How should I interpret MGIDI index scores and trait contributions in multi-trait selection?

Question

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1. What the student is asking (plain‑language restatement)

The student is using the Multi‑trait Genotype–Ideotype Distance Index (MGIDI) to rank genotypes (e.g., plant varieties) on the basis of several traits that have different desired directions (some to be maximised, some to be minimised).
They want to know:

Question	In everyday words
Q1	Does a smaller MGIDI value always mean the genotype is “closer” to the ideal (the ideotype)?
Q2	How should we read the contribution of each trait to the total MGIDI value?
Q3	If a trait contributes a lot, does that mean the genotype is bad for that trait, or just that the trait has a large influence on the distance?
Q4	What do the factor scores and loadings that appear in the MGIDI formula really represent?
Q5	Should we rely only on the MGIDI ranking to pick genotypes, or also look at the individual trait values?

Below is a step‑by‑step walk‑through that answers each of these points and shows how the pieces (standardisation, factor analysis, Euclidean distance) fit together.

2. Full derivation & interpretation of MGIDI

2.1 The basic idea: distance to an ideotype

Ideotype definition – For each trait (j) we specify a target value (t_j) that reflects the breeding goal (e.g., highest possible grain yield, lowest disease score).
Genotype vector – For genotype (i) we have the observed raw measurements (\mathbf{x}{i}= (x{i1},x_{i2},\dots ,x_{ip})^\top) for the (p) traits.

The MGIDI tries to summarise, in a single scalar, how far genotype (i) lies from the ideal point (\mathbf{t}=(t_1,\dots ,t_p)^\top) in a multivariate sense.

2.2 Step 0 – Pre‑processing of the data

Operation	Why we do it
Direction standardisation (multiply traits to be minimised by (-1))	Guarantees that for all traits “large = good”.
Centering & scaling (usually Z‑score: (\tilde{x}{ij}= (x{ij}-\bar{x}_j)/s_j))	Removes units, makes traits comparable, and puts them on the same metric. The same transformation is applied to the ideotype: (\tilde{t}_j = (t_j-\bar{x}_j)/s_j).

After this step the data matrix (\tilde{\mathbf X}) and the ideotype vector (\tilde{\mathbf t}) are dimensionless and have mean 0, variance 1 (per trait).

2.3 Step 1 – Factor analysis (FA)

Why FA?
- Many traits are correlated. Using them directly would give redundant weight to the same underlying biological factor (e.g., “plant vigor”).
- FA creates a smaller set of latent variables (factors) that capture most of the variance while being (approximately) orthogonal.
FA model

[ \tilde{\mathbf X}= \mathbf{F}\mathbf{L}^{\top}+ \mathbf{E}, ]

where

(\mathbf{F}) is the (n \times m) matrix of factor scores ( (m) = number of retained factors, usually chosen by eigenvalues > 1 or a cumulative variance threshold).
(\mathbf{L}) is the (p \times m) matrix of loadings (correlations between original traits and each factor).
(\mathbf{E}) is the residual matrix.

Interpretation of loadings
- A loading close to +1 (or –1) means that the trait strongly contributes to that factor, with the sign indicating the direction of the relationship.
- Loadings near 0 mean the trait is almost unrelated to the factor.
Thus, each factor can be thought of as a synthetic trait that is a weighted combination of the original ones.

2.4 Step 2 – Ideotype scores on the factor space

Because the factor analysis is a linear transformation, the ideotype vector can also be projected onto the factor space:

[ \mathbf{f}_t = \tilde{\mathbf t}\,\mathbf{L}, ]

giving the ideotype factor scores (\mathbf{f}t = (f{t1},\dots ,f_{tm})).

2.5 Step 3 – Euclidean distance in the factor space (the MGIDI)

For genotype (i) we have its factor scores (\mathbf{f}i = (\tilde{x}{i\cdot})\mathbf{L}).
The MGIDI is defined as the Euclidean distance between (\mathbf{f}_i) and the ideotype factor scores:

[ \boxed{\text{MGIDI}i = \bigl| \mathbf{f}_i-\mathbf{f}_t \bigr| = \sqrt{\sum{k=1}^{m}\bigl(f_{ik}-f_{tk}\bigr)^2}}. ]

Because Euclidean distance is always non‑negative, the smallest possible value is 0 (genotype = ideotype).

2.6 Step 4 – Decomposing the distance into trait contributions

Although the distance is computed in the factor space, we can trace back how each original trait influences it:

Compute the squared contribution of each factor:

[ C_{ik}^{(\text{factor})}= (f_{ik}-f_{tk})^{2}. ]

Express each factor as a linear combination of traits using the loadings. The contribution of trait (j) to factor (k) for genotype (i) is

[ c_{ijk}= L_{jk}^{2}\, C_{ik}^{(\text{factor})}, ]

where (L_{jk}) is the loading of trait (j) on factor (k).
(The squaring of the loading follows from the fact that distance is quadratic.)

Sum over factors to obtain the overall contribution of trait (j):

[ \boxed{C_{ij}= \sum_{k=1}^{m} L_{jk}^{2}\,(f_{ik}-f_{tk})^{2}} . ]

These (C_{ij}) values are always positive and add up to (\text{MGIDI}_i^{2}).

