# Statistics in Psychology Using R and SPSS (Rasch/Statistics in Psychology Using R and SPSS) || Definition - Character, Chance, Experiment, and Survey

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P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come4Definition character, chance,experiment, and surveyIn this chapter, quantitative and qualitative, and continuous and discrete characters, as wellas factors, will be distinguished, and different scale types of observed values/outcomes willbe described. The term chance will be defined more precisely. Studies will be divided intoexperiments and surveys.Carrying out a study for the gain of scientific insight within psychology entails conductingobservations with respect to at least a single specific character. For this, there are basicallytwo strategies. Both of which are based on the principle of chance. Fundamentally chanceis the phenomenon in gambling that is responsible for the outcome of the game (e.g. theoutcome when throwing a die, namely the number 1, 2, . . . , 6, or the outcome of a lottery).For empirical scientific purposes, one could also say that chance means all influences onevents or observations of interest, which are either not ascertainable or which we dont wantto ascertain.Bachelor The term chance should not be confused with its meaning in everyday use; therechance often means seldom or unexpected. In this book an event that happensby chance is an event that can, but doesnt have to happen.Bachelor Example 4.1 Arbitrariness and randomnessThe reader might ask within his/her wider range of friends (preferably exactly18 persons) for the first number between 3 and 20 that comes to mind. If chancewere the only cause of their respective responses, we would basically expect thatevery number between 3 and 20 would occur the same number of times, whichis about once. Experience has shown, however, that up to half of the personsStatistics in Psychology Using R and SPSS, First Edition. Dieter Rasch, Klaus D. Kubinger and Takuya Yanagida. 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to ComeDEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEY 31will say 17; also other prime numbers and odd numbers tend to occur relativelymore frequently than other numbers. Subsequent arguments explain that peoplelook for a random number and that they try not to pick any typical number like4 or 8. This example documents that, in this case, arbitrariness is at work,not chance.In contrast to events that are dependent on chance, thus being random events, other events wecall (strictly) deterministic.Master Example 4.2 Drawing from a deck of playing cardsWhen drawing from a set of well-mixed playing cards, the result is a randomevent, for example the 10 of diamonds. If somebody searches for the queen ofhearts and takes it, then this result is an arbitrary event because it has beendetermined by the will of that person. Finally if somebody draws the first cardfrom a newly unwrapped deck, it is a systematic event because, when packed,decks usually have the ace of spades on top.MasterDoctorWhen several isolated observations (occurrences, events; for example the throwingof two dice and the observation of the resulting number of spots on top) occur,we talk about chance if there is either no connection at all or an irrelevant (inter-)connection between them. If there is a great mass of events (for example theopening of buds on a blossoming apple tree) then chance is the product of anaccumulation of non-ascertainable influences, resulting without any rule in achain of events where one cannot predict in which order the buds will open.MasterDoctorStudents and lay people (professionals in neighboring disciplines), especiallypsychotherapists with a particular school of thought, often express the followingopinion: Chance-based coincidences dont happen. What they mean is that allevents happening around one person have their reason (cause) in that personshistory; and if not in that persons own history then in that of some other players.In the end this belief in causality implies that the past and the future of the universeare principally predictable. The past could be completely reconstructed and thefuture could be predicted up to the smallest detail. Despite the fact that this attitudetowards life may motivate a client to strengthen his/her internal locus of controlof reinforcement belief,1 such belief in causality is rather crude.Werner Heisenbergs Uncertainty Principle shows that the reference to chancedoes not have its origin in a lack of theoretically available information, but thatchance is a principle (Heisenberg, 1927); this principle includes the notion thatit is generally impossible to determine simultaneously both the position andmomentum of an elementary particle in some accurate manner, and this is not aquestion of methodological failings, but a principle.1 Greatly simplified, locus of control of reinforcement belief means: by whom or what a person thinks that his/herlife course is determined.