whole field is divided into a required number of equal plots and then the sample\/population (treatments) are randomized in those plots.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">If there are 5 sample\/population (treatments) A, B, C, D and E and 4 replication to each, the number of plots will be 20 and each sample\/population (treatment) will be allotted to four plots selected at random by means of random numbers.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">3.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Randomization of sample\/population (treatments)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Here since the number of units is 20, a two digit random number table will be consulted and a series of 20 random numbers will be taken excluding those which are greater than 20. Suppose the random numbers are 4, 18, 2, 14, 3, 7, 13, 1, 6, 10, 17, 20, 8, 15, 11, 5, 9, 12, 16, 19.<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">After this the plots will be serially numbered and the sample\/population (treatment) A will be allotted to the plots bearing the serial numbers 4, 18, 2 and 14, sample\/population (treatment) B will be allotted to the plots bearing the serial numbers 3, 7, 13 and 1, and so on for the other sample\/population (treatments) C.D. and E.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>In a similar way the randomization can be done for any number of sample\/population (treatments).<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>4.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Structure of Analysis of Variance<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">If we suppose the number of sample\/population (treatments) to be n, and the number of replications to be r for every sample\/population (treatment), the total number of experimental units will be nr = N. If the sample\/population (treatments) have varying number of sample\/population (treatments), r1, r2, r3\u2026\u2026..rn then the total number of units (=N) will be given by<\/p>\n<p>&nbsp;<\/p>\n<p>N\u00a0\u00a0 = r1 + r2 + r3 \u2026\u2026+rn Whereas r= number of replication<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In this design, the total number of degrees of freedom will be divided into two parts representing the independent comparisons. These two will be the two independent sources of variation.<\/p>\n<p>&nbsp;<\/p>\n<p>(a)\u00a0\u00a0\u00a0 Between sample\/population (treatments)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(b)\u00a0\u00a0 Within sample\/population (treatments) (Within sample\/population\/treatment components provides a basis for the estimation of error)<\/p>\n<p>&nbsp;<\/p>\n<p>Thus the structure of analysis of variance will be as follows:<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-332\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164.png\" alt=\"\" width=\"570\" height=\"174\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164.png 570w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164-300x92.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164-65x20.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164-225x69.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-164-350x107.png 350w\" sizes=\"auto, (max-width: 570px) 100vw, 570px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Whereas n= number of sample\/population (treatments)<\/p>\n<p>N= Total number of observations<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Here, since the total number of observations is N, the total degrees of freedom will be (N-1). Similarly, population\/sample (treatment) degrees of freedom will be (n-1), one less than the number of population\/sample (treatments), and the remaining (N-n) degrees of freedom will be for \u2018Within sample\/population (Treatments)\u2019 or \u2018Error\u2019.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>5.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Standard Errors<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">The standard error of the difference between the two population\/sample (treatment) means based on r1 and r2 replications is estimated by the following relation:<\/p>\n<p>1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1<\/p>\n<p>( . \u00a0. )\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 . = \u221a\u00a0\u00a0 ( 1 + 2)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Where <strong>V<\/strong><strong>E<\/strong> <strong>is the pooled error variance or error mean square.<\/strong> If r1 = r2= r the formula reduces to \u221a2VrE\u00a0 . The degree of freedom for t-test are the error degrees of freedom.<\/p>\n<p>&nbsp;<\/p>\n<p>Where r is the number of replication.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>6.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Merits and Demerits of Complete Randomized Design Merits<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">(i)\u00a0 In this design any number of sample\/population (treatments) and replicates may be used. The number of replicates may be used. The number of replicates can also be varies at will form population to population (treatment to treatment).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(ii) The statistical analysis of the data is very easy and it remains easy even if the numbers of replicates are not the same for all the sample\/population (treatments).