Summarizing a pattern involving many numbers Generalization example

Summarizing a pattern involving many numbers: Generalization, example, exception (“GEE”) Jane E. Miller, Ph. D The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Overview • Before watching this podcast, watch podcasts on – Reporting one # – Comparing two numbers or series of numbers. • • Common problems with summarizing patterns Systematic approach to summarizing patterns Illustrative example Suggestions for wording The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Describing complex patterns • Relationships among three variables, e. g. , – Multiple-line or clustered bar chart – Table with several panels or nested columns • Want to summarize the big picture – Convey the pattern that addresses the underlying research question for which the numbers are evidence The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Common problems in summarizing patterns among many numbers • Two opposite types of problems: – Report every one of the numeric values in the text. • Too many numbers! – Very difficult to organize lots of individual numeric values in prose. • Lose the forest for the trees. – Hard to see the overall pattern. – Report an arbitrary number from the table or chart. • Unlikely to be representative of the overall pattern. • Too few numbers. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

The “Generalization, example, exception” approach The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Generalization, example, exception (“GEE”) • Generalization: – Try to find and describe an overall pattern, rather than going through every cell in the table or point on the chart. • The detailed numbers are what the table or chart is for! • In your write-up, digest and synthesize the patterns. • Example: – Give 1 or 2 representative numeric contrasts to illustrate generalization. • Exception(s): – If parts of the table or chart deviate from your generalization, describe that departure. – Give a numeric example to illustrate the exception. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Complementary use of prose with table or chart • Create a table or chart to show all of the values involved in the pattern. • Provide your readers with the data to verify your summary. • Tables and charts are much better than prose for organizing a lot of individual numbers. • Write a narrative description of the pattern. • Refer to the table or chart by name as you report numeric examples to document that pattern. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• Topic: prevalence rates of drug and alcohol abuse for – Four substances (x-axis) – Three age groups (legend) • Aim to summarize: – Does the same age pattern apply to all four substances? – Is the most common substance consistent across age groups? The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

“Generalization” • Point (and paragraph) #1 – “For every type of substance studied, abuse declined with increasing age. ” – Compare within clusters (type of substance abuse) across bar colors (age groups). • Point (and paragraph) #2 – “For two of the three age groups, binge alcohol use was most common, followed by heavy alcohol use, marijuana use, and cocaine use. ” – Compare across clusters (substance) within bar color (age groups). The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

“Example” • [Generalization from previous slide] – “For every type of substance studied, abuse declined with increasing age. ” • Associated example: – “For example, rates of binge alcohol use declined from 42% of 18– 25 years olds, to 33% of 26– 34 year olds, to 18% of those aged 35 and older. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

“Exception” • One age group doesn’t follow the same rank order of substances as the other two. – Look at the blue bars. • Wording for the exception: – [Revised from second generalization] “For the older two age groups, binge alcohol use was most common, followed by heavy alcohol use, marijuana, and cocaine use. In the youngest age group, the pattern was similar except that marijuana use slightly exceeded heavy alcohol use. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Exceptions in direction • Generalization: Among the two oldest age groups, heavy alcohol use > marijuana use • Exception: Among the youngest adult age group, marijuana use > heavy alcohol use The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

• In the US, for both men and women, whites outlive blacks [generalization]. • The black disadvantage in life expectancy is larger among men than among women. • 4. 7 year gap among men • 3. 3 year gap among women Life expectancy at birth (years) Exceptions in magnitude 90 expectancy Black White Life by race and gender, US 2010 81. 3 80 70 76. 5 78. 0 71. 8 60 50 40 30 20 10 0 Male Female The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Exceptions in statistical significance • Examples: – All but one association shown in a table are statistically significant. – Only one of many associations shown on a chart is statistically significant – The association between an independent variable and a dependent variable is statistically significant only for some values of a second independent variable. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

GEE for other disciplines • Interpretations of historical events or scientific concepts. – Is there consensus in the literature, or are there conflicting views? – Consensus = generalization. – Conflicting views = exceptions. • Narrative perspective in literary works. – Were most of an author’s pieces written in the first person? Third person? – Quotations = examples. • Want representative examples, not arbitrary choices! The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Wording for a GEE The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Wording for a generalization • If a pattern characterizes many of the numbers you are summarizing, introduce the generalization with phrases such as: – “In general, . . . ” – “Typically, . . . ” – “By and large. . . ” – If the pattern characterizes all groups, say so! – “In every region studied, . . . ” – “Throughout the entire period of observation. . . ” – “Regardless of [characteristic], . . . ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Including measures of size in the generalization • Convey approximate share of values encompassed by the generalization using phrases such as – “Virtually all” – “In the majority of cases, ” – “Roughly three-quarters of. . . ” • Can include the numeric illustration in the sentence with the generalization by placing the numeric value in parentheses after the pertinent phrase. E. g. , – “Virtually all respondents (98%). . . ” – “In a majority of cases (59%). . . ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Wording to present the example • To document the pattern characterized by a generalization or an exception, – Introduce numeric evidence. – Refer to the table or chart where all of the numbers in the pattern can be found. • E. g. , – “For example, as shown in Figure X. . . [numeric fact, with its Ws attached]. ” – “For instance, . . . [numeric contrast, with the W’s specified]. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Wording to introduce exception(s) • To introduce exceptions to the general pattern you have just described, use phrases such as – “An exception [to that pattern] is. . . ” – “On the other hand, . . . ” – “However, . . . ” • E. g. , “In seven out of 10 years studied, [general pattern and example]. However, in the other three years, [contrasting pattern = exception]. ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Wording for exceptions in direction • Some exceptions are literally the opposite of the pattern described in the generalization. E. g. , – A falling rather than rising trend. – A deficit rather than an excess for one group compared to another. • Having already described the direction of the general pattern, introduce such exceptions using expressions such as – “On the contrary, . . . ” – “Conversely, . . . ” The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Wording for exceptions in direction 80 Labor force participation by race 70 Labor force participation, % • E. g. , “Among males, the labor force participation rate for whites exceeded that for blacks. In contrast, among females, blacks had a higher labor force participation rate than whites. ” and gender, US Black 60 White 50 40 30 20 10 0 Male Female The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Summary: Advantages of a GEE • Avoids reporting every number from a complex pattern in the text. – That is what the table or chart is for. • Summarizes the patterns. – How do numbers relate to one another? – How do patterns relate to the original question? – The forest, not the trees! The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Suggested resources • Chapter 2 and Appendix A 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”) • Chapter 9 in WA# or chapter 14 in WAMA, section on “generalization, example, exceptions revisited” • Miller, J. E. 2006. “How to Communicate Statistical Findings: An Expository Writing Approach. ” Chance 19 (4): 43– 49. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.

Suggested online resources • Podcasts on – Reporting one number – Comparing two numbers or series of numbers – Choosing tools to present numeric information – Steps for conducting a GEE 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. – Question #9 in problem set for chapter 2 – Question #9 in problem set for chapter 6 – Suggested course extensions for • chapter 2 – “Reviewing” exercise #7 – “Writing” and “revising” exercise #4 • chapter 14 – “Applying statistics and writing” exercises #3 and 4 – “Revising” exercise #3 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.