Linear Models III Thursday May 31, 10:15-12:00

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Linear Models IIIThursday May 31, 1015-1200

• Deborah Rosenberg, PhD
• Research Associate Professor
• Division of Epidemiology and Biostatistics
• University of IL School of Public Health
• Training Course in MCH Epidemiology

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Ordinal and Nominal Outcomes

• Outcomes with More than 2 Categories
• Examples of Outcomes which might be suited for
  ordinal or nominal regression
• Ordinal or Nominal bmi categories
• Nominal cause of death categories
• Ordinal or nominal severity of illness categories
• Ordinal or nominal categories of program
  participation

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• The primary motivation for using a logistic model
  with an ordinal outcome is to accommodate a truly
  ordinal variable that has a "ceiling" and "floor"
  effect and one in which the intervals between
  each response category can be somewhat arbitrary
  that is, it is not a continuous variable.
• Modeling an ordinal outcome as a continuous
  variable can yield biased results because it will
  yield predicted values outside the range of the
  ordinal variable.

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• An ordered outcome may reflect an underlying
  continuous variable for which we have no data or
  for which we don't know the "real" threshold
  values.
• For example, a Likert scale for satisfactionvery
  dissatisfied to very satisfiedor for
  agreementstrongly disagree to strongly agreehas
  response categories reflecting a continuous scale
  for which there is no data.

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Modeling Ordinal Outcomes

• Some other ordinal variables that may reflect an
  underlying continuous construct that cannot be
  measured as such. The ordered values are
  intended to reflect distinct threshold values.
• Examples of ordinal variables of this type
• access to care index
• reports of experience of life stress
• assessment of overall health status
• satisfaction with care

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• To appropriately model an outcome as ordinal, the
  proportional odds assumption must hold.
• The proportional odds assumption
• if an independent variable increases (or
  decreases) the odds of being in category 1 v. the
  remaining categories, then it also similarly
  increases (or decreases) the odds of being in
  category 2 and 1 combined v. the remaining
  categories, in categories 3, 2, and 1 combined v.
  the remaining categories, etc.

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• The null hypothesis for the proportional odds
  assumption is that the odds ratios for the
  association between a risk factor and an ordinal
  outcome are constant regardless of how the
  category boundaries are drawn.
• If the proportional odds assumption holds, then
  the association between an independent variable
  and the outcome can be expressed as a single
  summary estimatea common odds ratioacross all
  categories.

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• The proportional odds assumption can be tested
  with a chi-square statistic a score test. A
  nonsignificant result means that the null
  hypothesis will not be rejected and that the
  cumulative logit model is appropriate a
  significant result means that the proportional
  odds assumption may not hold.

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• For an ordered outcome with k categories
• Both the numerator and denominator change
• http//www.indiana.edu/7Estatmath/stat/all/cat/2b
  1.html

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Ordinal and Nominal Outcomes

• Odds Among the exposed a / bcd
• Odds Among the exposed ab / cd
• Odds Among the exposed abc / d

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• Given k categories of an ordered outcome
  variable, a cumulative logit model yields k-1
  intercept terms. Each intercept corresponds to a
  category combined with all adjacent lower-ordered
  categories.
• Since proportional odds are assumed, and
  therefore a common odds ratio, the effect of each
  covariate is reflected in a single beta
  coefficient.

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• Suppose an outcome variable has 4 categories and
  we are modeling one independent variable. The
  cumulative logit model will look as follows
• ln(Odds) b0,1 b0,12 b0,123 b1
• The odds ratio is the same regardless of
  category

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Ordinal and Nominal Outcomes

• A stratified approach to mimic a cumulative logit
  model for a 4 category variable, would mean
  creating new dichotomous variables something like
  the following
• if ordvar 1 then ordvar1 1
• else if ordvar . then ordvar1 0
• if 1ltordvarlt2 then ordvar2 1
• else if ordvar . then ordvar2 0
• if 1ltordvarlt3 then ordvar3 1
• else if ordvar . then ordvar3 0

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Ordinal and Nominal Outcomes

• Mimicking Cumulative Logit with Binary Logistic
  Models
• proc logistic The OR from each model
• model ordvar1 factors will be approx. the
  same if
• run the proportional odds
• proc logistic assumption holds.
• model ordvar2 factors
• run
• proc logistic Note that all observations
• model ordvar3 factors are used in each
  model.
• run

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Ordinal and Nominal Outcomes

• The Cumulative Logit Model
• If the proportional odds assumption does not
  hold, it might be because the outcome variable is
  nominal rather than ordinal, or it might be that
  we have mis-specified the categories, failing to
  pinpoint important thresholds on the underlying
  continuum.
• The score test is quite sensitiveit is up to the
  analyst to examine the pattern of ORs for
  different dichotomous cutpoints and decide
  whether it is reasonable to use a cumulative
  logit model.

