Karen A. F. Copeland

Comparing Two Populations

This chapter considers comparing responses from two experimental groups. The methods for comparing two groups depend on the type of populations (continuous or discrete) as well as how the experiment was performed. The chapter explains paired samples; it requires a careful experimental setup, but when appropriate, it provides an efficient method for comparing two population means. The experimenter starts with pairs of experimental units. In each pair one unit is randomly assigned to one treatment and the other unit is assigned to the other treatment. Furthermore, there are many situations in which the pairing of observations is either not reasonable or simply not possible. Thus, one is often interested in comparing two populations by considering independent (i.e., unrelated) random samples from the two. Such an experiment is performed by starting with a single pool of experimental units and randomly assigning units to one treatment group or the other. This random assignment is important because it ensures that any initial differences among the experimental units will be randomly scattered over both treatment groups, thus avoiding bias.

One-Factor Multi-Sample Experiments

This chapter focuses on general case of a multi-sample experiment. One of the first things one would want to do with a multi-sample set of data is find a useful way to graph it. Some possibilities are: a scatter plot with the treatment on the horizontal axis, side-by-side box plots, or an error-bar chart. Furthermore, in one-factor multi-sample experiment the first question one is interested in answering is whether the factor has any effect. in order to check if the factor has an effect one needs to check if the means for the different levels are all the same because it is assumed that different levels of the factor have the same variance. A good way to do this is using the analysis of means (ANOM). This test not only answers the question of whether there are any differences among the factor levels, but it also tells us which levels are better and which are worse.

Inference for Regression Models

This chapter considers inference for the estimated model coefficients and for values predicted from the model, and also introduces a test for lack of fit of the model.

Inference for a Single Population

This chapter combines the two types of models in order to determine the precision of estimates. In order to study the precision of an estimate, one needs to focus on the distribution of the estimator used to obtain the estimate. This chapter briefly discusses inference for parameter estimates from a single population and explains an important statistical result called the central limit theory (CLT). CLT is a statement about the (asymptotic) behavior of the sample mean, not of the sample itself. The CLT does not say, “large samples are normally distributed.” As the sample size gets larger, a histogram of the sample will start to look like the density function of the underlying distribution—which is not necessarily a normal density function. The CLT tells that for large enough samples the distribution of the sample mean will be approximately normal. Using the properties of expected values and variances together with the expected value and variance of the underlying distribution, one can determine the expected value and variance for the approximate normal distribution.

Experiments with Two Factors

This chapter assumes that multi-factor experiments are complete and balanced. In addition to the improved efficiency obtained from the use of multi-factor designs, one is also afforded the opportunity to study possible interaction between the factors. Factorial designs can estimate interactions, and if none are present they allow the effect of every factor to be evaluated as if the entire experiment were devoted entirely to that factor. An interaction plot can only indicate the possible presence of an interaction. Whether the differences in the slopes of the line segments in an interaction plot are statistically significant depends on the experimental error. In order to indicate the magnitude of the experimental error, confidence intervals are sometimes drawn around the end points of the line segments in an interaction plot.

Response Surface Methods

The responses of an experiment when considered as a function of the possible levels of the factors are called a response surface, and designs used to study a response surface are called response surface designs. Response surfaces are usually much too complex to be able to easily model the entire surface with a single function, so a simple model (one that is linear in the factor levels) is used to find the local slope (the slope in a specified area) of the surface and point the direction to the maximum (minimum) response. Then, when the desired area (say of the maximum) is reached, a more complex model with quadratic terms is used to provide a more accurate representation of the response surface in the area of its maximum value.

Models for the Ramdom Error

Models for the random error are called random variables. A random variable is a function that maps experimental outcomes to real numbers. For example, Y - the number of heads obtained in tossing a coin once, is a random variable that maps the experimental outcomes of a head or a tail to the numbers 1 and 0, respectively. If the experimental outcome is already a real number, then the outcome itself is a random variable. Every random variable has associated with it a distribution function that assigns probabilities to the possible values of the random variable. Distribution functions are specified by either a probability density function or a probability mass function, which emulate a histogram (that has been scaled so its area is equal to one) for a random sample of y values. If the response variable Y is continuous, then its probabilities are specified by a probability density function f (y), which is continuous. If the response variable Y is discrete, then its probabilities are specified by a probability mass function P(Y = k), which is discrete.

Multi-Factor Experiment

An experiment need not be limited to only one or two factors. If there are many factors that potentially affect the response, it is more efficient to design an experiment to examine as many of them as possible rather than running individual experiments for each factor. A well-designed experiment can detect significant factors even if there are only a few experimental units in each treatment group. To detect high-order interactions with any assurance, the experiment must be quite large. Often, one assumes that high-order interactions will not be important, and this assumption reduces the necessary size of the experiment. When this is the case, the variances associated with the high-order interactions are not considered separately. Instead, these variances are lumped in with the random variability. This type of data is generally analyzed on a computer. A multi-factor experiment can have blocks as well as treatments. Just as before, the purpose of blocking is to account for any effects that cannot (or should not) be held constant in the experiment. This reduces the random variability associated with the experimental units, and increases the power of the hypothesis tests. It is usual to assume that blocks and treatments do not interact.

Models for Experiment Outcomes

A way to summarize data is with a mathematical model. A mathematical model is simply a mathematical expression for the relationship among several variables. In addition, to infer the rate of degradation at normal temperatures from the rate at high temperatures, a mathematical model is needed. In this case that mathematical model is based on two other mathematical models. One needs a model for: the relationship between the amount of chemical product, known as the reactant, and the length of time it is stored at a particular temperature; and for the relationship between the degradation rate and the temperature. The chapter also considers single-factor experiments, that is, experiments in which only the levels of a single factor are changed. It is reasonable to suppose that the observations may be centered at different locations for different levels of the factor. Furthermore, models for bivariate data are also explained. When two pieces of information (e.g., height and weight) are obtained at each experimental trial (e.g., for each person sampled), it is referred to as bivariate data. The two pieces of information are generically labeled as the independent variable (denoted with x), and the dependent or response variable (denoted with Y). This reflects the idea that Y (e.g., weight) depends on x (e.g., height), so that knowing x should tell us something about Y.

Applied Linear Statistical Models

(1997). Applied Linear Statistical Models. Journal of Quality Technology: Vol. 29, No. 2, pp. 233-233.