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Factorial experiments are described by two things: the number of factors, and the number of levels of each factor. For example, a 2×3 factorial experiment has two factors, the first at 2 levels and the second at 3 levels. Such an experiment has 2×3=6 treatment combinations or cells.
This example of design experiments is attributed to Harold Hotelling, building on examples from Frank Yates. [21] [22] [14] The experiments designed in this example involve combinatorial designs. [23] Weights of eight objects are measured using a pan balance and set of standard weights. Each weighing measures the weight difference between ...
Associated with a factorial experiment is a collection of effects.Each factor determines a main effect, and each set of two or more factors determines an interaction effect (or simply an interaction) between those factors.
The alias structure determines which effects are confounded with each other. For example, the five-factor 2 5 − 2 can be generated by using a full three-factor factorial experiment involving three factors (say A, B, and C) and then choosing to confound the two remaining factors D and E with interactions generated by D = A*B and E = A*C.
Replication in statistics evaluates the consistency of experiment results across different trials to ensure external validity, while repetition measures precision and internal consistency within the same or similar experiments. [5] Replicates Example: Testing a new drug's effect on blood pressure in separate groups on different days.
Before performing a Yates analysis, the data should be arranged in "Yates' order". That is, given k factors, the k th column consists of 2 (k - 1) minus signs (i.e., the low level of the factor) followed by 2 (k - 1) plus signs (i.e., the high level of the factor). For example, for a full factorial design with three factors, the design matrix is
For example, if there are 3 levels of the primary factor with each level to be run 2 times, then there are 6! (where ! denotes factorial) possible run sequences (or ways to order the experimental trials). Because of the replication, the number of unique orderings is 90 (since 90 = 6!/(2!*2!*2!)). An example of an unrandomized design would be to ...
The factor model must then be rotated for analysis. [4] Canonical factor analysis, also called Rao's canonical factoring, is a different method of computing the same model as PCA, which uses the principal axis method. Canonical factor analysis seeks factors that have the highest canonical correlation with the observed variables.