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Ordinal and Nominal Outcomes

• The Generalized Logit Model
• In contrast to the cumulative logit model, in a
  generalized logit model, the outcome categories
  are like dummy variablesmutually exclusive
  categories compared to a common reference group.

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Ordinal and Nominal Outcomes

• The Generalized Logit Model
• For a nominal outcome with k categories
• Fixed denominator (reference category)
• http//www.indiana.edu/7Estatmath/stat/all/cat/2b
  1.html

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Ordinal and Nominal Outcomes

• Odds Among the exposed a / d
• Odds Among the exposed b / d
• Odds Among the exposed c / d

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Ordinal and Nominal Outcomes

• The Generalized Logit Model
• Given k categories of an outcome variable, a
  generalized logit model yields k-1 intercept
  terms. Each intercept corresponds to a single
  category.
• Since proportional odds are not assumed, odds
  ratios can vary across categories, and therefore
  the effect of each covariate is reflected in k-1
  slope parameters.

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Ordinal and Nominal Outcomes

• The Generalized Logit Model
• Suppose an outcome variable has 4 categories and
  we are modeling one independent variable. The
  generalized logit model is as follows
• ln(Odds) b0,1 b0,2 b0,3 b1,1 b1,2 b1,3
• 1. The odds ratios are
• distinct for each category
• 2. 3.

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Ordinal and Nominal Outcomes

• The Generalized Logit Model
• Each slope parameter tests the odds of being in
  one outcome category compared to the odds of
  being in the reference category
• Compared to those without Factor A, individuals
  with factor A have ___ times the odds of having
  the outcomecategory 1
• Compared to those without Factor A, individuals
  with factor A have ___ times the odds of having
  the outcomecategory 2
• Compared to those without Factor A, individuals
  with factor A have ___ times the odds of having
  the outcomecategory 3

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Ordinal and Nominal Outcomes

• A stratified approach to mimic generalized logit
  model for a 4 category variable, would not
  require creation of new variables, but would mean
  running models like the following

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Ordinal and Nominal Outcomes

• proc logistic Mimicking Generalized Logit
• where ordvar in(1,4) with Binary
  Logistic Models
• model ordvar factors
• run
• proc logistic The ORs from the
• where ordvar in(2,4) models will differ.
• model ordvar factors
• run
• proc logistic Note that different
• where ordvar in(3,4) subsets of
  observations
• model ordvar factors are used in each
  model.
• run

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Example 1.

• The Association of Smoking and Fetal/Infant Death
• in Preterm Deliveries
• Crude OR1.07

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Example 1.

• The Association of Smoking and Fetal/Infant Death
  in Preterm Deliveries
• Crude Logistic Model with Dichotomous Outcome

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Example 1.

• Cumulative Logit Odds of type of death among
  smokers
• and the OR for smoker v. nonsmoker
• Odds46 / (331135)0.04 Odds(4633) /
  11350.07
• OR 1.04 OR 1.07

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Example 1.

• Cumulative Logit Model with 3 Categories
• Ordered Value outcome5
  Frequency
• 1 fetal death gt20 wks
  332
• 2 neonatal death 0-28 days
  229
• 3 survivor gt28 days
  8520
• Probabilities modeled are cumulated over the
  lower Ordered Values.
• Score Test for the Proportional Odds Assumption
• Chi-Square DF Pr gt ChiSq The
  proportional
• 0.0400 1 0.8414 odds
  assumption
• holds

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Example 1.

• Cumulative Logit Each intercept corresponds to a
  category plus all categories with lower ordered
  values v. the remaining categories.
• The odds ratio is an average of the cumulative
  logits
• 46 / (331135) e-3.28030.0635 0.04
• (4633) / 1135 e-2.72910.0635 0.07

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Example 1.

• Generalized Logit Model with 3 Categories
• In a generalized logit model, each intercept and
  slope correspond to a single category.
• Is 1.07 a reasonable summary of 1.047 and 1.096?

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Example 2.