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come32 DEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEYThe two strategies for obtaining observations of a character are the following:1. The behavior of certain chosen persons (in general: research units) is (psychologically)examined with respect to some traits/aptitudes without them being influenced by theexaminer. This is what we call a survey. If sampling by chance, that is random sampling,occurs, we speak of a random sample. Since we dont look at samples other thanrandom samples in this book, we often only talk about samples. Rarely is there acomplete inventory ascertainment of the population or, in other words, a census of thepopulation/universe, as with statistical almanacs.2. The persons (research units) receive systematic treatments after they have been selectedfor a sample; for example they are first randomly allocated to several groups, thenevery group receives a different (psychological) intervention, and finally they arepsychologically examined. This procedure is referred to as an experiment.For psychological experiments we can give the following simplified definition: in an experi-ment the researcher observes the behavior (verbal and nonverbal actions, reactions) of his/herparticipants (persons; very often then called subjects) under strictly established and controlledconditions which were varied by him/her intentionally and specifically in order to test the con-ditions influence on the assessed behavior. Thereby randomization of the subjects conditionassignment is indispensable, and replication of the procedure must be essentially possible;controlling conditions aims for a reduction in variability of confounding sources.In both cases, survey and experiment, the collected data for the character of interest(for example (test) scores on a psychological assessment tool; in general observed val-ues/outcomes) are recorded and then statistically analyzed. If an experiment or a surveyconsists of several consecutive steps, where every step is based on the result of the precedingone, then we call this procedure sequential.MasterDoctorGenerally the observed data depend on chance in two ways: on the one hand theresearch units have been chosen at random and assigned to an experimental con-dition by chance, respectively. On the other hand the same research units observedunder nearly identical sampling circumstances dont lead to the same outcomes,but by chance to (slightly) different ones. The latter is due to measurement errors(see in Section 2.3 the quality criterion of reliability).Once the data have been collected by means of random sampling and randomization within anexperiment, respectively, we can analyze them with statistical methods. Then we reflect uponall the results concerning our scientific question that we could possibly obtain. First we lookat the underlying population: a population is the set of objects (persons,2 schools, hospitals,etc.), for which an empirical study using a subset (sample) is supposed to make a conclusionregarding certain characters. There must be an operational definition for the population thatallows for an assertion of whether a certain object belongs to or not.2 In the case of experiments we talk about subjects, within psychological assessment about testees/examinees.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to ComeDEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEY 33MasterDoctorExample 3.1 continuedThe psychological consequences of a hysterectomy are to be assessed.It is important, while formulating the scientific question, to clearly identifythe group of people or, in other words, the population that one wants to observeor whose aptitudes/traits one wants to quantify. In this example all hysterectomypatients could be the population, or those aged below 40 years, or those whocome from Central Europe, or those aged below 40 years that live in urban areas,etc. Therefore populations must be defined exactly with regards to space, time,and content.Often not only the objects, that are the (potential) research units, are called population orsample, respectively, but also the set of (potential) outcomes themselves.Real populations are finite; the population size is termed N the sample size is termed n.In principle, it is possible to carry out a census, meaning that one examines all elements ofthe population. In statistics, populations are often assumed to be infinite (in practice we canuse conclusions that are valid under the assumption of infinite populations for N > 1000 andnN < 0.1).3Master Research units can be distinguished more accurately for both experiments andsurveys. In an experiment, the object from a given population that is randomlyassigned to a certain condition (such conditions might be particular treatments) iswhat we call an experimental unit. In a survey, the element of the population thatbecomes part of the sample is termed survey unit. Of course also the experimentalunits have to be (randomly) selected as a sample from a well-defined population.Up to now we have talked about characters without properly defining what is meant by theterm: a character is that specific feature, which is the research objective of a study; it