Estimate risk ratio after xtlogit

[Chris Labos]

Dear All,

I'm hoping someone call tell if there is a way to estimate risk ratios after using xtlogit. The data is below. To give it some context. Pregnant patients are being assessed for the binary outcome PET. Six different biomarkers are being assessed although I only included one here to keep the dataset small std_PAPPA2ngml. Basic baseline variables include age, BMI pre-pregnancy, race, blood pressure (clinicMAP), and a biometric measurement called cfpwv.

* Example generated by -dataex-. For more info, type help dataex
clear
input int id byte(Trimester PET age) double bmi_pre_preg byte race_bin double(clinicMAP cfpwv) float(std_PAPPA2ngmL std_spRatio)
 1 1 1 29 25.15589569 0                 95         7.2   -.5222185     .3073503
 1 2 1 29 25.15589569 0                 95         7.3  -.25407675    -.3172096
 1 3 1 29 25.15589569 0              98.75           .    3.066324     .3035415
 2 1 0 39 54.55568942 0  96.66666666666667           .   -.6136905    .10374988
 2 2 0 39 54.55568942 0  91.33333333333333       7.442   -.6163323   -.27824256
 2 3 0 39 54.55568942 0                 98 7.814666667   -.3911197    -.2422454
 4 2 0 38 21.06263959 0                  .        6.55   -.3379536    -.3029306
 4 3 0 38 21.06263959 0                  .        7.15   1.1556487    -.2357633
 5 1 0 31           . 0                  .           .  -.50372595            .
 6 1 0 40  17.5432526 0                 71         4.7  -.37295735  .0023030285
 6 2 0 40  17.5432526 0  67.66666666666667        5.95 -.018627167   -.29787156
 6 3 0 40  17.5432526 0  73.66666666666667           .  -.20520365    -.2924929
 7 1 1 41    35.15625 1  89.66666666666667         7.1   -.4858939    .04625331
 7 2 1 41    35.15625 1  86.66666666666667 6.697333333  -.26101145    -.3174112
 7 3 1 41    35.15625 1  95.33333333333333           .   4.1877637   -.08966266
 8 1 0 37 25.23634033 0  87.33333333333333         7.9   -.5304741   -.04076392
 8 2 0 37 25.23634033 0  78.33333333333333         6.1    -.434379     -.309964
 8 3 0 37 25.23634033 0  77.33333333333333        8.05  -.08566263   -.25711867
 9 1 1 39 33.25488986 0  77.33333333333333        7.05   -.3980543   -.00615445
 9 2 1 39 33.25488986 0  73.33333333333333         6.5  -.18737157   -.29306722
 9 3 1 39 33.25488986 0                 97         9.9    6.701427    .54593694
10 1 0 37  24.6097337 0  80.33333333333333         5.9   -.4000357    .04421359
10 2 0 37  24.6097337 0  81.33333333333333           6  -.21544057   -.25934732
10 3 0 37  24.6097337 0  83.66666666666667         6.3 -.008390245     -.305451
11 1 0 39 24.30461788 0  75.66666666666667         5.3  -.53344613    .07865868
11 2 0 39 24.30461788 0  70.33333333333333           .  -.53707856    -.3097735
11 3 0 39 24.30461788 0  75.66666666666667         6.4   -.1682186   -.27875698
13 1 0 36  22.4996371 0  83.33333333333333         6.8   -.3815431    .00715128
13 2 0 36  22.4996371 0  73.66666666666667        6.65  -.13816822    -.2855877
13 3 0 36  22.4996371 0  68.33333333333333        8.85    4.111152     .9033498
14 1 0 46     26.5625 0  81.66666666666667         6.8   -.4941495   -.13494419
14 2 0 46     26.5625 0  79.33333333333333         7.7    -.373948            .
14 3 0 46     26.5625 0                 76         7.1    .7025816    -.1696765
15 1 0 34 32.10720958 0                 88           .   -.6093976   -.09368258
15 2 0 34 32.10720958 0  79.33333333333333           .  -.52684164    -.3123173
15 3 0 34 32.10720958 0                 90           .   .14912653    -.3024947
17 1 0 40 27.00513097 1  84.33333333333333        6.65   -.4865543     .0984958
17 2 0 40 27.00513097 1                 82         7.2   -.3815431    -.2449288
17 3 0 40 27.00513097 1  99.33333333333333         6.1  -.13915884   -.28574044
18 1 0 35 24.10694088 0  64.66666666666667        6.75   -.4115935   -.16467524
18 2 0 35 24.10694088 0  70.66666666666667         6.6   -.3775805    -.3052855
18 3 0 35 24.10694088 0  67.33333333333333        6.75    .1554008   -.22982918
19 3 0 40 27.88518739 0  92.33333333333333         6.6  -.05924468   -.25337964
20 1 0 41 20.28240971 0  71.33333333333333         5.2   -.4307465     .8040594
20 2 0 41 20.28240971 0  65.33333333333333        5.25   -.4267838   -.27968565
20 3 0 41 20.28240971 0  75.33333333333333         5.5    .4985033   -.19184238