• The Association of Maternal Risk and Fetal/Infant
  Death in Preterm Deliveries

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Example 2.

• The Association of Maternal Risk and Fetal/Infant
  Death in Preterm Deliveries
• Crude Logistic Model with Dichotomous Outcome

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Example 2.

• Cumulative Logit Model with 3 Categories
• Ordered Value outcome5
  Frequency
• 1 fetal death gt20 wks
  418
• 2 neonatal death 0-28 days
  261
• 3 survivor gt28 days
  9549
• Probabilities modeled are cumulated over the
  lower Ordered Values.
• Score Test for the Proportional Odds Assumption
• Chi-Square DF Pr gt ChiSq The
  proportional
• 10.7077 1 0.0011 odds
  assumption
• does not hold.

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Example 2.

• Cumulative Logit Model with 3 Categories
• The odds ratio is an average of the cumulative
  logits
• e-3.17500.0473 0.04
• e-2.66290.0473 0.07

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Example 2.

• Generalized Logit Model with 3 Categories
• Is 1.048 a reasonable summary of 0.86 and 1.5?

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Example 3. LBW

• Modeling a 3 category birthweight variable
• /cumulative logit /
• proc logistic orderformatted
• model bwcat smoking late_no_pnc
• run

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Example 3. LBW
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Example 3. LBW

• /mimicking cumulative logit with binary models/
• proc logistic orderformatted
• model vlbw smoking late_no_pnc
• run
• vlbw v.
• mlbw and normal
• proc logistic orderformatted
• model lbw smoking late_no_pnc
• run
• vlbw and mlbw v.
• normal
• Both models include all observations in the sample

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Example 3. LBW

• / generalized logit /
• proc logistic orderformatted
• model bwcat(ref'normal bw') smoking
  late_no_pnc
• / linkglogit
• run

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Example 3. LBW

• vlbw v. normal and mlbw v. normal

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Example 3. LBW

• / mimicking generalized logit with binary
  models/
• proc logistic orderformatted
• where bwcat 2 or bwcat 0
• model bwcat(ref'normal bw') smoking
  late_no_pnc
• / linkglogit
• run
• proc logistic orderformatted
• where bwcat 1 or bwcat 0
• model bwcat(ref'normal bw') smoking
  late_no_pnc
• / linkglogit
• run

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Example 3. LBW

• Generalized logit approach using binary models
  with only a subset of observations in each model
• vlbw v.
• normal
• mlbw v.
• normal

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Example 3. LBW

• Generalized logit models can get complicated,
• but custom estimates can still be obtained in the
  usual way.
• proc logistic orderformatted
• where 2ltmomagelt3
• class parityrisk(ref'no hx preterm') /
  paramref
• model bwcat smoking late_no_pnc matrisk
  momage
• parityrisk smokingparityrisk /
  linkglogit
• contrast 'sm-risk, hxpreterm' smoking 1
  matrisk 1
• smokingparityrisk 1 0 / estimateexp
• contrast 'sm-risk, primips'smoking 1 matrisk 1
  
• smokingparityrisk 0 1 / estimateexp
• contrast 'sm-risk, lorisk multips' smoking 1
  matrisk 1
• smokingparityrisk 0 0 / estimateexp
• run

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Example 3. LBW

• The tests for the constructs in the model are all
  statistically significant

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Example 3. LBW

• Not all beta coefficients are statistically
  significant.

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Example 3. LBW

• Parity-specific contrasts of the joint effect of
  smoking and having some antepartum medical risk,
  adjusting for entry into prenatal care and
  maternal age.
• Should we leave the smokingparityrisk term in
  the model?

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Example 4. Prenatal Care

• Should we consider the categories ordinal or
  nominal?

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Example 4. Prenatal Care

• The Overlapping dichotomous Contrasts
• No Pnc v. Any PNC, OR 3.2 Inad/No
  v. Adeq/Adeq/Inter, OR2.7
• Inter/Inad/No v. Adeq/Adeq, OR1.8 All others
  v. Adeq, OR0.60

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Example 4. Prenatal Care

• Non-overlapping dichotomous contrasts

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Example 4. Prenatal Care

• Cumulative Logit
• The null hypothesis of
• proportional odds is rejected.
• Any association is
• obscured by averaging
• across levels of APNCU.