is(in-)directly deduced from the scientific question. Apart from socio-graphic factors, in psy-chology these are mostly traits, aptitudes and all sorts of behavior patterns (see Example 1.1).Characters have thus various measurement values. The result of ascertainment of a char-acter, that is the outcome, is in psychology often called an investigational result, also obser-vational value or observed (measurement) value. Since the result of a statistical analysis of astudy can also be called an investigational result (when indicated result of the experiment)it is preferable to speak of outcomes as observed values, or for short, simply as observations.By the way, in psychology there is the special case of a test score, which is the (action or)reaction shown in a psychological test (in general: a psychological assessment tool) that hasbeen scored in a certain way.Values of characters are ascertained by means of a measuring instrument (in a broadersense) with the help of a certain scale. More precisely: an unambiguous assignment rule3 N stands for the extent of finite populations on the one hand, but on the other hand often also stands for the totalnumber of research units in a study that consists of several groups. However n is always the number of research unitssampled from a population as a single sample.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come34 DEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEYof numbers or symbols to the different shapes of a character is termed scaling in generaland a scoring rule in psychology. The set of numbers or symbols that are available, andbetween which a more, or less, differentiated relational system is defined, is called a scale.Therefore a scale mirrors (mostly in numerical terms) the (empirically detectable) relationsbetween research units. It makes the relations accessible to mathematical operations. The termscaling includes measurement of a character. The multitude of characters that are potentiallyinteresting to us now differ regarding their respective scale type.Master If we empirically compare (put in relation to one another) two objects, letssay two sticks, then we expect from the scaling that is available to us, in thiscase ideally (metric) measurement, that the assigned scale values (measurementvalues) represent the same relation. If we observe that stick Z is double the lengthof stick U by holding stick U two times against stick Z then an adequate scale(cm scale) will represent this; for example U : Z = 20 : 40 cm. If we empiricallycompare two objects, for example two pupils, and observe that pupil V canregularly correctly answer the questions concerning the subject material that theteacher addresses to him (and attends to his schoolwork according to instructionsand excels at written tests), whereas pupil W answers the questions concerningsubject material either incorrectly or not at all (and sometimes only attends tohis schoolwork after being asked to do so several times and sometimes failswritten tests), then we expect from the chosen scale, the assignment of grades, acorrespondingly identical relation of measurement values; for example grade Afor pupil V and grade D for pupil W.Example 4.3 Typical empirical relationsIn psychology we can transform empirical relations into numerical ones as in the followingexamples:1. Spouse B earns twice as much as spouse C.2. In a perception experiment (signal detection with presentation time appropriate to thesubjects age), senior citizen D makes, in comparison to senior citizen E, three timesmore mistakes than senior citizen F, in comparison to senior citizen E.3. The level of education (assessed by the highest completed schooling) of person G ishigher than person Hs.4. Patient I is never married, patient J married, patient K divorced, and patient L widowed.Every assignment of numbers to persons can logically only mirror those relations that areempirically given and observable; and those numbers are transformable as long as theyadequately depict the empirical relations.Below we will have a look at four types of scales, which differ with respect to the number andthe kind of represented relations between the research units. A superior (higher-order) scaletype also includes the relational qualities of the lower-order scale types.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to ComeORDINAL SCALE 354.1 Nominal scaleThe nominal scale depicts the fewest empirical relations. The scale values consist of signs(symbols, letters, names) that even if they are sometimes numbers can only expressdiscrepancies. In particular, there is no indication of order between the signs.Master One recognizes a nominal-scaled character by the fact that the feasible scalevalues are arbitrary as long as the discrepancies between them remain. The sex ofpersons, for example, is assessed by a nominal scale. It does not matter whetherfemale persons are characterized by a Venus symbol, f for female or 0, and malepersons by the Mars symbol, m for male or 1.4Example 4.3 continuedIn Case 4 (patient I is never married, patient J married, patient K divorced, and patient Lwidowed), we