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(iii) The method of analysis remains simple when the results from some units are missing or rejected<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">(iv) The relative loss of information due to missing data is smaller than that with any other design.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v) This design is especially useful in small experiments where the supply of experimental material is scarce and homogeneous, as the whole of the material is utilized in the experiment.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(vi) The design provides maximum number of degrees of freedom for the estimation of error as compared with other design, for a given number of sample\/population\/treatments and a given number of experimental units.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Demerits<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">There is one and the main objection in this design and that is on the grounds of accuracy. Since there is no restriction on the randomization of the population\/sample (treatments), we cannot be sure about the fact that the units receiving one population\/sample (treatment) are similar to those receiving the other population\/sample (treatment) and, therefore, the whole of the variation among the units enters into the experimental error.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>7.\u00a0<\/strong><strong>Applications of Complete Randomization<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>This design is especially advantageous and appropriate under the following circumstances:<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>(i)\u00a0 Where the experimental material is limited in quantity and homogeneous.<\/p>\n<p>&nbsp;<\/p>\n<p>(ii) Where it is expected that some of the units will be destroyed or will fail to respond.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(iii) In small experiments where the increased accuracy from the alternative design is not sufficient to exceed in importance the loss of error degrees of freedom.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>8.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Randomized Block Design<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">In this design the whole experimental material is divided into homogeneous groups, each of which constitutes a single replication. Each of these groups is further divided into a number of experimental units which are equal in all respects. The sample\/population (treatments) are applied to these units by any random process. In case of field experiments, if it is observed that the fertility gradient of the field is in one direction, the whole field may be divided into a number of equal plots. The number of plots in each\u00a0<span style=\"font-size: 1em;text-align: initial\">block is equal to the number of sample\/population (treatments), so that each block is a replicate of each sample\/population (treatment).<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>The following important points are to be kept in mind for this design:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(1) In this design the number of blocks must be equal to the number of replications fixed for each sample\/population (treatment).<\/p>\n<p>&nbsp;<\/p>\n<p>(2)\u00a0\u00a0 The number of plots in each block should be equal to the number of sample\/population (treatments).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(3)\u00a0\u00a0 An important and essential point, on which the attention is kept, is that the experimental errors within each block are to be kept as small as practically possible and the variation from block to block as great as possible. In this way all the Population\/sample (treatments) which are assigned to one block, experience the same type of environmental effects, and are, therefore, comparable.<\/p>\n<p>&nbsp;<\/p>\n<p>(4)\u00a0\u00a0 Randomization of population\/sample (treatments) in each block should be afresh.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Experiments other than the field experiments<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In other types of experiments the replicates can be identified with the sources of variation corresponding with position, time, classification of the experimental units, etc. For instance, if the experimental material is cows, they can be divided into groups according to breed, age, weight, location number etc. If in a herd the other sources of variation except the \u2018lactation number\u2019 are constant the cows can be grouped according to \u2018lactation number\u2019. One group will consist of cows of one \u2018lactation number\u2019, the other group will consist of cows of the 2nd \u2018lactation number\u2019, and so on. Here the \u2018lactation number\u2019 will be the replicates and a cow will be the experimental unit.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>9.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Randomization of Population\/Sample (Treatments)<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">The sample\/population (treatments) are assigned to the units (plots) within each group (block) entirely at random with the help of random numbers. It is important to note that for every group the randomization should be afresh. The same set of random numbers should not be used for all the groups.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>10.