To express them as percentages:

[ \%C_{ij}= \frac{C_{ij}}{\text{MGIDI}_i^{2}}\times 100. ]

3. Answering the specific questions

Q1 – Does a lower MGIDI always mean a genotype is closer to the ideotype?

Yes. By definition MGIDI is a Euclidean distance; the smaller the distance, the nearer the genotype’s factor scores are to the ideotype’s factor scores. A value of 0 would mean the genotype exactly matches the ideotype on every trait (after standardisation).

Note: Because we work in a transformed (standardised + factor‑reduced) space, “closer” is measured on a relative scale, not on the raw trait units.

Q2 – How to interpret the contribution of individual traits?

The absolute contribution (C_{ij}) tells how much trait (j) participates in the squared distance for genotype (i).
The percentage contribution (\%C_{ij}) tells the proportion of the total distance that is attributable to that trait.

If a trait accounts for, say, 45 % of the MGIDI of a genotype, it is the dominant source of the genotype’s deviation from the ideal.

Q3 – Does a large contribution mean the genotype is “bad” for that trait?

Not necessarily. A large contribution can arise from two sources:

Source	What it means
Large residual ((f_{ik}-f_{tk})) for a factor that heavily loads on the trait	The genotype truly deviates from the desired direction for that trait (or a group of correlated traits).
Large loading (L_{jk}) (the trait is strongly tied to that factor)	Even a modest residual in that factor can translate into a large contribution because the trait is a major component of the factor.

Hence, a high percentage contribution flags the trait as an important driver of distance, but you still need to look at the direction of the deviation (e.g., is the trait too low or too high?) to decide whether the genotype is “poor” for that trait.

Q4 – How to understand the factor scores & loadings?

Item	Interpretation
Loadings ((L_{jk}))	Correlation (or weight) of original trait (j) with latent factor (k). A loading of +0.8 means trait (j) moves in the same direction as factor (k); –0.8 means it moves oppositely.
Factor scores ((f_{ik}))	The coordinate of genotype (i) on factor (k). It is a synthetic trait that aggregates the information of all original traits that load on that factor.
Ideotype factor scores ((f_{tk}))	The coordinate of the ideal genotype on factor (k). Because the ideotype is defined on the original trait scale, we project it onto the factor space using the same loadings.

In practice, you can inspect the loading matrix to give each factor a biological label (e.g., “yield‑related”, “stress tolerance”) and then interpret a genotype’s distance in terms of those synthetic dimensions.

Q5 – Should selection be based solely on MGIDI ranking?

MGIDI is a powerful summary index, but it should not be the only decision tool. Recommended workflow:

Rank by MGIDI – eliminates genotypes that are far from the ideotype on the overall multivariate scale.
Inspect the trait‑percentage contributions – identify which traits are responsible for the remaining distance.
Check raw (or standardized) trait values for the top‑ranked genotypes, especially for traits that are of critical importance (e.g., disease resistance may be non‑negotiable even if its contribution to MGIDI is modest).
Apply any hard constraints (e.g., a genotype must have grain yield ≥ X). These constraints can be enforced before or after the MGIDI calculation.

Thus, MGIDI is best used in conjunction with trait‑level diagnostics, not as a black‑box selector.

4. Worked numeric illustration (optional but clarifies the math)

Trait (after direction standardisation)	Mean	SD	Desired (ideotype)
Yield (kg) (higher‑better)	5000	400	6000
Plant height (cm) (lower‑better)	150	10	130
Disease score (scale 1–9, lower‑better)	4	1	1

Standardise

[ \tilde{x}{ij}= \frac{x{ij}-\bar{x}_j}{s_j}, \qquad \tilde{t}_j = \frac{t_j-\bar{x}_j}{s_j} ]

Suppose for genotype A we have raw values (6000, 140, 2). After standardisation:

[ \tilde{\mathbf{x}}_A = (2.5,\; -1.0,\; -2.0), \qquad \tilde{\mathbf{t}} = (2.5,\; -2.0,\; -3.0). ]

Factor analysis (say we keep 2 factors). The loading matrix (rounded) is

[ \mathbf{L}= \begin{bmatrix} 0.80 & 0.10
0.30 & -0.90
0.50 & 0.20 \end{bmatrix}. ]

Factor scores

[ \mathbf{f}_A = \tilde{\mathbf{x}}_A\mathbf{L} = (2.5\cdot0.80+(-1.0)\cdot0.30+(-2.0)\cdot0.50,\; 2.5\cdot0.10+(-1.0)(-0.90)+(-2.0)\cdot0.20) = (0.65,\; 0.55). ]

[ \mathbf{f}_t = \tilde{\mathbf{t}}\mathbf{L} = (2.5\cdot0.80+(-2.0)\cdot0.30+(-3.0)\cdot0.50,\; 2.5\cdot0.10+(-2.0)(-0.90)+(-3.0)\cdot0.20) = (-0.85

Original question: How should I interpret MGIDI index scores and trait contributions in multi-trait selection? on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.