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### Analytic Strategy

### There is no free lunch!

PART 4 Non-linear models

- Logistic regression
- Other non-linear models
- Generalized Estimating Equations (GEE)
- Examples
- Crossover study
- British Social Attitudes Survey

BIO656--Multilevel Models

Models for Clustered Data

Inferential goals

- Marginal mean/Population Averaged
- Average response across “the population”
- Mean, conditional on
- Other responses in the cluster
- Unobserved random effects

BIO656--Multilevel Models

Interpreting Linear Model Coefficients

Same interpretation for conditional (cluster-specific)

and population-averaged inferences

- Unit change in dependent variable for a unit change in regressor
- Multi-level models specify correlations and latent effects:
- The random intercept model produces an

equal-correlation model (correlation

- The latent intercepts can be estimated and used for prediction

BIO656--Multilevel Models

Marginal Models

Inferential Target

- Marginal mean or population-averaged response for different values of predictor variables

Examples

- Difference in mean alcohol consumption for two age groups
- Rate of alcohol abuse for states with addiction treatment programs compared to those without

Public health assessments

BIO656--Multilevel Models

Conditional Models

Conditional on other observations in cluster

- Probability that a person abuses alcohol given family membership or given the number of family members that do
- Probability that a person will abuse next year, if abuses this year
- A person’s average alcohol consumption given the average in the neighborhood

BIO656--Multilevel Models

Conditional Models

Conditional on random effects

- Average consumption, conditional on a latent tendency
- Probability that a person abuses alcohol, conditional on a latent tendency

Can be thought of as

conditional on unmeasured covariates

BIO656--Multilevel Models

The basic, conditional logistic model

- Conditional on a random effect, you have the logistic regression:

logit(P) = log{P/(1-P)} = u + + X

u ~ (0, 2)

Implications

- Generally, the population averaged (marginal) model will not have the logistic shape
- In any case, the slope on a covariate will have a different impact in the conditional and marginal models

BIO656--Multilevel Models

BIO656--Multilevel Models

Conditional Logistic and Marginal ShapesU is a two-point mixture at 2

u=0

b =

BIO656--Multilevel Models

Adjust the conditional slope to closely match the marginal curve

- Assume that there is a population relation that is logistic with term X
- How far off is the marginal curve produced from the conditional logistic curve with term X?
- Let * be the slope needed in the conditional logistic so that the marginal curve produced from it comes close to the population relation
- “Comes close” means to track the middle part of the population curve

BIO656--Multilevel Models

Non-linear model coefficients

- Usually, population-averaged (marginal) and conditional models have different shapes
- Condition logistic is not population logistic
- But, conditional probit is population probit
- In any case, population-averaged and cluster-specific coefficients have different magnitudes and interpretations because they address different questions
- For example, when u is a two-point, 50/50 mixture at 2, = 4 and * = 8.

Need to consider impact on

probabilities not just on odds ratios

BIO656--Multilevel Models

SHAPE & SLOPE CHANGES

- For linear models, regression coefficients in random effects models and marginal models are identical:

average of linear model = linear model of average

- For non-linear models, coefficients have different meanings and values:

average of non-linear model

non-linear model of average

coefficient value and meaning in average model

coefficient value and meaning in conditional model

BIO656--Multilevel Models

Conditional Logistic and Marginal Shapes

Log(odds | u) = u -2.0 + 0.4X

Population prevalences

X = 1

X = 0

Cluster-specific probabilities

BIO656--Multilevel Models

Logistic Regression Example Cross-over trial

2 observations per person (before/after)

Response

1=not alcohol dependent; 0 = AlcDep

(so a high probability is good!)

