ICAR AICE-JRF/SRF Linear Mixed Models, Variance Components & Spatial Analysis of Field Trials: practice questions

1 · Fixed, random and mixed models revisited · easy

In the mixed model y = X beta + Z u + e, with u ~ N(0, G) and e ~ N(0, R), what is the variance of y?

1. G + R
2. Z R Z' + G
3. Z G Z' + R
4. X G X' + R

2 · Maximum likelihood and REML · medium

In a balanced one-way random model, how does ML differ from REML for the between-group component?

1. ML ignores the within-group variance completely in this model
2. ML is always zero in this model
3. ML gives exactly the ANOVA estimator
4. ML divides the between-group SS by a instead of a - 1

3 · Henderson's mixed-model equations; BLUE and BLUP · medium

In the mixed model, what is beta-hat from the mixed-model equations?

1. The BLUP of the random effects
2. The ML estimate of variance
3. The OLS estimate, ignoring the covariance matrix V
4. The BLUE of beta, equal to the GLS estimator

4 · Spatial analysis of field trials: nearest-neighbour and AR1 x AR1 models · easy

Why is spatial analysis used in field trials?

1. Fertility varies smoothly and irregularly, not only between blocks
2. Blocks are never useful in field trials
3. It removes the need for replication
4. Plots far apart are always more alike than adjacent plots in a block

5 · Trend and spline adjustment (e.g. SpATS approach) · medium

Can a spline surface and an AR1 x AR1 residual be combined in one model?

1. Only if the trial has a single row
2. Only when no genotypes or treatments are fitted in the model
3. No, they are mathematically incompatible
4. Yes, for large-scale trend plus local correlation

6 · Model selection with likelihood-ratio tests and information criteria · hard

Why is the naive chi-square test conservative when testing a variance component equal to zero?

1. Variance components are always large
2. REML likelihoods cannot be compared between any two models at all
3. The null value lies on the boundary of the parameter space
4. The chi-square has too many df

7 · Model selection with likelihood-ratio tests and information criteria · medium

Comparing AR(1) and Toeplitz residual structures with the same fixed effects, which tool is appropriate?

1. Bartlett's test on residuals
2. REML-based AIC or BIC
3. A t test on the variety means
4. ML-based F tests on fixed effects

8 · Maximum likelihood and REML · easy

What does REML stand for?

1. Repeated measures linear model
2. Restricted (residual) maximum likelihood
3. Random effects minimum loss
4. Regression estimate of mean level

9 · Henderson's mixed-model equations; BLUE and BLUP · easy

What does BLUP stand for?

1. Bayesian likelihood under priors
2. Best linear unbiased prediction
3. Balanced linear unit prediction
4. Block level uniform precision

10 · Covariance structures for repeated measures · easy

How many covariance parameters does compound symmetry have?

1. t(t + 1)/2
2. 2
3. t
4. 1