Comparing two numbers or series of numbers Jane
Comparing two numbers or series of numbers Jane E. Miller, Ph. D The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Why report and interpret? • Interpret the comparison to answer your question. – 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 other questions. E. g. , – Compare with other times, places, or groups. – Conduct additional calculations with your numbers. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. Very low: Almost – A 30% difference could mean: eradicated LBW • 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 high: Serious LBW problem The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Specify direction and magnitude of the association • 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 won Florida by a mere 537 votes. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 • How steeply? The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Direction and magnitude: Trend • Direction: Is it level, rising, or falling? • Magnitude: How rapidly? • “The population of City A rose gradually over the period, while the population of City C decreased rapidly and B remained stable. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Vocabulary to express direction and size • Verbs to describe change – Mundane: increased, declined – More interesting: rocketed, plummeted • Adverbs to modify boring verbs – Increased dramatically – Barely budged • Adjectives – Mundane: level, rising, smaller – More interesting: erratic, minuscule The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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] and averaging more than ten degrees above normal [documents that it was hot]. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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, or units. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Improved comparison statements • Better: “Privately insured children stayed longer than publicly insured children. ” – Topic, reference group, and direction but not size or units. • Best: “Children with private insurance stayed on average 7. 2 days longer than those with public insurance. ” – Topic, reference group, direction, magnitude, and units. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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, 2 nd 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. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Suggested online resources • Podcasts on – Reporting one number – Summarizing a pattern involving many numbers – Choosing a reference category – Interpreting multivariate coefficients The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Suggested practice exercises • Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2 nd 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 The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Contact information Jane E. Miller, Ph. D jmiller@ifh. rutgers. edu Online materials available at http: //press. uchicago. edu/books/miller/multivariate/index. html The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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