One-Way Random Effects Model (Consider Balanced Case First) in the One-Way Layout There Are a Groups of Observations, Where Each
One-way Random Effects Model (Consider Balanced Case First) In the one-way layout there are a groups of observations, where each group corresponds to a different treatment and contains experimental units that were randomized to that treatment. For this situation, the one-way fixed effect anova model allows for distinct group or treatment effects and esti- mates a population treatment means, which are the quantities of primary interest. Recall that for such a situation, the one-way (fixed effect) anova model takes the form: P ∗ yij = µ + αi + eij; where i αi = 0. ( ) However, a one-way layout is not the only situation where we may en- counter grouped data where we want to allow for distinct group effects. Sometimes the groups correspond to levels of a blocking factor, rather than a treatment factor. In such situations, it is appropriate to use the random effects version of the one-way anova model. Such a model is sim- ilar to (*), but there are important differences in the model assumptions, the interpretation of the fitted model, and the framework and scope of inference. Example | Calcium Measurement in Turnip Greens: (Snedecor & Cochran, 1989, section 13.3) Suppose we want to obtain a precise measurement of calcium con- centration in turnip greens. A single calcium measurement nor even a single turnip leaf is considered sufficient to estimate the popula- tion mean calcium concentration, so 4 measurements from each of 4 leaves were obtained. The resulting data are given below. Leaf Calcium Concentration Sum Mean 1 3.28 3.09 3.03 3.03 12.43 3.11 2 3.52 3.48 3.38 3.38 13.76 3.44 3 2.88 2.80 2.81 2.76 11.25 2.81 4 3.34 3.38 3.23 3.26 13.21 3.30 • Here we have grouped data, but the groups do not correspond to treatments.
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