Analysis of factorial experiments for survival data with long-tailed distributions

Two left-truncated survival data sets are collected in one-way factorial designs to examine the quality of products. We cannot specify our survival function completely, and only know that the tail has a power functional form of its argument. Thus, our problem is a left-truncated one with incomplete survivor functions. One of our data sets is the case where the usual analysis of variance (ANOVA) may be adapted. The other is a repeated measurement case. We note that the likelihood function is expressed as a product of conditional and marginal likelihood functions. Estimates of power parameters are always obtained by the conditional likelihood. Location parameters describing treatment eff ects are included in the marginal likelihood only, and their estimates are undetermined, because of missing values resulting from left truncation. However, in the ANOVA case, we show that a common structure of power parameters and some simple assumptions about the missing values enable us to construct an approximate F test for treatment effects through the marginal likelihood. This result is extended to a regression case. With the data in repeated measurements, a systematic variation of the power parameters and an apparent deviation from our presupposed model make an application of the ANOVA mentioned impossible, and compel us to generalize our model. By using the ratio of those generalized models, we show that a descriptive model for evaluating treatment effects can be constructed.