Third Variable Effects

1
Third Variable Effects

• Confounding and Effect Modification
• --------------------------------------------------
  -----------
• Describing and interpreting relationships when
  two measured variables affect the risk of disease

2
Lecture Organization

• Philosophy of Data Analysis (Very Brief)
• Two Dichotomous Predictors Relating to
  Dichotomous Outcome
• Philosophy of Multivariate Data Analysis for
  Technological Inference and for Scientific
  Inference

3
Philosophy of Data Analysis

• Be clear about your objectives before your
  analysis. Do you want to
• 1 Make inferences about the shape of variable
  relationships in your statistical target
  population.
• 2 Make inferences about which causal models or
  causal effects.
• Realize that for 2 you will always be using
  methods where the reality you are working with is
  inconsistent with the assumptions required by the
  methods.

4
Philosophy of Data Analysis

• Because reality is always inconsistent with
  assumptions
• There are no cookbook approaches to insuring that
  you come up with the right answer.
• Some feeling for how inconsistencies will affect
  inferences will improve your art of data
  analysis.
• The more you understand about the mathematics
  behind the causal processes you explore and the
  mathematics behind the methods you use, the
  better you can judge the quality of your
  inferences

5
Designation of Exposure Variables

• Var A, Var B,
• A exposed to A, B exposed to B
• A- unexposed to A, B- unexposed to B
• The designation of which is the variable of
  interest and which is a potential confounding or
  effect modifying variable is arbitrary.

6
Symbolic Representation of Risk Measures

• Always make refer to the high risk category
• Undefined for crossover relationships
• R(A,B-) Risk of disease in individuals exposed
  to A but not to B
• R(A) Risk in individuals exposed to A
  regardless of their exposure status to B
• RR(AB) Risk or Rate Ratio to A (in that
  subset of individuals who are exposed to) B
  (given B)
• RD(BA-) Risk Diffierence to B given A-

7
Confounding

• A non-causal effect of a third variable on the
  relationship between the variable of interest and
  the disease.
• Relationship may be expressed as a risk ratio or
  a risk difference
• Positive confounding Risk ratio or risk
  difference is increased from the true value by
  the third variable
• Negative confounding Risk difference or Risk
  Ratio are brought closer to the null value.

8
Confounding

• Positive confounding results when
• The relationship between the variable of interest
  and the third variable is positive (A is more
  frequent in the presence of B or vice versa) and
  the third variable has a positive relationship
  with disease
• The relationship between the variable of interest
  and the third variable is negative (A is less
  frequent in the presence of B or vice versa) and
  the third variable has a negative relationship
  with disease
• Negative confounding has unequal signs (-)

9
Positive Confounding
10
Mutual Positive Confounding
11
The sign of a confounded relationship is the
product of the signs of the relationships
generating the confounding
12
Crossover Joint Effects Make Confounding Hard to
Evaluate
13
Stratification describes effect modification and
controls confounding
14
Confounding Effect Modification or Joint Effects

• Whether a third variable modifies the effect of a
  primary variable of interest does not determine
  whether there will be confounding.
• 3rd Var may be both confounding modifying,
  confounding but not modifying, or just modifying
• Cross over effect modification (a variable has
  or - effects depending on level of 3rd variable),
  makes it hard to determine the presence of
  confounding
• Stratification in that case is needed to describe
  effect modification. This controls confounding.

15
Implications of Effect Modification for
Confounding

• The presence of effect modification by a third
  variable means that the third variable has an
  effect and therefore could be a confounder.
• The presence of effect modification means that
  the first task is to describe that modification.
• Describing effects at stratified levels of a
  third variable also controls for confounding
  because within a strata there is no variation
  which can be associated with the exposure
  variable of interest.

