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 anodized 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 full-scale 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 tour rows tot his title which nave been planted wit n the same s The plants on a particular row can be expected to have identical environmental conditions, I. E. , sunlight, water, etc. Each row is divided into four segments. The four fertilizers are randomly assigned to the four segments of each row. Table 2. 1 : Randomized complete block design set up for rice fertilizer example. Segment 2 3 4 Row 1 ROW 2 ROW 3 Row 4 One plant from each row/segment 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. 2. 1. 2. Model The model used for a randomized complete block design is where is the true overall mean, is the true fixed effect of the tit treatment of the axed factor, is the true random effect of the Jet block, and is the true error for each individual of the Jet block receiving the tit treatment. In SIPS, the last effect is constrained to be zero to be used as a baseline for the other effects. 2. 1. Example Data Set: The simulated results of the rice fertilizer example are as follows (measurements are lengths of fruiting period in days). Table 2. 2: Lengths of fruiting period for each of the fertilizer/row combinations for the rice fertilizer example Segment Fl: 1336 IF: 16. 27 IF: 13. 60 a: 16. 27 A: 16. 01 a: 1543 M :11. 53 a: 1343 a: 15. 66 Fl: 13. 54 a: 13. 06 IF: 16. 84 IF:14. 36 a: 14. 75 a: 12. 37 Fl: 14. 60 etc. See the video Testing RCA tort the setup and discussion tot the analyses. 2 1 4 Hypotheses In this example, we are interested in testing if there is an effect due to Fertilizer.
Thus the null hypothesis is: . The alternative hypothesis is that at least 1 pair of the are not equal. We could test: . However, this factor is not important other than reducing the overall variability. The F ratio and significance value for testing are 17. 8 and . 000. Thus, we will reject the null hypothesis and say that there are differences in the Fertilizers. The appraise comparisons indicate that all of the populations unequal. The best estimates of and are . 2907 and 1. 403; where as the true values were . 25 and 1. 2. 1. 5 Simulation Using the RCA. As file, you can increase the number of blocks and see what effect that has on your estimates of and . 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 and to 1 and 3 and see what effect occurs. Try different combinations. 2. 1. 6 Matrix Notation In this example, the matrix notation is: here: length x z and the Variance of (CB) is: Now and since this is a symmetric matrix we will give the lower triangle part of that matrix.
Given the V(Y) below, we see that In other words, Yes in the same block are correlated. Thus V(Y) 2. 2 Randomized Complete Block Design with Subliming 2. 2. 1 Examples Subliming in the randomized complete block design occurs when there is more than one individual in each treatment/block combination. Examples of Randomized Complete Block Design with Subliming Internet Advertising An internet advertising company wishes to compare worldwide internet usage time or four age groups: < 20 years, 20-40 years, 40-60 years, and > 60 years.
There are many doctors which may also intelligence internet usage time, but in this case the only other easily selected information about the individuals surveyed is the country of use. The company selects five countries that they expect will represent most other countries well. A question about internet usage time is sent to twenty individuals within each age category of each country. The average daily internet usage time of each individual is the response. The factor of interest is age. Since the only age levels f interest are the four age groups 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 age/country combination are assumed to be homogeneous. The result is a randomized complete block design with subliming since there are 20 individuals in each age/country combination. Rice fertilizer Consider the rice fertilizer example of Section 2. 1. 1 . Suppose that instead of sampling a single plant from each fertilizer/row segment, three plants are sampled room each segment.
The three samples in each segment are subleases. 2. 2. 2. Model The model used for a randomized complete block design with subleases is fixed factor, is the true random effect of the Jet block, is the random effect of each treatment/block combination, and is the true error for the kith individual of the Jet block receiving the tit treatment. Again, we assume that is constrained to be zero. 2. 2. 3 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 2. Three lengths of fruiting period for each of the fertilizer/row combinations for the rice fertilizer example Segment 13. 7, 14. 0, 14. 5 17. 5, 17. 4, 181 14. 9, 15. 2, 14. 9 16. 6, 17. 2, 16. 2 14. 8, 16. 4, 14. 3 16. 2, 15. 7, 15. 5 15. 3, 15. 4, 15. 9 12. 5, 11. 8, 11. 6 15. 6, 15. 0, 14. 8 12. 6, 12. 5, 11. 6 14. 0, 14. 7, 14. 4 15. 3, 14. 7, 14. 7 14. 3, 13. 4, 13. 9 8. 9, 8. 7, 9. 5 13. 5, 14. 1, 13. 9 12. 3, 12. 3, 13. 0 etc. See the video RCA_sub for the setup and discussion of the analyses. 2. 2. 4 Hypotheses Again, in this example, we are interested in testing I t there is an detect due to
Fertilizer. Thus the null hypothesis is:. The alternative hypothesis is that at least 1 pair of the are not equal. We could test: and. However, these factors are not important other than reducing the overall variability. The F ratio and significance value for testing are 5. 06 and . 025. Thus, we will reject the null hypothesis and say that there are differences in the Fertilizers. The appraise comparisons indicate that all of the populations unequal. The best estimates of, and are . 21 , 1. 70 and 1. 53; where as the true values were . 25, 1 and 4. 2. 2. 5 Simulation Using the RCA_sub. S file, you can increase the number of blocks and see what effect that has on your estimates of , and . 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, and to 1 and 3 and 2 and see what effect occurs. Try different combinations. 2. 2. 6 Matrix Notation To illustrate this we will use 2 Fertilizers, 2 Rows and 2 Plants per Fertilizer, Row combination. Here the model is: Now: and the Variance of 0 is: add some more: In other words, Yes in the same block are correlated and Yes in the same Fertilizer/Row combination are correlated.