only need arbitrary names (nominal) for I, J, K, and L, because the only thingthat matters is that the discrepancies are visible; e.g. the names n, m, d, and w are just aspossible as 1, 2, 3, and 4 or a, b, c, and d.Bachelor Example 4.4 Marital statusIn Example 1.1 we find the nominal-scaled character marital status of themother. This characters different shapes cannot at all be empirically relationallyordered; there is no general chronological order in the course of a persons life.In Example 5.2 we will identify the absolute frequency of the different scalevalues found in the n = 100 children; in Table 5.3 one can see that 9 mothers arenever married, 65 are married, 22 are divorced, and 4 are widowed. Of course wecan report these results in any other sequence. In any case, the value widowedis not logically relegated to divorced.When a (nominal-scaled) character has only two values, we also talk about alternative char-acters or dichotomous/binary characters.4.2 Ordinal scaleThe ordinal scale creates symbols or numbers that express a rank order of the charactersdifferent shapes. In ordinal-scaled characters the number of possible scale values is generallyindependent from the set of research units (cf. e.g. the grading scale). Also, full rankings(where every research unit out of n gets a rank from 1 to n and thus the number of signs isdependent on the set of research units) that are derived from metric data (which is how physicsmainly scales; see below for more details) represent ordinal scales.4 By the way, experience has shown that when female researchers use numbers they tend to code female as 1 andmale as 0, and male researchers the other way around, although a relation of order does not exist for psychologicalquestions.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come36 DEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEYMaster One recognizes an ordinal-scaled character by the fact that the scale values maybe arbitrarily chosen as long as the empirical rank order of them is captured.For Lecturers:Most statistics text books within psychology claim that a rank-scaled characteris recognizable in that its scale values may be transformed monotonically (e.g.y y2). For students this is indeed a very easily understood interpretation, but itis by no means exact. It is only correct if one postulates that negative values donot occur see for instance, however, the scale values 1, 0, and 1 for theresponse options in a questionnaire: no, I dont know, and yes; they do notallow the monotonic transformation y y2.Example 4.3 continuedIn Case 3 (the level of education of person G is higher than that of person H), the possible scalevalues of G and H are located on an ordinal scale because only the relation > is relevant; e.g.the scale values 5 and 4 would be as appropriate as the (admittedly strange-looking) values10.3 and 8.6.BachelorMasterSchool grades form an ordinal scale; in ascending order the characters differentshapes are, for instance, needs improvement, satisfactory, great, and excel-lent or 4.0 to 1.0 (in descending order of achievement).Bachelor Example 4.5 The scale type of the character social status in Example 1.1In our example, the character social status is operationally defined via theoccupation of the father, or in the case of single mothers via their occupation. Thevarious occupations have been ordered (in groups) as follows: upper classes,middle classes, lower middle class, upper lower class and lower classes.However the way the character social status has been observed here leads to anadditional scale value, which is single mother in household. It does not fit intothis order either at the beginning, or at the end, or anywhere in between. Whenconsidered like that, this is a nominal-scaled character. If we want to convey thebasically given (rank-) order to the recipient of the results, we have to omit all thecases with the scale value single mother in household.Nominal scale and ordinal scale are often summarized by the term non-metric scales. Andone terms both types as qualitative data. In contrast, data that stem from a scaling of oneof the two scale types discussed below, interval scale and ratio scale, are called quantitativedata.Doctor Within psychology one usually counts ordinal-scaled characters as quantitativecharacters too. This, after all, is because scale values in rank scales mirror grada-tions or gradual discrepancies, which means that they express an order regardinga more or less quantity. This differs from nominal-scaled qualitative characters,P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to ComeINTERVAL SCALE 37where the scale values have a different qualitative meaning; they are not one-dimensional but represent several dimensions that cannot be related to each other.Below we will use the term quantitative character, divergently from its usage inpsychology, but in concordance with the terminology in other natural sciencesand mathematical statistics, only for interval- and ratio-scaled characters.4.3 Interval scaleThe interval scale consists of numbers. The positive differences between them are interpretedas