16
CLASS EXERCISE

• Predict the direction of Bs confounding of As
  association with disease
• Var A is three times more frequent in the
  presence of Var B than in its absence, the
  RR(BA-) 2, RR(BA) 4
• Var A is more frequent in absence of B than in
  its presence, RR(BA-) 0.2, RR(BA) 0.4
• Var A is three times more frequent in the
  presence of Var B than in its absence, the
  RR(BA-) 0.2, RR(BA) 0.4

17
Additive Reference Point for Describing Joint
Effects
18
Additive Reference Point for Describing Joint
Effects
R(A,B) - R(A-,B-) R(A,B-) - R(A-,B-)
R(A-,B) - R(A-,B-)
19
Multiplicative Reference Point for Describing
Joint Effects
20
Multiplicative Reference Point for Describing
Joint Effects
R(A,B) R(A,B-) R(A-,B) ------------
------------ ------------R(A-,B-)
R(A-,B-) R(A-,B-)
21
No Additional Effect Reference Point for Joint
Effects
22
Reference Points for Describing Joint Effects
23
Describing Joint Effects
24
Three Different Modification Terminologies

• One reference point
• Effect Modification (not really effect but
  association)
• Two reference points
• Association modification and effect modification
• Multiple (usually 3) reference points
• Joint Effects

25
One Reference PointEffect Modification

• Most common but inadequate
• Risk or Odds ratio is the outcome used to assess
  modification
• Modification is deviation from multiplicativity
• Positive modification is when the presence of the
  high risk category of a third variable increases
  the Risk or Odds Ratio.
• Negative modification is when the presence of the
  high risk category of a third variable decreases
  the risk or odds ratio. This is very common
  because most joint effects are less than
  multiplicative.

26
Two Reference Points Association Effect
Modification

• Effect measured by RD, Association by RR
• Less common and a bit better
• Association modification deviation from
  multiplicativity
• Positive association modification is when
• RR(AB-)ltRR(AB)
• Negative association modification is when
• RR(AB-)gtRR(AB)
• Effect modification deviation from additivity

27
Multiple Reference Points Joint Effects

• Even less common but best Joint effects
  described in relation to multiplicativity,
  additivity, and no additional effects.
• Better because it is more descriptive.
• It uses more points of reference
• Better because it recognizes that two predictors
  of disease will always have some joint effects
  and the job is to describe those joint effects.
• The other approaches require a decision as to
  when deviation from one particular point is
  significant.

28
Describing Effect Modification Vs Describing
Joint Effects

• Effect Modification describes deviation from only
  one point on the joint effects scale
• Most authors make that the point of
  multiplicativity
• Some find the point of additivity more meaningful
• Joint effects are described as
• Greater than multiplicative
• Multiplicative
• Greater than additive but less than
  multiplicative
• Additive
• Less than additive
• Crossover

29
Analysis of Confounding Joint Effects (Effect
Modification)

• Describe Joint Effects
• Relevant for any and all three variable
  relationships where two variables are predictors
  of the outcome
• Control for confounding
• Needed only when the conditions for confounding
  are found
• Third variable is associated with exposure of
  interest in the population of reference
• Third variable is a predictor of disease in the
  population of reference
• Third variable is not causally intervening

30
How to Describe Joint Effects Given Strata
Specific Risks

• 1 Identify the high risk categories
• 2 Stratify
• 3 Compare absolute risks, RRs, and RDs
• a If R(A,B) lt MaxR(A,B-),R(A-,B) joint
  effects are crossover. You are done. If not,
• b If R(A,B) lt R(A,B-) R(A-,B) - R(A-,B-)
  joint effects are less than additive. If not
• c If R(A,B) lt R(A,B-)R(A-,B)/R(A-,B-) joint
  effects are lt additive and gt multiplicative
• d Otherwise gt multiplicative

31
CLASS EXERCIZE From which studies can one get
absolute risks?

• Cohorts with stratified exposures are followed
  prospectively
• Cases and controls are selected from a population
  that has been followed prospectively
• Cases and controls are selected from clinic
  patients
• Retrospective histories are taken from a cross
  section of the population

