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Comparing two numbers or series of numbers. Jane E. Miller, PhD. Overview. Build on principles for reporting one number Report and interpret Aside on types of variables Direction and magnitude Of a cross-sectional comparison Of a trend Mathematical and verbal approaches Reference groups.
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Comparing two numbers or series of numbers Jane E. Miller, PhD
Overview • Build on principles for reporting one number • Report and interpret • Aside on types of variables • Direction and magnitude • Of a cross-sectional comparison • Of a trend • Mathematical and verbal approaches • Reference groups
Apply principles for reporting a number • Set the context (W’s) using topic sentence for each paragraph or section. • Once W’s are reported, don’t need to repeat them unless they change. • E.g., if same place and time pertain to all numbers you compare, state them in the topic sentence. • Convey other W’s (e.g., gender, race) in the sentences reporting those numbers. • Specify the units as you report the numbers. • System of measurement. • Scale and level of aggregation.
Report and interpret • Give both the raw numbers and the result of the comparison. • Report means including a number in a sentence, table, or chart. • Interpret means explaining to your readers what that number means in the context of the question at hand. E.g., • Compare it to other numbers. • Relate it to hypotheses. • A “naked” number is not informative.
Why report and interpret? • Interpret the comparison to answeryourquestion. • Answer the “word problem” behind your analysis. • Provide a context so readers can understand the point you are making with the numbers. • Report numbers to give readers the raw data to answer otherquestions. E.g., • Compare with other times, places, or groups. • Conduct additional calculations with your numbers.
Why not just report? • Often, readers need a context in which to understand the numbers. • Is a given number big or small? • Are things changing? Stable? • Does a particular value exceed an important cutoff? • Every number you include should be chosen for a specific reason. • Convey that reason (or question) to your audience. • Write the solution, not the problem set.
Why not just interpret? • Readers should have access to the “raw numbers” upon which calculations are based. • Suppose you read that the prevalence of low birth weight (LBW) is 30% lower than five years ago, but the LBW rate is not given for either year. • A 30% difference could mean: • 1.0 LBW infant vs. 1.3 LBW infants per 1,000 live births • 400 LBW infants vs. 520 LBW infants per 1,000 live births Very low: Almost eradicated LBW Very high: Serious LBW problem
Which group is bigger, faster, smarter? Instead of: “In Florida, Bush received 2,912,790 votes to Gore’s 2,912,253.” Write: “In Florida, Bush defeated Gore.” How much bigger, faster, smarter? Instead of: “In Florida, Bush defeated Gore.” Write: “In the closest election in US history, Bush wonFlorida by a mere 537 votes.” Specify direction and magnitude of the association
Aside: Types of variables • Variables come in two broad types, also known as “levels of measurement” • Continuous • Categorical • Type of variable affects many aspects of writing about numbers, including pertinent and valid: • Types of calculations and statistics, • Types of charts, • Wording of sentences to present and compare numbers.
Categorical variables • Categorical variables are those that are classified into categories. • There are two types of categorical variables. • Nominal • Named categories with no inherent numeric order • e.g., gender, race, religion • Ordinal • Ordered categories • e.g., Likert scale, income group, letter grade, self-rated health
Continuous variables • Continuous variables • Measured in numeric units, but not grouped. • Two types of continuous variables: • Interval • Zero is not lowest possible value • e.g., temperature °Fahrenheit • Ratio • Zero is lowest possible value • e.g., temperature °Kelvin, height, weight
Why does type of variable matter? • Type of variable affects how you write about direction of association. • For nominal variables, must name the reference group. • Doesn’t make sense to say “as gender increases.” • For ordinal, interval, or ratio variables, can say “as variable A increases, variable B [increases/decreases.]” • Positive or negative association. • Type of variable also affects which statistics and types of chart make sense.
Specify direction and magnitude • For a cross-sectional comparison • Direction • Which value is bigger? • Magnitude • How much bigger? • For a trend • Direction • Is the trend rising, falling, or level? • Magnitude • Howsteeply?
Direction and magnitude:Cross-section • E.g., across groups or experimental conditions. • Of two values: • Direction: Which is bigger? • Magnitude: How much bigger? • “Company B’s policy costs ten dollars per month more than Company A’s.”
Direction and magnitude: Trend • Direction: Is it level, rising, or falling? • Magnitude: How rapidly? • “The population of City A rosegraduallyover the period, while the population of City C decreasedrapidly and B remained stable.”
