Can Hazard Ratios be presented as Probabilities?

[Chris Boulis]

I understand if a layperson read a paper that reported hazard ratios, they may struggle with interpretation. Hence, I have been reading the literature to find an alternative that may be more intuitive to the less academically-minded and some research (e.g., Spruance et al, 2004, p. 2791; Moser & McCann, 2008, p. 250) suggests converting Hazard Ratios to probabilities using the formula P = HR / (1 + HR).

I'm using the Cox proportional hazards model to report on relationship dissolution for different types of couple households observed using annual survey data from 2001 to 2018. I differentiate couples by their different belief systems (religious or non-religious) and this (belief) data is collected about every four years over this time period (2004, 2007, 2010, 2014, 2018).

(1) Is it reasonable to present the results as probabilities (possibly in addition to the HRs) and if so, is there a suggested way to code that rather than convert each using the above formula?
(2) If I wanted to report the probabilities of 'survival' for different couples between two time periods, e.g., 2004 and 2014 that remain in tact after a 10-year period, how could I do that? (An example can be seen in Lehrer and Chiswick (1993, p. 393-394).

Example results from the Cox model:

Couple type (differentiated by belief system)	Hazard ratio
[1] No religion	1.00
[2] Catholic	0.42***
[3] Protestant(a)	0.40***
[4] Protestant(b)	0.48**
[5] OTHER	0.11***
[6] Protestant (diff)	0.59**
[7] male Cath/female Prot	0.59**
[8] female Cath/male Prot	0.73
[9] male Cath/female None	0.98
[10] female Cath/male None	0.90
[11] male Prot/female None	1.29
[12] female Prot/male None	0.75
N=57,095

^* p < 0.10, ^** p < 0.05, ^*** p < 0.01

References

1) HRs to Probabilities:
Spruance, S. L., Reid, J. E., Grace, M., & Samore, M. (2004). Hazard Ratio in Clinical Trials. Antimicrobial agents and chemotherapy, 48(8), 2787-2792. https://doi.org/10.1128/AAC.48.8.2787-2792.2004.
Moser, B. K., & McCann, M. H. (2008). Reformulating the hazard ratio to enhance communication with clinical investigators. Clinical trials (London, England), 5(3), 248-252. https://doi.org/10.1177/1740774508091452

2) Using probabilities to reflect dissolution of different couple types
Lehrer, E. L., & Chiswick, C. U. (1993). Religion as a Determinant of Marital Stability. Demography, 30(3), 385-404. https://doi.org/10.2307/2061647

Stata v.15.1. Using panel data (HILDA survey data).

[Tiago Pereira]

I cannot see why the hassle with conversions to probability if the hazard ratio conveys the relative treatment/exposure effect and you can estimate the survival function for each group, which is a probability.

From HR to probability, it is a big leap, even for communication purposes.
If we want to communicate with the lay audience and still be technically correct, oversimplifications can be risky.

Denoting the probability of the event = HR / (1 + HR), we may convey that the hazard rate can be easily converted to a cumulative probability of the event. This doesn't seem right; the authors had something different in mind when they proposed.

I think the originally proposed idea was to compare two randomly chosen participants, one from Group A and one from Group B - at a given time. A hazard ratio of 4, for example, would imply that the "probability" of a randomly chosen participant from Group A having the event first compared to a randomly selected participant from Group B is 80% - given that both did not have the event thus far.

Many books and articles define hazard rate as "the probability that the event "E" occurs at a time "t", given that "E" has not happened before time "t". The hazard ratio is simply the ratio of two hazard rates. Somehow, I feel this is the best way to convey the idea - making it clear that hazard rate and cumulative risk/probability transmit a similar notion but are different.

[Chris Boulis]

Hi Tiago Pereira thank you for your reply. I accept your points, however, I am still confronted with the layperson will struggle to interpret HRs - for example (according to #1), a Catholic couple is about 58% less likely to 'fail' (or break up) compared to an unaffiliated couple. If one was to explain that to a family member, many would struggle to make sense of it.

Can you shed light on the code needed to achieve (2) in #1 - where I estimate the probability of dissolution (or /breakup) for each couple type over a 10-year period (2004 to 2014) based on HILDA survey data? I could add these results in an additional column next to the HRs for each couple type and explain how they need to be interpreted and note the limitations.

Kind regards, Chris

[Tiago Pereira]

Chris, hi.

I suggest:

https://stats.oarc.ucla.edu/stata/se...tata-survival/
https://www.stata.com/manuals/ststcurve.pdf

You can start with a simple Kaplan-Meier plot, and then add covariates via a Cox regression. There is a plenty of examples in the links above. If your follow-up is sufficiently long (e.g., 10-11 years) and not all couples have been censored, you will be able to estimate the survival function up to 10-11 years easily. The analysis will provide precisely the information you have been seeking: the probability of each couple type "surviving" (not experiencing a breakup) for a duration of up to 10 years.

Hope this helps.

[Maarten Buis]

The difficulty is that there is not one probability. Instead you have the probability of the partnership "surviving " after 1 year, 2 years, 3 years, 4 years, etc. stcurve allows you to visualize these probabilities for different groups after a Cox model.

sysuse cancer, clear
stcox i.drug age
stcurve, at(drug=(1 2 3)) survival

[Chris Boulis]

Hi Tiago Pereira. Thank you for the links - I've been working through the material. Graphing appears to be fine (see below). I tried using

estat concordance

but this didn't work as I have time varying covariates. Eventually I found

sts list

which provides either the hazard function or survivor function in terms of the proportion of relationships that failed at the end of each year for each couple type in my dataset and satisfies the need in #1. I'll add this information in an extra column to the main results (hazard ratios) with an explanation about how the results were obtained. It is also worth noting that the results obtained from "sts list" are consistent with the information in the following graph.

stcurve , survival at1(relig=1) at2(relig=2 at3=0) at3(relig=3 at3=0) at4(relig=4 at3=0) /// 
lpattern(dash solid solid solid) connect(1) /// 
lcolor(brown blue dkgreen orange_red) /// 
graphregion(color(white) icolor(white)) plotregion(style(none) color(white) icolor(white))  ///
title("No attend", size(medsmall)) ///
ytitle("Survival", size(small)) ylabel(.86(.02)1.0 , angle(horizontal)) yscale(noline) ylabel(, labsize(small)) xlabel(0(1)20) xlabel(, labsize(vsmall)) /// 
xtitle("Duration (in Years)", size(small)) legend(region(lwidth(none)) size(small) cols(4) pos(6) rowgap(1) span ///
order(4 "same" 3 "diff" 2 "mixed" 1 "None")) name(religat0, replace)

Hi Maarten Buis. Yes I understand "stcurve" allows you to visualise the probabilities at each period (wave), but is there a way to access the actual probability values as can be done with "sts list"?

Kind regards,
Chris