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Rob Woodruff

Battelle Memorial Institute, Health & Analytics

Email: woodruffr@battelle.org

Cynthia Ferre

Centers for Disease Control and Prevention

Conditional Stereotype Logistic RegressionA new estimation command

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Overview

• What is it?- Stereotype Logistic Regression

- Conditional on what?• What‘s it good for?• Syntax and Examples

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Constrained Multinomial Logistic Regression

• Multinomial Model

-Categorical Outcome Variable

-Vector of Explanatory Variables

-Related through the m logits:

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Constrained Multinomial (continued)

-The stereotype model imposes the constraints:

Note: The phi’s are scalar quantities

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It’s all about the phi’s

• Full multinomial has m(p+1) parameters• Stereotype model has m-1 + m + p = 2m-1+p

• The phi parameters give a way to quantify ordinality of the outcome variable. If

Then we have evidence of ordinal effect.

• Also allow tests of distinguishability of outcome categories

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So what’s the condition?

• The multinomial and stereotype logistic regression models are implemented in Stata by mlogit and slogit

• Assume independence of observations, not true for matched case-control data

• For matched case control study, only independence of matched groups (strata, panels, clusters, etc)

• For 1:M matching, condition on stratum total for outcome variable and focus instead on conditional likelihood

Do I have to?

Why condition on this particular event?

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Conditional vs. Unconditional Likelihood

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Conditional vs. Unconditional Likelihood

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CSTEREO

cstereo commandBasic syntax:

. cstereo depvar indepvars [if] [in], group(varname) [options]

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Example with Real Data:Preterm Birth and Vitamin D

• 1:2 (some 1:1) Pooled, Matched Case-Control Study of 2,583 Mothers in 870 matched groups

• A case defined as gestational age at delivery of <37 weeks

outcome4=3 (<32 weeks), outcome4=2, (32-35 weeks), outcome4=1 (36 weeks) and outcome4=0 (control: 37+ weeks)

• Primary exposure variable of interest: Vitamin D levels, ohd25_total: blood serum concentration of (25)OHD in ng/ml

• Sample of other covariates measured:

edu = 0/1 indicator of post-high school education

vitamin = 0/1 indicator of vitamin use during pregnancy

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Example Continued (nolog option):

P-value: .03138281

Chi2 value on 4 degrees of freedom: 10.604861

Log-Likelihood from Conditional Multinomial Model: -835.83679

all negative outcomes.

note: 77 groups (139 obs) dropped because of all positive or

. cstereo outcome4 ohd25_total edu vitamin, group(matchgroup) nolog

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Example Continued:

_cons .9398113 1.206139 0.78 0.436 -1.424178 3.3038

phi2

_cons .8764578 1.268331 0.69 0.490 -1.609424 3.36234

phi1

vitamin .1301369 .1954516 0.67 0.506 -.2529413 .5132151

edu -.4010391 .431587 -0.93 0.353 -1.246934 .4448559

ohd25_total -.0073684 .0144916 -0.51 0.611 -.0357714 .0210346

xb

outcome4 Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -841.13922 Prob > chi2 = 0.6048

Wald chi2(3) = 1.85

Number of obs = 2322

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Interpretation of cstereo output:

• Estimated beta coefficient of ohd25_total = -0.0074 with 95% confidence interval (-0.0358, 0.0210)

• Odds ratio of being in <32 weeks gestational age compared to control is exp(-0.0074) = 0.993 (0.965, 1.021)

• Now for odds ratios for the 32-35 weeks and 36 week case categories, we need the products of the parameters:

• For standard errors, use Delta Method via nlcom

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Interpretation continued:

_nl_1 -.0069249 .0072757 -0.95 0.341 -.021185 .0073351

outcome4 Coef. Std. Err. z P>|z| [95% Conf. Interval]

_nl_1: [xb]ohd25_total*[phi2]_cons

. nlcom [xb]ohd25_total*[phi2]_cons

Exponentiating gives the odds ratio of being in the 32-35 weeks case category compare to controls of 0.994 with a 95% C.I. of (0.983, 1.004)

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Constraints:

• Are the 36 week and 32-35 weeks case categories distinguishable?

all negative outcomes.

note: 77 groups (139 obs) dropped because of all positive or

. cstereo outcome4 ohd25_total edu vitamin, group(matchgroup) nolog constraints(1)

. constraint 1 [phi1]_cons=[phi2]_cons

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Constraint Output

_cons .9417836 1.24291 0.76 0.449 -1.494276 3.377843

phi2

_cons .9417836 1.24291 0.76 0.449 -1.494276 3.377843

phi1

vitamin .1294806 .1888154 0.69 0.493 -.2405908 .4995519

edu -.3909924 .4289348 -0.91 0.362 -1.231689 .4497043

ohd25_total -.0068382 .013172 -0.52 0.604 -.0326548 .0189784

xb

outcome4 Coef. Std. Err. z P>|z| [95% Conf. Interval]

( 1) [phi1]_cons - [phi2]_cons = 0

Log likelihood = -841.1454 Prob > chi2 = 0.6293

Wald chi2(3) = 1.73

Number of obs = 2322

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Constraint Output

• The log-likelihood from the constrained model is -841.145 compared to -841.139 for the unconstrained stereotype model

• Difference of 0.006 gives a chi2 value of 0.012 on 1 degree of freedom

• P-value = 0.91

• Unconstrained stereotype model does not fit significantly better than the constrained and the two case categories are indistinguishable

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Relationship to Other Models for Ordered/Categorical Outcomes

• Constrained Multinomial

• Not as parsimonious as the proportional odds model (ologit) but not valid in outcome dependent sampling

• Adjacent category model is (basically) a constrained stereotype model. Also valid under outcome dependent sampling

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Limitations

• Convergence Issues

• Currently only a one dimensional stereotype model

• Cannot currently force an ordering on the stereotype parameters

• Additional dependence structure

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References:• Ferre C, et al; Maternal 25-Hydroxyvitamin D Status and the Risk of Preterm Delivery: A Multi-Center Nested Case Control Study; preprint

• Mukherjee B, Liu I, Sinha S; Analysis of matched case-control data with multiple ordered disease states; Statistics in Medicine 2007

• Ahn J et. al.; Missing Exposure Date in Stereotype Regression Model; Biometrics 2011

• Andersen EB; Asymptotic Properties of Conditional Maximum-Likelihood Estimators; Journal of the Royal Statistical Society 1970

• Liang KY, Stewart WF; Polychotomous Logistic Regression Methods for Matched Case-Control Studies with Multiple Case or Control Groups; American Journal of Epidemiology 1987

• Scott AJ, Wild CJ; Fitting Regression Models to Case-Contro Data by Maximum Likelihood; Biometrika 1997

• Anderson JA; Regression and Ordered Categorical Variable; Journal of the Royal Statistical Society 1984\

• Greenland S; Alternative Models for Ordinal Logistic Regression; Statistics in Medicine 1994