III Introduction to experimental designs

Transcription

1 III Introduction to experimental designs 1 Terminology Definition 24. Experiment. An investigation in which the investigator applies some treatments to experimental units and then observes the effect of the treatments on the experimental units by measuring one or more response variables. Definition 25. Treatment. A condition or set of conditions applied to experimental units in an experiment. Definition 26. Experimental Unit. The physical entity to which a treatment is randomly assigned and independently applied. Definition 27. Observational Unit. The unit on which a response variable is measured. There is often a one-to-one correspondence between experimental units and observational units, but that is not always true. 52

2 TERMINOLOGY 53 L L L L M M M M H H H H Figure III.1: Exercise 10: design of the soil moisture experiment Exercise 10. Soil moisture. An experiment was conducted to study the effects of three soil moisture levels on gene expression in maize seedlings (see Figure III.1). A total of 36 seedlings were grown in 12 pots with 3 seedlings in each pot. The 3 soil moisture levels (low, medium, and high) were randomly assigned to the 12 pots with 4 pots for each soil moisture level. After three weeks, RNA was extracted from the above-ground tissues of each seedling. Each of the 36 RNA samples was hybridized to a microarray slide to measure gene expression. 1. Name the treatments in this experiment. 2. Name the experimental units in this experiment. 3. Name the observational units in this experiment. 4. Name the response variable or variables in this experiment.

3 RANDOMISATION 54 Exercise 11. Rats. An experiment was conducted to gauge the effects of a drug and feed consumption on gene expression in rats. A total of 40 rats were housed in individual cages. Half of the 40 rats were randomly assigned to a calorie-restricted diet where daily feed rations contained approximately 50% of the calories normally consumed by rats of the type used in the experiment. The other 20 rats were provided with access to feeders that were always kept full so that their calorie intake was completely unrestricted. Within each diet group, four doses of an experimental drug (0, 10, 20, and 30 mg/kg body weight) were randomly assigned to rats with 5 rats per dose within each diet group. At the conclusion of the study, gene expression was measured for each rat using microarrays. 1. Name the treatment factors used in this experiment. 2. Name the levels of each factor. 3. Name the treatments used in this experiment. 4. Was a full factorial treatment design used? 5. Name the experimental units used in this experiment. 6. Name the observational units used in this experiment. 2 Randomisation Definition 28. Randomisation is the random assignment of treatments to experimental units. From

4 BLOCKING AND RANDOMIZED COMPLETE BLOCK DESIGN 55 Definition 29. Completely Randomized Design (CRD). Experimental design in which, for a given number of experiment units per treatment, all possible assignments of treatments to experimental units are equally likely. Exercise 12. In Figure III.2, establish a strategy to assign to 8 plants to either of two treatments completely at random. Figure III.2: Completely Randomized design (CRD) for Exercise Blocking and Randomized Complete Block Design Definition 30. Block. A group of experimental units that, prior to treatment, are expected to be more like one another (with respect to one or more response variables) than experimental units in general (in short, groups of similar experimental units). Definition 31. Randomized Complete Block Design (RCBD). Experimental design in which separate and completely randomized treatment assignments are made for each of multiple blocks in such a way that all treatments have at least one experimental unit in each block. The idea is to block out the effect of nuisance factors.

5 BLOCKING AND RANDOMIZED COMPLETE BLOCK DESIGN 56 B" A" E" C" D" F" E" A" B" A" E" B" D" F" C" E" A" B" B" E" A" C" D" F" Figure III.3: Diagram for Exercise 13. Exercise 13. An experiment has been conducted to compare 6 different kinds of treatments (A, B, C, D, E and F). The treatments have been allocated to Blocks (the dashed-line boxes) as in Figure III.3. Let s assume that the goal of the experiment is to compare all treatments with equal accuracy. What is wrong with the experiment above? How could we improve on this design? Note: there may not be a single correct answer to this question. We just want you to think about how we could do better. If you want, see if you can come up with an arrangement of the treatments to the blocks that avoids or minimizes the problem raised in the first part of this question. However, dont spend too much time on this!

