Chapter 1 - Research Problems, Approaches, ond Questions I i c s >5 fi ÍO . CO w Q, g ÍOS o 8 C Wl o 3 a < í E S = e-5 Gl T" ň t- C uj i s c c O QJ o> & w s =>3 X.. »•£ o. « o>5 T_ d .8 |l 5a OS >l O vi ?s 'üfi Oj "O íl Š O 4* Dl Q> 2» H.C ? 5 o s; og :>■* A Sample Research Problem - Tlie High School and Beyond (HSB) Study Imagine lhal you arc interested in (lie general problem of what factors influence mathematics achievement at the end of high school. You might have some hunches or hypotheses about such factors based on your experiences and your reading of the research and popular literature. Some factors that might influence mathematics achievement arc commonly called demographics; c.g., gender, ethnic group, and mother's and father's education. A probable influence would be the mathematics courses that (lie student has taken. We might speculate that grades in math and in other subjects could have an impact on math achievement.* However, other "third" variables, such as students' 10 or parent encouragement and assistance, could be the actual causes of high matli achievement. Such extraneous variables could influence what courses one took, die grades one received, and might be correlates of the demographic variables. We might wonder how spatial performance scores such as pattern/mosaic score and visualization score might enter into a more complete understanding of the problem and whether those skills seem to be influenced by the same factors as math achievement. Finally, students' attitudes about mathematics might be factors affecting these math achievement scores. Before wc stale the research problem and questions in more formal ways, we need to step back and discuss the types of variables and the approaches that might be used to study the above problem. Think about what arc the independent/antecedent (presumed causes) variables and what arc the dependent/outcomes variable(s) in the above problem. Hopefully, it is obvious that math achievement is the primary dependent variable. Given the above research problem, which focuses on achievement tests at the end of Iho senior year, the number of math courses taken is best considered to be an antecedent or independent variable in tins study. What about father's and mother's education and gender? How would you classify etliuic group in terms of the type of variable? What about grades? Like IQ and parent encouragemeni they would be independent variables, but, as with any study, wc were not able to measure all the variables that might be of interest. Visualization and mosaic pattern scores could probably be cither independent or dependent variables depending upon the specific research question. Finally, die math attitude questions and the resulting composite or scale scores derived from them also could be either independent or dependent variables, but probably independent/antecedent variables in this study. Note that student's class or grade level is not a variable in tiiis study because all the participants arc high school seniors (i.e., it does not vary; it is the population ofinterest). As wc have discussed, independent variables can be active (given to the participant or manipulated by the investigator) or attributes of the participants or their environments. Are there (regression) approach lo study was conducted by ll« National Opinion Research Center (1980). 'Ilw example, discussed hero and throughout the book, is based oil 15 variables obtained from n random sample i>f 75 out of 28,240 high scliool seniors. These variables include achievement scorn, grades, and demographies. The raw data for (be 13 variables were obtained from an appendix iu Hinfcle, Wnntnia, and Jura (19W). Nolc lhal additional variables (ethnicily and ninth altitudes) with realistic bul fictitious data have been added lo tlie USB data sel in order lo provide examples of common additional types of analysis (eg-, summalcd scales and Cronbach's alpha). ■.:'■ CSMOtR I - RMcaith huhlcna, Appro*ebťl. mní <>»Miiim 5) \\{ Assignment H, we will answer additional basic association«! research questions (using iv.ii'iMii pioduct-mumcnt coiieliilion coefficients) such as, "Is thcie a posilivc tttOt uiioii/relatioruhip Ikjiwccti grades in high school and .1 in.'Mi achievement?" I lir. a.Mi'iuncnl also will pmdm ■■ .1..... ,.r.i.., :;i.itm >■: .ill I.. ...... . CO «t co n oX c w o w Ä < S E £ = d ra ** C UJ i s C c 2 S 5 J ^ •• 2 -c ä g »-5 O) 3s S.S Ol C s S 5Í ■a s So *í o a; D»á K z£ ? s o s; o g 2-Š CHAPTER3 Measurement and Descriptive Statistics According lo S. S. Slcvcns (1951). "In ils broadest sense measurement is Hie assignment of numerals lo objects or events according to rules" (p.l). As wc have seen in chapter 1, the process of research begins with a problem that is made Up of I question about (he relationship between two, or usually more, variables. Measurement is introduced when these variables are