23 1 0 36  25.0995016 0  84.66666666666667           .   -.5978398    .41634265
23 2 0 36  25.0995016 0  84.33333333333333         7.5  -.42876515    -.2650159
24 1 0 37 18.59012493 0                 68         6.2   -.3574368   -.05839132
24 2 0 37 18.59012493 0                 74           .  -.05197978    -.3012516
24 3 0 37 18.59012493 0        76.88888888        5.25    2.592122   -.17694044
27 1 0 39 30.50508507 1  87.66666666666666           .   -.4726849    -.1043352
27 2 0 39 30.50508507 1  91.66666666666666         6.7   -.4172074    -.3009494
27 3 0 39 30.50508507 1  88.83333333333333        6.75  -.09986226    -.3029617
28 1 0 36 18.77834467 0  69.66666666666667         5.4   -.3524835    .06102021
28 2 0 36 18.77834467 0  85.33333333333333 5.166666667  -.23129132    -.3130392
28 3 0 36 18.77834467 0  78.16666666666667        6.25   .02661346    -.3232161
29 1 0 37 23.63281255 1                 78        7.15   -.4419741   -.19482003
29 2 0 37 23.63281255 1                 85        5.45   -.3779107     -.301454
29 3 0 37 23.63281255 1  84.66666666666667       6.475    .1950276   -.15213487
31 1 0 36 21.60410478 0  83.33333333333333       5.962   -.4664107    .05515991
31 3 0 36 21.60410478 0  82.83333333333333         5.6    1.515923   -.20182908
32 1 0 38 26.77706407 0  79.66666666666667        6.55  -.54995733   -.13579708
32 2 0 38 26.77706407 0  82.66666666666667         6.2   -.3353118    -.3085924
32 3 0 38 26.77706407 0                 78        6.35    .6946563   -.13677949
33 1 0 38 32.88888889 1                 74        7.85   -.3194611   -.04061984
33 2 0 38 32.88888889 1                 86         7.8     -.37692   -.30236775
33 3 0 38 32.88888889 1  76.66666666666667           .   .11049034   -.23288435
34 1 0 33 25.99591237 0  86.66666666666667         6.3   -.3455488    .19505447
34 2 0 33 25.99591237 0                 86         7.3  .010762734   -.26586282
34 3 0 33 25.99591237 0  90.66666666666667        8.25   1.7318892    .19765627
35 1 0 42 38.32001657 1  87.33333333333333           .   -.6100581     .9908481
35 3 0 42 38.32001657 1  78.33333334666666           . -.018296935    -.3013093
36 1 0 36 21.92687484 0  84.33333333333333 5.966666667   -.6117092     .2410784
36 2 0 36 21.92687484 0  76.33333333333333 5.833333333  -.50603753    -.3125821
36 3 0 36 21.92687484 0                 75       6.387   -.3280469   -.30805385
38 1 0 31 30.83653053 0  89.66666666666667         8.4  -.54929686    .06234656
38 2 0 31 30.83653053 0 101.33333333333333        7.85   -.4944797    -.3068409
38 3 0 31 30.83653053 0  95.16666666666667           .   1.1536673   -.06227478
39 1 0 26 20.22790628 0                  .           5  -.50075394    .07180576
39 2 0 26 20.22790628 0                  .        5.55  -.04240326    -.3112041
40 2 0 43 20.33672338 0                 89        5.85    .3733485    -.3101761
40 3 0 43 20.33672338 0  80.16666666666667         5.2   1.2183913   -.28347412
41 1 0 37        22.9 0                 74           .    -.456504            .
42 1 0 42 23.80869017 0  80.66666666666667       6.564   -.3438976 -.0020568385
42 2 0 42 23.80869017 0  83.33333333333333         6.3   -.3613995   -.29500842
42 3 0 42 23.80869017 0  73.66666666666667         6.8   .20757617    -.2118544
43 1 0 33       26.75 0  95.66666666666667           .  -.57670546            .
44 1 0 37 30.87360693 0                 72        6.35    -.484573    .52756953
44 2 0 37 30.87360693 0               70.5         7.2   -.3138473   -.30360615
45 1 0 39 24.79963244 0  83.33333333333333       6.504   -.4274443   -.14511436
45 2 0 39 24.79963244 0                 85         7.8    -.468392   -.28935882
45 3 0 39 24.79963244 0  93.33333333333333         7.2   .07053323    -.3061145
46 1 0 40 21.96712018 0  78.66666666666667        5.65   -.2256775  -.020348873
46 2 0 40 21.96712018 0                 72           .    -.162935    -.3131802
46 3 0 40 21.96712018 0  72.88888888666666         7.3   1.6589098   -.29485402
47 1 0 38 23.03004535 0 105.66666666666667       6.309    -.459476    .06723389
47 2 0 38 23.03004535 0  97.33333333333333         5.3   -.3336607   -.28034976
48 1 0 40 21.92687484 0                  .           7  -.56316626     .8659224
48 2 0 40 21.92687484 0                  .         5.8  -.18472977    -.3192273
end
label values race_bin race_bin
label def race_bin 0 "White or other", modify
label def race_bin 1 "Black", modify

