1. Types of quantitative comparisons Jane E. Miller, PhD The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

2. Types of quantitative comparisons Level Rank Difference Ratio Percentage difference Percentage change The Chicago Guide to Writing about Numbers, 2 nd edition.

3. Overview For each type of quantitative comparison: – How it is calculated – How to write clear sentences to report and interpret results of that calculation – Advantages: what questions it can answer – Disadvantages: what questions it can’t answer How to chose among the ways to compare numbers, based on the question you seek to address The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

4. Coordinating calculations with your writing An important aspect of working with quantitative comparisons is coordinating them with the ways you report and interpret them in prose form. Before you calculate, think about how you prefer to phrase your comparisons, then perform the corresponding calculations accordingly. Doing so will avert the confusion of interpreting – Ratios that you have accidentally described “upside down.” – Subtraction you have inadvertently explained “backward.”

5. Level Level = value of the measure for one observation. Example sentences: – “Michael Phelps’ time in the 200 meter individual medley (IM) at the 2012 Olympic Games was 1:54.27.” – “The average price of a gallon of gasoline in New Brunswick, New Jersey, in January, 2015 was $1.99.” – “In 2011, the infant mortality rate in the U.S. was 6.1 deaths per 1000 live births.” Checklist for reporting the level – Context (W’s) – Units – See podcast on reporting one number The Chicago Guide to Writing about Numbers, 2 nd edition.

6. Disadvantages of reporting level alone Difficult to assess what the number means in the context of the topic under study. – Is it high or low? – Fast or slow? – Cheap or expensive? To assess those aspects, need a comparison. The Chicago Guide to Writing about Numbers, 2 nd edition.

7. Rank Rank = order, compared to other values in a list, e.g., – A series of other values observed in the same time or the same place – A well-established standard – An historic high or low value Examples: – “Michael Phelps placed first in the 200 meter IM at the 2012 London Olympic Games.” – “The cost of gasoline is the lowest observed in several years.” – “In 2009, the U.S. ranked 29th in the world in terms of infant mortality.” The Chicago Guide to Writing about Numbers, 2 nd edition.

8. When is rank useful? When all that matters is the order, not the size of the difference between groups. – In an election, all that matters is who came in first. It doesn’t matter whether the margin of victory is 1 vote or 1 million votes! The Chicago Guide to Writing about Numbers, 2 nd edition.

9. Disadvantages of using rank alone Rank tells you where the value falls relative to other values. Doesn’t tell you how close the other values are. – Could be the fastest, but only by some microscopic amount. – Could be the least expensive, but with 15 other brands each costing only a few cents more. Need other measures to tell you how big a difference, which is often critical for interpreting what that value means in context. The Chicago Guide to Writing about Numbers, 2 nd edition.

10. Difference The value of interest (X), subtracted from some other value (Y), or Y minus X. Difference = Y - X Examples – “Michael Phelps’ 200 meter IM time was 0.63 seconds faster than that of the second place swimmer.” – “In January 2015, gasoline cost about $2.00 per gallon less than it had a month earlier.” Checklist for comparing two numbers – Report and interpret. – Specify direction and magnitude. The Chicago Guide to Writing about Numbers, 2 nd edition.

11. When is the difference useful? When the difference itself is of interest. – How much more will something cost? Is that amount within your budget? – How much time will be saved if a certain change is made? When you do not need to assess size of difference compared to anything else. The Chicago Guide to Writing about Numbers, 2 nd edition.

12. Disadvantages of difference alone Is a difference of $2.00 a lot or a little? – It depends on the level. If the price of a gallon of gasoline decreased by $2.00, you would consider that a big change. If the price of a car decreased by $2.00, you probably wouldn’t even notice! – Why? Because the gallon of gas cost about $3.75 before the change, whereas the car probably cost at least $15,000. To address the question of whether a given difference is big or small, you need a measure that looks at the difference or change relative to the initial value. The Chicago Guide to Writing about Numbers, 2 nd edition.

13. Ratio The value of interest (X) divided by some other value (Y). Ratio = X/Y – If X and Y are measures of risk, the ratio is the relative risk. Examples – “Infant mortality among black infants is more than twice as high as among white infants.” – “The cost of gasoline is less than half of what it was last year.” The Chicago Guide to Writing about Numbers, 2 nd edition.

14. Avoid use of jargon When interpreting a ratio, use intuitive phrasing that a layperson can understand. Poor: “The ratio is 2.25.” – Can’t tell What variable is being studied. What groups are being compared. Which group has the higher value. Better: “The infant mortality rate (IMR) for black infants is more than twice as high as for white infants.” See podcast on writing about ratios. The Chicago Guide to Writing about Numbers, 2 nd edition.

15. Percentage difference Percentage difference = difference (from subtraction) divided by the initial value multiplied by 100. percentage difference = [(Y-X)/X] * 100 Example: The infant mortality rate (IMR) for blacks = 11.5 deaths per 1000 live births; the IMR for whites = 5.1. – Calculate: [(11.5 - 5.1)/5.1] * 100 = 125% – Write: “Black infants are 125% more likely than their white counterparts to die before their first birthday.” The Chicago Guide to Writing about Numbers, 2 nd edition.

16. Percentage change Calculated the same way as percentage difference. – Compares values from two points in time. Subtract the earlier value (T1) from the later value (T2) Divide by the value for the earlier time point and multiply by 100: Percentage change = [(T2 - T1)/T1] * 100 Direction of percentage change – A positive percentage change means an increase over time. – A negative percentage change means a decrease over time. The Chicago Guide to Writing about Numbers, 2 nd edition.

