1. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting multivariate OLS and logit coefficients Jane E. Miller, PhD

2. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overview What elements to report for coefficients Coefficients on – Continuous independent variables (IVs; predictors) – Categorical independent variables Ordinary least squares (OLS) and logit coefficients Topic sentences for paragraphs reporting multivariate coefficients

3. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Report and interpret results Report detailed multivariate results in tables. – Coefficients. – Inferential statistical results: standard error or test statistic, p-value or symbol. – Model goodness of fit statistics. Interpret coefficients in the text. – Refer to associated table.

4. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. What to report for coefficients Topic – Independent variable (IV) – Dependent variable (DV) Direction (AKA “sign”) Magnitude (AKA “size”) Units or categories Statistical significance Most authors remember to report statistical significance, so I have listed that element last!

5. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting coefficients Poor: “The effect of public insurance was –7.2 (p < 0.05).” – Reports the coefficient without interpreting it. Without units or reference group, the meaning of “– 7.2” cannot be interpreted. Better: “Children with private insurance stayed on average 7.2 days longer than those with public insurance (p < 0.05).” – Interprets the β in intuitive terms, mentioning the topic, units, categories, direction, magnitude and statistical significance.

6. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. More examples of interpreting βs Poor: “Insurance and length of stay were associated (p < 0.05).” – Topic and statistical significance, but not direction or size. Better: “Privately-insured children stayed longer than publicly insured children (p < 0.05).” – Statistical significance and direction, but not size. Best: “Children with private insurance stayed on average 7.2 days longer than those with public insurance (p < 0.05).” – Topic, direction, magnitude, units, categories, and statistical significance.

7. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpretation of βs depends on types of variables in your models The type of dependent variable: – Continuous dependent variable Ordinary least squares (OLS) – Categorical dependent variable Logistic (logit) regression model Type of independent variable: – Continuous – Categorical

8. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting coefficients from ordinary least squares (OLS) models

9. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Coefficients for OLS models For ordinary least squares (OLS) models, the coefficient (β) is a measure of difference in the DV for a 1-unit increase in the IV. – For unstandardized coefficients, difference in the same units as the dependent variable. Can be explained using wording for results of subtraction. For standardized coefficients, β measures difference in standardized units (multiples of standard deviations). – See podcast about resolving the Goldilocks problem using model specification.

10. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting βs: continuous predictors The unstandardized coefficient on a continuous predictor in an OLS model measures – The difference in the dependent variable for a one- unit increase in the independent variable. – Effect size is in original units of the DV. Example topic: Mother’s age as a predictor of birth weight: – Dependent variable = birth weight in grams. – Independent variable = mother’s age in years. – Both are continuous variables.

11. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: Mother’s age as a predictor of birth weight Poor: “Mother’s age and child’s birth weight are correlated (p<0.01).” – Names the dependent and independent variables and conveys statistical significance, but not direction or magnitude of the association. Better: “As mother’s age increases, her child’s birth weight also increases (p<0.01).” – Concepts, direction, and statistical significance, but not size. Best: “For each additional year of mother’s age at the time of her child’s birth, the child’s birth weight increases by 10.7 grams (p<0.01).” – Concepts, units, direction, size, and statistical significance.

12. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting βs: categorical predictors The β on a categorical IV in an OLS model measures the difference in the DV for the category of interest compared to the reference category. – A “1-unit increase” does NOT make sense. Example: gender – Dummy variable (AKA “binary variable”) coded 1 = boy 0 = girl = omitted (reference) category

13. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: Gender as a predictor of birth weight Poor: “The β for ‘BBBOY’ is 116.1 with an s.e. of 12.3 (table 15.3).” – Uses a cryptic acronym rather than naming the independent variable or conveying that it is categorical. – Doesn’t convey the dependent variable. – Reports the same information as the table (size of coefficient and standard error), but does not interpret them. – The direction of the effect cannot be determined because categories and units are not specified. – To assess statistical significance, readers must calculate test statistic and compare it against critical value.

14. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Gender as a predictor of birth weight, cont. Slightly better: “Gender is associated with a difference of 116.1 grams in birth weight (p < 0.01).” – Concepts, magnitude, units, 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 < 0.01).” – Concepts, reference category and units, direction, magnitude, and statistical significance.

15. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Identifying the reference category For categorical variables, mention identity of reference category. – E.g., effect size is relative to whom? Example for 2-category comparison: – “Boys weighed 116 grams more than girls.” Example for multicategory comparison: – “Compared to white infants, black and Hispanic infants weighed 62 and 16 grams less on average.” – OR “Mean birth weight was 62 and 16 grams less, for black and Hispanic infants, respectively, when each is compared to white infants.”

16. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interpreting coefficients from logistic regression models

17. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Logit models for categorical dependent variables Logit = log[p/(1 – p)] = log(odds of the category you are modeling) – p is the proportion of the sample in the modeled category β measures the log relative-odds of the outcome for different values of the independent variable Exponentiate the logit coefficient e β = relative odds, or “odds ratio” Compares the odds of the outcome for different values of the independent variable

18. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: Logit model of LBW Low birth weight (LBW) = birth weight <2,500 grams Log-odds = log[p LBW /(1 – p LBW )] – Where p LBW is the proportion of the sample that is LBW. Log relative odds of LBW = comparison of log- odds of LBW for different values of the independent variable. e β = relative odds of LBW for different values of the independent variable.

19. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Wording for odds ratios βs for logit models are in the form of ratios. For suggestions on how to phrase descriptions of ratios with minimal jargon, see – Table 5.3 in The Chicago Guide to Writing about Numbers OR – Table 8.3 in The Chicago Guide to Writing about Multivariate Analysis

20. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Phrases for ratios Type of ratioRatio example Rule of thumbWriting suggestion < 1.0 (e.g., 0.x) % difference = ratio  100 0.80[Group] is only x% as ___ b as the reference value. “Males were only 80% as likely as females to graduate from the program.” Close to 1.01.02Use phrasing to express similarity between the two groups. “Average test scores were similar for males and females (ratio = 1.02 for males vs. females).” >1.0 (e.g., 1.y) % difference = (ratio – 1)  100. 1.20[Group] is 1.y times as ___ as the reference value. “On average, males were 1.20 times as tall as females.” OR [Group] is y% ___er than the reference value. OR “Males were on average 20% taller than females.” 2.34 [Group] is (2.34 – 1)  100, or 134% more ___ than the reference value. “Males’ incomes were 134% higher than those of females.” Close to a multiple of 1.0 (e.g., z.00) 2.96[Group] is (about) z times as ___. “Males were nearly three times as likely to commit a crime as their female peers.” See tables in WA#s or WAMA

21. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Odds ratios for categorical independent variables Odds ratio of the outcome for the category of interest compared to the reference category. “Infants born to smokers had 1.4 times the odds of low birth weight (LBW) as those born to nonsmokers (p < 0.01).” – Concepts, reference category, direction, magnitude, and statistical significance.

22. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Odds ratios for continuous independent variables Odds ratio of the outcome for a one-unit increase in the independent variable. “Odds of LBW decreased by about 0.8% for each 1 year increase in mother’s age (NS).” – Concepts, units, direction, magnitude, and statistical significance.

23. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Topic sentences for paragraphs reporting multivariate results Start each paragraph of the results section with a restatement of topic addressed by analysis to be reported in that paragraph. – Can paraphrase title of table or chart that reports the detailed statistical results. Topic sentence should mention: – Dependent variable. – Independent variable(s). Use summary phrase rather than long list of variables. – Type of analysis.

24. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example topic sentences “Multivariate logistic regression results show that insurance is a powerful predictor of length of stay (table X).” [Next sentence goes into detail about direction, size, and statistical significance.] – Mentions type of analysis, dependent variable, and independent variable. “As shown in figure Y, race and income level interact in their effect on risk of asthma.”

25. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Summary Report detailed multivariate results in tables. Interpret coefficients in prose. Specify direction, magnitude, and statistical significance of associations. – Units for continuous variables – Categories for nominal or ordinal variables Write about concepts, not acronyms. – Introduce concepts under study in topic sentences.

26. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested resources Miller, J. E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. – Chapter 5, on creating effective multivariate tables – Chapter 8, on wording for results of subtraction (OLS βs) ratios (logit βs) – Chapter 9, on writing about βs from OLS and logit models – Chapter 10, on the Goldilocks problem for choosing a fitting contrast size for interpreting coefficients

27. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested online resources Podcasts on – Comparing two numbers or series – Choosing a reference category – Defining the Goldilocks problem – Resolving the Goldilocks problem: Presenting results – Differentiating between statistical significance and substantive importance

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. – Problem sets for chapters 9 and 15 – Suggested course extensions for chapters 9 and 15 “Reviewing,” “writing” and “revising” exercises.

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