STAT IN RESEARCH II
STAT IN RESEARCH II STAT 652
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Chapter 2 Randomized Complete Block Design 21 Randomized Complete Block Design 211 Examples When examining the effect of a factor it is often helpful to remove the effect of excess variation through the use of blocking A blocking variable is one that may affect the variation of the response but is unrelated to the primary hypothesis of interest The desired result is to have homogeneous experimental units within each block so that when the blocking effect is removed through modeling all individuals can be considered homogeneous before receiving the treatments The term randomized complete block design usually refers to a design where there is a single fixed factor of interest and a single random blocking effect The number of experimental units in each block is such that within each block each of the treatments can be randomly assigned the same number of times Some examples of a randomized complete block designs follow Examples of Randomized Complete Block Design Snakebite Venom To compare the immune response of mice to the venom of four poisonous snakes venom is taken from adult male coral copperhead sidewinder and pit viper snakes One mouse from each of seven litters are randomly assigned to the four snakes Each receives minute amounts of the venom of the corresponding snake by injection The increase in antibody activity as measured from a blood sample is the response The four treatments that constitute the fixed factor are the four snakes of interest The seven litters represent all litters and thus make up a random effect Litter is a blocking effect since it is not of primary interest to determine the variation in antibody activity between litters but instead to remove the added variation that comes with differing litters Recyclable Scrap Metal Before beginning a fullscale operation to promote recycling of metals in a large county a recycling agency conducts a study to compare recycling opportunities in the six largest cities of that county The primary question of interest is which of the cities should be the major focus of the agency To answer this question the agency wishes to compare the amount of scrap metal wasted by individuals in each of the cities Ten days of the year are randomly selected for scrap metal examination On each of the ten days one randomly chosen garbage truck load of equal size from each city is scoured for recyclable scrap metal The material is then weighed for each load Only the six cities are of interest These make up the fixed factor The day of collection is a random effect since it is a sample of all possible days Further day of collection is a blocking effect since the variation among days is not of primary interest Rice fertilizer A rice farmer has a choice among four fertilizers To compare the fertilizers he randomly selects four rows of his field which have been planted with the same seed The plants on a particular row can be expected to have identical environmental conditions ie sunlight water etc Each row is divided into four segments The four fertilizers are randomly assigned to the four segments of each row Table 21 Randomized complete block design set up for rice fertilizer example Segment Row 1 F1 F4 F3 F2 Row 2 F4 F3 F1 F2 Row 3 F4 F1 F3 F2 Row 4 F3 F2 F4 F1 One plant from each rowsegment combination is chosen The response of interest is the length of the fruiting period measured in days The four fertilizers constitute the fixed factor The row is the random block since the rows represent all other rows 212 Model The model used for a randomized complete block design is Y1 u04 Bjsw i1a j1b where u is the true overall mean a is the true fixed effect of the ith treatment of the fixed factor B N N0o is the true random effect of the jth block and 51 N N0oz is the true error for each individual of the jth block receiVing the ith treatment In SPSS the last effect an is constrained to be zero to be used as a baseline for the other effects 213 Example Data Set The simulated results of the rice fertilizer example are as follows measurements are lengths of fruiting period in days Table 22 Lengths of fruiting period for each of the fertilizerrow combinations for the rice fertilizer example Segment 1 2 3 4 Row 1 F1 1336 F4 1601 F3 1566 F21436 Row2 F4 1627 F3 1543 F1 1354 F2 1475 Row3 F4 1360 F11153 F3 1306 F2 1237 Row4 F3 1627 F2 1343 F4 1684 F1 1460 etc See the video RCB for the setup and discussion of the analyses 214 Hypotheses Testing In this example we are interested in testing if there is an effect due to Fertilizer Thus the null hypothesis is H0 a1 a2 a3 a4 The alternative hypothesis is that at least 1 pair of the 1 s are not equal We could test H0 0 0 However this factor is not important other than reducing the overall variability The F ratio and signi cance value for testing H0 a1 a2 05 a are 178 and 000 Thus we will reject the null hypothesis and say that there are differences in the Fertilizers The pairwise comparisons indicate that all of the populations unequal The best estimates of o2 and o are 2907 and 1403 where as the true values were 25 and 1 215 Simulation Using the RCBsps file you can increase the number of blocks and see what effect that has on your estimates of o2 and 6123 Then increase the number of treatments and see what effect that has Keeping the number of blocks and treatments at the original level change the variance of