Power of logistic regression with measurement error in predictor variable and varying number of obse

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Power of logistic regression with measurement
error in predictor variable and varying number of
observations per subject

• Olga Melnichouk, Salomon Minkin, Lisa J. Martin,
  Norman F. Boyd
• Princess Margaret Hospital, Ontario Cancer
  Institute
• Department of Epidemiology and Statistics

April 2009
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Outline

• What motivated our calculations? Proposed
  case-control study of serum biomarkers and risk
  of breast cancerCanadian Diet and Breast
  Cancer Prevention Study ? nested case-control
  study.
• Measurement error and its effect on estimated
  regression coefficient and power.
• Measurement error model and simulation study.
• Results and Discussion.

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Canadian Diet and Breast Cancer Prevention Study
Design
Eligible Subjects Identified
Pre-randomization
assessment
Intervention
Comparison
(n2,341)
(n2,349)
Annual Visits

Demographic data

Anthropometric data

3 day food records

non fasting serum
Follow up until Dec 2005
(7-17 years per subject)
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Canadian Diet and Breast Cancer Prevention Study

• Randomized trial of intervention with a low-fat
  high-carbohydrate diet.
• Begun in Toronto in 1988.
• Participating centers Toronto, Hamilton,
  Kitchener-Waterloo, London, Sarnia, and Windsor
  (Ontario), Vancouver and Surrey (British
  Columbia).
• In total, 4690 women with extensive mammographic
  density were recruited and randomized to an
  intervention or comparison group.
• The intervention group received intensive
  individual counseling to reduce fat intake to a
  target of 15 calories, and increase carbohydrate
  to 65 of calories.

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Canadian Diet and Breast Cancer Prevention Study
Nested case-control study

• 251 (projected 320) case subjects
• Individually matched with 2 controls selected
  from all trial subjects - who had the same or
  longer follow-up time and - who had not within
  that time developed breast cancer.
• Additional matching criteria age (within
  1 year), date of randomization (within 1
  year), center of randomization.
• In addition to epidemiologic data and food
  records, a blood sample was obtained at baseline
  and annually thereafter, ? varying number of
  blood samples per subject.

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Proposed case-control study

• Most of the published studies of serum biomarkers
  and risk of breast cancer are based on
  biomarkers measured in one blood sample.
• Variability of a single measurement (Table 1).

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Proposed case-control study

• To examine association of long-term exposure to
  serum biomarkers and risk of breast cancer?
  Measure biomarkers in multiple blood samples per
  subject.? Use their average as a surrogate of a
  long-term exposure.
• What do we gain if we measure biomarkers in
  multiple blood samples?? Simulation study

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Measurement error

• Regression analysis ? predictor of interest is
  measured exactly.
• Regression analysis with measurement error ?
  available predictor is related to predictor of
  interest but with additional variability.
• Two issues 1) attenuated estimate of the
  regression coefficient (in the model with a
  single predictor)2) power loss.

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Measurement error

• To correct for measurement errora) conduct a
  validation study regress gold standard measure
  (no measurement error) on exposure measured with
  errorb) obtain multiple measurements per
  subject of the error-prone exposure (reliability
  study).
• In both cases, we use predicted values as a
  surrogate of the true exposure.
• External or internal reliability study ? naive
  estimate of the regression coefficient can be
  corrected.
• If a reliability study is part of the main study
  (internal study) ? we have more information about
  the true exposure ? should power increase?

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Measurement error model

• errors are non-differential

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Number of blood samples per subject

Figure 1. (A) Distribution and (B) cumulative
distribution of the number of available blood
samples per subject for the projected 311 case
subjects.
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Measurement error model reliability
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Measurement error model
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Measurement error model
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Measurement error model
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Measurement error model
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Simulation study plan
Continued on next page
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Simulation study plan
In each repeat, significance testing H0
ßZ0, H0 ßZ_bar0, H0 ßZ_bar_t0, H0 ßX0.
To account for extra variability due to
estimation of R, correct SE of ßZ_bar_t
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Power probability that the null hypothesis is
rejected. Empirical power proportion of
simulated data sets in which a regression
coefficient was significantly different from zero
at 5 level of significance.
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Estimate of the regression coefficient average
of the estimated regression coefficients in 1000
repeats.
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Table 2. Number of measured blood samples.
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Simulation study parameters
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Results power
true exposure measured without error ?solid
line transformed average ?long dashed
line average ?dotted line one observation
measured with error ?short dashed line
           
     


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Results regression coefficient
true exposure measured without error ?solid
line transformed average ?long dashed
line average ?dotted line one observation
measured with error ?short dashed line
           
     





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Results biomarkers
April 7, 2009
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Discussion

• Measuring serum biomarkers in repeated blood
  samples
• reduces bias? valid estimates of the effect of
  these biomarkers on breast cancer risk can be
  obtained
• increases power? study with relatively moderate
  sample size will detect important
  associations? improved precision of the effect
  estimates.

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References

• BG Armstrong, AS Whittemore, and GR Howe (1989)
  Analysis of case-control data with covariate
  measurement error application to diet and colon
  cancer. Stat Med 81151-1163.
• AS Whittemore (1989) Errors-in-variables
  regression using Stein estimates. Am Statistician
  43226-228.
• D Thomas, D Stram, J Dwyer (1993) Exposure
  measurement error influence on exposure-disease
  relationships and methods of correction. Annu Rev
  Public Health 1469-93.
• MY Kim and A Zeleniuch-Jacquotte (1997)
  Correcting for measurement error in the analysis
  of case-control data with repeated measurements
  of exposure. Am J Epidemiol 145 1003-1010.

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Thank you!

• Questions?