Chapter 1 The Completely Randomized Design with a Numerical Response A Completely Randomized Design CRD is a particular type of comparative study The word design means that the researcher has a very specific protocol to follow in conducting the study The word randomized refers to the fact that the process of randomization is part of the design The word completely tells us that complete randomization is required in contrast to some form of incomplete randomization such as the randomized pairs design we will study later in these notes What is a numerical response See the following section 1 1 Comparative Studies So what is a comparative study Let s look at its two words beginning with the word study According to dictionary com http dictionary reference com the fifth definition of study is Research or a detailed examination and analysis of a subject phenomenon etc This reasonably well fits what I mean by a study Next again according to dictionary com the first definition of compare the root word of comparative is To examine two or more objects ideas people etc in order to note similarities and differences Because of time limitations for the most part in these notes we will restrict attention to exactly two as opposed to two or more things being compared In the examples of the first two chapters Dawn wants to compare two flavors of cat treats Kymn wants to compare two settings on an exercise machine Sara wants to compare two golf clubs and Cathy wants to compare two routes for jogging In the practice and homework problems of these first two chapters you will be introduced to several other comparative studies Indeed a large majority of the chapters in this book are devoted to comparative studies Why Two reasons 3 1 Comparative studies are extremely important in science 2 The discipline of Statistics includes several good ideas and methods that help scientists perform and analyze comparative studies Next some terminology the two things being compared are called the two levels of the study factor For our examples we have the following study factors and levels Dawn s study factor is the flavor of the cat treat with levels equal to chicken flavored and tuna flavored For Kymn s study her exercise apparatus is called an ergometer which requires two choices by its operator Kymn s study factor is the machine setting with first level defined as small gear with the vent closed her second level is large gear with the vent open Sara s study factor is the golf club she used with levels 3 Wood and 3 Iron Cathy s study factor is the route for her one mile run with levels at her local high school and through the park The remaining components of a comparative study are The units that provide the researcher with information The response which is the particular information from the unit of interest to the researcher One of the following methods The researcher identifies each unit with its level of the study factor or The researcher assigns each unit to a level of the study factor I choose to introduce you to the units and the response for each of our studies in the various sections below I do want to say a bit about the method in the last bullet of the above list Examples of identifying sometimes called classifying are comparing men and women comparing old people with young people comparing residents of Wyoming with residents of Wisconsin Our development of randomization based inference beginning with Chapter 3 in Part I of these notes will not consider any studies that involve identifying units with levels As the last sentence implies randomization based inference is restricted to studies in which the researcher has the option of assigning units to levels In fact as the name suggests we attend only to those studies in which the researcher exercised the option of assignment by using a method called randomization You will learn about randomization in Chapter 3 4 1 2 Dawn s Study Various Tools Dawn completed my class several years ago In this section you will be introduced to Dawn s project The choice of a project topic or indeed any research should begin with a curiosity about how the world operates Here is Dawn s motivation for her study as she wrote in her report I decided to study my cat s preference to two different flavors of the same brand name cat treats I was interested in this study because I figured that Bob my cat would prefer the tuna flavored treats over the chicken flavored because Bob absolutely loves canned tuna with a passion My interest came when I looked at the ingredients on the two labels I noticed that the percentage of real chicken in the chicken flavored treats was larger than the percentage of real tuna in the tuna flavored treats Thus Dawn had a pretty good idea of what she wanted to study Her next step was to operationalize the above notions into the standard language of a comparative study We know her study factor and its levels from the previous section Now we need to specify the definition of the units and the response A unit consisted of presenting Bob with a pile of ten treats all of the same flavor The flavor of the treats in the pile determined the level of the study factor for that unit either chicken level 1 or tuna level 2 The response is the number of treats that Bob consumed in five minutes The technical term unit is not very descriptive In this course there will be two types of units trials and subjects Dawn s units are trials Essentially we have trials if data collection involves doing something repeatedly In Dawn s case this something is setting out a pile of ten treats And then doing it again the next day By contrast in many studies the different units are different people When the units are people or other distinct objects we call them subjects As we will see later in these notes sometimes the distinction between trials and subjects is blurry fortunately this doesn t matter Dawn decided to collect data for 20 days with one trial per day Dawn further decided to have 10 days assigned to each level of her study factor Sometimes as here I will speak of assigning units to levels Other times I will speak of assigning levels to units These two different perspectives are equivalent Dawn had to decide of course which days would be assigned to chicken flavored and which days would be assigned to tuna flavored She made this decision by using a method called randomization Randomization is very important We will learn what it is in Chapter 3 In fact the word randomized in
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