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Responder endpoint and continuous endpoint, logistic regression or ANOVA?

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## Responder endpoint and continuous endpoint, logistic regression or ANOVA?

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**Presentation title**Responder endpoint and continuous endpoint, logistic regression or ANOVA? DSBS 24 OCT 2013 Søren Andersen**Presentation title**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**Presentation title**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**Presentation title**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**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 probabilitieswereestimated 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**Estimated p’s of 3 treatments across subgroups**Presentation title**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)**Presentation title**Comparing logit and probit treatment differences**Presentation title**Estimated p’s of 3 treatments across subgroups ANCOVA and probit**Presentation title**Comparison on “latent scale” of parameter estimates**Presentation title**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 = 0.2695 To obtain the same precision of estimate from probit and logit as for ANCOVA twice as many observations areneeded**Presentation title**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”**Presentation title**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)