Evaluation strategies for case series: is Cox regression an alternative to the self controlled case series method for terminal events?

Abstract

In this paper, we deal with the analysis of case series. The self-controlled case series method (SCCS) was developed to analyse the temporal association between time-varying exposure and an outcome event. We apply the SCCS method to the vaccination data of the German Examination Survey for Children and Adolescents (KiGGS). We illustrate that the standard SCCS method cannot be applied to terminal events such as death. In this situation, an extension of SCCS adjusted for terminal events gives unbiased point estimators. The key question of this paper is whether the general Cox regression model for time-dependent covariates may be an alternative to the adjusted SCCS method for terminal events. In contrast to the SCCS method, Cox regression is included in most software packages (SPSS, SAS, STATA, R, …) and it is easy to use. We can show that Cox regression is applicable to test the null hypothesis. In our KiGGS example without censored data, the Cox regression and the adjusted SCCS method yield point estimates almost identical to the standard SCCS method. We have conducted several simulation studies to complete the comparison of the two methods. The Cox regression shows a tendency to underestimate the true effect with prolonged risk periods and strong effects (Relative Incidence >2). If risk of the event is strongly affected by the age, the adjusted SCCS method slightly overestimates the predefined exposure effect. Cox regression has the same efficiency as the adjusted SCCS method in the simulation.

Appendix

1.1 A.1 Application of Cox regression in the context of a case series analysis with SPSS

We used SPSS for our example. The SPSS syntax to compute the general exposure effect is

TIME PROGRAM.

compute risk_T=0.

do if (T_>=vac1 and T_<(vac1+delta)).

compute risk_T=1.

else if (T_>=vac2 and T_<(vac2+delta)).

compute risk_T=1.

else if (T_>=vac3 and T_<(vac3+delta)).

compute risk_T=1.

else if (T_>=vac4 and T_<(vac4+delta)).

compute risk_T=1.

end if.

COXREG

/death/STATUS=dummy(1)

/METHOD=ENTER risk_T

/PRINT=CI(95)

/CRITERIA=PIN(.05) POUT(.06) ITERATE(20).

The explanations of the variables are:

vac1:

through vac4 are the numbers of days from birth to vaccination 1 through 4.

risk_T :

is the risk status (risk_T=1 at risk, otherwise risk_T=0).

delta :

is the duration of the risk period.

T_:

is the internal time variable.

death :

is the number of days from birth to event.

dummy :

is a dummy variable always equal 1.

1.2 A.2 Definition of the age classes

We derived the age classes from the empirical age distribution of 689 SUD cases which were reported by German local health authorities in the framework of the German TOKEN study (www.token-studie.de) between 2005 and 2008. The 689 cases are divided into 21 age groups. Age class k has as its upper limit the k×4.76th percentile of the distribution. This ensures that each class nearly has the same number of cases. The definition the second interval (41.00–51.00] is 41.00<age _i ≤51.00. The round bracket indicates that the 41.00 is not included in the interval.

Definition of the age classes
ID	Age	ID	Age
1	[30.00–41.00]	11	(129.62–140.38]
2	(41.00–51.00]	12	(140.38–159.00]
3	(51.00–59.00]	13	(159.00–179.90]
4	(59.00–67.00]	14	(179.90–206.00]
5	(67.00–73.81]	15	(206.00–231.00]
6	(73.81–78.57]	16	(231.00–257.19]
7	(78.57–88.00]	17	(257.19–286.95]
8	(88.00–101.10]	18	(286.95–347.71]
9	(101.10–117.00]	19	(347.71–421.00]
10	(117.00–129.62]	20	(421.00–512.00]
		21	(512.00–730.00]

1.3 A.3 COX regression with time-dependent covariates; the Bias of risk estimation

The conditional-hazard function at time t reads

In the following we assume that R(t) is binary, with R(t)=1 indicating the status of risk present and R(t)=0 outside a risk period. We look at the conditional-hazard ratio (HR) of the risk status R(t)=1 versus R(t)=0 and find

That is, the conditional-hazard ratio is no longer a constant. Instead, the unconditional value exp[β] is multiplied by a function

Thus, the application of the COX model to the conditional situation T≤L will generally introduce a bias into the estimation of β or exp(β). However, it can be shown that the direction of the risk effect is the same for the unconditional and the conditional situation. That is, if the unconditional HRexp(β)>1, then the conditional-hazard ratio HR(t∣X=x,T≤L)≥1, too, and exp(β)<1 implies HR(t∣X=x,T≤L)≤1. To see this, we first assume β>0, that is, exp(β)>1. In this case

An upper limit is easily obtained as

That is

Similarly, for exp(β)<1

There are three remarks.

• 1. As a result, although the estimation of β may be biased with regard to its numerical value, it is not biased with respect to its sign. Also, since β=0, the conditional-hazard ratio HR(t∣X=x,T≤L) is 1, independently of t and X, significance testing of H ₀:β=0 is valid within the COX model.

• 2. The inequalities held independently of any influences of covariates X on the risk distributions $(\overline{R}\mid R(t)=r(t), X=x)$ after time t.

• 3. The lower limit of HR(t∣X=x,T≤L)=1 for exp[β]>1 is reached, if, for any time point t, R(t)=1 implies R(τ)=1 for all τ>t, and, also, R(t)=0 implies R(τ)=0 for all τ>t. This describes the situation, in which a case is under risk either throughout the observation time or never. Clearly, this is a situation in which β is not estimable.