Risk ratios (AKA relative risk) are relatively common in certain areas, e.g., epidemiology
If the baseline probability (i.e., “risk”) in the population is well-known—risk ratios can be informative
Sometimes used to portray results from randomized control trials (e.g., vaccine efficacy rates are a version of a risk ratio)
Risk ratios (RRs) are a way to present changes in probabilities in terms of multiplicative—rather than additive effects
Recall my caution above about not being able to interpret an odds ratio (OR) in terms of “doubling the chance” or making something “half as likely to occur”
These are risk ratio statements—multiplicative effects in the metric of probabilities
“It is well known… that most people misinterpret odds ratios as risk ratios…” (Norton et al. 2013)
Calculating risk ratios
Recall the formula for the marginal effect at the mean (MEM):
Risk ratios are generally used to interpret group differences (e.g., men compared to women; participants in the treatment condition compared to participants in the control condition; etc.)
Below, I represent the grouping variable as g (which equals 0 or 1)
cogB -- Dementia-range cognition
-----------------------------------------------------------
| Freq. Percent Valid Cum.
--------------+--------------------------------------------
Valid 0 no | 14366 96.10 96.10 96.10
1 yes | 583 3.90 3.90 100.00
Total | 14949 100.00 100.00
-----------------------------------------------------------
marstatB -- Currently married
-----------------------------------------------------------------------------
| Freq. Percent Valid Cum.
--------------------------------+--------------------------------------------
Valid 0 not currently married | 5863 39.22 39.22 39.22
1 currently married | 9086 60.78 60.78 100.00
Total | 14949 100.00 100.00
-----------------------------------------------------------------------------
rename marstatB married
library(haven) # import Stata data and labelslibrary(dplyr) # data manipulationlibrary(sjmisc) # frequency tableslibrary(marginaleffects) # predictions and marginal effects# Health and Retirement Study 2020hrs <-read_dta("https://tdmize.github.io/data/data/cda_hrs.dta")hrs <- hrs[complete.cases(hrs[c("cogB", "marstatB", "collegeB", "woman", "age","nlivsib")]), ]frq(hrs, cogB, marstatB)
Dementia-range cognition (cogB) <numeric>
# total N=14949 valid N=14949 mean=0.04 sd=0.19
Value | Label | N | Raw % | Valid % | Cum. %
------------------------------------------------
0 | no | 14366 | 96.10 | 96.10 | 96.10
1 | yes | 583 | 3.90 | 3.90 | 100.00
<NA> | <NA> | 0 | 0.00 | <NA> | <NA>
Currently married (marstatB) <numeric>
# total N=14949 valid N=14949 mean=0.61 sd=0.49
Value | Label | N | Raw % | Valid % | Cum. %
---------------------------------------------------------------
0 | not currently married | 5863 | 39.22 | 39.22 | 39.22
1 | currently married | 9086 | 60.78 | 60.78 | 100.00
<NA> | <NA> | 0 | 0.00 | <NA> | <NA>
* Risk ratio, averaging over observed valuesestrestore cogmod
(results cogmod are active now)
margins collegeB, post
Predictive margins Number of obs = 14,949
Model VCE: OIM
Expression: Pr(cogB), predict()
-------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
--------------+----------------------------------------------------------------
collegeB |
no bachelo.. | 0.048 0.002 23.964 0.000 0.044 0.052
bachelor's.. | 0.012 0.002 6.758 0.000 0.008 0.015
-------------------------------------------------------------------------------
avg_predictions(cogmod, variables ="collegeB", by ="collegeB")
collegeB Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
no bachelor's 0.0482 0.00201 23.96 <0.001 419.2 0.04428 0.0522
bachelor's or higher 0.0119 0.00176 6.76 <0.001 36.1 0.00844 0.0153
Type: response
Older adults without a college degree have a 0.048 probability of having very low dementia-level cognition, compared to only a 0.012 probability for older adults with college degree
MEs and RRs are based on same information (two predicted probabilities)
One subtracts the two (MEs)
One divides them to get the multiplicative effect (RR)
# no bachelor's - bachelor's or higheravg_predictions(cogmod, variables ="collegeB", by ="collegeB",hypothesis =~ revpairwise)
Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
(no bachelor's) - (bachelor's or higher) 0.0363 0.00268 13.6 <0.001 136.8 0.0311 0.0416
Type: response
This replicates what mecompare and avg_comparisons calculates
AME: Older adults without college degrees have a 0.036 higher probability of having very low dementia-level cognition than do older adults with a college degree
How effective is the vaccine? If you are young, it reduces your chance of dying from COVID by approximately 100%
It is less effective at preventing death for older adults
These are risk ratios
They are useful for determining if a vaccine works and how effective it is among different groups
Risk difference (ME) for COVID death
But how many people did the vaccine prevent from dying?
