Overview The level of data is part of what determines which statistical procedure is appropropriate to one's research. That is, particular data levels constitute assumptions for particular procedures. If a procedure is used which assumes a higher level of data than one has (ex., using ordinal data with interval procedures like regression), standard errors will be underestimated and Type I errors (false positive findings) will be higher than the significance level (ex., .05) would suggest. If a procedure assumes a lower level of data than one has, power will be lower than it should be due to loss of information and Type II errors (false negative findings) will be increased. Ideally, the researcher selects a procedure which matches his or her data level. As a general rule, when variables are of different levels, one must select the procedure matching the lower level of data. However, violation of data level assumptions may or may not be critical. Treating nominal-level data as interval leads to nonsensical results, but treating ordinal data as interval is commonly done in social science and usually does not alter substantive research conclusions.

Contents Key concepts and terms Frequently asked questions Bibliography




Key Concepts and Terms

• Levels of Measurement
  • Nominal data has no order, and the assignment of numbers to categories is purely arbitrary (ex., 1=East, 2=North, 3=South, etc.). Because of lack of order or equal intervals, one cannot perform arithmetic (+, -, /, *) or logical operations (>, <, =) on nominal data.

  • Ordinal data has order, but the intervals between scale points may be uneven. Rank data are usually (see below) ordinal, as in students' rank in class. The interval distance from the top student to the second-highest student may be great, but the interval from the second-ranked student to the third-ranked may be very close. Because of lack of equal distances, arithmetic operations are impossible with ordinal data, which are restricted to logical operations (more than, less than, equal to). For instance, given a person of rank 50 and a person of rank 25 in a school class of 100, where rank 100 is highest achievement, one cannot divide 25 into 50 to conclude that the first person has twice the achievement of the second. However, one can say the first person represents more achievement than the second person.

  • Interval data has order and equal intervals. Counts are interval, such as counts of income, years of education, or number of Democratic votes. Ratio data are interval data which also have a true zero point. Temperature is not ratio because zero degrees is not "no temperature," but income is ratio because zero dollars is truly "no income," For most statistical procedures the distinction between interval and ratio does not matter and it is common to use the term "interval" to refer to ratio data as well. Occasionally, however, the distinction between interval and ratio becomes important. With interval data, one can perform logical operations, add, and subtract, but one cannot multiply or divide. For instance, if a liquid is 40 degrees and we add 10 degrees, it will be 50 degrees. However, a liquid at 40 degrees does not have twice the temperature of a liquid at 20 degrees because 0 degrees does not represent "no temperature" -- to multiply or divide in this way we would have to be using the Kelvin temperature scale, with a true zero point (0 degrees Kelvin = -273.15 degrees Celsius). Fortunately, in social science the issue of "true zero" rarely arises, but researchers should be aware of the statistical issues involved.

  
  

• Data Levels and Measures of Central Tendency
  The mode is used for nominal data, the median for ordinal data, and the mean for interval or ratio data. Note, however, that when interval or ratio data are skewed, the mean will be a "bad average" and for such data the median is often used instead.







Frequently Asked Questions




• Is it okay to use ordinal variables in procedures like regression or path analysis which assume interval data?
  In a review of the literature on this topic, Jaccard and Wan (1996: 4) summarize, "for many statistical tests, rather severe departures (from intervalness) do not seem to affect Type I and Type II errors dramatically." Standard citations to literature showing the robustness of correlation and other parametric coefficients with respect to ordinal distortion are Labovitz (1967, 1970) and Kim (1975). Others are Binder (1984) and Zumbo and Zimmerman (1993). Use of ordinal variables such as 5-point Likert scales with interval techniques is the norm in contemporary social science. Use of scales with fewer values not only violates normality assumptions but also runs a heightened risk of confounding difficulty factors as discussed in the section below on use of dichotomies. There is widespread agreement that the greater the number of points on an ordinal scale, the less the likelihood of substantive errors of interpretation when using ordinal data for interval procedures.

