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I am looking at patient data with clinical scores for each that run from zero to 6 (integers, where zero is best and scoring 6 on symptoms is worst). There are follow up scores on each patient (at varying time intervals) - even the number of follow-ups is unequal among patients. The average number of follow-ups is around 4 ranging from 2 through 13. Each follow-up has a timestamp.

I wish to look at trends in the follow-up - whether the scores improve/worsen or stay stable to group the patients for further analysis.

Is there any specific metric that can capture in essence, the trend of ordinal scores?

Another issue is that the range of scores is 0-6 while the range of time is in months to years. The wide range in time is causing an issue in using measures like slope or r^2

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I think you should change the word "categorical" in the title with a more precise "ordinal". In fact it would be impossible to find a "trend" in a unordered categorical variable.

Anyway, I think you should build an ordered logit/probit model, this is necessary to evaluate distances between different observed scores, unless you want to assume equally spaced score levels, in that case you can turn your variable to numeric.

In your ordered logit model, you will regress score against time, using a linear fit, so that you can get a single number indicating a trend. However, it's worth noting that:

  • a linear fit is almost always a good first stab indicator of what's going on in data, but it isn't necessarily good enaugh for your needs. This holds true with correlation, and with this approach as well. In particular, linear fit won't be able to show even likely phenomena like healing after deterioration, or temporal autocorrelation causing drops of patients conditions. You could even need to switch to a completely different model to fully understand the progression of the disease.
  • you may want to use a random effects model. If you use fixed effects, the estimation of each trend will be unbiased, but it could be too variable when the number of checks is low; if you use random effects, this variabilty will be controlled and all trends will be schrinked towards the common mean. Shrinking will be very weak for those individuals who where examinated most times, so that their trend is estimated more precisely.
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