Statistical Techniques IIEXST7015Experimental Design Examples19a_ExpDesignExamples 1Example 2: Snedecor & Cochran, 1980 (pg 248)Examine the calcium content of Turnip leaves. There are 4 plants, 3 leaves from each plant and 2 samples of 100 mg from each leaf.This experiment was probably done not so much to test leaves as to estimate the relative variability among the 3 levels of sampling. Example 219a_ExpDesignExamples 2Example 2 (continued)We want to test plants with the error term leaf(plant) and not the error term sample(leaf plant). This is a CRD with a nested error term (2 levels). Yij= µ+ τi+ βj+ εij A fully nested design is one place where the TYPE I SS can be used in Designed experiments. TYPE III are used for everything else. 19a_ExpDesignExamples 3Example 2 (continued)There are two ways to do this. The TEST statement - where you specify the H=effect to be tested and the E= effect to use as error term. You can also specify a Type SS (HTYPE and ETYPE). A test statement was used for this problem.The RANDOM statement, with the test option, will determine the error term to be used and correct for unbalanced data. The test option was not requested for this example. 19a_ExpDesignExamples 4Example 2 (continued)Note that SAS always tests by default with the residual error term. This is not correct in this example. The correct test was requested with the test statement, and Type I SS used. TEST H=PLANT E=LEAF(PLANT) / HTYPE=1 ETYPE=1;19a_ExpDesignExamples 5Example 2 (continued)The Random statement in PROC GLM would specify which of the components in the model are random. Note that SAS will provide you with the EMS with numeric coefficients for the RANDOM statement (with or without the test option. 19a_ExpDesignExamples 6Example 2 (continued)PROC MIXED is a newer alternative to solving this type of problem. This procedure works differently from the usual least squares procedures. It is not a least squares solution, it is an iterative solution (maximum likelihood). It estimates the random variance components (σ2) instead of the EMS. Then it tests any fixed effect components to the model. 19a_ExpDesignExamples 7Example 2 (continued)PROC MIXED works a little differently from PROC GLM. In PROC MIXED the fixed components ONLY go in the model and the random effects go in the RANDOM statement. There is not a "test" option in PROC MIXED. 19a_ExpDesignExamples 8Example 2 (continued)In the Turnip calcium example there are no fixed components, so only random components are estimated. Confidence limits for the random variance components were requested. 19a_ExpDesignExamples 9Example 3 (continued)Example 3: Snedecor & Cochran, 1980 (pg 293)This is an example of a CRD with different numbers of observations at different levels. Wheat yields were available for 6 districts in the midwest. There were UP TO 10 farms per district and UP TO 3 fields per farm. The model for this design is Yij= µ + τi+ βj+ εij 19a_ExpDesignExamples 10Example 3 (continued)In this example the TEST statement was requested, but it actually gives the wrong answers because the unbalanced design has no clear error term estimated (see EMS coefficients). The TEST option on the RANDOM statement will make a simple algebraic adjust for unbalanced designs. Occasionally negative F tests can result. 19a_ExpDesignExamples 11Example 3 (continued)The TEST statement produced a test of DISTRICTS with a P-value of 0.3056. The TEST option on the RANDOM statement causes the tests to be calculated (with the appropriate error terms). This test also adjusts the tests to account for the unequal coefficients on the EMS. The P-value was 0.4601. 19a_ExpDesignExamples 12Example 3 (continued)The MIXED model again estimates the variance components with confidence intervals. Note that the confidence intervals to not include zero (a negative value should not be possible with variance components), but it is very wide and overlaps with the other variance components. Note that algebraic calculations of the Variance components from GLM give different results. 19a_ExpDesignExamples 13Example 4: Snedecor & Cochran, 1980 (pg 256)The experiment tests the failure of soybean seeds to germinate after treatment with one of 4 fungicides and a control. Randomized block design without replication within blocks. Five blocks and four treatment levels in each blocks. Example 419a_ExpDesignExamples 14Example 4 (continued)The model for this design is Yij= µ + τi+ βj+ εij The error term is the cell to cell variation estimated by the "interaction" term. Since there is no error for testing the interaction we ASSUME that this variation represents only random variation and that there is no real interaction. 19a_ExpDesignExamples 15Example 4 (continued)The GLM tests for this model are correct because the lone error term is used for both of the sources in the model. Note that the output from the test option on the RANDOM statement provides the same results. The coefficients on the BLOCK and TREATMENT EMS are both 5. The 5 blocks are the treatment reps and the 5 treatments are the block reps. 19a_ExpDesignExamples 16Example 4 (continued)The PROC MIXED was fitted with the treatments as random effects. The output gives the same test as the GLM. This is often true for simple designs. A histogram is the only graphic output needed to express the difference between the main effects. 19a_ExpDesignExamples 17Example 5Example 5: Snedecor & Cochran, 1980 (pg 267)The experiment tests efficacy of fumigants on wire worms (nematodes). There are two fumigants (C and S) and a control (0). There are 5 blocks. This is a Randomized block design with replication within blocks. Five blocks with three treatment levels in each blocks and 4 replicates for each treatment in each block. 19a_ExpDesignExamples 18Example 5 (continued)The model for this design is Yij= µ + τi+ βj+ (τβ)ij+ εijk If the replicates are replicated experimental error we can use it to test the interaction. If they are sampling units within plots the test makes less sense. 19a_ExpDesignExamples 19Example 5 (continued)In this case the test was done with the TEST statement instead of the RANDOM statement test option. The TEST statement should be adequate since the appropriate error term can be specified (see EMS) and the design is balanced. The MIXED model analysis of this simple problem will also give the same result. A histogram shows the differences