lecture1 (2024)

Lecture1
Types of scales & levels of measurement

Discrete andcontinuous variables
Daniel's text distinguishes between discrete and continuous variables. Theseare technical distinctions that will not be all that important to us in thisclass. According to the text, discrete variables are variables in which thereare no intermediate values possible. For instance, the number of phone callsyou receive per day. You cannot receive 6.3 phone calls. Continuous variablesare everything else; any variable that can theoretically have values in betweenpoints (e.g., between 153 and 154 lbs. for instance). It turns out that this isnot all that useful of a distinction for our purposes. What is really moreimportant for statistical considerations is the level of measurementused. When I say it is more important, I've really understated this.Understanding the level of measurement of a variable (or scale or measure) isthe first and most important distinction one must make about a variable whendoing statistics!

Levels ofmeasurement
Statisticians often refer to the "levels of measurement" of avariable, a measure, or a scale to distinguish between measured variables thathave different properties. There are four basic levels: nominal, ordinal,interval, and ratio.

Nominal
A variable measured on a "nominal" scale isa variable that does not really have any evaluative distinction. One value isreally not any greater than another. A good example of a nominal variable issex (or gender). Information in a data set on sex is usually coded as 0 or 1, 1indicating male and 0 indicating female (or the other way around--0 for male, 1for female). 1 in this case is an arbitrary value and it is not any greater orbetter than 0. There is only a nominal difference between 0 and 1. With nominalvariables, there is a qualitative difference between values, not a quantitativeone.

Ordinal
Something measured on an "ordinal" scaledoes have an evaluative connotation. One value is greater or larger or betterthan the other. Product A is preferred over product B, and therefore A receivesa value of 1 and B receives a value of 2. Another example might be rating yourjob satisfaction on a scale from 1 to 10, with 10 representing completesatisfaction. With ordinal scales, we only know that 2 isbetter than 1 or 10 is better than 9; we do not know by how much. It may vary. The distance between 1 and 2 maybe shorter than between 9 and 10.

Interval
A variable measured on an interval scale givesinformation about more or betterness as ordinalscales do, but interval variables have an equal distance between each value.The distance between 1 and 2 is equal to the distance between 9 and 10.Temperature using Celsius or Fahrenheit is a good example, there is the exactsame difference between 100 degrees and 90 as there is between 42 and 32.

Ratio
Something measured on a ratio scale has the sameproperties that an interval scale has except, with a ratio scaling, there is anabsolute zero point. Temperature measured in Kelvin is an example. There is novalue possible below 0 degrees Kelvin, it is absolute zero. Weight is anotherexample, 0 lbs. is a meaningful absence of weight. Your bank account balance isanother. Although you can have a negative or positive account balance, there isa definite and nonarbitrary meaning of an accountbalance of 0.

One can think of nominal, ordinal,interval, and ratio as being ranked in their relation to one another. Ratio ismore sophisticated than interval, interval is more sophisticated than ordinal,and ordinal is more sophisticated than nominal. I don't know if the ranks areequidistant or not, probably not. So what kind of measurement level is thisranking of measurement levels?? I'd say ordinal. In statistics, it's best to bea little conservative when in doubt.

TwoGeneral Classes of Variables (Who Cares?)
Ok, remember I stated that this is the first and most important distinctionwhen using statistics? Here's why. For the most part, statisticians orresearchers wind up only caring about the difference between nominal and allthe others. There are generally two classes of statistics: those that deal withnominal dependent variables and those that deal with ordinal, interval,or ratio variables. (Right now we will focus on the dependent variable andlater we will discuss the independent variable). When I describe these types oftwo general classes of variables, I (and many others) usually refer to them as"categorical" and "continuous." (Sometimes I'll use "dichotomous"instead of "categorical" ). Note also, that"continuous" in this sense is not exactly the same as"continuous" used in Chapter 1 of the text when distinguishingbetween discrete and continuous. It’s a much looser term. Categorical anddichotomous usually mean that a scale is nominal. "Continuous"variables are usually those that are ordinal or better.

Ordinal scales with few categories(2,3, or possibly 4) and nominal measures are often classified as categoricaland are analyzed using binomial class of statistical tests, whereas ordinalscales with many categories (5 or more), interval, and ratio, are usuallyanalyzed with the normal theory class of statistical tests. Although the distinction is a somewhat fuzzyone, it is often a very useful distinction for choosing the correct statisticaltest. There are a number of specialstatistics that have been developed to deal with ordinal variables with just afew possible values, but we are not going to cover them in this class (see Agresti, 1984, 1990; O’Connell, 2006; Wickens,1989 for more information on analysis of ordinal variables).

General Classes ofStatistics (Oh, I Guess I Do Care)
Ok, so we have these two general categories (i.e., continuous and categorical),what next…? Well this distinction (as fuzzy as it may sound) has very importantimplications for the type of statistical procedure used and we will be makingdecisions based on this distinction all through the course. There aretwo general classes of statistics: those based on binomial theory andthose based on normal theory. Chi-square and logistic regression dealwith binomial theory or binomial distributions, and t-tests,ANOVA, correlation, and regression deal with normal theory. So here's a tableto summarize.

SurveyQuestions and Measures: Some Common Examples
In actual practice, researchersand real life research problems do not tell you how the dependent variableshould be categorized, so I will outline a few types of survey questions orother measures that are commonly used.

Yes/NoQuestions
Any question on a survey that has yes or no as a possible response is nominal,and so binomial statistics will be applied whenever a single yes/no questionserves as the dependent variable or one of the dependent variables in ananalysis.

Likert Scales
A special kind of survey question uses a set ofresponses that are ordered so that one response is greater than another. Theterm Likert scale is named after the inventor,Rensis Likert, whose nameis pronounced "Lickert." Generally, thisterm is used for any question that has about 5 or more possible options. Anexample might be: "How would you rate your department administrator?"1=very incompetent, 2=somewhat incompetent, 3=neither competent, 4=somewhatcompetent, or 5=very competent. Likert scales areeither ordinal or interval, and many psychometricianswould argue that they are interval scales because, when well constructed, thereis equal distance between each value. So if a Likertscale is used as a dependent variable in an analysis, normal theory statisticsare used such as ANOVA or regression would be used.

PhysicalMeasures
Most physical measures, such asheight, weight, systolic blood pressure, distance etc., are interval or ratioscales, so they fall into the general "continuous "category. Therefore, normal theory type statistics are also used when a such a measure serves as the dependent variable in ananalysis.

Counts
Counts are tricky. If a variable is measured by counting, such as the case if aresearcher is counting the number of days a hospital patient has beenhospitalized, the variable is on a ratio scale and is treated as a continuousvariable. Special statistics are often recommended, however, because countvariables often have a very skewed distribution with alarge number of cases with a zero count (see Agresti,1990, p. 125; Cohen, Cohen, West, & Aiken, 2003, Chapter 13). If a researcher is counting the number ofsubjects in an experiment (or number of cases in the data set), a continuoustype measure is not really being used. Counting in this instance is reallyexamining the frequency that some value of a variable occurs. For example,counting the number of subjects in the data set that report having beenhospitalized in the last year, relies on a dichotomous variable in the data setthat stands for being hospitalized or not being hospitalized (e.g., from aquestion such as "have you been hospitalized in the last year?").Even if one were to count the number of cases based on the question "howmany days in the past year have you been hospitalized," which is acontinuous measure, the variable being used in the analysis is really not thiscontinuous variable. Instead, the researcher would actually be analyzing adichotomous variable by counting the number of people who had not beenhospitalized in the past year (0 days) vs. those that had been (1 or moredays).