Predictors

period (Pd = 0 or 1)

treatment group (Trt = 0 or 1)

Parameter of interest

- Treatment vs placebo after/before log(OddsRatio)
- A positive slope favors the treatment

BIO656--Multilevel Models

Baseline/Follow-up Model

i = period, j = person; logit(P) = log(P/[1-P])

Population level (no individual effects)

logit(Pij) = + 1PDij + 2TRij + 3PDijTR2ij

= + 1PDi + 2TRj + 3PDiTRj

logit(P2j) - logit(P1j) = 1+ 3TRj

(3 is the treatment effect)

Person-level (individual intercept)

logit(Pij) = uj + * + *1PDi + *2TRj + *3PDiTRj

uj ~ (0, 2)

BIO656--Multilevel Models

Results for population-level regressions(logistic without multi-level component)

Similar estimates; wrong standard error for Std. Logistic

BIO656--Multilevel Models

The effect of accounting for correlation

- Treatment effect estimates are the same for marginal logistic and correlation accounted logistic
- But, SEs are 0.38 and 0.23 respectively
- Why is the second smaller than the first?

Answer

- The treatment effect is estimated by contrasting (differencing) period 2 and period 1
- The positive, within-person correlation produces a smaller variance of this difference than does assuming independence

BIO656--Multilevel Models

Population-level vs Random Intercept logistic regressionslog(OR)(se)

BIO656--Multilevel Models

Marginal Logistic versusRandom Intercept Logistic

Unconditional Logistic (Population-level inference):

The population AlcnonDep(after/before), treatment/placebo

prevalence odds ratiois exp(0.57) = 1.77

Conditional, RE Logistic (Individual-level inference):

An individual’sAlcnonDep (after/before), treatment/placebo

prevalence odds ratio is exp(1.80) = 6.05

Ratio: (Conditional)/(Marginal)

6.05/1.77 = 3.42 (= e1.23; 1.23 = 1.80-0.57)

Different questions; different (but compatible) answers

BIO656--Multilevel Models

Consequence of Conditional/Marginal Slope Differences

- A population-level analysis that does not build on a multi-level model (that does not include the random effect) can understate the individual-level (cluster level) risk or benefit
- Understate environmental risk
- Understate benefits of lowering blood pressure
- .........

BIO656--Multilevel Models

and conditional ORs

logit(pr(Y = 1 | X, u) = u + log(3)X

u = log(3) with probability 1/2

3.00= (.5/.5)(.25/.75) = (.9/.1) (.75/.25)

BIO656--Multilevel Models

u as a missing covariate

- Without knowing u, a marginal logistic regression predicts 0.50 and 0.70 for X=0 and X=1 respectively
- The log(OR) slope on X is 0.847 = log(2.333)
- If we know u, a logistic regression with it as a covariate (conditional on it) predicts as in the table
- The log(OR) slope on X is 1.099 = log(3.00)

BIO656--Multilevel Models

(Y1, Y2) are in the same cluster

The RE model produces the

following 22 table for X = 0

5/16 = [(3/4)(3/4) + (1/4)(1/4)]2

pr(Y2 =1 | Y1 = 0) = 3/8 = 3/(3+5)

pr(Y2 =1 | Y1 = 1) = 5/8 = 5/(3+5)

BIO656--Multilevel Models

(Y1, Y2) are in the same cluster

The RE model produces the

following 22 table for X = 1

13/100 = [(1/2)(1/2) + (1/10)(1/10)]2

pr(Y2 =1 | Y1 = 0) = 17/30 = 17/(17+13)

pr(Y2 =1 | Y1 = 1) = 53/70 = 53/(17+53)

BIO656--Multilevel Models

Updating the distribution of u

For X = 1 (you can try it for X = 0)

pr(u = +log(3) | Y = 0) = pr(u = +log(3), Y = 0)/pr(Y = 0)

= (1/2)(1/10)(3/10) = 1/6 < 0.5

pr(u = +log(3) | Y = 1) = pr(u = +log(3), Y = 1)/pr(Y = 1)

= (1/2)(9/10)(7/10) = 9/14 > 0.5

pr(u = +log(3)) = (1/6)(3/10) + (9/14)(7/10) = 0.5

Can use these to get [Y2 | Y1]

BIO656--Multilevel Models

Marginal Multi-level, non-linear Models

GEE: Marginal mean as a function of covariates

- Working independence or other working model
- Followed by Robust SE
- “Cluster(id) in Stata
- “Robust” Option in SAS Proc Mixed or GenMod
- No “robustness” in BUGS