the distance between the characters different shapes. Besides the equality of the values,the equality of the distances is also given. However, there is no absolute zero point if theinterval scale had an absolute (that is to say natural) zero point then it would be a ratio scale!Master An interval-scaled character is recognized by the fact that the measurement valuesallow for a linear transformation of the form y a + by (b > 0) without violatingthe empirically given relations. One example for such a transformation is theconversion of temperatures from Fahrenheit into Celsius:Celsius = (Fahrenheit 32) 59or with b = 5 / 9, a = 160 / 9, Celsius = y, and Fahrenheit = x:y = 59x 1609Doctor The measurement of temperatures in degrees Celsius or Fahrenheit happens on aninterval scale. Both scales are convertible into each other by a linear transforma-tion. Particularly with the existence of two scales (Celsius and Fahrenheit) insteadof a single one, it is obvious that temperatures are usually described without anabsolute zero point; the numerically existing zero of the Celsius scale is not anatural zero because it has been arbitrarily located at the temperature wherewater passes from a fluid into a solid state. There is, however, a natural zero thatthe temperature cannot fall below; it is defined by the kelvin scale, which is aratio scale. Length and mass also have an absolute zero.Example 4.3 continuedIn Case 2 (in a perception experiment signal detection with presentation time appropriate tothe subjects age senior citizen D makes, in comparison to senior citizen E, three times moremistakes than senior citizen F, in comparison to senior citizen E), the possible measurementvalues for D, E, and F are scaled at an interval scale level. This is because the measurementvalue difference (concerning E) must always express the ratio 3 : 1, without the measurementvalues being able to take into account the chosen difficulty of the signal detection (i.e.depending on the presentation time); for example the test scores 7, 1, and 3 would be asadequate as 9, 3, and 5, or 20, 8, and 12. The reason why we cannot interpret the ratio of themeasurement values regarding content is that those ratios do not represent empirical relations.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come38 DEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEYIn the given signal detection experiment, only some of all possible signals are used, but wecould instead have used, for example, two signals that are very difficult to discern; then wemight really have observed the values 9, 3, and 5 instead of 7, 1, and 3. If we had used a feweasy, medium, and difficult ones in addition, then we could have got the results 20, 8, and 12,or would have expected a corresponding result if really fair measurements apply.Doctor The scientific attempt to scale intelligence is a good example for psychologists,showing how to adequately and not artificially interpret an interval-scaled char-acter. Described very simply, the intelligence quotient (IQ) is defined as follows.The test scores (mostly the number of solved problems) obtained in a pertinentintelligence test from a very large standardization sample are first averaged andsecond put in relation to the extent of all the differences between them: the meanestimated for the population is termed ; the corresponding extent of all differ-ences is termed (see more precisely in Chapter 6). Then the test score of everysingle testee, for example V with the test score yV, is linearly transformed:IQV =15 (yV )+ 100In this, the specification of the two constants 15 and 100 is arbitrary, especiallythe specification of the additional 100; this could have been specified differentlyin an equally plausible way, for example 0 instead of 100. Therefore it makessense to form the quotient between differences for example in such a way thatthe difference concerning the IQ between person V with IQV = 110 and personW with IQW = 120 is actually twice as much as the difference between personP with IQP = 90 and person S with IQS = 95. It is, however, not empiricallyfounded to judge that person W with IQW = 120 is one-third more intelligent thanperson P with IQP = 90, although arithmetically 120 / 90 = 4 / 3. If the additiveconstant were set to 0, then the test scores of the persons V , W, P, and S would be10, 20, 10, and 5; the quotient of the differences between V and W or P and Swould still be 2, but the quotient between W and P would have changed to ()2!Therefore the intelligence quotient is interval-scaled.4.4 Ratio scaleA ratio scale has all the relational qualities of an interval scale and in addition an absolutezero point. Therefore the equality of relations (proportions, quotients) is presupposed.Master A ratio-scaled character is recognizable by the fact that the character values canonly be transformed in the way y ay, (a = 0); otherwise the empiricallyascertained relations do not match.Example 4.3 continuedIn Case 1 (spouse B earns twice as much as spouse C), the feasible measurement values takenby B and C lie on a ratio scale, because there must always be the ratio 2 : 1; e.g. 10 familyallowances and 5 family allowances, $1750 and $875, 1120 and 560.