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Example 4. Prenatal Care

• Generalized
• Logit

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Example 4. Prenatal Care

• Women with a prior lbw delivery had more than 4
  times the odds of receiving no or inadequate
  prenatal care rather than adequate care compared
  to women with no history of lbw delivery.
• Compared to women without a history of lbw
  delivery, however, these high risk women also had
  more than twice the odds of appropriately
  receiving care beyond what is considered adequate
  for most women.

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Example 5.

• Outcome is a
• 3 level rating
• of MCH
• epidemiology
• functioning
• above average
• average
• below average

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Summary Ordinal and Nominal Outcomes

• Cumulative--Ordinal

• Generalized--Nominal

• Proportional odds assumptionassess the series of
  binary comparisons from collapsing categories
• k-1 intercepts
• 1 slope / 1 odds ratio

• No assumption of the shape of the association
• Categories compared to a reference group
• k-1 intercepts
• k-1 slopes / k-1 odds ratios

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Summary Ordinal and Nominal Outcomes

• Issues for categorizing an outcome variable are
  similar to those for defining categories for
  independent variables
• Conceptual meaning of the categories
• Statistical tests v. judgment about differences
  between categories
• Sample size and power

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Summary Ordinal and Nominal Outcomes

• Model Building
• Similar to beginning with examining dummy
  variables for an independent variable prior to
  deciding whether to use it in an ordinal form,
  sometimes it is useful to run a generalized logit
  model first, since it requires no assumption
  about the ordering of the categories, and
  empirically assess whether the variation in
  category-specific odds ratios is important or
  negligible.

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Summary Ordinal and Nominal Outcomes

• And even if the proportional odds assumption
  holds, reporting separate odds ratios for each
  categoryusing generalized logitmay be important
  in order to emphasize the similarity of the
  strength of the association across categories.
• In addition, the cumulative logit model will not
  only force the strength of association to be
  uniform, the predicted values will also be forced
  to be linear. Using generalized logit, the
  predicted odds and odds ratios will both more
  closely reflect the observed values.

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Summary Ordinal and Nominal Outcomes

• Why Not Just Always Run Stratified Models for
  Generalized Logit?
• For nominal outcomes, using a single model may be
  more efficient than using separate binary models
• With separate binary models, need to decide
  whether each model should include the same
  independent variables or whether different final,
  category-specific models make sense, each
  including only those variables which are risk or
  protective factors for a particular binary
  comparison

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Summary Ordinal and Nominal Outcomes

• Using a single multinomial model permits a
  unified profile of risk and protective factors
  across the categoriesboth significant and
  insignificant

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Summary Ordinal and Nominal Outcomes

• For a variable that is actually continuous, are
  there reasons to use a cumulative logit model
  instead of a continuous outcome model?
• For example, when would modeling ordinal
  categories of birthweight be preferable either to
  modeling birthweight continuously in grams or
  categorized into nominal groups?
• using a variable as ordinal (with fewer
  categories) as opposed to continuous will yield
  odds ratios instead of mean differences
• No assumption of normality required

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Summary Ordinal and Nominal Outcomes

• For a variable that meets the proportional odds
  assumption, is it still appropriate to choose to
  use a generalized logit approach?
• using ordinal as opposed to nominal categories
  will be more efficient if there is truly an
  ordinal effect
• Why "waste" degrees of freedom on multiple odds
  ratios, if the effect is constant across
  categories?

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Which Modeling Approach?

• Choosing the form of the outcome variable
• Stressful Life Events
• Any stressful life event (y/n) independent vars
• (dichotomous)
• Fin. Emot. Traum. Partner independent vars
• (Nominal - No stressful life events as the
  reference)
• Sum of stressful life events independent vars
• (continuous)
• Scale of stressful life events independent vars
• (ordinal)

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Which Modeling Approach?

• Choosing the form of the outcome variable
• Maternal Depression
• Any depression (y/n) independent vars
• PrePost Pre_Only PP_Only independent vars
• (Nominal - No depression as the reference)
• Severe Moderate Mild independent vars
• (Ordinal or Nominal)
• Depression Severity Scale independent vars
• (ordinal)

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Which Modeling Approach?

• Choosing the form of the outcome variable
• Breastfeeding
• Ever Breastfed (yes v. no) independent vars
• Exclusive BFgt2 mos. (yes v. no) independent
  vars
• Exclusive gt2 mo. Exclusive BFlt2 mo.
  independent vars
• Never Breastfed as reference
• BFlt2 mo. BF 2-6 mo. BF gt 6 mo. independent vars
• Never Breastfed as reference
• Breastfeeding duration in weeks independent
  vars