32
CLASS EXERCISE Describe the joint effects of A
B from prospective cohort data
33
CLASS EXERCISE Describe the joint effects of A
B from prospective cohort data
34
CLASS EXERCISE Determine if there is effect
modification (1 ref) its direction
35
CLASS EXERCISE Determine if the risk differences
or ratios change across strata
36
CLASS EXERCISE Determine if the conditions for
confounding exist
37
CLASS EXERCISE Calculate Crude RR and RD for A
and B and compare to the stratified
38
Stratifying on A or on B Shows JEff Relationship
to Additivity

• RD(AB) RD(AB-) RD(BA-)
• R(A,B) - R(A-,B-)
• R(A,B-) - R(A-,B-) R(A-,B) - R(A-,B-)
• R(A,B) - R(A,B-) R(A-,B) - R(A-,B-)
• RD(BA) RD(BA-)
• R(A,B) - R(A-,B) R(A,B-) - R(A-,B-)
• RD(AB) RD(AB-)

39
Stratifying on A or B shows Multiplicativity
Equally

• RR(AB) RR(AB-) RR(BA-)
• R(A,B) ? R(A-,B-)
• R(A,B-) ? R(A-,B-) R(A-,B) ? R(A-,B-)
• R(A,B) ? R(A,B-) R(A-,B) ? R(A-,B-)
• RR(BA) RR(BA-)
• R(A,B) ? R(A-,B) R(A,B-) ? R(A-,B-)
• RR(AB) RR(AB-)

40
The Point of No Additional Effects is Not
Symmetrical

• The RD to B may crossover from to - going from
  the absence to the presence of A while the RD to
  A may not crossover (or vice versa)
• R(A,B) 0.03
• R(A,B-) 0.04
• R(A-,B) 0.02
• R(A-,B-) 0.01
• RD(BA-) is but RD(BA) is -.
• RD(AB-) is RD(AB) is .

41
CLASS EXERCIZE

• How will increased or decreased association
  between A and B affect
• 1 Confounding?
• 2 Joint effects of A and B?

42
How association between predictor variables
alters confounding and joint effects

• Confounding is dependent upon this association
  but joint effects are not affected by it
• The following slides compare crude and stratified
  RDs when A B are negatively associated and
  when they are positively associated.

43
Negative Association Between A B (OR 1/16)
44
Positive Association Between A B (OR 1/16)
45
CLASS EXERCIZE

• Suppose you have the crude relationships between
  A and B (a 2 by 2 table), the crude relationships
  between A and D (a 2 by 2 table), and the crude
  relationships between B and D (a 2 by 2 table),
  from these data can you
• 1. Describe joint effects
• 2. Control for confounding

46
Crude effects of A B and their association
47
Multiple stratifications are consistent with the
marginal tables
48
Multiple stratifications are consistent with the
marginal tables
49
How to Describe Joint Effects Given Case-Control
Data

• Use ORs to approximate RRs and compare ORs
  across strata of third variable
• Increased OR in the high risk third variable
  category means gt multiplicative joint effects,
  decreased means lt multiplicative
• If lt multiplicative, examine the ratio of risk
  differences to see relationships to additivity
• OR of 1 sets no additional effect point (again
  must examine from both points of view)

50
The Ratio of Risk Differences From Case-Control
Data

• R(B,A-) - R(B-,A-)
• OR(BDA-)-1 _at_ RR(B for DA-)-1
  ------------------------------
• 
  R(B-,A-)
• OR(BDA)-1 R(B,A) - R(B-,A)
  R(B-,A-)
• ---------------- -------------------
  -------- -------------
• OR(BDA-)-1 R(B,A-) - R(B-,A-)
  R(B-,A)
• OR(BA)-1 R(B,A) - R(B-,A)
  RD(BA)
• OR(AB-) ---------------
  ---------------------------- --------------
• OR(BA-)-1 R(B,A-)
  - R(B-,A-) RD(BA-)

51
Class Exercize

• You do a case control study of endometrial
  carcinoma with two risk factors, A B, which you
  classify dichotomously. You observe
• Describe the joint effects of these two factors

52
Assess Joint Effects From Case Control Data

• OR(BA)-1 R(B,A) - R(B-,A)
  RD(BA)
• OR(AB-) ---------------
  ---------------------------- --------------
• OR(BA-)-1 R(B,A-)
  - R(B-,A-) RD(BA-)

