ANOVA: Analysis of Variance

Analysis of Variance: Designed Experiments

• When the data were obtained according to certain specified sampling procedures, they are easy to analyze and also may contain more information pertinent to the population parameters than could be obtained using simple random sampling• Design of an Experiment – the procedure for selecting sample data

Experimental Design: Terminology

• Factors – independent variables that may be related to a response variable y• Level – the value – that is, the intensity setting – assumed by a factor in an experiment• Treatments – the combination of levels of the factors for which y will be observed. The term treatments is used to describe the factor-level combinations to be included in an experiment because many experiments involve treating or doing something to alter the nature of the experimental unit, the object upon which a measurement is made.

Example:Suppose that an experiment is conducted to measure the hardness y of a new type of plastic as a function of two factors, the pressure and temperature at the time of moulding. If the hardness of the plastic is measured at pressures 200, 300, and 400 pounds per square inch (psi) and at temperatures 200 and 300 degrees Fahrenheit (F), then pressure is at three levels and temperature is at two levels. The combinations of levels of the factors for which y will be observed are called treatments. For example, if the hardness of the new plastic is measured for each of the six pressure-temperature combination, (200psi, 200F), (300psi, 200F), (400psi, 200F), (200psi, 300F), (300psi, 300F), (400psi, 300F), then the experiment would involve six treatments.

Definition 14.1. The independent variables that are related to a response variable y are called factors. Definition 14.2 The intensity setting of a factor is called a level. Definition 14.3 A treatment is a particular combination of levels of the factors involved in an experiment.

The design of an experiment involves the following four steps: 1. Select the factors to be included in the experiment and identify the parameters that are the object of the study. Usually, the target parameters are the population means associated with the factor – level combinations (i.e. treatments).2. Decide how much information you want to acquire – that is, decide upon the magnitude of the standard error(s) that you desire.3. Choose the treatments (the factor – level combinations) to be included in the experiment and determine the number of observations to be made for each treatment.4. Decide how the treatments will be assigned to the experimental units.

Completely Randomized Designs

Definition 14.4 A completely randomized design to compare p treatment means is one in which the treatments are randomly assigned to the experimental units, or in which independent random samples are drawn from each of the p populations.

Example:

Six different machines are being considered for use in manufacturing rubber seals. The machines are being compared with respect to tensile strength of the product. A random sample of 4 seals from each machine is used to determine whether the mean tensile strength varies from machine to machine. The following are the tensile-strength measurements in kilograms per square centimeter x 10-1.

Perform the analysis of variance at the 0.05 level of significance and indicate whether or not the mean tensile strengths differ significantly for the 6 machines.

Sum of Squares: Unequal Samples