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to ComeRATIO SCALE 39Doctor As previously mentioned, the measurement of temperature in kelvin uses aratio scale. In psychology, ratio-scaled characters can be found principally wherephysical measurements such as age of a person are given. However, from a psycho-logical point of view, not all physical measurements represent empirical relationsthat correspond to a ratio scale. For example, measurements of reaction times thatare mandatorily obtained by the means of a specific psychological assessment toolwithin the personel recruitment for a job in surveillance, are indeed fundamentallybased on a metric scale (time scale of the clock); the psychological quality of thenumerical relations of the measurement values is ambiguous, yet: extreme values(for example missed stimuli or very delayed reactions) can as concerns theachievement potential of a testee hardly establish the same ratio as the concretenumerical ratio of the observed reaction times. Even the quotients of differencesin reaction times must be viewed critically regarding their content. The choiceof whether the researcher insinuates that the data are ratio-, interval-, or onlyordinal-scaled appears to be based solely on his/her subjective opinion; orientedon methodic fundamentals, it is, however, possible to empirically determine thescale type of data by implementing the scaling techniques of psychometrics (seeKubinger, 2009b).For Lecturers:An example taken from sports illustrates how arbitrarily and improperly non-or pre-scientific scaling procedures in everyday life are applied. In the (alpine)ski world cup, best-ranked athletes are rewarded with world cup points and thesum of gained points determines the winner at the end of the season. It wouldbe easy to contrast racer R with racer S; that is to say to determine the empiricalrelation of their speed over the whole season by matching corresponding videorecordings (provided that both have contested the same races). The rules, however,provide another scaling which represents numerical relations that can contradictthe empirical ones; from that point of view the scaling is unfair: The winner ofa race gets 100 points, the second 80, the third 60, the fourth 40, etc. SupposingR is 0.01 seconds slower in each of five races than the winner S, and always insecond place; and wins the sixth race, being 2 seconds faster than S, who placesfourth. S would then win the world cup with 550 points, 50 points ahead of R with500 points, although the matching of the video recordings shows that, calculatedover the whole season, R was 1.95 seconds faster than S. Such an unfair scalingis irrelevant as long as athletes and viewers freely accede to such authoritarianrules for the sake of amusement in psychological assessment in case work,however, there exists a demand for fairness and objectivity. In those instancessuch an unfair scaling would not be justifiable, for ethical reasons!Master A superior scale can always be downgraded into a lower-ranked one. Thus, asmeasures of achievement in a speed-skating competition, instead of stating thetimes of the three best racers, one can award a gold medal for the shortest time,and silver and bronze for the second and third best times. In long jumping this isP1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come40 DEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEYapplicable in an analogous way, except that the gold medal here is awarded for thehighest numerical value reached. From this one can see how senseless it wouldbe to position medal ranks on a number line. In one example gold is on the leftside of the others and in the other it is on the right side. If one doesnt know whatis behind the ranks in a quantitative way, then not even the direction is clear!Doctor For example, concerning the measurement of temperature, temperaturesbelow 40 degrees Fahrenheit and above 120 degrees Fahrenheit might happen,although with normal outdoor thermometers we cant measure them. Statementsof temperature are partly made using the measurement values, that is to say quan-titatively, but partly in the form of below 40 degrees and above 120 degrees.This corresponds to a mixture of ordinal and interval scale. Such cases occur dur-ing the measurement of reaction times (at least in the upper level) as well as duringthe measurement of lifetime, especially when the researcher cannot wait for thedeath of the research units of a sample (e.g. for scientific questions of survival afterclinical treatments). Statistically such data are analyzed either at a lower-rankedscale that is to say that the scale values are represented by an ordinal scale orthe given sample is treated as a censored sample (for a precise description of thisprocedure see Rasch, Herrendorfer, Bock, Victor, & Guiard, 2008).For their authorized use, statistical analysis methods hardly ever need the high requirementsof the ratio scale for the character of interest. Instead they require just an interval scale. That iswhy we dont have to distinguish