operationally defined by certain ndes which determine how the participants' responses will be translated into numerals. These numbers can represent nonordercd categories in which the numerals do not indicate a greater or lesser degree of the characteristic of the variable. Stevens went on to describe four scales or levels of measurement that he labeled: nominal, ordinal, interval, and ratio. Stevens and most writers since then have argued that the level or scale of measurement used to collect data is one of the most important determinants of the types of statistics that can be done appropriately with that data. As implied by the phrase "levels of measurement," these types of measurements vary from the most basic (nominal) to the highest level (ratio). However, since none of the statistics that are commonly used in social sciences or education require the use of ratio scales we will not discuss them to any extent. nominal Scales/Variables These are the most basic or primitive forms of scales in which the numerals assigned to each category stand for the name of the category, hut have no implied order or value. Males may be assigned the numeral 1 and females may be coded as 2. This docs not imply that females arc higher than males or (hat two males equal a fcinalc or any of the other typical mathematical »scs of (he numerals. The same reasoning applies to many other true nominal categories such as ethnic groups, type of disability, section number in a class schedule, or marital status (e.g., never married, married, divorced, or widowed). In each of these cases the categories arc distinct and nonovcrlapping, but not ordered, (hus each category in (he variable marital status is different from each other but there is no necessary order to the categories. Thus, the four categories could be mimbered 1 for never married, 2 for married, 3 for divorced, and 4 for widowed or the reverse, or any combination of assigning a number to each category. What this obviously implies is that you must nor treat the numbers used for identifying the categories in a nominal scale as if they were numbers that could be used in a formula, added together, subtracted from one another, or used to compute an average. Average marital status makes no sense. However, ifoneasksa computer lo do average marital status, it will blindly do so and give you meaningless information. The important thing about nominal scales is to have clearly defined, nonovcrlapping or mutually exclusive categories which can he coded reliably by observers or by self-report. Qualitative or naturalistic researchers rely heavily, iľnot exclusively, on nominal scales and on the process of developing appropriate codes or categories for bchnviors, words, etc. Although using qualitative/nominal scales does dramatically reduce (ho types of statistics that can be used with your data, it docs not altogether eliminate the use of statistics to summarize your data and ■•' make inferences. Therefore, even when the data are nominal or qualitative allegories, one's research may benefit from the use of appropriate statistics. We will return shortly to discuss the types of statistics, both descriptive and inferential, that are appropriate for nominal data. IHelwlonums Variables It is oflcn hard to tell whether n dichotomous variable, one with (wo values or categories (e.g.. Yes or No, Pass or Fail), is nominal or Ordered and researchers disagree. We arguo that, although some such dichotomous variables arc clearly nominal (e.g., gender) and others arc clearly ordered (e.g., math grades-high and low), all dichotomous variables form a special case. Statistics such as the mean or variance would be meaningless for a three or more category nominal variable (e.g., cllinic group or marital status, as described above). However, such statistics do have meaning when there are only two categories. For example, in the 1ISB dal a the average gender is 1.55 (with males = I and females = 2). This means that 55% of the participants were females. Furthermore, we will sec in Chapter 12, multiple regression, that dichotomous variables, called dummy variables, can be used as independent variables along with oilier variables that arc interval scale. Thus, it is not necessary to decide whether a dicliotomous variable is nominal, and it can be treated as if it were interval scale. Tabic 3.1. Descriptions of Scales of Measurement With Dichotomous Variables Added Scale___________________Description_____________________________________________ Nominal - 3 or more unordered or nominal categories Dichotomous " 2 categories cither nominal or ordered (special case) Ordinal = 3 or more ordered categories, but clearly unequal intervals between categories or ranks Interval = 3 or more ordered categories, and approximately equal intervals between categories Ratio - 3 or more ordered categories, with equal intervals between categories