53
Observation of Less Than Additive Joint Effects
From a Case Control Sample
54
Observation of Additive Joint Effects From a Case
Control Sample
55
Observation of Multiplicative Joint Effects From
a Case Control Sample
56
The goals and tradeoffs of analyzing risk factor
associations with disease

• Valid estimations of RR or RD to single factor
• Stratified more valid than crude
• Precise estimations of RR or RD to single factor
• Stratified less precise than crude
• Describe Joint Effects
• Needs stratified

57
Why do we want to describe joint effects?

• To indicate which statistical models are most
  appropriate to control confounding
• Logistic regression assumes multiplicative joint
  effects
• Linear regression assumes additive joint effects
• Infer causal relationships between the two
  variables in the process of generating disease
• Determine if third variables need to be taken
  into account to generalize from one population to
  another

58
What do we mean by no interaction between
variables?

• Statistical
• No significant logistic regression interaction
  term
• ORs are the same across strata
• No significant linear regression interaction term
• RDs are the same across strata
• Causal
• One variable changes the risk of disease
  associated with the causal action of the other
• Examined statistically by seeing if the
  probabilities of escaping causal effects are
  independent

59
Causal Theory for Independent Happenings

• Epidemiology needs theory on how patterns of
  exposures in populations lead to patterns of
  disease.
• Two major lines of theory
• Outcomes of individual multivariate exposures
  assuming that outcomes are independent between
  individuals
• Theory about the dependence of outcomes on
  multivariable relationships based on an
  understanding of pathophysiology
• Transmission Theory
• Theory about dependence of exposure outcome
  between individuals generated by infection
  transmission
• Similar theory is needed in social epidemiology

60
An Important Caveat About Causal Theory in This
Lecture

• All theory of joint effects discussed here
  assumes that the outcome of exposure in one
  individual does not influence the outcome of
  exposure in another individual.
• This is not the case when the population
  processes causing disease involve interactions
  between individuals.
• Infectious Diseases
• Social Factors

61
Simple Independent Action Action in Independent
Causal Pathways

• The probabilities of escaping the action of both
  variables and of background factors multiply
• R(A,B) 1 - 1 - RD(AB-)/(1-R(A-,B-)
• 1 - RD(BA -)/(1-R(A-,B-)1 - R(A-,B-)
• Note that to get the effect of the risk factors
  on the total population, one must divide by the
  probability of escaping background factors

62
Simple Independent Action

• The Simple Independent Action Forumula can be
  recast as
• R(A,B) R(A,B-) R(A-,B) - R(A-,B-)
• RD(AB-)RD(BA-)
• -
  --------------------------------------------------
  -----------------
• 1 - R(A-,B-)

63
Simple Independent Action

• When disease risks are small, for all practical
  purposes this is the same as additive risks.
• The additive scale defines causal independence
• No interaction on the multiplicative scale may be
  consistent with causal interaction
• Negative interaction on a multiplicative scale
  may be consistent with no causal interaction
• When risks are high, this is a fourth point on
  which joint effects should be described.

64
A Four Point Joint Effects Scale
65
Causal Interpretation of Multiplicative Effects

• Various models can generate multiplicative
  relationships under special conditions
• The model which always generates multiplicative
  relationships is where the two risk factors have
  distinct causal actions but those actions act in
  the same pathway leading to disease.
• Assuming that there are no dependencies between
  outcomes as occurs with infectious diseases

66
Two Causal Determinations

• Do two different factors have the same causal
  action?
• e.g. do both factors increase estrogen exposure?
• If actions are different, do they act in the same
  pathogenetic process?
• If they act in the same process, does one have to
  act before the other (e.g. initiator, promotor)
  or does it make no difference (e.g. accumulate
  critical number of genomic alterations)

67
Three Causal Models From Two Decisions

• Same causal action May run from no additional
  effect to greater than multiplicative
• Seeing change on the joint effects scale as one
  changes cutpoints for exposure measurement
  supports this causal model
• Complementary Causal Action Multiplicative if
  there is only one pathway, between additive and
  multiplicative if there are several pathways
• Simple Independent Action Action in different
  causal pathways. e.g. two different agents

68
Causal Interpretation of Crossover Joint Effects

• The same risk factor has both causative and
  protective causal pathways
• This is the only way one can get crossover
• Only one factor may have such dual pathways in
  which case crossover only occurs from one point
  of view.

