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### PROJECTS ARE DUE

### Alternatives to the Gaussian Distribution for Random Effects

### The t-distribution

### Informative Sample Size(Similar to informative Censoring)See Louis et al. SMMR 2006

### General Setting

By midnight, Friday, May 19th

Electronic submission only to tlouis@jhsph.edu

Please name the file:

[myname]-project.[filetype]

or

[name1_name2]-project.[filetype]

BIO656--Multilevel Models

Efficiency-Robustness Trade-offs

- First, we consider alternatives to the Gaussian distribution for random effects
- Then, we move to issues of weighting, starting with some formalism
- Then, move to an example of informative sample size
- And, finally give a basic example that has broad implications of choosing among weighting schemes

BIO656--Multilevel Models

BIO656--Multilevel Models

Broader tails than the Gaussian

So, shrinks less for deviant Y-values

The t-prior allows “outlying” parameters and

so a deviant Y is not so indicative of a

large, level 1 residual

BIO656--Multilevel Models

Creating a t-distribution

- Assume a Gaussian sampling distribution,
- Using the sample standard deviation produces the t-distribution
- Z is t with a large df
- t3 is the most different from Z for t-distributions with

a finite variance

BIO656--Multilevel Models

Estimated Gaussian & Fully Non-parametric priors for the USRDS data

BIO656--Multilevel Models

BIO656--Multilevel Models

BIO656--Multilevel Models

Choosing among weighting schemes“Optimality” versus goal achievement

BIO656--Multilevel Models

Inferential Context

Question

What is the average length of in-hospital stay?

A more specific question

- What is the average length of stay for:
- Several hospitals of interest?
- Maryland hospitals?
- All hospitals?
- .......

BIO656--Multilevel Models

“Data” Collection & Goal

Data gathered from 5 hospitals

- Hospitals are selected by some method
- nhosp patient records are sampled at random
- Length of stay (LOS) is recorded

Goal is to: Estimate the “population” mean

BIO656--Multilevel Models

Procedure

- Compute hospital-specific means
- “Average” them
- For simplicity assume that the population variance is known and the same for all hospitals

How should we compute the average?

- Need a goal and then a good/best way to

combine information

BIO656--Multilevel Models

“DATA”

BIO656--Multilevel Models

Weighted averages & Variances (Variances are based on FE not RE)

Each weighted average is mean =

Reciprocal variance weights minimize variance

Is that our goal?

BIO656--Multilevel Models

There are many weighting choices and weighting goals

- Minimize variance by using reciprocal variance weights
- Minimize bias for the population mean by using population weights (“survey weights”)
- Use policy weights (e.g., equal weighting)
- Use “my weights,” ...

BIO656--Multilevel Models

When the model is correct

All weighting schemes estimate the same quantities

same value for slopes in a multiple regression

So, it is clearly best to minimize variance by using

reciprocal variance weights

When the model is incorrect

Must consider analysis goals and use appropriate weights

Of course, it is generally true that our model is not correct!

BIO656--Multilevel Models

Weights and their properties

- But if m1 = m2 = m3 = m4 =m5 = m =

then all weighted averages estimate the population mean: = kk

So, it’s best to minimize the variance

But, if the hospital-specific mk are not all equal, then

- Each set of weights estimates a different target
- Minimizing variance might not be “best”
- For an unbiased estimate of setwk = pk

BIO656--Multilevel Models

The variance-bias tradeoff

General idea

Trade-off variance & bias to produce low

Mean Squared Error (MSE)

MSE = Expected(Estimate - True)2

= Variance + (Bias)2

- Bias is unknown unless we know the mk

(the true hospital-specific mean LOS)

- But, we can study MSE (m, w, p)
- In practice, make some “guesses” and do sensitivity analyses

BIO656--Multilevel Models

Variance, Bias and MSE as a function of (the ms, w, p)

- Consider a true value for the variation of the between hospital means (* is the “overall mean”)

T = (k - *)2

- Study BIAS, Variance, MSE for weights that optimize MSE for an assumed value (A) of the between-hospital variance
- So, when A = T, MSE is minimized by this optimizer
- In the following plot, A is converted to a fraction of the total variance A/(A + within-hospital)
- Fraction = 0 minimize variance
- Fraction = 1 minimize bias

BIO656--Multilevel Models

The bias-variance trade-offX-axis is assumed variance fractionY is performance computed under the true fraction

Assumed

k

BIO656--Multilevel Models

Summary

- Much of statistics depends on weighted averages
- Weights should depend on assumptions and goals
- If you trust your (regression) model,
- Then, minimize the variance, using “optimal” weights
- This generalizes the equal m case
- If you worry about model validity (bias for mp),
- You can buy full insurance by using population weights
- But, you pay in variance (efficiency)
- So, consider purchasing only the insurance you need by

using compromise weights

BIO656--Multilevel Models

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