As an aside, I did use multiple imputation to deal with some missing data although I don't think it's relevant to my specific question. I will however include the code here for the sake of completeness.

mi set wide  
mi register imputed  clinicMAP cfpwv
mi register regular age bmi_pre_preg race_bin std_* id Trimester PET
mi impute mvn   clinicMAP  cfpwv = age  race_bin Trimester PET, add(25) rseed(12345)

And as such I used xtlogit to estimate the association between my biomarker std_PAPPA2 and the outcome PET. Trimester denotes the trimeter in pregnancy so most patients had three different measurements. So trimester is the panel variable.

mi xtset Trimester
mi estimate, or post: quietly xtlogit PET std_PAPPA2ngmL age bmi_pre_preg race_bin clinicMAP std_spRatio cfpwv

Again, you can likely omit the mi estimate to answer the fundamental question I had which is:

Can you estimate a risk ratio after xtlogit rather than the odds ratio? I'm not certain that is easy to calculate but I thought I would ask if such a thing is possible using a post-estimation command or something else.

My thanks for any assistance than can be provided and apologies in advance for rather convoluted setup.

Much obliged,

Christopher

[Clyde Schechter]

Can you estimate a risk ratio after xtlogit rather than the odds ratio?

No.

Added: To put a more positive spin on things, you can do that after a Poisson regression (which, by the way, will not give you an odds ratio.) In fact, any model that assumes that a constant odds ratio exists is inherently incompatible with the existence of a constant risk ratio, and vice versa (unless the OR = 1 and the RR = 1.)

[Chris Labos]

Fair enough. I assumed not, but thought worthwhile to ask others. I become plagued by self doubt when reading reviewer comments. Many thanks for the answer.

I am curious about:

In fact, any model that assumes that a constant odds ratio exists is inherently incompatible with the existence of a constant risk ratio, and vice versa (unless the OR = 1 and the RR = 1.)

