Lecture 4W-InterpretingRegression
- 1. Lecture 4W: Explaining Regression Results Describing Results in Everyday English
- 2. Explaining Regression Results • Some things are technical, precise ▫ Everyone who does same command should get same table • Other things more open to interpretation ▫ Should we care about our results?
- 3. Regression Table: Tons of Info • Covered regression coefficients Monday • Will cover rest of output today
- 4. Picking Up From Monday • Start by looking at bottom table ▫ Column for coefficients usually first • y = a + bx + E • conrinc = -11679.16 + 1036.512 * prestg10 + E
- 5. Protip: Focus on IV More Than Constant • Easy to overemphasize constant, particularly when it’s negative ▫ Constant usually matters less
- 6. Describing Coefficients in Words • Number under coeff for prestg10. (1036.512) ▫ Slope of the line. • If someone scores 1 point higher on occupational prestige, how much more money would we expect them to earn, given these results?
- 7. Explaining the 1036.512 Coefficient • If someone scores 1 point higher on occupational prestige, how much more money would we expect them to earn, given these results? ▫ We would expect them to earn 1 * 1036.512 = $1036.512 more per year. • If Mary’s occupational prestige is 10 points higher than Jess, how much more would we expect her to earn? ▫ 10 * 1036.512 = $10365.12 more per year.
- 8. A “One-Unit Increase” • Common way to describe a regression coefficient in English. ▫ A one unit increase in occupational prestige leads to a 1036.512 unit increase in annual income. ▫ Better than saying “the regression coefficient is 1036.512.” • Don’t round until every calculation is done
- 9. Why One Unit Increase vs. Ten? • Start with one unit since it is the easiest to calculate. • Only time you calculate just to show you can is homework problems. ▫ When we talk about interpreting results and making arguments, up to you to pick resonable number of units as part of the argument
- 10. Predicting Scores • We can also use this equation to predict how much money someone would make, based on their occupational prestige. ▫ conrinc = -11679.16 + 1036.512 * prestg10 + E • Assume occupational prestige = 50. ▫ Income = -11679.16 + 1036.512 * 50 = $40146.44 • What about someone with 80 occ. prestige? ▫ Income = -11679.16 + 1036.512 * 80 = $71241.80
- 11. Prediction and Exceptions • Reminder: most people not exactly on the regression line ▫ Exceptions do not invalidate the pattern
- 12. Constants can be weird • Imagine using age to predict income. Toddlers may be predicted to have negative income. ▫ It’s because there are no toddlers in our sample. ▫ The constant may be nonsense if we never see x = 0 in our sample.
- 13. Statistical Significance in Regression • Goal is to see whether change in independent variable leads to change in dependent variable ▫ Is relationship relatively unlikely to appear just from random chance? • Null hypothesis: regression coefficient = 0
- 14. Calculating Statistical Significance • Each variable has it’s own standard error term. • Use standard error to get a t statistic for each term. ▫ We don’t care about constant though
- 15. Computing SE for Regression Coeff. • Where σε 2 is the variance in error term εi • sx 2 is the sample variance of x, sx is the sample standard deviation of x. 2 2 2 2 2 2 2 2 2 22 2 )1()( )( )( 1 )( )( xi b ii i a snxx bV xx x nxxn x aV
- 16. SE Formula Implications • In general, lower SE shows better estimates ▫ A worse regression model means bigger error term, higher SE for any variable ▫ Large N reduces SE
- 17. P>|t| is p-value for a Variable • Read across to get the appropriate p-value • Would we reject the null hypothesis? ▫ Yes, p < 0.05
- 18. What Does p-value Tell Us? • A low p-value tells us a relationship is unlikely to happen by random chance. ▫ We can be very confident that people with higher prestige jobs tend to make more money. • However, p-value does not tell us whether the relationship has any real world meaning.
- 19. Is the following important? • If we survey everyone in LA, people born in January may make $10/year more than December babies. ▫ With millions in the data set, p < 0.001 • But should we care about $10 a year? • Common problem when people who know a little stats encounter “big data”
- 20. Statistical vs. Substantive Significance • Ideally we want both. • Statistical significance is based strictly on p- values. • Substantive significance is based on our knowledge of the world. What is worth telling people about? ▫ These judgments won’t come from a statistics class! ▫ Often worth discussing substantively significant variables that don’t quite reach p < .05
- 21. Two Main Criteria for Substantive Significance Effect Size Personal Interest • Always need to explain regression coefficient in a sentence (or more). • Is number large enough for us to care about relationship? ▫ If not, need to offer a reasonable benchmark • Is number nonsense? • Could be intellectual or personal interest • May feel any statistically significant relationship is important
- 22. Balancing Effect Size and Interest Interest in Variable/Relationship Effect Size Not Sig. Significant
- 23. Common Problems: Strike Zone Metaphor Too low: not enough explanation Too high: of threshold for effect size Outside: Too Much Spin Outside: Too Much Spin JUST THROW STRIKES!
- 24. Describe our results • Would you say the effect age has on income is substantively significant? Why or why not?
- 25. Is There Substantive Significance? • Arguments for yes: ▫ It is statistically significant and ▫ Some may feel $573 more per year is enough • Arguments for no: ▫ Some may feel $573 is not enough. ▫ It doesn’t make sense. People retire! • Hold off claims about other variables
- 26. Note on r-squared • We were initially scheduled to cover r-squared today, but I wanted to spend more time on substantive significance because it is the hardest concept. • r-squared appears on HW 2 but will be pushed to Monday’s lecture. If necessary I will pare down on other concepts.
Editor's Notes
- #8: Ask them to do question 2. Note that I ask many versions of this on HW.
- #10: Entirely about explanation, and the burden is on person doing stats to make it clear to the reader.
- #11: 1: Note that E goes away in prediction. 2: Again, ask to do #2
- #13: Also common with year
- #19: Clarify p = .000 is an abbreviation in practice, last minute?
- #22: Explain coefficient needs to consider scales of X and Y
- #25: MAKE THIS IN CLASS PARTICIPATE