  Researchers should be aware that there is an opposing viewpoint. Thomas Wilson (1971), for instance, concluded, "the ordinal level of measurement prohibits all but the weakest inferences concerning the fit between data and a theoretical model formulated in terms of interval variables." The researcher should attempt to discern if the values of the ordinal variable seem to display obvious marked departures from equal intervalness and qualify his or her inferences accordingly.

• Is it okay to use dichotomous variables in procedures like regression or path analysis which assume interval data?
  If one assumes the dichotomy collapses an underlying interval variable, there will be more distortion with dichotomization than with an ordinal simplification. Dichotomization of continuous variables attenuates the resulting correlations. Moreover, the degree of attenuation will be affected by the cutting points used in the dichotomization. If the underlying correlations are high (over .7), the cutting points will have a non-trivial effect. Note also that categorical variables with similar splits will necessarily tend to correlate with each other, regardless of their content (see Gorsuch, 1983). This is particularly apt to occur when dichotomies are used. The correlation will reflect similarity of "difficulty" for items in a testing context, hence such correlated variables are called difficulty factors. The researcher should examine the factor loadings of categorical variables with care to assess whether common loading reflects a difficulty factor or substantive correlation. Nonetheless, it is common to use dichotomies in interval-level techniques like correlation and regression.

• What is attenuation of correlation?
  When an interval variable is binned into a categorical variable or dichotomized (ex., reducing income to high/medium/low income categories), information is lost and the variance of the variable is reduced. This usually leads to a lowering of correlation, though in a small minority of instances correlation may remain the same or even increase.
  

• Can Likert scales be considered interval?
  Likert scales (ex., strongly agree, agree, etc.) are very commonly used with interval procedures, provided the scale item has at least 5 and preferably 7 categories. Most researchers would not use a 3-point Likert scale with a technique requiring interval data. The fewer the number of points, the more likely the departure from the assumption of normal distribution, required for many tests. Here is a typical footnote inserted in research using interval techniques with Likert scales:
  "In regard to our use of (insert name of procedure), which assumes interval data, with ordinal Likert scale items, in a recent review of the literature on this topic, Jaccard and Wan (1996: 4) summarize, "for many statistical tests, rather severe departures (from intervalness) do not seem to affect Type I and Type II errors dramatically." Jaccard, James and Choi K. Wan (1996). LISREL approaches to interaction effects in multiple regression. Thousand Oaks, CA: Sage Publications. "

  Likert scales are ordinal but their use in statistical procedures assuming interval level data is commonplace for the reason given above. Note, though, that under certain circumstances, Likert and other rank data can be interval. This would happen, for instance, in a survey of childrens' allowances if all children in the sample got allowances of $5, $10, or $15 exactly, and these were measured as "low," "medium," and "high." That is, intervalness is an attribute of the data, not of the labels. In most cases, of course, Likert and rank variables are ordinal but the extent to which they approach intervalness depends on the correspondence of the ordinal labels to the empirical data.

• I have categorical data but I want to do interval procedures like factor analysis and regression. What do I do?
  One option, of course, is to use dummy variables, leaving out one category. Today, many procedures in SPSS and other statistical packages support automatic conversion of categorical variables to dummies. See the section on regression for a discussion of the use of dummy variables. Altenratively, the SPSS CATEGORIES module provides a variety of procedures for handling categorical variables. One may use PRINCALS, for instance, to get interval-level factor scores for categorical variables, then use these scores in interval procedures.
  1. PRINCALS, OVERALS, and CATREG are analogous to principal components, canonical correlation, and regression. All three procedures allow the user to specify a transformation type of nominal, ordinal, or numeric, on a variable-by-variable basis. All three procedures use an alternating least squares algorithm.

  2. CORRESPONDENCE performs a singular value decomposition of a transformed two-way array of numbers. This enables the user to represent the information in the array in a low-dimensional approximation. The output includes information that can be rendered in a biplot, which produces revealing insights into the data.

