grass -- should marijuana be made legal
-----------------------------------------------------------------------
| Freq. Percent Valid Cum.
--------------------------+--------------------------------------------
Valid 0 Not be legal | 930 24.66 24.66 24.66
1 Should be legal | 2841 75.34 75.34 100.00
Total | 3771 100.00 100.00
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recode grass 2 = 0
(0 changes made to grass)
lab def glab 0"not be legal" 1 "should be legal"lab val grass glabfre grass
grass -- should marijuana be made legal
-----------------------------------------------------------------------
| Freq. Percent Valid Cum.
--------------------------+--------------------------------------------
Valid 0 not be legal | 930 24.66 24.66 24.66
1 should be legal | 2841 75.34 75.34 100.00
Total | 3771 100.00 100.00
-----------------------------------------------------------------------
library(haven) # import Stata data and labelslibrary(dplyr) # data manipulationlibrary(sjmisc) # frequency tableslibrary(questionr) # odds ratiosgss <-read_dta("https://tdmize.github.io/data/data/cda_gss22.dta")# Drop any missing data on model vars; keep 2021 and 2022gss <- gss[complete.cases(gss[c("grass", "age", "conserv", "faminc", "college","woman", "race4")]), ]gss <- gss |>filter(year >=2020, year <=2023)frq(gss, grass)
should marijuana be made legal (grass) <numeric>
# total N=3771 valid N=3771 mean=0.75 sd=0.43
Value | Label | N | Raw % | Valid % | Cum. %
---------------------------------------------------------
0 | Not be legal | 930 | 24.66 | 24.66 | 24.66
1 | Should be legal | 2841 | 75.34 | 75.34 | 100.00
<NA> | <NA> | 0 | 0.00 | <NA> | <NA>
IVs: Age, conservative political views (binary), household income, college degree, gender, race-ethnicity
# Treat binary/nominal variables as factors using their value labelsgss <- gss |>mutate(conserv =as_factor(conserv), college =as_factor(college),woman =as_factor(woman), race4 =as_factor(race4))glm1 <-glm(grass ~ age + conserv + faminc + college + woman + race4,data = gss, family =binomial(link ="logit"))odds.ratio(glm1, level = .95)
Examples odds ratio interpretations in binary logit
Example binary/nominal IV interpretations:
Conservative’s odds of supporting marijuana legalization are lower than non-conservative’s by a factor of 0.250, holding other variables constant
Hispanic adults have odds of supporting marijuana legalization that are lower than White adults’ by a factor of 0.289 (other variables held constant). Black and White adults odds of supporting marijuana legalization are not significantly different… [and so on for other categories]
Example continuous IV interpretations:
Controlling for other factors, a one-year increase in age decreases the odds of supporting marijuana legalization by a factor of 0.966
After accounting for other factors, family income is not associated with the odds of supporting marijuana legalization
Interpretation tip: A non-significant effect has a confidence interval that includes positive, null, and negative values. I do not think it is appropriate to interpret the direction of a null effect.
Considerations when interpreting odds ratios (ORs)
ORs are multiplicative effects—so you cannot say it “increases the odds by _____” (as you would with an additive effect)
Instead, you must specify that it increases or decreases the odds by a factor of/ # times which indicates effect is multiplicative instead of additive
You also cannot say things such as it “doubles the chance” of something happening or “makes it half as likely”—these are probability statements
Odds are not in the probability metric
You must interpret ORs in terms of what they suggest in terms of a multiplicative change in the odds
E.g., An OR=2 does not necessarily “double the chance” of observing an outcome
The substantive impact in terms of probability depends on what the starting value is (see the next few sections for examples)
In general, I do not find odds ratios interpretations very satisfying
Odds are less intuitive than probabilities; a ratio of two odds is much less intuitive than probabilities
The same size OR can have drastically different effects on the probability depending on the starting value
How OR’s and Pr(y) are related
The odds and probability are linked nonlinearly (see below)
The odds (often denoted \(\Omega\) [Omega]) are:
The probability of something happening (probability often denoted \(\pi\) [pi])
Compared to the probability of it not happening (1 – \(\pi\))
\[\Omega = \frac{\pi}{1 - \pi}\]
If we know the odds, can convert to probability:
\[\pi = \frac{\Omega}{1 + \Omega}\]
Effect sizes with multiplicative effects on the odds
Think about doubling the odds from 1/10 to 2/10
Probability of 0.091 becomes 0.167
An increase of 0.076
Think about doubling the odds from 1/1 to 2/1
Probability of 0.500 becomes 0.667
An increase of 0.167
Think about doubling the odds of winning the lottery from 1/100,000,000 to 2/100,000,000
Probability of 0.00000001 becomes 0.00000002
An increase of 0.00000001
The effect size depends on the starting probability
“Doubling the odds” means something fundamentally different across the two examples despite being the same multiplicative effect size
