Statistical methods for data where only a minimum/maximum value is known Is there a branch of statistics that deals with data for which exact values are not known, but for each individual, we know either a maximum or minimum bound to the value?
I suspect that my problem stems largely from the fact that I am struggling to articulate it in statistical terms, but hopefully an example will help to clarify:
Say there are two connected populations $A$ and $B$ such that, at some point, members of $A$ may "transition" into $B$, but the reverse is not possible. The timing of the transition is variable, but non-random. For example, $A$ could be "individuals without offspring" and $B$ "individuals with at least one offspring". I am interested in the age this progression occurs but I only have cross-sectional data. For any given individual, I can find out if they belong to $A$ or $B$. I also know the age of these individuals. For each individual in population $A$, I know that the age at transition will be GREATER THAN their current age. Likewise, for members of $B$, I know that the age at transition was LESS THAN their current age. But I don't know the exact values.
Say I have some other factor that I want to compare with the age of transition. For example, I want to know whether an individual's subspecies or body size affects the age of first offspring. I definitely have some useful information that should inform those questions: on average, of the individuals in $A$, older individuals will have a later transition. But the information is imperfect, particularly for younger individuals. And vice versa for population $B$.
Are there established methods to deal with this sort of data? I do not necessarily need a full method of how to carry out such an analysis, just some search terms or useful resources to start me off in the right place!
Caveats: I am making the simplifying assumption that transition from $A$ to $B$ is instantaneous. I am also prepared to assume that most individuals will at some point progress to $B$, assuming they live long enough. And I realise that longitutinal data would be very helpful, but assume that it is not available in this case.
Apologies if this is a duplicate, as I said, part of my problem is that I don't know what I should be searching for. For the same reason, please add other tags if appropriate.
Sample dataset: Ssp indicates one of two subspecies, $X$ or $Y$. Offspring indicates either no offspring ($A$) or at least one offspring ($B$)
 age ssp offsp
  21   Y     A
  20   Y     B
  26   X     B
  33   X     B
  33   X     A
  24   X     B
  34   Y     B
  22   Y     B
  10   Y     B
  20   Y     A
  44   X     B
  18   Y     A
  11   Y     B
  27   X     A
  31   X     B
  14   Y     B
  41   X     B
  15   Y     A
  33   X     B
  24   X     B
  11   Y     A
  28   X     A
  22   X     B
  16   Y     A
  16   Y     B
  24   Y     B
  20   Y     B
  18   X     B
  21   Y     B
  16   Y     B
  24   Y     A
  39   X     B
  13   Y     A
  10   Y     B
  18   Y     A
  16   Y     A
  21   X     A
  26   X     B
  11   Y     A
  40   X     B
   8   Y     A
  41   X     B
  29   X     B
  53   X     B
  34   X     B
  34   X     B
  15   Y     A
  40   X     B
  30   X     A
  40   X     B

Edit: example dataset changed as it wasn't very representative
 A: This is a case of censoring/coarse data. Assume you think that your data arises from a distribution with nicely behaved continuous (etc.) pdf $f(x)$ and cdf $F(x)$. The standard solution for time to event data when the exact time $x_i$ of an event for subject $i$ is known is that the likelihood contribution is $f(x_i)$. If we only know that the time was greater than $y_i$ (right-censoring), then the likelihood contribution is $1-F(y_i)$ under the assumption of independent censoring. If we know that the time is less than $z_i$ (left-censoring), then the likelihood contribution is $F(z_i)$. Finally, if the time falls into some interval $(y_i, z_i]$, then the likelihood contribution would be $F(z_i)-F(y_i)$.
A: This problem seems like it might be handled well by logistic regression.
You have two states, A and B, and want to examine the probability of whether a particular individual has switched irreversibly from state A to state B. One fundamental predictor variable would be age at the time of observation. The other factor or factors of interest would be additional predictor variables.
Your logistic model would then use the actual observations of A/B state, age, and other factors to estimate the probability of being in state B as a function of those predictors. The age at which that probability passes 0.5 could be used as the estimate of the transition time, and you would then examine the influences of the other factor(s) on that predicted transition time.
Added in response to discussion:
As with any linear model, you need to ensure that your predictors are transformed in a way that they bear a linear relation to the outcome variable, in this case the log-odds of the probability of having moved to state B. That is not necessarily a trivial problem. The answer by @CliffAB shows how a log transformation of the age variable might be used.