6 REPLICATION 57 Exercise 14. Puppies. An investigator wants to examine the effectiveness of 2 drugs A and B for controlling heartworms in puppies. Veterinarians gave conjectures that the effectiveness of the drugs may depend on a puppy s diet. Three different diets are combined with the two drugs. Also, the effectiveness of the drugs may depend on a transmitted inherent protection against heartworm obtained from the puppy s mother. 1. What are the factors in this experiment, how many treatments are compared? 2. What is the blocking factor? 3. Describe the design in a table. 4 Replication Definition 32. Replication consists in applying a treatment independently to two or more experimental units. Exercise 15. Dairy cattle. Suppose an experiment is to be conducted to study the effects of 5 treatments (A, B, C, D, and E) on gene expression in dairy cattle. A total of 25 GeneChips and a total of 25 cows, located on 5 farms with 5 cows on each farm, are available for the experiment.

7 REPLICATION 58 Design 1: To reduce variability within treatment groups, randomly assign the 5 treatments to the 5 farms so that all 5 cows on any one farm receive the same treatment. Measure gene expression using one GeneChip for each cow. Design 2: Randomly assign the 5 treatments to the 5 cows within each farm so that all 5 treatments are represented on each farm. Measure gene expression using one GeneChip for each cow. For each design, answer the following questions 1. Represent the design in a table. 2. Name the observational units in each design. 3. Name the experimental units in each design. 4. Is blocking used for either design? If so, describe the blocks. 5. Describe the level of replication for each experimental design. 6. Which of the following designs is better from a statistical standpoint?

8 REPLICATION 59 Remark 20. Both blocking and randomization deal with nuisance factors, factors that are not of interest but might influence the outcome of experiment. Blocking is used when the nuisance factor is under our control. If nuisance factor is not under our control, use randomization. The general rule is: block what you can, randomize what you can not. Exercise 16. SNP wheat array experiment. In that experiment, 2 plants of each of 4 different genotypes are grown, and DNA samples are extracted from each plant and processed via 90k SNP arrays. Note that in reality, SNP arrays produce a proportion of missing values. Also, in reality it is not clear that all of the SNPs that are detected by the 90K array are actually different. However, to keep this example simple, lets assume that there are NO missing values, and that the SNPs being detetected are all UNIQUE. Thus, the final dataset will contain 2*4*90,000 = 720,000 SNP status values, corresponding to 90,000 unique SNPs detected. For the purposes of comparing genotypes, how many replicates are there in this experiment? Justify your answer. 2*4 = 8 90, *90,000 = 180,000 G2" G3" G4" G1" G4" G5" G3" G1" G1" G2" G5" G4" G2" G3" G1" G5" G5" G4" G2" G3" Figure III.4: Diagram for Exercise 17.

9 REPLICATION 60 Exercise 17. Chlorophyll content in plants. An experiment is designed to investigate chlorophyll content in the leaves of pre-anthesis (i.e. pre-flowering) wheat plants. Four plants (one plant per pot) of each of 5 Genotypes (G1, G2, G3, G4, G5) is randomised (i.e. allocated in a random manner) to a 4-row by 5-column layout on a growing table, as shown in Figure III.4. The dashed line shows that the table is located next to the wall of the glasshouse in which this experiment placed. Chorophyll content of a leaf can be measured using a SPAD meter a sensor is clamped to a leaf and light is shone through the leaf to provide a standardised measure of the greenness of the leaf. In this experiment, when the scientist has decided that the plants have reached the pre-anthesis stage of growth, they use a SPAD meter to obtain SPAD readings from 3 leaves from each plant. Hence, assuming no missing plants or missing observations, the scientist s final SPAD dataset contains 4*5*3 = 60 SPAD readings. 1. For the purposes of comparing genotypes, how many replicates does the scientist have in this experiment? Justify your answer. 5 4*3 = 12 5*3 = Now imagine that the researcher not only measures 3 leaves from each plant, but also measures each leaf 3 times. How many replicates are there for comparing genotypes? 3. If the scientist was concerned about the possibility of a trend in the growth of the plants due to the table s position next to the glasshouse wall, what could we do to account for this possibility? Does the above arrangement of plants on the table need any adjustment to allow this? Why or why not? Was my allocation of genotypes to positions on the table completely random?

10 CONFOUNDING What do you notice about the arrangements of the genotypes to the columns of the above design? Is this a good thing? 5. Can you think of any further practical problems or biases that might need to be addressed in this sort of experiment? How might you address them? 6. For both versions of the experiment described above - what are the experimental units? What are the observational units? 5 Confounding Confounding occurs when the effects of two or more explanatory variables (on a response variable of interest) cannot be distinguished from one another. Confounding can be problematic or useful depending on the situation. In Design 1 from Exercise 15, the effects of farms and treatments were completely confounded. This was very problematic. Exercise 18. Can you think of an example of experiment with a confounding factor?