The Null Distribution of the Heterogeneity Lod Score Does Depend on the Assumed Genetic Model for the Trait
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Bibliographic record
Abstract
It is well known that the asymptotic null distribution of the homogeneity lod score (LOD) does not depend on the genetic model specified in the analysis. When appropriately rescaled, the LOD is asymptotically distributed as 0.5 <i>χ</i><sup>2</sup>&#8320; + 0.5 <i>χ</i><sup>2</sup><sub>1</sub>, regardless of the assumed trait model. However, because locus heterogeneity is a common phenomenon, the heterogeneity lod score (HLOD), rather than the LOD itself, is often used in gene mapping studies. We show here that, in contrast with the LOD, the asymptotic null distribution of the HLOD does depend upon the genetic model assumed in the analysis. In affected sib pair (ASP) data, this distribution can be worked out explicitly as (0.5 – <i>c</i>)<i>χ</i><sup>2</sup>&#8320; + 0.5<i>χ</i><sup>2</sup><sub>1</sub> + <i>c</i><i>χ</i><sup>2</sup><sub>2</sub>, where <i>c</i> depends on the assumed trait model. E.g., for a simple dominant model (HLOD/D), c is a function of the disease allele frequency p: for <i>p</i> = 0.01, <i>c</i> = 0.0006; while for <i>p</i> = 0.1, <i>c</i> = 0.059. For a simple recessive model (HLOD/R), <i>c</i> = 0.098 independently of <i>p</i>. This latter (recessive) distribution turns out to be the same as the asymptotic distribution of the MLS statistic under the possible triangle constraint, which is asymptotically equivalent to the HLOD/R. The null distribution of the HLOD/D is close to that of the LOD, because the weight <i>c</i> on the <i>χ</i><sup>2</sup><sub>2</sub> component is small. These results mean that the cutoff value for a test of size <i>α</i> will tend to be smaller for the HLOD/D than the HLOD/R. For example, the <i>α</i> = 0.0001 cutoff (on the lod scale) for the HLOD/D with <i>p</i> = 0.05 is 3.01, while for the LOD it is 3.00, and for the HLOD/R it is 3.27. For general pedigrees, explicit analytical expression of the null HLOD distribution does not appear possible, but it will still depend on the assumed genetic model.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it