# 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

• This is an interesting situation. Can you provide your data? – gung - Reinstate Monica Mar 18 '16 at 12:32
• I would not be able to post the full dataset but could give an example set. – user2390246 Mar 18 '16 at 13:38

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 $T_i$ be the (unobserved) true event time for subject $i$. Let $C_i$ the inspection time for subject $i$ (in your case: age at inspection). If $C_i < T_i$, 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 $(l_i, r_i)$, which represents the interval in which we know $T_i$ to be contained. So if subject $i$ is right censored at inspection time $c_i$, we would write $(c_i, \infty)$. If it is left censored at $c_i$, we would represent it as $(0, c_i)$.

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

$\text{Odds}(t|X, \beta) = e^{X^T \beta} \text{Odds}_o(t)$

where $\text{Odds}_o(t)$ 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

$P(Y = 1 | T = t) = \frac{\exp(\beta_0 + \beta_1 \log(t))}{1 + \exp(\beta_0 + \beta_1\log(t))}$

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
>
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  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. • This looks very helpful. I will have a look at the resources you point to and a play with the icenReg package. I am trying to get my head around why logistic regression is less suitable - @EdM 's suggestion looks on the surface as if it should work. Does the bias arise because the "event" - here, having offspring - might have an effect on survival? So, if it decreases survival, we would find that among individuals of a given age, those that have not reproduced will be over-represented? – user2390246 Mar 18 '16 at 15:39 • @user2390246: You could use logistic regression for current status data. But then you have to do a lot of work getting the functional form of age, and it's interaction with other variables, correct. This is very much non-trivial. With survival based models, you can use a semi-parametric baseline (ic_sp in icenReg) and not worry at all about that. In addition, looking at the survival curves for the two groups answers your question correctly. Trying to recreate this from the logistic fit could be done, but again, much more work than using survival models. – Cliff AB Mar 18 '16 at 15:43 • I agree with @CliffAB on this. I had a hesitation about recommending logistic regression specifically because of the difficulty of getting the right functional form for the dependency on age. I haven't had any experience with current status data analysis; not having to figure out that form of the dependency on age is a big advantage of that technique. I will keep my answer up nevertheless so that those who later examine this thread will understand how this played out. – EdM Mar 18 '16 at 18:08 • It seems to me that your comment here is the crux of the matter. It would help if you could develop that in your answer. Eg, if you could use the OP's example data to build a LR model & an interval censored survival model, & show how the latter more easily answers the OP's research question. – gung - Reinstate Monica Mar 18 '16 at 23:30 • @gung: actually, I've taken a softer stance about logistic regression. I edited my answer to reflect this. – Cliff AB Mar 19 '16 at 14:42 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$F(z_i)-F(y_i)$. • There's no need for$f(x)$to be continuous. Or even well behaved. It could be a discrete survival model (so the pdf is undefined and a pmf is used instead) and the rest of what you said would be correct, with a slight adjustment (replace$F(y_i)$with$F(y_{i+})\$. – Cliff AB Mar 18 '16 at 15:55

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.