and a tnic zero Ordinal Scales/Variables O.e., Unequal Interval Scales) In ordinal scales (here are nol only mutually exclusive categories as in nominal scales, but tho categories are ordered from low to high in much the same way that one would rank the order in which horses finished a race (i.e., first, second, third, ...last). Thus, in an ordinal scale one knows which participant is highest or most preferred on a dimension but the intervals between the various ranks are not equal. For example, the second place horse may finish far behind the winner but only a fraction of a second in front of the (bird place finisher. Thus, in this case there 2-, are unequal intervals between first, second, and third place with a very small interval between second and third and a much larger one between first and second. interval ami Ratio Scales/Variables fie., Equal interval Scales) Interval scales have not only mutually exclusive categories thai arc ordered from low to high, but also the categories arc equally spaced (i.e.. have equal intervals between them). Most physical measurements (length, weight, money, etc.) are ratio scales because Ihcy not only have equal intervals between the vahies/categories, but also have a true zero, wlu'ch means in tlie above examples, no length, no weight, or no money. Few psychological scales have Ibis property of a true zero and thus even if they are very well constructed equal interval scales, it is not possible to say that one has no intelligence or no extroversion or no attitude of a certain type. While there are differences between interval and ratio scales, the differences arc not important for us because we can do all of the types of statistics that we have available with interval data. As long as the scale has equal intervals, it is not necessary to have a true zero. Distinguishing Between Ordinal and Interval Scales 1 ' It is usually fairly easy to tell whether three categories arc ordered or not, so students and researchers can distinguish between nominal and ordinal data, except perhaps when there are only two categories, and then it docs not matter. The distinction between nominal and ordinal makes a lot of difference in what statistics are appropriate. However, it is considerably harder to distinguish between ordinal and interval data. While almost all physical measurements provide cither ratio or interval data, the situation is less clear with regard to psychological measurements. When we come to the measurement of psychological characteristics such as altitudes, oflcn we cannot be certain about whether the intervals between the ordered categories are equal, as required for an interval level scale. Suppose we have a five-point scale on which we are (o rat« our altitude about » certain statement flom strongly agree as S to strongly disagree as 1. The issue is whether the intervals between a rating of I and 2,2 and 3, 3 and 4, and 4 and 5 arc all equal or nnt. One could argue that because Ihe numbers arc equally spaced on the page, and because they arc equally spaced in terms of their numerical valtiM, Ihe subjects will view them as equal intervals. However, especially if the in-between points arc identified (e.g., strongly agree, agree, neutral, disagree, and strongly disagree), it could be argued tliat the difference between strongly agree and agree is not Ihe same as between agree and neutral; this contention would be hard to disprove. Some questionnaire or survey items have response categories that arc not exactly equal intervals. For example, let's take the case where the subjects are asked to identify their age as one of five categories: 21 to 30,31 to 40,41 to 50,51 to 60, and 61 and above. It should be clear that the last category is larger in terms of number of years covered than the other four categories. Thus, the age intervals are not exactly equal. However, we would consider this scale and the ones above to be at least approximately interval. On the other hand, an example of an ordered scale that is clearly not interval would be one that asked how frequently subjects do something. 'Hie answers go something liko this: every day, once a week, once a mouth, once a year, once every 5 years. You can sec that the categories 27 become wider and wider and, therefore, are not equal intervals. There is clearly much more difference between 1 year and 5 years than there is lwtwcen 1 day and 1 week. Most of the above information is summarized in the top of Tabic 3.2. Tahle 3.2. Selection of Appropriate Descriptive Statistics for One Dependent Variable Level/Stale o' Measurement of \ ■rlabli Nomina Ordinal Interval or Ratio Characteristics of the - Qualitative data - Qu» n lila rive dala • Quantitative data Variable - Not ordered - Ordered dala - Ordcied da u - True categories: only - Rank order only • Iíqual intervals names labels between values Examples Gender, sciioot, 1st, 2nd, 3>