Conditional mean, as a function of marginal mean

and cluster-specific random effects

- Heagerty (1999, Biometrics)
- Heagerty and Zeger (2000, Statistical Science)

BIO656--Multilevel Models

Generalized Linear Models (GLMs)g(mean) = 0 + 1 X1 + ... + p Xp(always a marginal model)

BIO656--Multilevel Models

Baseline/Follow-up Model

i = period, j = person; logit(P) = log(P/[1-P])

Population level (no individual effects)

logit(Pij) = + 1PDij + 2TRij + 3PDijTR2ij

= + 1PDi + 2TRj + 3PDiTRj

logit(P2j) - logit(P1j) = 1+ 3TRj

(3 is the treatment effect)

Person-level (individual intercept)

logit(Pij) = uj + * + *1PDi + *2TRj + *3PDiTRj

uj ~ (0, 2)

BIO656--Multilevel Models

Marginal Generalized Linear Modelsvia Generalized Estimating Equations (GEE)

- Ordinary GLM (linear, logistic, Poisson,..)
- Population-average parameters
- Logit: Oij = logit(pij) = 0 + 1Xij
- Then, model association among observations

i and i’ in cluster j:

corr(log(Oij/ Oi’j))= function(G)

- Solve generalized estimating equation (GEE)
- Diggle, Heagerty, Liang and Zeger, 2002)
- Gives highly efficient and valid inferences

on population-average parameters

BIO656--Multilevel Models

Marginal Models for the Cross-Over Studylog(OR)

Estimation method has an effect

BIO656--Multilevel Models

Conditional (RE) Models for the Cross-Over Studylog(OR)

BIO656--Multilevel Models

Accounting for Clusteringvia Sample Reuse

Standard GEE: “Robust” option in SAS

Jackknife

- Compute hat
- Delete aperson (in general, a “unit”)
- Compute -i i = 1, ..., n
- Compute i* = nhat - (n-1) -i
- Compute the sampe (co)variance of the i*

Bootstrap

- Put each person’s data on a token
- Sample “n” tokens with replacement and compute estimates from the sample
- Do this “Nboot” times and compute sample (co)variance of the estimates
- Can get more sophisticated CIs, via BCa

BIO656--Multilevel Models

British Social Attitudes Survey: Conditional and Marginal MLMsNote:Subscript order reversed from our usual

Response

- Yijk = 1 if favor abortion; 0 if not
- district i = 1,…264
- person j = 1,…,1056
- year k = 1, 2, 3, 4

Levels

- Time within person
- Persons within districts
- Districts

BIO656--Multilevel Models

Covariates at the three levels

Level 1: time

- Indicators of time

Level 2: person

- Class: upper working; lower working
- Gender
- Religion: protestant, catholic, other

Level 3: district

- Percentage protestant (derived)

BIO656--Multilevel Models

Scientific Questions

Conditional Model

- How does a woman’s religion associate with her probability of favoring abortion?
- How does the predominant religion in a district associate with a woman’s probability of favoring abortion?

Marginal Model

- How does the rate of favoring abortion differ between Protestants and, otherwise similar, Catholics?
- How does the rate of favoring abortion differ between districts that are predominantly Protestant versus Catholic?

BIO656--Multilevel Models

Schematic of Marginal Random-effects Model

BIO656--Multilevel Models

Conditional Multi-level Model

Modeling the Population Expectation

We build a “regression model” for 2

Person and district random effects

BIO656--Multilevel Models

Conditional Multi-level Model Results

All of this is a “regression model” for 2

BIO656--Multilevel Models

Conditional model results

- How does a woman’s religion associate with her probability of favoring abortion?
- How does the predominant religion in a district associate with a woman’s probability of favoring abortion?

BIO656--Multilevel Models

Marginal Multi-level Model

If the conditional is logistic, can the marginal be logistic?