o2 and o to 1 and 3 and see what effect occurs Try different combinations 216 Matrix Notation In this example the matrix notation is lengthm mmixxhm 15mm m Where length I 47 X 4M 47 Z 4quot and the Variance of Zb is 000 000 000 000 000 000 000 000 VZb 000 000 000 000 111 111 111 111 D ID ID Il OOOOOOOOOOOO OOOOOOOOOOOOl D ID I Now VY MZbV and since this is a symmetric matrix we will give the lower triangle part of that matrix Given the VY below we see that 2 f 3939v COTYWYWFFb or J 0 forjij In other words Y s in the same block are correlated Thus VY is aquot 039 039 aquot 039 039 of aquot 039 039 aquot aquot aquot 039 0 0 0 a a a 0 0 0 0 a a a 0 0 0 0 a a a a 0 0 0 0 a 173 030 7 a 0 0 0 0 0 0 0 o a a 0 0 0 0 0 0 0 0 a a a 0 0 0 0 0 0 0 0 a a a a 0 0 0 0 0 0 0 0 a a a a a 0 0 0 0 0 0 0 0 0 0 0 o a a 0 0 0 0 0 0 0 0 0 0 0 0 a a a 0 0 0 0 0 0 0 0 0 0 0 0 a a a 7 0 0 0 0 0 0 0 0 0 0 0 0 a a a 22 Randomized Complete Block Design with Subsampling 221 Examples Subsampling in the randomized complete block design occurs when there is more than one individual in each treatmentblock combination Examples of Randomized Complete Block Design with Subsampling Internet Advertising An intemet advertising company wishes to compare worldwide intemet usage time for four age groups lt 20 years 2040 years 4060 years and gt 60 years There are many factors which may also in uence intemet usage time but in this case the only other easily selected information about the individuals surveyed is the country of use The company selects ve countries that they expect will represent most other countries well A question about intemet usage time is sent to twenty individuals within each age category of each country The average daily intemet usage time of each individual is the response The factor of interest is age Since the only age levels of interest are the four age groups 2 039 o39bo39 s considered this factor is fixed Additional variation is removed by considering country of use This is the blocking effect Because the five countries represent all countries it is a random blocking effect Individuals within each agecountry combination are assumed to be homogeneous The result is a randomized complete block design with subsampling since there are 20 individuals in each agecountry combination Rice fertilizer Consider the rice fertilizer example of Section 211 Suppose that instead of sampling a single plant from each fertilizerrow segment three plants are sampled from each segment The three samples in each segment are subsamples 222 Model The model used for a randomized complete block design with subsamples is KjkyqBj77UsUk 139 a jlb kln where u is the true overall mean a is the true fixed effect of the ith treatment of the fixed factor B N N0o is the true random effect of the jth block 771 N N0039 is the random effect of each treatmentblock combination andsyk N00392 is the true error for the kth individual of the jth block receiving the ith treatment Again we assume that an is constrained to be zero 223 Example Data Set If we sample three plants from each segment in the rice fertilizer example the simulated results are as follows measurements are lengths of fruiting period in days Table 23 Three lengths of fruiting period for each of the fertilizerrow combinations for the rice fertilizer example Segment 1 2 3 4 Row 1 F4 F3 F2 137 140 145 166172162 148164143 126125116 Row 2 F3 F1 F2 175 174 181 162 157 155 153154159 140 147 144 Row 3 F1 F3 F2 149152149 125118116 156150148 153 147 147 Row 4 F2 F4 F1 143 134 139 89 87 95 135141139 123 123 130 etc See the video RCBisub for the setup and discussion of the analyses 224 Hypotheses Testing Again in this example we are interested in testing if there is an effect due to Fertilizer Thus the null hypothesis isH0 a1 a2 a3 a4 The alternative hypothesis is that at least 1 pair ofthe 1 s are not equal We could test H0 0 0 anng 2 0 However these factors are not important other than reducing the overall variability The F ratio and signi cance value for testing H0 a1 a2 a3 Q are 506 and 025 Thus we will reject the null hypothesis and say that there are differences in the Fertilizers The pairwise comparisons indicate that all of the populations unequal The best estimates of oz of3 and 039 are 21 170 and 153 where as the true values were 25 1 and 4 225 Simulation Using the RCBisubsps file you can increase the number of blocks and see what effect that has on your estimates of o2 o and a Then increase the number of treatments and see what effect that has Keeping the number of blocks and treatments at the original level change the variance of 02 cg and a to 1 and 3 and 2 and see what effect occurs Try dAfferent combmauons 226 IVLatrix Nntztinn combmauon Here the model is leng39hm Wm WQ 312 a m quotm m m Now EHETiFEH iR uw ii am 7 7 7 7 EH an mm mm andthe Variance ofZb1s 11110000 11110000 11110000 VZb111100000 00001111 00001111 00001111 00001111 and the Variance ofZd is 11000000 11000000 00110000 V1261 0011000072 00001100 1 00001100 00000011 00000011 Now VY VZb VZdV and since this is a symmetric matrix we will give the lower triangle part of that matrix Given the VY below we see that 2 f 3939v 52 add some more In other words Y s in the same block are correlated and Y s in the same FertilizerRow combination are correlated 505020 5w 0 20 0 50 50 30 30 29 0 130 1430 59 30 0 130 50 20 0 3 a q Zozgzo 0 0 0 0 a q Z9 Z0 3 a q Zozgzo 0000 q Z0 q Z0 3 a q Lt q Z9 Z9 Z0 Z9 Z0 3 a q 10 10 10 DA