Simultaneous inference in general parametric models.
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References
Robust Regression and Outlier Detection
Approximation Theorems of Mathematical Statistics
Multiple Comparison Procedures
Related Papers (5)
Frequently Asked Questions (14)
Q2. What is the default re-parametrization used as elemental parameters in the R?
The so-called ”treatment contrast” vector θ = (µ, γ2− γ1, γ3− γ1, . . . , γq−γ1) is, for example, the default re-parametrization used as elemental parameters in the R-system for statistical computing (R Development Core Team, 2008).
Q3. What is the purpose of this paper?
In this paper the authors aim at a unified description of simultaneous inference procedures in parametric models with generally correlated parameter estimates.
Q4. What is the advantage of single-step procedures?
Single-step procedures have the advantage that corresponding simultaneous confidence intervals are easily available, as previously noted.
Q5. What is the p-value for a given family of null hypotheses?
That is, for a given family of null hypotheses H10 , . . . , H k 0 , an individual hypothesis H j 0 is rejected only if all intersection hypotheses HJ = ⋂ i∈J H i 0 with j ∈ J ⊆ {1, . . . , k} are rejected (Marcus et˜al., 1976).
Q6. What is the scalar test statistic for testing the global null hypothesis?
By construction, the authors can reject an individual null hypothesis Hj0 , j = 1, . . . , k, whenever the associated adjusted p-value is less than or equal to the pre-specified significance level α, i.e., pj ≤ α.
Q7. What is the simplest way to model the response?
The response is modelled by a linear combination of the covariates with normal error εi and constant variance σ 2,Yi = β0 +q ∑j=1βjXij +
Q8. What is the general framework for simultaneous inference?
The general framework described here extends the current canonical theory with respect to the following aspects: (i) model assumptions such as normality and homoscedasticity are relaxed, thus allowing for simultaneous inference in generalized linear models, mixed effects models, survival models, etc.; (ii) arbitrary linear functions of the elemental parameters are allowed, not just contrasts of means in AN(C)OVA models; (iii) computing the reference distribution is feasible for arbitrary designs, especially for unbalanced designs; and (iv) a unified implementation is provided which allows for a fast transition of the theoretical results to the desks of data analysts interested in simultaneous inferences for multiple hypotheses.
Q9. What is the p-value for the jth individual two-sided hypothesis?
In the present context of single-step tests, the (at least asymptotic) adjusted p-value for the jth individual two-sided hypothesis Hj0 : ϑj = mj, j = 1, . . . , k, is given bypj = 1− gν(Rn, |tj|),where t1, . . . , tk denote the observed test statistics.
Q10. What is the p-value for the global null hypothesis?
The resulting global p-value (exact or approximate, depending on context) for H0 is 1 − gν(Rn,max |t|) when T = t has been observed.
Q11. What is the way to test the global null hypothesis?
Another suitable scalar test statistic for testing the global hypothesis H0 is to consider the maximum of the individual test statistics T1,n, . . . , Tk,n of the multivariate statistic Tn = (T1,n, . . . , Tk,n), leading to a max-t type test statistic max(|Tn|).
Q12. What are examples of multiple comparison procedures?
Examples of such multiple comparison procedures include Dunnett’s many-to-one comparisons, Tukey’s all-pairwise comparisons, sequential pairwise contrasts, comparisons with the average, changepoint analyses, dose-response contrasts, etc.
Q13. Why is mcp() not available in multcomp?
Because it is impossible to determine the parameters of interest automatically in this case, mcp() in multcomp will by default generate comparisons for the main effects γj only, ignoring covariates and interactions.
Q14. What is the sequence of n needed to establish -convergence in (4)?
Then the authors haveãnRn = D −1/2 n S ⋆ nD −1/2 n= (anDn) −1/2(anS ⋆ n)(anDn) −1/2P −→ diag(Σ⋆)−1/2 Σ⋆ diag(Σ⋆)−1/2 =: R ∈ Rk,kwhere the convergence in probability to a constant follows from Slutzky’s Theorem (Theorem 1.5.4, Serfling, 1980) and therefore (4) holds.