17. Reporting numeric examples Once you have interpreted the result of the comparison, report the numbers to document that pattern. Sometimes the numbers can go in the same sentence with the description. – “Mortality is nearly 3 times as high among persons aged 85 and older (15.3 deaths per 1000 persons) as among those aged 75-84 (5.7 deaths per 1000).” OR – “Mortality is nearly 3 times as high among persons aged 85 and older than among those aged 75-84: 15.3 deaths per 1000 persons and 5.7 deaths per 1000, respectively.” The Chicago Guide to Writing about Numbers, 2 nd edition.

18. Reporting numeric examples, cont. Sometimes it is simpler to report the numbers in a separate sentence. – “Michael Phelps’ margin of victory in the 200 meter individual medley at the 2012 Olympic Games was 0.63 seconds. {Phelps finished in 1:54.27, followed by Ryan Lochte in 1:54.90.” The Chicago Guide to Writing about Numbers, 2 nd edition.

19. Choosing which types of quantitative comparisons to use in your writing The choice of which calculations to include depends on your topic and discipline. – Report results of a race, election, or marketing study in terms of rank and difference. – Describe time trends using difference or percentage change. To express large changes such as tripling, or halving using a ratio. – Describe variations in risk or probability in terms of ratios. – Report only one measure of rank (e.g., position, percentile, decile, or quartile).

20. Type of variable constrains your choice of quantitative comparison The level of measurement (a.k.a. type of variable) determines which types of quantitative comparisons make sense for that variable. For each of the variables in your analysis, need to know its level of measurement. – See suggested resources at the end of this lecture for readings and podcasts on that topic.

21. Categorical variables Categorical variables are grouped into categories or ranges. – Nominal variables have named categories with no inherent numeric order. e.g., gender, race, religion – Ordinal variables have ordered categories. They can, but do not have to, have numeric units. E.g., Age group (in years) and income range (in $) have numeric units Likert scale and letter grade do not have numeric units The Chicago Guide to Writing about Numbers, 2 nd edition.

22. Continuous variables Continuous variables are measured in numeric units but are not grouped. – Interval variables can take on negative values. e.g., temperature in °Fahrenheit – Ratio variables cannot have negative values. e.g., temperature on the Kelvin scale, weight in pounds, income in Euros

23. Quantitative comparisons for continuous variables Can do calculations with their values. Ratio variables – Rank, e.g., tallest person – Difference, e.g., 7 inches taller – Ratio, e.g., twice as tall Interval variables – Rank, e.g., coldest day in history – Difference, e.g., 10 degrees colder – CANNOT interpret a ratio of values e.g., if yesterday was -2 °F and today is 2 °F, what would a ratio of -1 mean? The Chicago Guide to Writing about Numbers, 2 nd edition.

24. Quantitative comparisons for categorical variables More limited comparisons Ordinal variables – Rank, e.g., “excellent health” is better than “poor health.” – CANNOT interpret difference or ratio of values e.g. what would “excellent health” minus “poor health” mean? Nominal variables – Can only ask “same or different” values – CANNOT rank, or calculate difference or ratio e.g., what would it mean to say Jewish is higher than Catholic? The Chicago Guide to Writing about Numbers, 2 nd edition.

25. Review: Which type(s) of comparison fits which level(s) of measurement? Can assess “same” or “different” values across cases for all types of variables. – Nominal, ordinal, interval, or ratio variables. Can rank values for any of the quantitative variables. – Ordinal, interval, or ratio variables. Can subtract values for either type of continuous variable. – Interval or ratio variables. Can divide values only for ratio variables.

26. Summary How to choose which type(s) of quantitative comparison best suit to the question at hand. – Each type of calculation answers some questions well but leaves other questions unanswered. – Pick types of quantitative comparison(s) that suit the levels of measurement of the variables in your analysis. How to report and interpret the results of those calculations clearly and correctly. – Convey the topic, not just the numbers. – Report direction and magnitude. – Avoid using jargon about the calculation. The Chicago Guide to Writing about Numbers, 2 nd edition.

27. Suggested resources Miller, J. E. 2015. The Chicago Guide to Writing about Numbers, 2nd Edition. University of Chicago Press. – Chapter 4, section on levels of measurement – Chapter 5, on types of quantitative comparisons Chambliss, D.F., and R.K. Schutt. 2012. Making Sense of the Social World: Methods of Investigation. 4th ed. Thousand Oaks, CA: Sage Publications. – Chapter 4, on levels of measurement The Chicago Guide to Writing about Numbers, 2 nd edition.

28. Suggested online resources Podcasts on – Reporting one number – Comparing two numbers or series of numbers – Getting to know your variables – Choosing a comparison group or value – Writing about ratios The Chicago Guide to Writing about Numbers, 2 nd edition.

29. Suggested practice exercises Study guide to The Chicago Guide to Writing about Numbers, 2nd Edition. – Exercises #1-6 and #8 in the problem set for chapter 5 – Suggested course extensions for chapter 5 “Reviewing” exercise #1 “Applying statistics” exercise #1 “Writing and revising” exercises #1 and 2 The Chicago Guide to Writing about Numbers, 2 nd edition.

30. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/numbers/index.html The Chicago Guide to Writing about Numbers, 2 nd edition.