\u00a0 <\/strong><strong>Structure of Analysis of Variance<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">Taking the case of agricultural experiment, if we suppose the number of sample\/population (treatments) to be n and the number of replications to be r, the total\u00a0<span style=\"font-size: 1em;text-align: initial\">number of degrees of freedom will be divided into three parts representing the independent comparisons:<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>(a)\u00a0\u00a0\u00a0 Between blocks<\/p>\n<p>(b)\u00a0\u00a0 Between sample\/population (treatments)<\/p>\n<p>(c)\u00a0\u00a0\u00a0 Random variation which provides a basis for the estimation of error. Thus the structure of analysis will<\/p>\n<p>be as follows:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-333\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165.png\" alt=\"\" width=\"590\" height=\"209\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165.png 590w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165-300x106.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165-65x23.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165-225x80.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-165-350x124.png 350w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/p>\n<p>(r-1)= Degree of freedom between blocks<\/p>\n<p>&nbsp;<\/p>\n<p>n-1= Degree of freedom between sample\/population (treatment)<\/p>\n<p>&nbsp;<\/p>\n<p>(n-1)(r-1)=Degree of freedom due to Error<\/p>\n<p>&nbsp;<\/p>\n<p>nr-1= Total degree of freedom<\/p>\n<p>&nbsp;<\/p>\n<p>VB= Mean Sum of Square (Blocks)<\/p>\n<p>&nbsp;<\/p>\n<p>VT= Mean Sum of Square (Sample\/Population\/Treatment)<\/p>\n<p>&nbsp;<\/p>\n<p>VE= Mean Sum of Square (Error)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Here, since the total number of observations is nr the total degrees of freedom will be (nr-1). Similarly, since the blocks and the sample\/population (treatments) are respectively r and n in number, their corresponding degrees of freedom will be (r-1) and (n-1).<\/p>\n<p>&nbsp;<\/p>\n<p><strong>11.\u00a0 <\/strong><strong>Standard Errors and Critical Difference<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>The standard error of the difference between the sample\/population (treatment) means based on r replications is estimated by the relation<\/p>\n<p>2<\/p>\n<p>( .\u00a0 . )\u00a0 \u00a0 = \u221a<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">Where VE is the pooled Error Mean Square.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Critical differences at 5% level of significance<\/span><span style=\"text-align: initial;font-size: 1em\">= ( .\u00a0 . )\u00a0\u00a0\u00a0\u00a0 \u00d7 5%<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">If some sample\/population (treatments) receive extra replications, the general formula for the standard error is<\/p>\n<p>1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1<\/p>\n<p>( .\u00a0 . )\u00a0 \u00a0 \u00a0 = \u221a\u00a0\u00a0 ( 1 + 2)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Where, r1 and r2 are the numbers of replications of the sample\/population (treatments) to be compared.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>12.\u00a0 <\/strong><strong>Advantages of the Randomized Block Design<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">(1)\u00a0\u00a0 When the material is heterogeneous, the residual variance can be reduced by choosing blocks or plots such that the plots within any block are fairly similar.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(2)\u00a0\u00a0 This design allows of any number of sample\/population (treatments) and any number of replications, but when the number of sample\/population (treatments) is very large (approximately 20 or more) the efficiency of error control decreases.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(3)\u00a0\u00a0 Although a reduction in the number of replications leads to a larger standard error, yet it furnishes a result of some value at least.<\/p>\n<p>&nbsp;<\/p>\n<p>(4)\u00a0\u00a0 By means of grouping more accurate results are usually obtained than with the completely randomized design.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(5)\u00a0\u00a0 If we find that the experimental error variance is larger for some sample\/population (treatments) than for others, we can obtain an unbiased error for testing any specific combination of the sample\/population (treatment) means.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>13.\u00a0 <\/strong><strong>Self-Check Exercise with solutions<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">Q.1. The following table gives the yields in pounds per plot, of five varieties of wheat after being applied to each of 4 plots, completely randomized.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-334\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166.png\" alt=\"\" width=\"613\" height=\"222\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166.png 613w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166-300x109.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166-65x24.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166-225x81.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-166-350x127.png 350w\" sizes=\"auto, (max-width: 613px) 100vw, 613px\" \/><\/p>\n<p>Analyse the data and state your conclusions.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Analysis<\/strong><\/p>\n<p>224)2<\/p>\n<p>= = 2508.8<\/p>\n<p>. . = (82 + 82 + \u2026 \u2026 \u2026 . +92 + 82) \u2212 . . = 207.2<\/p>\n<p>. . = (322+ 442+ 642+482+362) \u2212 . . = 155.2<\/p>\n<p>4<\/p>\n<p>. . = . . \u2212 . .