Questions on LMM

Question 1

Explain the difference between Uoj et Eij

Answer 1

Uoj explains the between subject-variance and Eij explains the within-subject variance

Question 2

True or false : Fixed factors are variables that we think directly affect the response

Answer 2

True

Question 3

What is the mean of each random effect?

Answer 3

The mean is 0

Question 4

Please tell what are the data designs :
1. Observations of entities that are contained in a larger entity
2. Multiple observations for each subject
3. Observations over a period of time for each subject
4. Observations of subjects contained in a larger entity over a period of time
5.

Answer 4

1. Clustered data
2. Repeated measures data
3. Longitudinal data
4. Clustered longitudinal data

Question 5

All of the general data are hierarchical, when one variable varies within another variable, we say that the effects of the first variable are..

Answer 5

Nested effects

Question 6

When a value of 1 variable varies independently of another variable (no hierarchy), we say that ..

Answer 6

this is crossed effects

Question 7

What is the output of a LMM?

Answer 7

1. Estimates of the fixed coefficients
2. Estimates of the random coefficients
3. Variance /Covariance parameters of the random effects
4. Standard errors/ t statistic

Question 8

Dans un LMM, Ui et Ei suivent quelle distribution? Quelles sont les paramètres de cette distribution?

Answer 8

Ui suit une multinormal avec moyenne 0 et variance D

Ei suit une multinormal avec moyenne 0 et variance Ri

Question 9

What are the 2 common structures of D?

Answer 9

1. Unstructured : Variances and covariances can be anything. Since it is a covariance matrix, it must be positive definite and symmetric
2. Diagonal / Variance components : All covariances are 0

Question 10

True or false : D can change by entities (heterogeneous)

Answer 10

Absolutely true. D can change for each level-2 entity.

Question 11

True : Like D, Ri need to be positive definite and symmetric

Answer 11

True

Question 12

What are the common structures of Ri? Please do an example of each type

Answer 12

1. Diagonal : Variance is constant and Covariance = 0
2. Compound symmetry
3. First order autoregressive AR(1)

Question 13

For which type of data the compound symmetry can be appropriate?

Answer 13

The compound symmetry structure can be appropriate for clustered data. Can be suitable too for repeated measures data

Question 14

For which type of data the AR(1) can be appropriate?

Answer 14

The AR(1) structure can be appropriate for longitudinal data, if the data is at equals time interval

Question 15

True or false : Unlike D, Ri is always homogeneous

Answer 15

False. Ri can change by entity too

Question 16

What is the method when you want to create the matrices for all observations ? For Y,X,Z,u,G,R,B?

Answer 16

1. Y = Stacked vertically
2. X = Stacked Vertically
   E = Stacked vertically
3. Z = Diagonal
4. G = Diagonal
5. R = Diagonal
   B = Doesn’t change

Question 17

True or false : Marginal models in LMM without random effects

Answer 17

TRUE

Question 18

Which distribution Ei follow in a marginal model?

Answer 18

Ei follows a Multivariate normal (0,Vi)

Question 19

True or false : In a marginal model, variances of residuals may vary, and covariance between residuals may not be 0

Answer 19

TRUE

Question 20

True or false : Marginal models are subject-averaged

Answer 20

False. Marginal model are population-averaged. LMM are subject averaged

Question 21

What population-averaged means?

Answer 21

No difference between predictions for different subjects

Question 22

What is the most popular marginal model?

Answer 22

Implied Marginal model. Marginal model implied in a LMM

Question 23

True or false : For the implied marginal model, the predicted mean is the same as the predicted overall mean of the underlying LMM

Answer 23

True bitch

Question 24

What are the 2 reasons why the implied marginal model is important ?

Answer 24

1. The parameters of the LMM are estimated using the implied marginal model
2. Even when the LMM cannot be estimated because D or Ri are not positive definite, the implied marginal model’s Vi might be positive definite. Thus the implied marginal model may help diagnose non-positive definiteness of D or help answer research questions.

Question 25

If the variance/covariance parameters are known (theta), we can estimate the fixed effects by using the generalized least square method and the estimators are the BLUE : Best linear unbiaised estimator of B . True?

Answer 25

True

Question 26

Since the variances/covariances are not known, we can have a EBLUE for the estimators of fixed effects

Answer 26

True

Question 27

What are the 2 techniques we are using the estimate the fixed coefficients and the variance/covariance of the random factors

Answer 27

1. ML estimation | 2. REML estimation

Question 28

When using ML, the estimates of theta (variance/covariance) are biaised, why?

Answer 28

Because they do not take into account the loss of degrees of freedom for estimating B

Question 29

Does REML give a biaised estimator for Theta?

Answer 29

No, REML gives an unbiaised estimator for the variance/covariance parameters for the random effectrs

Question 30

True or false : ML is invariant of a change in the parametrization of the fixed effects, but REML is not invariant

Answer 30

True

Question 31

True or false : If ML was used to fit models with different structure of fixed effects, likelihood ratio test and other test may not be compared

Answer 31

True

Question 32

Complete : Use REML fitting for models if you wish to compare models with different structures for the .... and use ML fitting if you wish to compare models with different structures for the...

Answer 32

1. Random effects | 2. Fixed effects

Question 33

True or false : The estimators of Variance(B) are unbiaised under ML and REML

Answer 33

False! The variance of B is biaised under the 2 methods

Question 34

How the Kenward-Rogers adjustment accounts for the biaised estimates of Var(B) under ML/REML?

Answer 34

1. Adjusting the number of degrees of freedom in t and F test 2. Inflating the Vi matrix

Question 35

What are the 3 algo that are used to optimize LMM?

Answer 35

1. Expectation maximization 2. Newton-Raphson 3. Fischer Scoring

Question 36

Give the 3 features of the Expectation Maximization

Answer 36

1. Converges slowly 2. Estimation of precision of estimators in optimistic (Underestimate variance of estimators) 3. Method is used only to find the good starting values for other algorithms

Question 37

Give the 4 features about the Newton Raphson

Answer 37

1. Each iteration takes longer than EM, but fewer is needed 2. The Hessian Matrix is used to estimate the variance and covariance of estimators 3. It is the most commonly used Algo 4. Preferred Algo

Question 38

What is the big difference between the Newton Raphson and Fischer Scoring?

Answer 38

Fischer Scoring uses the expected Hessian instead of the Observed Hessian

Question 39

Give the 4 features of Fischer scoring

Answer 39

1. More stable numerically than N-R 2. More likely to converge than N-R 3. Calculations are simplified relative to N-R 4. Not recommended for final estimate because may be difficult to determine the expected Hessian Matrix

Question 40

When problems arise with estimation of variance/covariance, what are the alternatives?

Answer 40

1. Change starting values/Change Algorithm 2. Rescale the covariates 3. Reduce number of variance parameters 4. FIt the implied marginal model 5. Fit a marginal model with unstructured Vi

Question 41

True or false : The algo (EM, N-R, Fischer) assure that the covariances matrices are positive definite at each iteration

Answer 41

False