Mantel-Haenszel Test Simulation

Ref:

- Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemother Rep 1966;50:719-748
- Hankey B, Myers M. Evaluating differences in survival between two groups of patients. J Chron Dis 1971;24:523-531

This program computes the power of the Mantel-Haenszel statistical test by simulating the experiment as prescribed by the user. This program extends the capabilities of options 1, 3 and 6 of the system (Power Program) by allowing the specification of multiple time points of observation (option 6) and further generalizing some model specifications(options 1 and 3). The program operation provides several different models for the control rates and postulated experimental benefit. Currently, there are six different models which can be selected by the user. In particular, note that the alpha level in this program is two-sided. Alpha levels specified in options 1,3 and 6 of the Power Program are one-sided because normal approximations are used.

Note: When comparing to other modules in the Power Program, the word 'option' is used. When referring to features of this particular application, the word 'model' is used.

Models 1,2,3 provide for the specification of control rates in up to 8 strata of participants entering the trial. Rates can be specified for 1 and/or 2 time points and are treated as binomial p's with the rate at time(2) treated as a conditional probability given one does not have the event at time(1).

For example, Models 1,2,3 in their most general form would require the specification of 16 binomial p's ( 8 strata times 2 times) for the control participants. The experimental rates are calculated according to the difference parameter identified by the particular model.

These three models differ only in how the alternative binomial p's are specified. That is, the alternative may be specified as having a constant relative risk, constant relative odds or constant difference with respect to the control. See option 6 (Power Program) for further examples of how these three 'difference' models are used. Additionally, these three models allow for specification of separate numerical values for the difference parameter at each of the two time points with the stipulation that the difference parameter fixed across each of the strata.

Model 4 provides the most flexible specification for an experiment that is to be evaluated at two time points. Model 4 requires the complete specification of each of the control p's along with each of the experimental p's. This model can be used in those situations where the experimental 'difference' does not follow the constant or fixed effect across strata as assumed in Models 1, 2,3. There is also no restriction that the direction of treatment effect be the same in all strata.

Models 5 and 6 provide for a more parametric specification of the rate associated with the 'disease' process. The main advantage of these two models is to explore the increase in power obtained when extended followup is used in the experiment. Model 5 assumes a completely exponential process which is a commonly used assumption in many trials of short duration. For those situations where the user is not comfortable with the strict exponential assumption, Model 6 provides for change-in-the-rate-over-time with the specification of the Weibull distribution for event times.

In both these models, the user can approximate the usual clinical trial that accrues participants uniformly over a period of time and then is accompanied by a followup period of fixed length after final accrual. (See options 1 and 3 in this system for further applications and descriptions of these trial designs.) In the current program one can ignore the accrual phenomenon by specifying an accrual time of zero with a positive followup time period. This program extends the capabilites of option 1 by allowing stratification, unequal allocation and Weibull hazard rates. This program extends the capabilites of option 3 by allowing a control and/or experimental group loss rates and Weibull hazard rates. Options 1 and 3 determine the total accrual as the number of entrants per unit of time multiplied by the accrual time. This option specifies the total accrued which is then split uniformly into the accrual time intervals. Two additional rates can also be specified: Drop-in rate is the rate at which the control subjects switch to the intervention, Drop-out rate is the rate at which the intervention subjects go off the active intervention.

Models 5 and 6 assume that the total specified sample is accrued uniformly over the specified accrual period. The user can specify up to 8 strata frequencies which add to 1 and each with different rate parameters. The relative risk parameter is assumed fixed for each stratum and is assumed to be the ratio of the control-to-experimental exponential rate parameters (lambdas). There are three ways to specify control event rates where the design employs more than one stratum:

- Additive exponential (minimum hazard, delta) assumes that the strata proportions are specified so that the hazard of the first stratum is a minimum and that hazards through the remaining strata increase by adding delta to the rate of the previous stratum.
- Multiplicative exponential (minimum hazard, ratio) assumes that the strata proportions are specified so that the hazard of the first stratum is a minimum and that hazards through the remaining strata increase by multiplying ratio to the rate of the previous stratum.
- Strata specific rates indicate individual rates will be specified for each stratum.

For unequal total sample for experimental and control groups, the same ratio of allocation is assumed for each stratum. Additionally, an exponential loss rate can be specified separately for the experimental and control groups. This rate is assumed to be the same across all strata. The analytical procedure divides the total accrual plus followup time into 10 equally spaced intervals to perform the Mantel-Haenszel test.

The program performs Monte Carlo simulation using the number of simulations specified by the user. The Mantel-Haenszel test used does not employ the continuity correction recommended for the use of the test in 'real life' situations.

A feature of the program allows computation of the adjusted and/or unadjusted Mantel-Haenszel test statistic. Comparison of the two methods allows one to evaluate the benefit, or lack thereof, for the stratified analysis. Increased sample size and/or number of simulations will increase the execution time of the program so be patient.