Ventajas y dificultades del uso del modelado y simulación

Interest lies in estimating the marginal causal effect of BV plus FOLFIRI on the risk of a TE during treatment for two subgroups: (i) those whose treatment was a second line of treatment and (ii) those whose treatment was a first line of treatment. We use the following logistic regression model to relate the treatment and subgroup variables to the response:

P (Y = 1|X, S; β) = expit(β0+ β1X + β2S + β3XS) , (5.2)

where X denotes the treatment variable in a randomized setting. See Section 5.2 for de- tails on variable selection and definitions of other variables. Since we are working with observational data with an incomplete subgroup variable, we do not assume that the miss- ingness is ignorable when we condition on treatment and the subgroup variable alone; also treatment selection is highly correlated with baseline characteristics that may be associ- ated with the response therefore confounding may be an issue in this study. To account for the non-ignorably missing data and confounding, we use the approaches introduced in Chapters 2, 3 and 4 to estimate the marginal causal effects.

We begin by using the doubly weighted estimating equation approach introduced in Chapter 2 (see equation (2.18)). We compare the doubly weighted estimating equation approach to the following possibly misspecified models: complete case analysis without weights (equation 2.25); complete case analysis with a weight for confounding, with the addition of Ri in the numerator of the weight to ensure that only complete cases are

included in the model (equation (2.11)); and complete case analysis with a weight for missingness (equation (2.15)).

In an application of the doubly weighted estimating equation method, the median (range) of the product of the double inverse probability weights is 1.99 (1.18, 7.15). See Table 5.13 for results; we see that in using the doubly weighted estimating equation ap- proach, there may be a subgroup effect, where patients receiving treatment as a second line of therapy may have a higher odds ratio of a TE (1.82, 95% CI 0.45 to 7.39) compared to those receiving treatment as a first line of therapy (1.15, 95% CI 0.50 to 2.61). We note however that the interaction effect is not statistically significant, and the causal effect

Table 5.13: Results of fitting a complete case estimating function with or without weights in an application of the methods proposed in Chapter 2 to estimate marginal causal pa- rameters in a logistic regression setting. See Section 5.2 for notation and equation (5.2) for the model of interest.

Method OR (95% CI a) OR (95% CIb)

Complete case unweighted method

Treatment effect, for S = 0 1.04 (0.48, 2.25) 1.04 (0.49, 2.64)

Treatment effect, for S = 1 1.82 (0.46, 7.26) 1.82 (0.45, 6.67)

Complete case, IPW for confounding

Treatment effect, for S = 0 1.14 (0.44, 2.98) 1.14 (0.52, 3.07)

Treatment effect, for S = 1 1.89 (0.39, 9.09) 1.89 (0.46, 6.91)

Complete case, IPW for missingness

Treatment effect, for S = 0 1.05 (0.49, 2.27) 1.05 (0.50, 2.67)

Treatment effect, for S = 1 1.75 (0.44, 6.97) 1.75 (0.43, 6.40)

Complete case, doubly weighted estimating equation c

Treatment effect, for S = 0 1.15 (0.50, 2.61) 1.15 (0.52, 3.08)

Treatment effect, for S = 1 1.82 (0.45, 7.39) 1.82 (0.44, 6.61)

Abbreviatons: OR odds ratio; CI confidence interval; IPW inverse probability weight

a _{Confidence interval derived using methods proposed in Section 2.3.2.}

b _{The 2.5}th _{and 97.5}th _{percentiles from 5000 bootstrap samples.} c_{Interaction p-value = 0.41.}

is not statistically significant in either subgroup. We also report the confidence intervals (CIs) obtained from bootstrapping and we find that the bootstrap CIs are comparable to those obtained using the standard error derivation in Section 2.3.2. In the estimation of bootstrap CIs, for approximately 5% of the bootstrap samples, there is not enough data to obtain an estimate of the interaction effect between the treatment and subgroup variables. We omit these bootstrap sample estimates to obtain the 2.5th and 97.5th percentiles of the estimates.

Table 5.14: Results of applying the doubly weighted EM-type algorithm introduced in Chapter 3 to the BV plus FOLFIRI data to estimate marginal causal parameters in a logistic regression setting. See Section5.2 for notation and equation (5.2) for the model of interest.

OR (95% CIa)

Doubly weighted EM-type algorithm

Treatment effect, for S = 0 (exp(β1)) 1.32 (0.51, 3.39)

Treatment effect, for S = 1 (exp(β1+ β3)) 2.66 (0.10, 68.29)

Abbreviatons: OR odds ratio; CI confidence interval; EM expectation-maximization

a _{Confidence interval calculated using the method proposed}

in Section3.4.4.

The interaction p-value is 0.70.

weights for missingness and confounding. The median (range) of the product of the inverse probability weights for confounding and missing data is 1.99 (1.18, 118.0). The effect estimates are similar in comparison to the doubly weighted estimating equation approach, although the confidence intervals are much wider. We note that there is a violation of the assumption that S and Z1 are independent (see Table 5.6), therefore the results from the

application of the weighted EM-type approach are not as useful as the results from the other marginal models.

See Table5.15for results of an application of the multiple imputation approach with an inverse probability weight for confounding. Following the imputation model recommenda- tions given in Section 4.1.2, three interaction terms are included in the imputation model for S: Y W , Y Z1 and W Z1, and the number of imputations is K = 5. The effect estimates

are very similar to those obtained in an application of the doubly weighted estimating equation method.

Table 5.15: Results of applying the weighted multiple imputation method introduced in Chapter 4 to the BV plus FOLFIRI data to estimate marginal causal parameters in a logistic regression setting. See Section5.2 for notation and equation (5.2) for the model of interest.

OR (95% CIa)

Weighted multiple imputation

Treatment effect, for S = 0 (β1) 1.16 (0.43, 3.12)

Treatment effect, for S = 1 (β1+ β3) 1.71 (0.36, 8.24)

Abbreviatons: OR odds ratio; CI confidence interval

a _{Confidence interval calculated using the method proposed}

in Section4.4.2.

The interaction p-value is 0.67.