We simultaneously model the underlying random effects

structure, but we are still fitting the marginal model

Person and district random effects

BIO656--Multilevel Models

Marginal Multi-level Model Results

All of this is a “regression model” for 2

BIO656--Multilevel Models

Marginal model results

How does the rate of favoring abortion differ between

protestants and otherwise similar catholics?

How does the predominant religion in a district

influence the probability of favoring abortion?

BIO656--Multilevel Models

Refresher: Forests & Trees

Multi-Level Models:

- Explanatory variables from multiple levels
- Family
- Neighborhood
- State
- Interactions

Must take account of correlation among responses from same clusters:

- Marginal: GEE, MMM
- Conditional: RE, GLMM

BIO656--Multilevel Models

Key Points

“Multi-level” Models:

- Have covariates from many levels and their interactions
- Acknowledge correlation among observations from within a level (cluster)

Conditional and Marginal Multi-level models have different targets; ask different questions

- When population-averaged parameters are the focus, use
- GEE
- Marginal Multi-level Models

(Heagerty and Zeger, 2000)

BIO656--Multilevel Models

Key Points (continued)

- When cluster-specific parameters are the focus, use random effects models that condition on unobserved latent variables that are assumed to be the source of correlation
- Warning: Model Carefully. Cluster-specific targets often involve extrapolations where there are no actual data for support
- e.g. % protestant in neighborhood given a random neighborhood effect

BIO656--Multilevel Models

Recap

Population-averaged parameters

- GEE
- Marginal multi-level models

Cluster-specific parameters and latent effects

- Random Effects models
- built up from latent effects (variance components)
- Possibly, overlay “Time Series” Models
- to induce additional correlation

Warning

- Inferences on latent effects can be very model-dependent

BIO656--Multilevel Models

Working Independence versus modeling correlationLongitudinal Example

Generate data in clusters (i.e., a person)

- 5 observations per cluster

Response is a linear function of time

Yit = 0 + 1t + eit

The residuals are first-order autoregressive, AR(1)

eit =ei(t-1) + uit(the u’s are independent)

corr(ei(t+s) , eit) = s

Estimate the slope by

- OLS: assumes independent residuals
- Maximum likelihood: models the autocorrelation

BIO656--Multilevel Models

Comparisons

Compare the following reported Var(1)

- That reported by OLS (it’s incorrect)
- That reported by a robustly estimated SE for the OLS slope (It’s correct for the OLS slope)
- That reported by the MLE model
- It’s correct if the MLE model is correct

You can use any working correlation model,

but need a robust SE to get valid inferences

BIO656--Multilevel Models

Variance of OLS & MLE Estimates of b versus , the first-lag Correlation

MLE reported variance

OLS reported variance

True variance of OLS

BIO656--Multilevel Models

Use a model that fits the observed data well

Directly model observeds or check fit by aggregating a random effects model

“Good” models (candidate models) will give similar observed-datapredictions

Then, “speculate” on latent effects models by finding several that fit the observed data

See if these give similar messages and produce similar individual-levelpredictions

Yes a sturdy finding; No additional info needed

Note: > 0 indicates that there is unexplained,

individual-level heterogeneity

BIO656--Multilevel Models

MLMs

Models are multi-level because they

- Include covariates from many levels

(and their interactions)

- Structure correlation among

observations within a cluster

Conditional and marginal models

- Have different goals
- Ask different questions
- Can/should get different answers

BIO656--Multilevel Models

Benefits & Drawbacks of working non-independence

Benefits

- Efficient estimates
- Valid standard errors and sampling distributions
- Protection from some missing data processes
- The MLM/RE approach allows estimating conditional-level parameters, estimating latent effects and improving estimates

Drawbacks

- Working non-independence imposes more strict validity requirements on the fixed effects model (the Xs)
- Can get valid SEs via working independence with robust standard errors
- At a sacrifice in efficiency

BIO656--Multilevel Models

Working independence models

(coupled with robust SEs!!!)

are sturdy, but inefficient

Fancy models are potentially

efficient, but can be fragile

BIO656--Multilevel Models

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