Transcript ANCOVA and logistic regression of responder rates

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Responder endpoint and
continuous endpoint,
logistic regression or
ANOVA?
DSBS 24 OCT 2013
Søren Andersen
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Slide no 2
Example and problem
• HbA1c is analysed with an ANCOVA model and in
addition the ”responder rate” (HbA1c < 7%) is analysed
by a logistic regression model
• Well documented that dichotomising reduces sensitivity
• Results presented as difference in HbA1c and as odds
ratio
• Difficult to compare the results
• Difficult to interpret odds-ratio for probalities p (from
logistic regression model) in [0.2; 0.8], no
interpretation as relative risk
• Example: Old study with Liraglutide
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Outline
• Comparisons on probability scale
• Show no difference between logit and probit in estimated
responder probabilities (and in treatment differences in
responder probabilities)
• Compare responder probabilites derived from ANCOVA
with responder probabilities from logit and probit
• Comparisons on continuous scale
• Compare estimates from logit and probit to estimates
from ANCOVA
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Suggestion: use probit instead of logit
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A probit model for binary data is very similar to a logit
model. Very difficult to discriminate between the two.
Pro logit:
• a logit model is very useful for retrospective studies
(not the case here)
• a logit model is convenient for calculation of
conditional probabilities
• a logit model offers interpretation in terms of oddsratio
• Technical point: simple sufficient statistics
Pro probit:
• offers interpretation in terms of a latent normal
variable (threshold model)
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Comparions of logit and probit estimates
of probabilities
• Logit and probit model with effects of
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Country (17)
Pre-treatment (2)
Treatment (3)
Base line HbA1c
• responder probabilities were estimated for all countries
(17) and pre-treatment (2), treatments (3) and 3
values of base line HbA1c (mean +- std)
• In all 17 x 2 x 3 x 3 = 306 probabilities
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Estimated p’s of 3 treatments across subgroups
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Presentation of results from probit and
logit models
• Present differences in estimated proportions between
two treatment groups, Lira and Comparator – not
constant
• Depend on proportion in the Lira (or Comparator)
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Slide no 11
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Comparing logit and probit treatment
differences
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Estimated p’s of 3 treatments across
subgroups ANCOVA and probit
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Comparison on “latent scale” of
parameter estimates
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Comparison of estimates of treatment
difference
• From ANCOVA :
0.2367 (residual s = 0.81)
• From probit:
0.3379 (”residual s = 1”)
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0.3379*0.81 = 0.2758
• From logit:
0.5440
convert to probit: 0.5440*0.607 = 0.3302
convert to ANCOVA: 0.3302*0.81 = 0.2695
To obtain the same precision of estimate from probit and
logit as for ANCOVA twice as many observations are
needed
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Slide no 21
Conclusions
• Dichotomising reduces sensitivity (in the example
sample size doubles)
• Communicate results from logit/probit as difference in
proportions if OR markedly different from RR
• Compare results from ANCOVA and logit/probit on
probability scale and on ”latent scale”
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Slide no 22
Composite responder endpoint?
• Responder: (HbA1c < 7) & (change in weight < 0), i.e.
two binary response B1 and B2 combined
• Why composite? Why collapse 3 categories of the B1 x
B2 outcome?
• For quantitative responses we test for each parameter:
H0: no difference in HbA1c, H0: no difference in
chg_bw
• Analyse B1 and B2 separately or
• Analyse the full response pattern B1 x B2, as marginal
B1, B2 conditional on B1 (or other way round)