1. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Differentiating between statistical significance and substantive importance Jane E. Miller, PhD

2. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overview Substantive significance defined Quick review of statistics – What questions can they answer? – What questions can’t they answer? How to implement a balanced presentation of multivariate results. Both – Statistical significance – Substantive importance

3. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Objective of most research papers Few people who write about multivariate analysis are focused solely on statistical mechanics such as developing new computer algorithms or formal statistical tests. – Some statisticians and methodologists will have those interests. Most of us are interested in studying some relationship among social science or health concepts. – Test a hypothesis, derived from theory or previous empirical studies. – Inferential statistics are a necessary tool for hypothesis testing in quantitative research.

4. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. What is substantive significance? Substantive significance of an association between two variables. – “So what?” – “How much does it matter?” Real-world relevance to topic In various disciplines, substantive significance = – “clinically… – “economically… – “educationally… – …meaningful” variation.

5. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: BMI & mortality Body mass index (BMI) shows a statistically significant positive association with mortality. But is that gradient substantively significant? – Is it worth designing an intervention to decrease BMI as a way of decreasing mortality?

6. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Key criteria for assessing substantive significance Is the association causal? – Will changing the hypothesized cause lead to change in the purported effect? – Will weight loss (reduced BMI) yield lower mortality? Is the effect big enough to matter? – Is the excess mortality among overweight or obese persons large enough to justify a program? Can the hypothesized cause be changed? – Is BMI malleable?

7. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example prose “For every hour a boy played a video game, he read just two minutes less than a boy who didn’t play video games. Notably, non-gaming boys didn’t read much at all either, spending only eight minutes a day with a book.” From a NYT summary of Cummings and Vandewater, 2007. “Relation of Adolescent Video Game Play to Time Spent in Other Activities,” Archives of Pediatrics and Adolescent Medicine.

8. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Quick review of statistical significance testing

9. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Start with a hypothesis In the gaming example, the authors hypothesized that the more time adolescents spent on video games, the less time they spent on homework. – So far, description is purely in terms of the concepts under study. – No statistical jargon, yet… To formalize this for statistical testing – Homework time = dependent variable (Y) – Gaming time = independent variable (X i ) – H a = gaming time is negatively associated with homework time. In other words, X i is inversely associated with Y

10. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Contrast it against the null hypothesis The assumption of “no difference between groups” is called the null hypothesis (H 0 ). In the study on effects of gaming on homework time – H 0 : time among gamers = time among non-gamers OR – time among gamers - time among non-gamers = 0 – In words, the null hypothesis states that there is no difference in the amount of time spent on homework by gamers versus non-gamers.

11. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. What ? does inferential statistics answer? “How likely would it be to obtain a difference at least as large as that observed between groups in the sample if in fact there is no difference between groups in the population?” The p-value tells us the probability of falsely rejecting the null hypothesis. – Conventional levels of “statistical significance” : p<.05 – Strictly speaking, p<.05 tells us that for a large sample such as that used in the gaming study (N~1,400), the estimated coefficient on time spent gaming is at least 1.96 times its standard error.

12. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. What questions DOESN’T it answer? Whether the relationship is – Causal Association ≠ causation – In the expected direction The difference could be statistically significant but in the opposite of the hypothesized direction. – Big enough to matter in the real-world context Each hour spent gaming reduced reading time by 2 minutes. Is that enough to induce genuine concern from parents or teachers? – Malleable

13. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Conclusion: Don’t stop at “p<.05”! “p<.05” answers only part of what we want to know about our research question. – It is a necessary but not sufficient part of statistical analysis. Also need to consider questions about – Substantive significance Direction Size – Causality Non-causal associations should not be used to inform policy or program changes. Confounding or spurious associations should be ruled out. – Often why we estimate a multivariate model.

14. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Substantive significance overlooked Many statistics textbooks show how to assess and present statistical significance. Few if any show how to assess and present substantive significance.

15. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Balance presentation of statistical and substantive significance How to include both: – Inferential statistics for formal hypothesis testing. – Interpretation of substantive significance of findings in the context of the specific research question. Critical for policy-makers and others not formally trained in statistics.

16. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Principles for presenting results Name the specific variables. Avoid – Writing about “my dependent variable” or “the coefficient.” – Using acronyms from your database  Report numbers in tables. – Complete set of coefficients, standard errors, goodness-of-fit statistics. Interpret numbers in text. – Incorporate units and categories for variables into the prose description.

17. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. What to report for coefficients Direction (AKA “sign”) – For categorical independent variables (IV), which category has higher value of the dependent variable (DV)? – For continuous IVs, is the trend in the DV up, down, or level? Magnitude – How big is the difference in the DV across values of the IV? Statistical significance

18. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Gender as a predictor of birth weight Poor: “Boys weigh significantly more at birth than girls.” – Concepts and direction but not magnitude. – Statistical significance is ambiguous: Is the term “significant” intended in the statistical sense or to describe a large difference? Slightly better: “Gender is associated with a difference of 116.1 grams in birth weight (p<.01).” – Concepts, magnitude, and statistical significance but not direction: Was birth weight higher for boys or for girls? Best: “At birth, boys weigh on average 116 grams more than girls (p<.01).” – Concepts, reference category, direction, magnitude, and statistical significance.

19. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Substantive issues for coefficients on continuous predictors A β on a continuous independent variable measures the change in the dependent variable for a 1-unit increase in that independent variable. – For some variables, a 1-unit increase is too small to be substantively meaningful. E.g., a $1 increase in annual per capita income in the US today. – For other variables, a 1-unit increase is too big to be plausible. E.g., a 1-unit increase in a variable measured as a proportion. “The Goldilocks problem” Need to look at distribution of values in your data.

20. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Solutions to Goldilocks problems In those cases, to assess whether a coefficient is “big” or “small,” need a different sized contrast. Important for comparing coefficients across variables. See related podcasts on the Goldilocks problem. Identifying a Goldilocks problem Solutions: – Defining variables – Specifying models – Interpreting results

21. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Substantive significance in the discussion Place findings back in the broader perspective of the original research question. Do they correspond to your hypothesis in terms of – Direction (sign) of the effect? – Size? – Was the effect size attenuated when potential confounders or mediators were introduced into the model? What is the evidence for a causal relationship? – If not causal, what explains the association? – If causal, what are the implications for policy, programs, etc.?

22. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Substantive issues from gaming study “But the meaning of the finding [that girls who are gamers spend less time than non-gamers on homework] is not clear, as high-academic achievers often spend less time on homework as well.” – Places the finding in broader context by discussing other correlates of homework time. “Although only a small % of girls played video games, our findings suggest that gaming may have different social implications for boys than for girls.” – Raises the question of selection effects: which girls play video games, and do their other characteristics affect how they spend their time?

23. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Relate findings to previous studies’ Are your findings consistent with the published literature on the subject in terms of statistical significance, sign, and approximate size? If not, why not? – Different sample (place, time, subgroup) – Different data source or study design – Different model specification Included potential confounders not previously analyzed. Tested for possible mediating effects of 1+ factors.

24. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Statistical significance in the discussion Describe in words, not #s. – No detailed standard errors, p-values, or test statistics. Focus on the purpose of the statistical tests – Did the main variable of interest increase proportion of variance explained by the model? – Did some other variable “explain” the association between your key variable and the outcome?

25. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Summary Emphasize the substantive issues behind the statistical analyses. – Design the specification to match topic and data. – Choose plausible, relevant numeric contrasts. Aim for a balanced presentation of statistical significance and substantive importance. – Use prose to ask and answer research question. – Use tables to report comprehensive, detailed statistics. – Use charts if needed to convey complex patterns.

26. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested resources Chapter 3 (Statistical significance, substantive significance, and causality) in – Miller, J.E., 2004. The Chicago Guide to Writing about Numbers OR – Miller, J.E., 2013. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. (“WAMA II”) Miller J.E. and Y.V. Rodgers, 2008. “Economic Importance and Statistical Significance: Guidelines for Communicating Empirical Research.” Feminist Economics. 14(2):117-149. Chapter 10 (Goldilocks problem) in WAMA II.

27. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested online resources Podcasts on – Comparing two numbers or series – Reporting coefficients from OLS and logit models – Defining the Goldilocks problem – Resolving the Goldilocks problem: Presenting results

28. 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, 2nd Edition. – Questions #2 and #4 from the problem set for chapter 3 – Suggested course extensions for chapter 3 “Reviewing” exercises #1–4 “Writing and revising” exercises #1 and #2

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