I'm not sure I completely understand the point here. I know you can estimate the risk ratio after a simple logistic regression but not in more complex models. I would be grateful for an explanation for my own personal edification if nothing else.

Sincerely,

Christopher

[Clyde Schechter]

No, you cannot estimate the risk ratio after a simple logistic regression. Let's say that the probability of y = 1 when x = 0 is p. If we have a risk ratio of r associated with an increase of 1 in x, then the subsequent probabilities of y are pr, pr² pr³, etc. This is a simple geometric series: that is what is implied by a constant risk ratio.Evidently if r > 1, this sequence must eventually stop, unless p = 0, because we cannot have p > 1.

Now, what happens if instead we let r be an odds ratio instead? Then p(x = 1) is pretty complicated: Odds(x=1) = rp/(1-p). Therefore p(x=1) = Odds(x=1)/(1+Odds(x=1)) = (after some algebra) rp/(1+(r-1)p). So this is clearly not a geometric series. There is no value of r other than 1 that can turn p(x=1) into a simple multiple (risk ratio) of p(x=0) for all possible values of p(x=0). Consequently if there is a constant odds ratio (other than 1), there cannot be a constant risk ratio. The proof of the converse is similar.

[Chris Labos]

I think I understand. So you're saying that the usual conversion formula of RR = OR / (1 – p + (p x OR)) is (I'm struggling for the right word here, variable? not-constant?).

So the conversion from odds ratio to risk ratio will vary at different parts along the curve? So then, (and apologies for the tangent but I find this interesting) what do you make of the various user written commands in stata that convert odds ratio to risk ratios after logistic regression? Invalid? Or are they just estimating the risk ratio from a glm model rather than "converting" the odds ratio. I'm thinking of programs like -oddsrisk-

Thanks and again sorry for the digression.

Christopher

[Clyde Schechter]

I think I understand. So you're saying that the usual conversion formula of RR = OR / (1 – p + (p x OR)) is (I'm struggling for the right word here, variable? not-constant?).

It very explicitly depends on p. So with the same value of OR, you will get different values of RR at different values of p.

So the conversion from odds ratio to risk ratio will vary at different parts along the curve?

Exactly so.

So then, (and apologies for the tangent but I find this interesting) what do you make of the various user written commands in stata that convert odds ratio to risk ratios after logistic regression? Invalid? Or are they just estimating the risk ratio from a glm model rather than "converting" the odds ratio. I'm thinking of programs like -oddsrisk-

I'm not familiar with any of those commands, so I can't really comment on them. I do think, however, that the very notion of converting between risk ratios and odds ratios is a questionable practice. At best, because the relationship explicitly depends on the baseline probability, any such conversion is only applicable to populations having exactly the prevalence in the specific data set being studied. As research typically hopes to create generalizable knowledge, this falls short. More important, I think that we need to take models seriously. If we believe that a logistic regression model is a decent approximation to the real data generating process, then an inescapable consequence of that is that there is no constant RR that applies. If we have some other model which entails a constant RR, then the non-existence of a constant OR is an inescapable consequence of that. More important still, once one has a handy formula, there is a risk of misapplying it outside its constraints. For example, in a case-control study, the RR is not estimable at all (and the application of a statistical model that attempted to do so would be a serious error). Yet I have seen people attempt to translate the OR from a logistic regression (or other analysis that produces an OR) into an RR.

And, yes, I, too, have been asked by reviewers to translate ORs into RRs. It's annoying at best. Nowadays, being pretty senior in my career, I either ignore those requests or explicitly reject them and state my reasons in a response to the reviewer. When I was younger, I would do it, but generally only in the reviewer response and not in the manuscript itself, or, if really pressured to included it in the manuscript, put a clear disclaimer about its lack of generalizability with it.

[Chris Labos]

Very interesting Clyde. Thanks for the explanation. Indeed the reason I started this thread was because I was specifically asked to translate the odds ratios into risk ratios by a reviewer. I appreciate the time taken to reply. I don't post here often, but when I do, I generally learn something new which is always appreciated.

Best,

Christopher