<\/p>\n<p>= 207.2 \u2212 155.2 = 52.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Table for Analysis of Variance<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-335\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167.png\" alt=\"\" width=\"657\" height=\"131\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167.png 657w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167-300x60.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167-65x13.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167-225x45.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-167-350x70.png 350w\" sizes=\"auto, (max-width: 657px) 100vw, 657px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Here, F-test indicates that there are significant differences between the sample\/population (treatment) means, since the observed value of the variance ratio is highly significant at 0.1% level of significance. Now we wish to know as to which variety is the best and also which varieties show the significant differences among themselves. This can be done with the help of critical difference and confidence interval.<\/p>\n<p>&nbsp;<\/p>\n<p>Standard error of the difference between two sample\/population (treatment) means<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">= \u221a 2 = \u221a 2 \u00d7 3.47 = 1.317<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">= ( .\u00a0 . )\u00a0 \u00d7 5% .\u00a0 . = 15<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">= 1.317 \u00d7 2.131 = 2.81<\/span><\/p>\n<p>= ( . . ) \u00d7 5% . . = 15<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Here, F-test in the analysis of variance indicates significant differences between the varieties and therefore we are justified in comparing the individual varieties with the help of critical difference.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Summary of results<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In agricultural experiments it is advisable to express the results in the commercial units of measurement like pounds per acre, or quintals per hectare, etc. This can be done by calculating the appropriate conversion factor, depending upon the area of each plot and unit of measurement and multiplying each sample\/population (treatment) mean by this conversion factor.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-336\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168.png\" alt=\"\" width=\"669\" height=\"126\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168.png 669w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168-300x57.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168-225x42.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-168-350x66.png 350w\" sizes=\"auto, (max-width: 669px) 100vw, 669px\" \/><\/p>\n<p style=\"text-align: justify\">The varieties which do not differ significantly have been underlined by a bar. This method of underlying the sample\/population (treatments) which do not differ significantly is the concise way of indicating the significance and non significance of individual comparisons.<\/p>\n<p>&nbsp;<\/p>\n<p>Q.2. The yields of 6 varieties of crop in lbs., along-with the plan of the experiment, are given bellow. The number of blocks is 5, plot size is 1\/20 acre and the varieties have been represented by A, B, C, D, E and F.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-337\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169.png\" alt=\"\" width=\"428\" height=\"363\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169.png 428w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169-300x254.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169-65x55.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169-225x191.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-169-350x297.png 350w\" sizes=\"auto, (max-width: 428px) 100vw, 428px\" \/><\/p>\n<p>Test the significance of the variation due to strains.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Analysis<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Tabulation:<\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td>Varieties<\/td>\n<td><\/td>\n<td><\/td>\n<td>Blocks<\/td>\n<td><\/td>\n<td><\/td>\n<td>Variety<\/td>\n<td>Variety<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>I<\/td>\n<td>II<\/td>\n<td>III<\/td>\n<td>IV<\/td>\n<td>V<\/td>\n<td>Totals<\/td>\n<td>means<\/td>\n<\/tr>\n<tr>\n<td>A<\/td>\n<td>20<\/td>\n<td>26<\/td>\n<td>30<\/td>\n<td>28<\/td>\n<td>23<\/td>\n<td>127<\/td>\n<td>25.4<\/td>\n<\/tr>\n<tr>\n<td>B<\/td>\n<td>9<\/td>\n<td>12<\/td>\n<td>10<\/td>\n<td>9<\/td>\n<td>7<\/td>\n<td>47<\/td>\n<td>9.4<\/td>\n<\/tr>\n<tr>\n<td>C<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<td>16<\/td>\n<td>14<\/td>\n<td>14<\/td>\n<td>71<\/td>\n<td>14.2<\/td>\n<\/tr>\n<tr>\n<td>D<\/td>\n<td>17<\/td>\n<td>10<\/td>\n<td>20<\/td>\n<td>23<\/td>\n<td>20<\/td>\n<td>90<\/td>\n<td>18.0<\/td>\n<\/tr>\n<tr>\n<td>E<\/td>\n<td>28<\/td>\n<td>26<\/td>\n<td>23<\/td>\n<td>35<\/td>\n<td>30<\/td>\n<td>142<\/td>\n<td>28.4<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>70<\/td>\n<td>62<\/td>\n<td>56<\/td>\n<td>64<\/td>\n<td>75<\/td>\n<td>327<\/td>\n<td>65.4<\/td>\n<\/tr>\n<tr>\n<td>Totals<\/td>\n<td>156<\/td>\n<td>151<\/td>\n<td>155<\/td>\n<td>173<\/td>\n<td>169<\/td>\n<td>804<\/td>\n<td>G.M.=26.