  3. HOMALS performs multiple correspondence analysis of two or more categorical variables treated as if the categories are unordered. This can be viewed as a type of principal components analysis of categorical variables. The output includes a plot of the category quantifications, which gives insight into the relationships between the variables in the analysis. The category quantifications are centroids of the objects that share that category. When viewed in the same space, the category quantifications and the object scores comprise one type of biplot.

• When I have a mixture of categorical and interval independents, what procedure should I use?
  There is no simple answer to this question, but the general linear model (GLM) is a procedure with this capability. GLM evolved from the regression model but uses maximum likelihood estimation (MLE) rather than ordinary least squares estimation (OLS). In SAS, all the GLM procedures automatically recode categorical variables. SPSS, as of version 8.0, does not implement full GLM but some procedures, such as logistic regression, do recode categorical variables automatically (after you designate them "categorical," of course). Note that automatic recoding usually selects the last category as the reference (omitted) category when creating sets of dummy variables. Some researchers prefer to recode manually so as to use a middle category or use some other criterion for selecting the reference category.




Bibliography

• Achen, Christopher and W. Phillips Shively (1995). Cross-level inference. Chicago: University of Chicago Press
• Agresti, Alan (2002). Categorical Data Analysis, 2nd Edition. NY: Wiley.
• Agresti, Alan (2007). An Introduction to Categorical Data Analysis, 2nd Edition. NY: Wiley.
• Binder, Arnold (1984). Restrictions on statistics imposed by method of measurement: Some reality, some myth. Journal of Criminal Justice, Vol. 12: 467-481. Defends robustness of parametric techniques even when using ordinal data.
• Epstein, David & Sharyn O'Halloran (1997). " Who needs ecological regression? Measuring the constitutionality of majority-minority districts." Midwest Political Science Conference, 1997. Available at http://wizard.ucr.edu/polmeth/working_papers97/epste97.html.
• Gorsuch R. L. (1983). Factor Analysis. Hillsdale, NJ: Lawrence Erlbaum. Orig. ed. 1974. Cited with respect to difficulty factors in use of dichotomous and categorical data.
• Jaccard, James & Choi K. Wan (1996). LISREL approaches to interaction effects in multiple regression. Thousand Oaks, CA: Sage Publications.
• Kim, J. O. (1975). "Multivariate analysis of ordinal variables." American Journal of Sociology, Vol. 81: 261-298.
• King, Gary (1997). A solution to the ecological inference problem. Princeton: Princeton University Press.
• Labovitz, S. (1967). "Some observations on measurement and statistics." Social Forces, Vol. 46: 151-160.
• Labovitz, S. (1970). "The assignment of numbers to rank order categories." American Sociological Review, Vol. 35: 515-524.
• Langbein, Laura Irwin & Allan J. Lichtman (1978). Ecological inference. Series: Quantitative Applications in the Social Sciences, No. 10. Thousand Oaks, CA: Sage Publications. Explores ways of using aggregate data for individual-level models. There are no good answers, but the authors outline "ecological regression" as the best of bad alternatives.
• Liu, I. & Agresti, A. (2005).The analysis of ordered categorical data: An overview and a survey of recent developments. TEST, Vol. 14: 1-73.
• Robinson, William S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, Vol. 15: 351-357. Coined the term "ecological fallacy" and helped overturn the then-common practice of using aggregate data for individual-level models.
• Wilson, Thomas. (1971). Critique of ordinal variables. Ch. 24 in H. M. Blalock, ed., Causal models in the social sciences. Chicago, Aldine.
• Zumbo, Bruno D. and Donald W. Zimmerman (1993). Is the selection of statistical methods governed by level of measurement?. Canadian Psychology, Vol. 34: 390-399. Defends robustness of parametric techniques even when using ordinal data.

Copyright 1998, 2008, 2009 by G. David Garson..
Last update 5/12/2009.