Presentation title Date
Responder endpoint and continuous endpoint, logistic regression or
ANOVA?
DSBS 24 OCT 2013Søren Andersen
Presentation title Slide no 2Date
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
• 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 odds-ratio• Technical point: simple sufficient statistics
• Pro probit:• offers interpretation in terms of a latent normal variable
(threshold model)
Presentation title Slide no 7Date
Comparions of logit and probit estimates of probabilities
• Logit and probit model with effects of• 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
Presentation title Slide no 8Date
Estimated p’s of 3 treatments across subgroups
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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Comparing logit and probit treatment differences
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”)• 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 = 0.2695To obtain the same precision of estimate from probit and
logit as for ANCOVA twice as many observations are needed
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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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)
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