69
Assessing Variance of Joint Effect Descriptions

• Test for the significance of interaction terms in
  
• a multiplicative model (e.g. logistic
  regression),
• an additive model (e.g. linear regression),
• a no additional effect model (Test for any effect
  of second variable in high risk group of the
  first variable. Then switch variables)
• Confidence intervals around statistical
  interaction terms are useful only if the
  statistical model used is causally meaningful!

70
Insights about assessing variance on joint effects

• Truly additive or multiplicative relationships
  might generate data that deviate from these
  relationships on the basis of chance
• The observed risk R(A,B) might deviate from
  predicted on the basis of chance.
• The predicted R(A,B) might deviate from the
  true value because R(A,B-), R(A-,B), or
• R(A-,B-) differ from their true values on the
  basis of chance.

71
Variance in the predicted risk in the jointly
exposed

• R(A-,B-) 0.002 to 0.004
• R(A,B-) 0.006 to 0.009
• R(A-,B) 0.005 to 0.007
• Predicted Multiplicative R(A,B) could go from
  .006.005/.004 .0075 to .009.007/.002
  .0315
• The covariance of all three predictors and the
  observed narrows this range but this gives a
  quick, crude approach. If the CI of the observed
  R(A,B) fall outside of the range predicted for
  R(A,B), the data are inconsistent with the model

72
Comparing Additive and Multiplicative Predictions

• R(A-,B-) 0.002 to 0.004
• R(A,B-) 0.006 to 0.009
• R(A-,B) 0.005 to 0.007
• Predicted Multiplicative R(A,B) could go from
  .006.005/.004 .0075 to
• .009.007/.002 .0315
• Predicted Additive R(A,B) could go from
• .006 .005 - .004 .007 to
• .009 .007 - .002 0.014

73
Power to Describe Joint Effects

• If individual causal effects are weak and data is
  not voluminous, the no additional effects model,
  the additive model, and the multiplicative model
  may all be consistent with the data.
• Because deviation from an additive or
  multiplicative model is not statistically
  significant does not mean that the deviation will
  not cause severe distortion of risk estimates

74
Philosophy of Multivariate Data Analysis

• When independent variables are continuous, their
  joint effects can be changed by transforming
  their scale
• Take the log to go from multiplicative to
  additive
• Exponentiate for the reverse
• Causally meaningful scale is needed to interpret
  joint effects models of continuous variables.
• Dichotomize continuous variables to assess joint
  effects models

75
Philosophy of Multivariate Data Analysis

• Always check joint effect values to see if they
  are consistent with analytic model assumptions
• Statistically insignificant interaction terms do
  not provide an adequate basis to conclude that
  joint effects are consistent with your analytical
  model assumptions.
• Use methods that have the least dependence upon
  multivariate model form assumptions
• Rubins Propensity Score Stratification
• Robins inverse weighted confounder score

76
Why Describing Joint Effects is Often Not Done

• Epidemiologists do not have solid traditions in
  the development of multivariate theory to explain
  how patterns of exposures generate patterns of
  diseases in populations.
• Both Epidemiologists and Biostatisticians seem
  unaware of how sensitive conclusions about
  individual effects can be to assumptions of model
  form.
• Power is often lacking.

77
Why we need to assess joint effects more often

• Choosing the most effective disease control
  strategies hinges on understanding joint effects
• True for targetting different populations
• True for allocating resources to risk factor
  control
• To stimulate theory development regarding how
  patterns of exposure generate patterns of disease
• Without such theory epidemiology is mere risk
  factor collection and classification