between the two scale types in the following. We will onlytalk about whether a character meets the requirements of an interval scale or not, and mean bythis that the character at least meets these requirements. A ratio scale perhaps beyond that isnonessential for our observations. Therefore we will simply talk about quantitative characters.4.5 Characters and factorsIrrespective of the scale type, quantitative characters have to be divided into discrete andcontinuous characters. A quantitative character is discrete if it only has very specific mea-surement values; for example only natural numbers. If however, within a specific interval, allreal numbers are possible measurement values, then the character is continuous.For Lecturers:Lem (1971) made clear in one of his utopian novels just how extensive even theamount of natural numbers is. Here we give a modified account from a passageof his work:Ijon Tichy is in a cosmic hotel with an endless amount of rooms which have just been taken by an endless number of dentists that havecome from all over the universe in order to take part in a conference.He witnesses the arrival of an endless number of psychologists thatwant to take part in a cosmic psychology conference. Of course theyalso need rooms but the hotel manager explains that all rooms havebeen taken. Ijon Tichy suggests now that every dentist should justP1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to ComeREFERENCES 41move into the room with double the room number of the one theyoriginally had. Since, after the multiplication by two, these are alleven room numbers, all the rooms with uneven room numbers arefree: there is also an endless amount of them and the psychologistscan then move into those.Master Depending on the scale type that underlies the sampled data, different statisticalanalysis methods are applicable; that is to say, appropriate. For example, themean (Section 5.3.1) can only be used to suitably describe quantitative data ina short and concise way. The mean is therefore absolutely proper to representthe children from Example 1.1 regarding their test scores in the subtest EverydayKnowledge. The middle social status or moreover the middle marital statusof the mother are senseless quantities, as there are no empirical differences thatcan be observed; at best a rank order is given; the mean would unwarrantedlyinsinuate very specific quantities.If a character is designated to several (treatment) conditions within a study, then that char-acter is commonly termed a factor. Completely analogous to characters, one can talk aboutquantitative and qualitative factors. The values of a character that is a factor arent called scaleor measurement values, but instead factor levels or, for short, levels. A particular differencebetween character and factor is the following: Whereas a character is generally postulated asbeing determined by chance, a factor can also be modeled as being not random.Sometimes there are factors that are not (primarily) of interest for a specific question andtherefore are not part of the design of the study: they are confounding factors and are callednoise factors. If they were not held constant or adequately taken into account, the studysresults would be biased one says that the results have a bias.The choice of which statistical method will be used for the analysis of a study dependsclosely on which conditions are investigated and which noise factors have to be taken intoaccount, as well as if, when, and how the influence of unknown noise factors can be minimized.SummaryFor the research units in an empirical study (either a survey or an experiment) we get observed(measurement) values (within psychological assessment mostly test scores) for the characterin question. These stem from different scale types according to which empirical relations theyrepresent. These scale types distinguish between quantitative and qualitative characters; thelatter have to be divided into nominal- and ordinal-scaled ones.ReferencesHeisenberg, W. (1927). Uber den anschaulichen Inhalt der quantentheoretischen Kinematik undMechanik. Zeitschrift fur Physik, 43, 172198 (for an English translation see J. A. Wheeler &H. Zurek (1983). Quantum Theory and Measurement (pp. 6284)).Kubinger, K. D. (2009b). Psychologische Diagnostik Theorie und Praxis psychologischen Diagnos-tizierens ( 2nd edn) [Psychological Assessment Theory and Practice of Psychological Consulting].Gottingen: Hogrefe.P1: OTA/XYZ P2: ABCJWST094-c04 JWST094-Rasch September 20, 2011 2:11 Printer Name: Yet to Come42 DEFINITION CHARACTER, CHANCE, EXPERIMENT, AND SURVEYLem, S. (1971). Dzienniki Gwiadowe [Die Stern-Tagebucher des Ijon Tichy]. Frankfurt am Main:Suhrkamp (for an English translation see The Star Diaries: Further Reminiscences of Ijon Tichy(1985). Philadelphia: Harvest Books).Rasch, D., Herrendorfer, G., Bock, J., Victor, N., & Guiard, V. (2008). Verfahrensbibliothek Versuchs-planung und -auswertung. Elektronisches Buch [Collection of Procedures in Design and Analysis ofExperiments. Electronic Book]. Munich: Oldenbourg.

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