3.2: Experiments
- Page ID
- 58606
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the previous chapter we considered the development of empiricism and the scientific method. The key step of the scientific method is the test step where observation is used to provide evidence about the truth of the hypothesis specified in the previous step. The method that produces these observations is called an experiment.
An experiment is any process that produces observations that provide evidence that can be used to determine if a hypothesis is true or not.
There are many types of hypotheses and experiments. We will describe some of the more common types used in modern applications of statistics. In the most common formulation, researchers are interested in whether a certain condition is related to or associated with another condition. For example, a researcher might hypothesize that individual income is related to race or ethnicity. In this case individual income and race are the conditions that are of interest to the researcher in forming a hypothesis. Note that there is a natural direction to this hypothesis in that implicitly the idea that the race of an individual is at least associated with, to some extent, the income of the individual. To see the direction of this relationship, a researcher would hopefully not hypothesize that individual income influences the race of the individual. If it is observed that families of certain races tend to have lower measures of wealth than families of other races, then evidence has been observed that the race of a family is a condition that partially determines family wealth. As we shall see later in this book, this is only partial evidence of such a relationship.
In this type of formulation, race is known as a factor or a treatment, and family wealth is called the response. In the general formulation of these experiments, researchers are usually interested in how the response is related to the factor or factors observed in an experiment.
In an experiment the response is the observed condition that is of interest to the researcher.
In an experiment a factor or treatment is a condition that is hypothesized to have an effect on, or is associated with, the response.
Considering another example, suppose that a researcher hypothesizes that the political affiliation of an individual is a factor that partially determines the network that the individual trusts as their news source. In this case the response is the network of the news source of the individual and the factor is the political affiliation. Expanding on this example the researcher might also be interested in how gender and age also determine the network of the news source of an individual. For this example the response is still the network of the news source, but now there are three factors: political affiliation, gender, and age. Note that it is possible that an experiment may have more than one response as well, though we will not discuss these studies except in a few examples.
It can also happen that an experiment does not have any factors, and only responses. For example, a researcher might hypothesize that the average age of students at a community college is greater than 25 years. The researcher would only observe the response which in this case is age, and there are no factors. In some other cases researchers may observe more than one measurement but no directional relationship between the measurements is hypothesized. In this type of study, the researcher may be interested in how well the measurements mimic one another without hypothesizing that one condition is causing the other condition to behave in a certain way.