Example: Direction and magnitude • Poor: “Age and income are correlated.” Or “Age and income are associated.” • Neither direction nor magnitude is specified. • Better: “As age increases, income increases.” • Now we know direction but not magnitude. • Best: “Between the ages of 20 and 49, income increases roughly 10% for each 10 year increase in age, then levels off through age 64.” • Could also include statistical significance.
Calculations to express size • The math is straightforward, e.g., • Subtraction • Division • Percentage difference or change • The writing is more challenging • Often authors focus more on explaining the arithmetic than the answer to the word problem. • Use too much jargon • See suggested resources for references about how to present the answer to your calculation.
Combining calculations and vocabulary for an effective description • Start with a verbal description • Set the context • Provide a verbal sketch of the shape and size of the pattern • Document with numeric evidence • Specify units • Continue to use descriptive vocabulary
Vocabulary to express direction and size • Verbsto describe change • Mundane: increased, declined • More interesting: rocketed, plummeted • Adverbsto modify boring verbs • Increased dramatically • Barely budged • Adjectives • Mundane: level, rising, smaller • More interesting: erratic, minuscule
Illustrative sentences • “In the recent election, Candidate Q eked out a narrow victory over his rival.” • “Plants fed ‘worm tea’ compost grew rapidly over the course of the experiment, whereas those given only water grew more modestly.” • “Immediately after Hurricane Katrina, gasoline prices in the US spiked to an all time high. They remained volatile in the subsequent two months.”
Adding numeric evidence • “In the recent election, Candidate Q eked out a narrow victory over his rival, winning by only 231 votes out of a total of 2 million votes tallied.” • “Plants fed ‘worm tea’ grew rapidly over the course of the experiment, whereas those given only water grew more modestly. The ‘worm tea’ plants averaged 17 cm/week, versus only 13 cm/week for the water-only plants.” • “Last week was uniformly hot, with daily high temperatures within three degrees of one another [illustrates narrow range] andaveraging more than ten degrees above normal[documents that it was hot].”
Comparative writing • For every comparison specify what is being compared to what. • If all you write is “X is 20% higher”, the reader doesn’t know higher than what? • Especially if you are comparing several groups or places or time periods, omission of the referent can be very confusing.
Reference group for multiple comparisons • If you are comparing age distributions for two time periods in two regions, “The elderly age group is smaller” doesn’t tell your reader whether you mean: • Smaller than other age groups in the same region, or • Smaller than the same age group in the other region, or • Smaller than it used to be, in the same region.
Poor comparison statements • Poor version #1: “The difference was 7.2.” • Magnitude but not topic, units, or direction. • Poor version #2: “Insurance and length of stay were associated.” • Specifies topic but not direction,size, orunits.
Improved comparison statements • Better: “Privately insured children stayed longer than publicly insured children.” • Topic, reference group, and direction but not size orunits. • Best: “Children with private insurance stayed on average 7.2 dayslonger thanthose with public insurance.” • Topic, reference group, direction, magnitude, and units.
Summary • When comparing two numbers or series, always • Report and interpret values • Specify direction and magnitude of the association • Convey reference value • Use different, complementary approaches • Results of mathematical calculations to convey size • Vocabulary to express direction and size • Same principles apply for writing about coefficients from multivariate model
Suggested resources: general • For basic principles, see chapters 1 and 2 in • Miller, J. E. 2004. The Chicago Guide to Writing about Numbers(“WA#”) OR • Miller, J. E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. (“WAMA”) • For additional examples, see • Miller, J. E. 2006. “How to Communicate Statistical Findings: An Expository Writing Approach.” Chance 19 (4): 43–49. • Miller, J. E. 2010. “Quantitative Literacy across the Curriculum: Integrating Skills from English Composition, Mathematics, and the Substantive Disciplines.” The Educational Forum. 74 (4): 334–46.
Suggested resources: units and calculations • Units • Chapter 4 of WA# or WAMA • Types of quantitative comparisons and how to • Choose which one(s) to use for a specific task • Write about them • chapter 8 of WA# or chapter 5 of WAMA
Suggested online resources • Podcasts on • Reporting one number • Summarizing a pattern involving many numbers • Choosing a reference category • Interpreting multivariate coefficients
Suggested practice exercises • Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. • Questions #1, 7 and 8 in problem set for chapter 2 • Suggested course extensions for • Chapter 2 • “Reviewing” exercise #6 • “Writing” and “revising” exercise #1 • Chapter 14 • “Reviewing” exercise #3 • “Applying statistics and writing” exercises #1 and 2 • “Revising” exercises #1 and 2
Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html