EXST 7015 - Statistical Inference II, Fall 2011 Lab 11: Randomized Block Design and Nested Design OBJECTIVES: The objective of an experimental design is to provide the maximum amount of reliable information at the minimum cost. In statistical terms, the reliability of information is measured by the standard error of estimates (that is directly related with the population variance, inversely related to sample size). Properly applied experimental design may effectively reduce the population variance, and/or could structure data collection to reduce the magnitude of the experimental error. Usually data resulting from the implementation of experimental designs are described by linear model and analyzed by the analysis of variance as was introduced last week. Random block design is one of the simplest and probably the most popular experimental design. In this design the sample of experimental units is divided into groups or blocks and then treatments are randomly assigned to units in each block. In some experiments blocks may be of sufficient size to allow several units to be assigned to each treatment in a block. Such replication of treatments is referred to as randomized blocks with sampling. The linear model for data from such an experiment is yijk = μ + τi + βj + τβij + εijk (i = 1, 2, …,t; j = 1, 2, …, b; k =1, 2, …, k) Where yijk is the observed value of the response variable in the kth replicate of the treatment i in block j; μ is the overall mean; τi is the fixed effect of treatment i; βj is the effect of block j, a random variable with mean zero and variance σ2β; τβij is the experimental error, a random variable with mean zero and variance σ2τβ ; and εijk is the sampling error, which is the measure of variation among units treated alike within a block, a random variable with mean zero and variance σ2. As you might be aware that in the above model, if the block effect is fixed, the interaction is also fixed so that it becomes a two-factor factorial design and F ratios for all the tests using the sampling error in the denominator. Nested design refers to some experimental situations where experimental units may contain sampling units, which may, in turn, contain sample subunits. Since the design describes subsamplies nested within sample or experimental units, it is called nested or hierarchical design. The linear model for nest design is yijk = μ + αi + βj(i) + εk(ij) (i = 1, 2, …, a; j = 1, 2, …, b; k =1, 2, …, n) where yijk is the kth observed value for level i of factor A and level j of factor B which is nested in the ith level of factor A; μ is the overall mean; αi is the effect of the ith level of factor A; βj(i) is the effect of level j of factor B nested in the ith level of factor A; and εk(ij) variation among sampled units and is the random error. The subscrip j(i) is used to denoted that different j subscripts occur within each value of i; that is, they are “nested” in i. Likewise, the k subscript is “nested” in groups identified by the combined ij subscript (Keep in mind that it is not interaction term). In this week’s lab, PROC MIXED will be used to analyze the data from Random Block Design with Sampling and Nested Design. Simple CONTRAST will be introduced as well.LABORATORY INSTRUCTIONS Housekeeping Statements dm 'log; clear; output; clear'; options nodate nocenter pageno = 1 ls=78 ps=53; title1 'EXST7015 lab 11, Name, Section#'; ods rtf file = ‘c:/temp/lab11.rtf’; ods html file = ‘c:/temp/lab9.html’; Nested Design with Subsampling: The data set of nested design is from Statistical Methods II by Roger Cue of Department of Animal Science from McGill University. This experiment tests the effects of different treatments on the growth of apples. There are three types of treatments which are applied to 12 randomly selected apple trees (four trees per treatment). As you can see the trees are nested in the treatment. At the end of the experiment, six apples were randomly selected from each apple tree and the weight of each apple was recorded. PROC MIXED will be performed to test overall treatment effect, and simple CONTRAST will be used to test the difference between the treatments. data nested; title1 'ANOVA with nested design' input trt tree apple wt; cards; 1 1 1 313.063 1 1 2 329.132 1 1 3 334.278 1 1 4 330.088 1 1 5 334.987 1 1 6 325.075 1 2 1 333.936 1 2 2 326.155 1 2 3 352.854 1 2 4 350.791 1 2 5 318.560 1 2 6 323.473 1 3 1 345.494 1 3 2 349.296 1 3 3 339.190 1 3 4 338.942 1 3 5 331.370 1 3 6 339.097 1 4 1 340.840 1 4 2 336.798 1 4 3 313.810 1 4 4 333.880 1 4 5 343.068 1 4 6 319.171 2 1 1 349.271 2 1 2 336.695 2 1 3 352.797 2 1 4 348.486 2 1 5 352.0772 1 6 341.423 2 2 1 356.880 2 2 2 356.256 2 2 3 364.950 2 2 4 360.570 2 2 5 362.104 2 2 6 371.829 2 3 1 324.161 2 3 2 340.130 2 3 3 334.580 2 3 4 342.813 2 3 5 327.415 2 3 6 333.571 2 4 1 338.742 2 4 2 340.348 2 4 3 362.837 2 4 4 340.782 2 4 5 348.730 2 4 6 325.444 3 1 1 387.868 3 1 2 372.807 3 1 3 380.505 3 1 4 391.804 3 1 5 388.935 3 1 6 361.860 3 2 1 377.948 3 2 2 380.033 3 2 3 361.913 3 2 4 363.098 3 2 5 365.375 3 2 6 382.121 3 3 1 363.583 3 3 2 387.727 3 3 3 373.021 3 3 4 362.931 3 3 5 378.928 3 3 6 364.442 3 4 1 374.851 3 4 2 361.291 3 4 3 377.389 3 4 4 366.722 3 4 5 374.187 3 4 6 380.383 ; proc mixed; class trt tree; model wt = trt; random tree(trt); lsmeans trt / adjust=tukey pdiff; contrast ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5; estimate ' trt 1 - (trt 2+3)/2' trt 1 -.5 -.5; run; CONTRAST: Specify the contrast between treatment 1 and the average of treatments 2 and 3. Note that the Sums of Squares, Mean Squares and F-ratio are not the same as those for the comparisons between treatments.ESTIMATE: returns the estimate of the difference and its standard error. It is a standard t-test. Random Block Design with Sampling: The data set of random block design comes from your textbook