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><strong>Sum of Squares for different sources of variation<\/strong><\/p>\n<p>8042<\/p>\n<p>=\u00a0 30\u00a0 = 21547.2<\/p>\n<p>.\u00a0 . = (202 + 92 + \u2026 \u2026 \u2026 + 302 + 752) \u2212\u00a0 .\u00a0 . = 10646.8<\/p>\n<p>.\u00a0 .= (1562 + \u22ef + 1692) \u2212\u00a0 .\u00a0 . = 61.5 6<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">.\u00a0 .= (1272 + \u2026 \u2026 + 3272) \u2212\u00a0 .\u00a0 . = 10167.2 5<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">.\u00a0 .= (\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 .\u00a0 . ) \u2212\u00a0 .\u00a0 . (\u00a0 . ) \u2212\u00a0 .\u00a0 . (\u00a0 )<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">= 10646.8 \u2212 61.5 \u2212 10167.2 = 418.1<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Table of analysis of variance<\/strong><\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td>Sources<\/td>\n<td>of<\/td>\n<td>D.F.<\/td>\n<td>S.S.<\/td>\n<td>M.S.<\/td>\n<td>F (Cal.)<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>variation<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>5%<\/td>\n<td>1%<\/td>\n<td>0.1%<\/td>\n<\/tr>\n<tr>\n<td>Blocks<\/td>\n<td><\/td>\n<td>4<\/td>\n<td>61.5<\/td>\n<td>15.38<\/td>\n<td>97.25***<\/td>\n<td>2.71<\/td>\n<td>4.10<\/td>\n<td>6.46<\/td>\n<\/tr>\n<tr>\n<td>Varieties<\/td>\n<td><\/td>\n<td>5<\/td>\n<td>10167.2<\/td>\n<td>2033.44<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Error<\/td>\n<td><\/td>\n<td>20<\/td>\n<td>418.1<\/td>\n<td>20.91<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td><\/td>\n<td>29<\/td>\n<td>10646.8<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>***Significant at 0.1% level of significance.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It is clear from the table that this high observed value of F is significant at 0.1% level of significance which proves that there are significant differences between the variety means. Now, we have to test the significance of the difference between the individual varieties, and that will be done with the help of critical difference.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Summary<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Randomization, replication and error control are the three main principles of design of experiments.\u00a0\u00a0\u00a0 Randomization\u00a0\u00a0\u00a0 which\u00a0\u00a0\u00a0 defines\u00a0\u00a0 the\u00a0\u00a0\u00a0 manner\u00a0\u00a0\u00a0 of\u00a0\u00a0\u00a0 allocation\u00a0\u00a0\u00a0 of\u00a0\u00a0\u00a0 the sample\/population (treatments) to the experimental units, replication which specifies the number of units to be provided for each of the sample\/population (treatments) and error control which increases the precision by choosing appropriate type of experimental units and also their grouping. We\u00a0 have\u00a0 seen\u00a0 that\u00a0 in\u00a0 a\u00a0 completely\u00a0 randomized\u00a0 design\u00a0 no\u00a0 local\u00a0 control\u00a0 is\u00a0 adopted excepting that the experimental units should be homogenous. Usually when experiments require a large number of experimental units, completely randomized designs cannot ensure\u00a0\u00a0 precision\u00a0\u00a0\u00a0 of\u00a0\u00a0 the\u00a0\u00a0 estimates\u00a0\u00a0 of\u00a0\u00a0 sample\/population\u00a0\u00a0\u00a0 (treatment)\u00a0\u00a0 effects.\u00a0\u00a0 An improvement\u00a0 of completely\u00a0 randomized\u00a0 designs\u00a0 can\u00a0 be\u00a0 obtained\u00a0 by providing\u00a0 error control measures as randomized block design. The error control measures in this design\u00a0<span style=\"text-align: initial;font-size: 1em\">consist of making the units in each of these groups homogeneous. These groups are commonly known as blocks and the experimental units in the blocks are known as plots. This type of homogeneous grouping of the experimental units and the random allocation of sample\/population (treatments) separately in each block are the two main characteristic features of randomized block design.<\/span><\/p>\n<\/div>\n<p style=\"text-align: center\"><strong>Learn More:<\/strong><\/p>\n<ol>\n<li>http:\/\/www.statisticshowto.com\/experimental-design\/<\/li>\n<li>Chandel, S.R.S. (2006). In: A Handbook of Agricultural Statistics, Anchal Prakashan mandir, Kanpur.<\/li>\n<li>Sharma, J. K. (2014). In: Business Statistics, II eds., S. Chand &amp; Company, N Delhi.<\/li>\n<li>http:\/\/stattrek.com\/experiments\/experimental-design.aspx?Tutorial=AP<\/li>\n<li>Das, M.N. and Giri, N.C. (1991). In: Design and Analysis of Experiments. Wiley Eastern Limited, Second Eds., New Delhi.<\/li>\n<\/ol>\n","protected":false},"author":3,"menu_order":30,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-pankaj-madan"],"pb_section_license":""},"chapter-type":[],"contributor":[61],"license":[],"class_list":["post-327","chapter","type-chapter","status-publish","hentry","contributor-prof-pankaj-madan"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/327","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/327\/revisions"}],"predecessor-version":[{"id":339,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/327\/revisions\/339"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/327\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/media?parent=327"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapter-type?post=327"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/contributor?post=327"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/license?post=327"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}