Calculate the simple difference in deaths across each group
\[{deaths\ prevented}_{16-44} = 36 - 0 = 36\]
\[{deaths\ prevented}_{45-64} = 125 - 14 = 111\]
\[{deaths\ prevented}_{65+} = 554 - 124 = 430\]
About 111 people 45 - 64 years old died unnecessarily: their deaths could have been prevented by getting vaccinated
This risk difference is a marginal effect as it calculates the difference in deaths for each age group
Should you use the marginal effect (ME) or risk ratio (RR)?
Refer back to the results above
The vaccine is more efficacious for younger people
Despite being less efficacious for older people, the vaccine prevents far more deaths among older people than among younger people
The RR and ME point in opposite directions about the vaccine’s effect based on age
Why? Base rates of deaths are much higher for older people
Given this conflicting evidence, which would you use?
If you were in charge of giving out the first vaccines, when they were still limited, who would you prioritize access for?
Older or younger adults?
Why?
Do you want to decide if the vaccine works and whether it should be approved?
Use the RR
Do you want to help the most people?
E.g., Who should you give the first available doses to?
Use the ME: prioritizing older people getting vaccinated will prevent the most deaths despite the fact the efficacy RR is lower for older people
More thoughts on risk ratios
“For each set of 100,000 people who have one drink a day per year, 918 can expect to experience one of the 23 alcohol-related problems in any year. Of those who drink nothing, 914 can expect to experience a problem. This means that 99,082 are unaffected, and 914 will have an issue no matter what. Only 4 in 100,000 people who consume a drink a day may have a problem caused by the drinking, according to this study.” (Carroll 2018)
Recall the NY Times Upshot article about the risk of alcohol-related diseases based on # of drinks consumed
A large risk ratio does not necessarily translate into a large number of people affected
More thoughts on RRs vs MEs
RR effect sizes are harder to understand and often misinterpreted
Because negative effects are bounded between 0 and 1, easy to misunderstand effect size
E.g., 20 is the same size RR as 0.05 just in the opposite direction
There is an upper bound on effect size for common but not rare outcomes
Sometimes a multiplicate effect may be of theoretical/substantive interest—in this case, I would recommend using an RR instead of relying on the odds ratio (OR)
I would only consider RRs for summaries of an effect (e.g., to replace ORs in a regression table)
I find it odd that many scholars argue for multiplicative effects only with binary DVs
We can calculate multiplicative effects in linear models—but this is rarely done
Here, a linear model using HRS that predicts # of siblings (which is really a count)
# bachelor's or higher / no bachelor'savg_predictions(sibmod, variables ="collegeB", by ="collegeB",hypothesis = ratio ~ pairwise)
Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
(bachelor's or higher) / (no bachelor's) 0.721 0.0122 -22.9 <0.001 384.0 0.697 0.745
Type: response
Risk ratio: Older adults with college degrees have about 0.72 times fewer siblings than do older adults without college degrees
In general, I find graphs of predicted probabilities and marginal effects to be more helpful, more easily interpretable, and more satisfying summaries of effects
Marginal effects summarize effects in an intuitive and additive metric
MEs are the most similar method of interpretation to how the coefficients are interpreted in linear regression so the most familiar/easiest for most people to understand
Effect size is also easier to understand, as the impact of a multiplicative effect (e.g., OR or RR) is dependent on the starting point / base rate
