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- Garmo, Hans
(författare)
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Random railways and cycles in random regular graphs
- 1998
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In a cubic multigraph certain restrictions on the paths are made to define what is called a railway. Due to these restrictions a special kind of connectivity is defined. As the number of vertices tends to infinity, the asymptotic probability of obtaining an, in this sense, connected random cubic multigraph is shown to be 1/3.An equivalence relation on the tracks in the railway (edges in the multigraph) is defined in order to further study the properties of railways. The number of equivalence classes induced by this relation - the connectivity number - is investigated for a random railway achieved from a random cubic multigraph.As a result we obtain the asymptotic distribution of this connectivity number.In recent years the asymptotic distribution of Hamiltonian cycles in random r-regular graphs has been derived. As a generalization we investigate the asymptotic distribution of the number of cycles of length l in a random r-regular graph. The length of the cycles is defined as a function of the number of vertices n in the graph, thus, l = l(n), where l(n) → ∞ as n → ∞. The resulting limiting distribution turns out to depend on whetherl(n)/n → 0 or l(n)/n> → q, for 0 < q < 1.In the first case the limit distribution is a weighted sum of Poisson variables while in the other case the limit distribution is similar to the limit distribution of Hamiltonian cycles in a random r-regular graph.
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| 2. |
- Westerberg, Marcus, 1990-
(författare)
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Prostate cancer incidence, treatment and mortality : Empirical longitudinal register-based studies and methods for handling missing data
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The diagnostic activity for prostate cancer has increased substantially in Sweden, primarily due to increased use of prostate-specific antigen (PSA) testing in asymptomatic men, and this has led to a large increase in diagnoses. There have also been changes in the diagnostic workup, guidelines, treatment strategies, and more effective treatments have been introduced in different phases of the disease. This thesis aims to increase the understanding of consequences of changes in diagnostic activity and treatment, with a focus on empirical studies, methodological development, and handling of missing data.In paper I, the survival of men with metastatic prostate cancer was investigated across calendar time periods by use of Kaplan-Meier analyses and Cox regression. The median survival from diagnosis increased with six months comparing men diagnosed 1998-2001 with men diagnosed 2010-2015, while median PSA decreased.In paper II, a discrete time multivariate longitudinal model was combined with a proxy for the unobserved level of diagnostic activity to produce prognoses of incidence and mortality. Simulations indicated that a higher diagnostic activity was associated with fewer men diagnosed with metastatic disease and fewer prostate cancer deaths.In paper III, we looked for clinical variables predictive of the survival of men with castration-resistant prostate cancer (CRPC). A new data base was created including longitudinal data on prescriptions of hormonal treatment, PSA, and cause of death. We found that PSA doubling time and PSA at time of CRCP were highly predictive and could be used for treatment decision.In paper IV, we estimated annual incidence of metastatic prostate cancer using different methods for handling missing data in metastatic status (M stage). Missing data in M stage was high and varied over calendar time and risk groups, yet each method indicated a downward trend in incidence. Although men with unknown metastatic status cannot be assumed to have nonmetastatic disease in general, this may be reasonable among those with tumour characteristics that indicate a low risk of metastases.In paper V, the estimation of multivariate longitudinal models was considered in a context where some events are observed on a coarser level (e.g. grouped) at some time points, causing gaps in the data. The likelihood function, score and observed information were derived under an independent coarsening mechanism. A simulation study was conducted comparing properties of several estimators including direct maximum likelihood and Monte Carlo Expectation Maximisation.
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