| true count | category |
|---|---|
1 |
≤ 10 |
3 |
≤ 10 |
12 |
11-20 |
19 |
11-20 |
24 |
21-50 |
55 |
51-100 |
369 |
> 100 |
Analyzing ordered categorical phenotypes: challenges and pitfalls
Who is this talk for?
- Anyone who works with categorical phenotypes
- Hopefully you have a little experience with statistics
- It’s OK if you do more “applied” stats and don’t have a solid background in statistical theory … I don’t either!
Talk outline
- What is categorical phenotype data, and what are its advantages and disadvantages?
- How should we analyze categorical data?
- Practical recommendations
Image (C) Maria Gusarova
What is categorical phenotype data?
- It may be ordered or unordered
- unordered: color, etc. (not the focus of this talk)
- ordered: low/medium/high, disease score from 1-5, etc.
Image (C) Allison Horst
Why do we collect categorical phenotype data?
- If the “true” quantitative variable is very time-consuming to measure, many individuals can be phenotyped more quickly if you use categorical data
- Example: counting the number of hairs on a leaf vs. visually assigning low/medium/high)
- Or there is no easily determined single quantitative variable, so we have no choice
How do we analyze categorical phenotype data?
Option 1: Treat categorical variable as continuous
- I call this the “quick and dirty way”
Quick and dirty way: the upside
- Continuous model is easy to fit and often gives acceptable results, even though it violates a lot of assumptions
- Usually fine to use in preliminary screening phase
- A simulation study showed that it usually has similar predictions (Azevedo et al. 2024, Theor. Appl. Gen.)
- Acceptable as long as the categorical variable is estimating a true underlying continuous variable (disease intensity)
- Lumping multiple things together may give strange results (presence/absence of a disease and disease intensity)
Option 2: Account for the ordered categorical nature of the data
Image (C) Getty Images
Binomial distribution
- Many categorical phenotype datasets are binary
- Binomial distribution has one parameter, \(\theta\), the probability of a “success”
- Define one outcome as a success (1) and the other as a failure (0)
- Example: the outcomes of flipping a coin many times are distributed binomially with \(\theta = 0.5\)
How to analyze binary categorical data?
- A linear model requires the error, or residuals, to be normally distributed
- Residuals can’t be normally distributed if the data are all 0s and 1s
- Normal distribution (bell curve) is symmetrical and can take any \(x\) value
How to analyze binary categorical data?
- We model the probability of a success on the logit (log-odds) scale because log-odds can take any value from \(-\infty\) to \(+\infty\)
- Straight between ~0.25 and ~0.75, steeper closer to 0 and 1
- \(\text{logit} y = \mu + G_{ij} + E_{ij} + GE_{ij} + \epsilon\)
- Probit can be used instead of logit but the outcome is similar
Multinomial distribution
- Now instead of flipping a coin, we’re rolling a die
- More than two outcomes, each with a probability of occurring
- For fair six-sided dice the probabilities are all 1/6
- But they don’t have to be equal, they just have to sum to 1
- There are \(n-1\) parameters, where \(n\) is the number of outcomes (classes)
Cumulative logistic model: the downsides
- It is a “data hungry” model because you need to have some representation of all the categories in all the treatments
- 2.25× greater training population size needed compared to continuous phenotypes (Kizilkaya et al. 2014, Gen. Sel. Evol.)
- This is especially true for mixed models because you also need representation of all the categories in all the environments (blocks)
- Convergence of the model fitting algorithm is often a problem
Quasi-complete separation
- If you have zeros for some categories, you end up with complete or quasi-complete separation
- Some genotypes’ coefficients in the model can’t be estimated because those genotypes have no observations for one or more categories
- This becomes more and more likely, the more categorical levels you have
Figure from Schwendinger et al. 2021, Comp. Stat.
Making predictions in cumulative logistic model
- We can make several different kinds of predictions from a CLM
- Probabilities, or cumulative probabilities for each class within each genotype
- More honest but not as simple to interpret
Making predictions in cumulative logistic model
- Mean class predictions (weighted average by the probability of each class for each genotype)
- Similar-looking result as the quick and dirty way but better because the underlying model doesn’t assume it’s a continuous variable
- Hypothesis tests can be done on either kind of comparison
Bayesian CLM
- You can incorporate prior distributions on the fixed and random effect parameters, and the threshold probabilities
- This can allow you to get estimates, even when there is complete separation and a classical statistical approach can’t handle it/won’t converge
Image (C) Analytics Vidhya
Machine learning approaches
- Ordered categorical outcomes are supported by a lot of machine learning models such as random forest
- Regularization in machine learning models can work the same way as Bayesian priors, and fix the quasi-complete separation issue
Image (C) MathWorks
Practical recommendations: deciding on number of categories
- Number of categories should ideally be 3 to 5
- Binary loses too much information, and greater than 5 is too error-prone and data-hungry
- However: you can always lump categories together later, but you can’t split them!
- Category thresholds should be validated to minimize disagreement between raters
Approach with caution
- “All models are wrong, but some are useful”
- No matter what, some kind of assumption will have to be made
- Treating categorical variable like a number
- Lumping together categories
- Including prior distributions on the parameters or regularization
Further reading
- UCLA stats tutorials
- R tutorial on ordinal logistic regression
- Paper on ordinal GWAS with BGLR package
- Contact me for more!
