Responder endpoint and continuous endpoint, logistic regression or ANOVA?

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


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


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)


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


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)



Comparing logit and probit treatment differences

Estimated p’s of 3 treatments across subgroups ANCOVA and probit


Comparison on “latent scale” of parameter estimates



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”



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)

Responder endpoint and continuous endpoint, logistic regression or ANOVA?

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