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This may seem like a really simple question but I think my data straddles the line between interval and ordinal and I'm not sure quite how to treat it.

Simply, participants have to estimate the time between two events by selecting 1 of 4 intervals which they have been previously trained to identify. The 4 options are 200ms, 400ms, 600ms, and 800ms.

At face value the data seems to be interval; the options are ordered, and each one separated from its neighbour by the same meaningful interval.

However, does the fact that there are only 4 possible options mean that the data is more ordinal than interval?

Each participant is exposed to 8 presentations of each delay during the experimental phase, I don't know whether I can reasonably average their responses (which would be possible if the data was interval) or whether I should look at the number of times they selected each interval (like I would if it was ordinal).

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  • $\begingroup$ Your times given in the question are not intervals. There are several ways to express them as intervals though, which may make it more clear as to the type of data, and thus the most correct statistical method, to apply to the data... And what you you mean by "and each one separated from its neighbour by the same meaningful interval." ? $\endgroup$ – compbiostats May 28 '18 at 19:13
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    $\begingroup$ @JarrettPhillips OP is talking about an interval scale in the sense of Stephens's levels of measurement. $\endgroup$ – Kodiologist May 28 '18 at 19:29
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    $\begingroup$ Your data don't consist of the intervals--those are given, not measured, and they will have all the properties enjoyed by using seconds to express time intervals. Your data are simply counts. $\endgroup$ – whuber♦ May 28 '18 at 19:40
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    $\begingroup$ I would opt for an ordinal approach for practical reasons - it handles heavy ties in the data very accurately. $\endgroup$ – Frank Harrell May 28 '18 at 19:44
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    $\begingroup$ Thank you. Stephens' theory of scales pertains to measurements, not to data per se. You aren't measuring these durations: you are establishing them de facto. What you are measuring are the subjects' responses. Those responses are decidedly ordinal: they consist of a unique choice among four possibilities which have a natural, meaningful order for your study. But that's as far as it goes: based on the scale of measurement, one doesn't make any important or useful decisions about what statistical procedure to use or how to carry it out. $\endgroup$ – whuber♦ May 29 '18 at 14:14
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Well, this may be a different way of thinking about your problem, but what if you exploit the fact that:

200 = fx200 where f = 1
400 = fx200 where f = 2 
600 = fx200 where f = 3 
800 = fx200 where f = 4?

If I understand your problem correctly, each of your subjects will have to choose 8 different times among the set of 4 options {1x200, 2x200, 3x200, 4x200} and you are interested in summarizing the resulting information across these 8 times. Each time a subject chooses an option, all the information you need to fully specify that option is the multiplicative factor f.

As an example, subject #1 might make the following choices of f: 

1st time:  f = 1
2nd time:  f = 3
3rd time:  f = 1
4th time:  f = 2
5th time:  f = 2 
6th time:  f = 4 
7th time:  f = 1
8th time:  f = 3

So if you wanted to summarize the information corresponding to this subject, you could do it in a variety of ways, including:

  1. Typical value of f across the 8 repetitions (i.e., the average of the 8 values of f provided by the subject) - this gives you information about the typical estimate of time chosen by the subject;

  2. Number of times subject chooses a specific f value across the 8 repetitions (e.g., number of time subject chooses f = 4 - that is, the highest estimate of time - across the 8 repetitions);

  3. Number of times subject chooses a specific set of f values across the 8 repetitions (e.g., number of times subject chooses f = 3 or f = 4 - that is, the higher estimates of time - across the 8 repetitions).

I hope someone else on this forum will read my answer and confirm whether what I propose makes sense. Again, this answer assumes you are interested in summarizing information for each subject across 8 repetitions.

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  • $\begingroup$ To be precise with the method, participants are trained to discriminate between 4 durations, 200ms, 400ms, 600ms, 800ms. They are then presented with two events separated by a delay e.g. beep -delay- beep (we used 300ms, 500ms, or 700ms as the target delays but participants were naive to this) and then participants must select estimate the time between the two events by selecting one of the 4 durations they were trained on. Your summary of the possible ways to treat the data seems like the same options I've considered so far but where I'm getting stuck is which way is the most appropriate. $\endgroup$ – SLorimer May 29 '18 at 12:38
  • $\begingroup$ How you ultimately proceed has to reflect: how the participants themselves treated the "intervals" (e.g., did they make a qualitative statement about how much larger 400 is relative to 200 or did they estimate the actual amount of 400) , what research question(s) you are interested in and the limitations of the data (if any). If you are not sure about which approach is most suitable, you can look at the problem from multiple angles to get a flavour for what you learn from the data from different vantage points. Let's see if anyone else on this forum can add further insights. $\endgroup$ – Isabella Ghement May 29 '18 at 13:12
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That there are only a few legal options doesn't seem to invalidate the interval scale of the variable. In particular, the mean of this variable still makes sense to compute, because e.g. 612 ms is an understandable value.

Still, be sure not to take Stephens's levels of measurement to be stricter than they are. You can still compute modes and medians with interval-scaled data and these can be more appropriate than the mean depending on the distribution of the variable, your analytic goals, etc.

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  • $\begingroup$ I don't think "legal" makes for a good choice of adjective here. If your analysis is poor, the statistics police don't come knocking. No doubt you mean it in some broader sense of breaking some rule or guideline or boundary of good practice, but then it would be better to be clear about what kind of rule this might be and if possible, what the likely consequence would be. In short - it may be that you can make a clearer point by being more precise about what you think the specific problem is there. $\endgroup$ – Glen_b -Reinstate Monica May 29 '18 at 0:13
  • $\begingroup$ @Glen_b OED sense 4(b) "gen. Allowed by or in accordance with a particular set of rules; acceptable, permissible." The applicable rule here is that subjects were only offered four options. $\endgroup$ – Kodiologist May 29 '18 at 1:22
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    $\begingroup$ Thanks, @Kodiologist. The ultimate goal is to see whether estimates of delay differ as a function of condition; in one block of trials setup is button press--delay--beep; in the other block the setup is beep--delay--beep. The reason for my question in the first place was to decide whether the data can be analysed using mixed factor ANOVA on the average estimate of delay in each condition or whether something more complicated like a mixed effect model is required. $\endgroup$ – SLorimer May 29 '18 at 12:59

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