Model Equivalence in General Linear Models: Set-to-Zero, Sum-to-Zero Restrictions, and Extra Sum of Squares Method
Bibliographic record
Abstract
The paper is drawn from the authors' experience in teaching general and generalized linear fixed effects models at the university level. The steps followed include model specification, model estimation, and hypothesis testing in general linear model setting. Among these steps, estimation of model parameters such as the main effect least squares means and contrasts were among the most challenging for students. Since no unique solution exists, students are first exposed to the equivalence between two popular techniques that an over-parameterized model can be subjected to in order to obtain the parameter estimates. This is particularly important because existing software do not necessarily follow the same path to produce an Analysis of Variance (or Covariance) of the general, generalized linear fixed or mixed effects models. These steps are generally hidden from the users. It is therefore crucial for the students to understand the intermediary processes that ultimately produce the same results regardless of the software one uses. The equivalent techniques, the set-to-zero and sum-to-zero restrictions, used to obtain solution of the normal equations of the fixed effects model, are presented. The relationship between them is also presented and in the process, data analysis makes use of two important concepts: the generalized inverse and estimable function. The invariance property of estimable functions is also explained in details in addition to the extra sum of squares principle which is introduced to supplement the other concepts. To exemplify these ideas and put them in practice, a simple one-way treatment structure analysis of variance is performed.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.080 | 0.255 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.005 |
| Bibliometrics | 0.003 | 0.005 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.005 | 0.007 |
| Open science | 0.006 | 0.010 |
| Research integrity | 0.003 | 0.011 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".