This is referred to as current status data. You get one cross sectional view of the data, and regarding the response, all you know is that at the observed age of each subject, the event (in your case: transitioning from A to B) has happened or not. This is a special case of interval censoring.
To formally define it, let Ti be the (unobserved) true event time for subject i. Let Ci the inspection time for subject i (in your case: age at inspection). If Ci<Ti, the data are right censored. Otherwise, the data are left censored. We are interesting in modeling the distribution of T. For regression models, we are interested in modeling how that distribution changes with a set of covariates X.
To analyze this using interval censoring methods, you want to put your data into the general interval censoring format. That is, for each subject, we have the interval (li,ri), which represents the interval in which we know Ti to be contained. So if subject i is right censored at inspection time ci, we would write (ci,∞). If it is left censored at ci, we would represent it as (0,ci).
Shameless plug: if you want to use regression models to analyze your data, this can be done in R using icenReg
(I'm the author). In fact, in a similar question about current status data, the OP put up a nice demo of using icenReg. He starts by showing that ignoring the censoring part and using logistic regression leads to bias (important note: he is referring to using logistic regression without adjusting for age. More on this later.)
Another great package is interval
, which contains log-rank statistic tests, among other tools.
EDIT:
@EdM suggested using logistic regression to answer the problem. I was unfairly dismissive of this, saying that you would have to worry about the functional form of time. While I stand behind the statement that you should worry about the functional form of time, I realized that there was a very reasonable transformation that leads to a reasonable parametric estimator.
In particular, if we use log(time) as a covariate in our model with logistic regression, we end up with a proportional odds model with a log-logistic baseline.
To see this, first consider that the proportional odds regression model is defined as
Odds(t|X,β)=eXTβOddso(t)
where Oddso(t) is the baseline odds of survival at time t. Note that the regression effects are the same as with logistic regression. So all we need to do now is show that the baseline distribution is log-logistic.
Now consider a logistic regression with log(Time) as a covariate. We then have
P(Y=1|T=t)=exp(β0+β1log(t))1+exp(β0+β1log(t))
With a little work, you can see this as the CDF of a log-logistic model (with a non-linear transformation of the parameters).
R demonstration that the fits are equivalent:
> library(icenReg)
> data(miceData)
>
> ## miceData contains current status data about presence
> ## of tumors at sacrifice in two groups
> ## in interval censored format:
> ## l = lower end of interval, u = upper end
> ## first three mice all left censored
>
> head(miceData, 3)
l u grp
1 0 381 ce
2 0 477 ce
3 0 485 ce
>
> ## To fit this with logistic regression,
> ## we need to extract age at sacrifice
> ## if the observation is left censored,
> ## this is the upper end of the interval
> ## if right censored, is the lower end of interval
>
> age <- numeric()
> isLeftCensored <- miceData$l == 0
> age[isLeftCensored] <- miceData$u[isLeftCensored]
> age[!isLeftCensored] <- miceData$l[!isLeftCensored]
>
> log_age <- log(age)
> resp <- !isLeftCensored
>
>
> ## Fitting logistic regression model
> logReg_fit <- glm(resp ~ log_age + grp,
+ data = miceData, family = binomial)
>
> ## Fitting proportional odds regression model with log-logistic baseline
> ## interval censored model
> ic_fit <- ic_par(cbind(l,u) ~ grp,
+ model = 'po', dist = 'loglogistic', data = miceData)
>
> summary(logReg_fit)
Call:
glm(formula = resp ~ log_age + grp, family = binomial, data = miceData)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.1413 -0.8052 0.5712 0.8778 1.8767
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 18.3526 6.7149 2.733 0.00627 **
log_age -2.7203 1.0414 -2.612 0.00900 **
grpge -1.1721 0.4713 -2.487 0.01288 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 196.84 on 143 degrees of freedom
Residual deviance: 160.61 on 141 degrees of freedom
AIC: 166.61
Number of Fisher Scoring iterations: 5
> summary(ic_fit)
Model: Proportional Odds
Baseline: loglogistic
Call: ic_par(formula = cbind(l, u) ~ grp, data = miceData, model = "po",
dist = "loglogistic")
Estimate Exp(Est) Std.Error z-value p
log_alpha 6.603 737.2000 0.07747 85.240 0.000000
log_beta 1.001 2.7200 0.38280 2.614 0.008943
grpge -1.172 0.3097 0.47130 -2.487 0.012880
final llk = -80.30575
Iterations = 10
>
> ## Comparing loglikelihoods
> logReg_fit$deviance/(-2) - ic_fit$llk
[1] 2.643219e-12
Note that the effect of grp
is the same in each model, and the final log-likelihood differs only by numeric error. The baseline parameters (i.e. intercept and log_age for logistic regression, alpha and beta for the interval censored model) are different parameterizations so they are not equal.
So there you have it: using logistic regression is equivalent to fitting the proportional odds with a log-logistic baseline distribution. If you're okay with fitting this parametric model, logistic regression is quite reasonable. I do caution that with interval censored data, semi-parametric models are typically favored due to difficulty of assessing model fit, but if I truly thought there was no place for fully-parametric models I would have not included them in icenReg
.