Logistic Regression II

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Logistic Regression II
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How to calculate ORs from model with interaction
term

• Standard Wald
• Parameter DF Estimate Error
  Chi-Square Pr gt ChiSq
•  
• Intercept 1 -1.2858 0.6247
  4.2360 0.0396
• psa 1 0.0608 0.0280 11.6952
  0.0006
• race 1 0.0954 0.5421
  0.0310 0.8603
• psarace 1 -0.0349 0.0193
  3.2822 0.0700

Increased odds for every 5 mg/ml increase in
PSA If white (race0) If black (race1)
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ORs for increasing psa at different levels of
race.
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ORs for increasing psa at different levels of
race.
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OR for being black (vs. white), at different
levels of psa.
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Predictions

• The model

• Whats the predicted probability for a white man
  with psa level of 10 mg/ml?

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Predictions

• The model

• Whats the predicted probability for a black man
  with psa level of 10 mg/ml?

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Predictions

• The model

• Whats the predicted probability for a white man
  with psa level of 0 mg/ml (reference group)?

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Predictions

• The model

• Whats the predicted probability for a black man
  with psa level of 0 mg/ml?

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Diagnostics Residuals

• Whats a residual in the context of logistic
  regression?
• Residualobserved-predicted
• For logistic regression
• residual 1 predicted probability
• OR residual 0 predicted probability

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Diagnostics Residuals

• Whats the residual for a white man with psa
  level of 0 mg/ml who has capsule penetration?

• Whats the residual for a white man with psa
  level of 0 mg/ml who does not have capsule
  penetration?

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In SASrecall model with psa and gleason

• proc logistic data hrp261.psa
• model capsule (event"1") psa gleason
• output outMyOutdata lMyLowerCI
• pMypredicted uMyUpperCI resdevMyresiduals
• run
• proc gplot data MyOutdata
• plot Myresidualspredictor
• run

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Residualpsa
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Estimated probgleason
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Model-Checking Goodness of Fit

• Partition observations into groups by level of 1
  predictor (e.g. 8 groups by psa level)
• Sum the predicted probabilities for all ?n/8 in
  each of the groups. (expected)
• Count the total number of events (e.g.,
  capsule1) in each of the groups (observed)
• You have 8 observed vs. expected
• (observed-expected)2/expected has ? chi-square
  distribution with df groups(8)-number
  parameters in the model (2) 6
• Null Hypothesis model is a good fit

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Hosmer and Lemeshow Lack Fit Test

• Divide observations into (as best as possible)
  deciles from lowest to highest predicted
  probabilities
• May not be exactly even deciles because of ties
  and if total observations is not a multiple of 10
• Sum the predicted probabilities for all ?n/10 in
  each decile. (expected)
• Count the total number of events in each decile
  (observed)
• You have 10 observed vs. expected
• (observed-expected)2/expected has ? chi-square
  (df8) distribution
• Null Hypothesis model is a good fit

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In SAS

• proc logistic data hrp261.psa
• model capsule (event"1") psa race psarace
  /lackfit
• run

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Results lackfit option

• Partition for the Hosmer and Lemeshow Test
• 
  capsule 1 capsule 0
• Group Total Observed
  Expected Observed Expected
• 1 39 4
  9.80 35 29.20
• 2 38 11
  10.40 27 27.60
• 3 38 11
  11.10 27 26.90
• 4 40 13
  12.33 27 27.67
• 5 38 17
  12.46 21 25.54
• 6 38 12
  13.27 26 24.73
• 7 38 13
  14.61 25 23.39
• 8 38 22
  17.18 16 20.82
• 9 38 21
  21.84 17 16.16
• 10 32 27
  28.01 5 3.99
• Hosmer and Lemeshow
  Goodness-of-Fit Test

INDICATES GOOD FIT!
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In SAS

• proc logistic data hrp261.psa
• model capsule (event"1") psa /lackfit
• run

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Results lackfit option

• 
  capsule 1 capsule 0
• Group Total Observed
  Expected Observed Expected
• 1 40 4
  10.51 36 29.49
• 2 39 12
  11.02 27 27.98
• 3 38 9
  11.35 29 26.65
• 4 37 15
  11.65 22 25.35
• 5 40 17
  13.24 23 26.76
• 6 40 9
  14.04 31 25.96
• 7 38 13
  14.56 25 23.44
• 8 39 28
  17.44 11 21.56
• 9 38 21
  21.86 17 16.14
• 10 31 25
  27.32 6 3.68
• Hosmer and Lemeshow
  Goodness-of-Fit Test
• Chi-Square
  DF Pr gt ChiSq

INDICATES POOR FIT!
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Conditional Logistic Regression for Matched Data
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Recall Matching

• Matching can control for extraneous sources of
  variability and increase the power of a
  statistical test.
• Match M controls to each case based on potential
  confounders, such as age and gender.

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Recall Agresti example, diabetes and MI

• Match each MI case to an MI control based on age
  and gender.
• Ask about history of diabetes to find out if
  diabetes increases your risk for MI.

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odds(favors case/discordant pair)
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Conditional Logistic Regression
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The Conditional Likelihood each discordant
stratum (rather than individual) gets 1 term in
the likelihood
For each stratum, we add to the likelihood the
CONDITIONAL probability that the case got disease
and the control did not, given that we have a
case-control pair.
Note the marginal probability of disease may
differ in each age-gender stratum, but we assume
that the (multiplicative) increase in disease
risk due to exposure is constant across strata.
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Recall probability terms
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?The conditional likelihood
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Conditional Logistic Regression
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Example MI and diabetes
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Conditional Logistic Regression
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In SAS

• proc logistic data YourDatamodel MI (event
  "Yes") diabetesstrata PairIDrun

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ExamplePrenatal ultrasound examinations and risk
of childhood leukemia case-control study BMJ
2000320282-283

• Could there be an association between exposure to
  ultrasound in utero and an increased risk of
  childhood malignancies?
• Previous studies have found no association, but
  they have had poor statistical power to detect an
  association.
• Swedish researchers performed a nationwide
  population based case-control study using
  prospectively assembled data on prenatal exposure
  to ultrasound.

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ExamplePrenatal ultrasound examinations and risk
of childhood leukemia case-control study BMJ
2000320282-283

• 535 cases all children born and diagnosed as
  having myeloid leukemia between 1973 and 1989 in
  Swedish registers of birth, cancer, and causes of
  death.
• 535 matched controls 1 control was randomly
  selected for each case from the Swedish Birth
  Registry, matched by sex and year and month of
  birth.

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115
85
235
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But this type of analysis is limited to single
dichotomous exposure

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• Used conditional logistic regression to look at
  dose-response with number of ultrasounds
• Results
• Reference OR 1.0 no ultrasounds
• OR .91 for 1-2 ultrasounds
• OR.64 for gt3 ultrasounds
• Conclusion no evidence of a positive association
  between prenatal ultrasound and childhood
  leukemia even evidence of inverse association
  (which could be explained by reasons for frequent
  ultrasound)

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Extension 1M matching

• Each term in the likelihood represents a stratum
  of 1M individuals
• More complicated likelihood expression!
• Just as easy to implement in SAS as well see
  Wednesday