Marketing Mix Modelling has been used since the 1950s, leveraging statistical inference to attribute media investments to sales. Typically, regression models have been used to model the relationship between the two. However, the media landscape evolves at an increasingly rapid pace, driving the need for more refined models which are able to accurately capture these changes. One class of such models are Bayesian structural time series, which are the focal point in this thesis. This class of models retains the relationship between media investments and sales, while also allowing for model parameters to vary over time. The effectiveness of these models is evaluated with respect to prediction accuracy and certainty, both in and out-of-sample. A total of four different models of varying degrees of complexity were investigated. It was concluded that the in-sample performance was similar across models, yet when it came to out-of-sample performance models with time-varying performance outperformed their static counterparts, with respect to uncertainty. Furthermore, the functional form of the intercept influenced the uncertainty of the forecasts on extended time horizons. / Marketing mix modellering har använts sedan 1950-talet för att dra slutsatser om hur mediainvesteringar påverkar försäljning, med hjälp av statistisk inferens. Vanligtvis har regressionmodeller använts för att modellera relationen mellan de två. Men medielandskapet utvecklas allt snabbare, vilket kräver mer sofistikerade modeller som kan fånga upp dessa förändringar på ett mer precist sätt. En klass av sådana modeller är Bayesianska strukturella tidsseriemodeller, som är fokus för detta arbete. Denna klass av modeller bibehåller den strukturella relationen mellan mediainvesteringar och försäljning, samtidigt som de också tillåter modellparametrarna att variera över tid. Effektiviteten hos modellerna bedöms med avseende på noggrannhet och säkerhet, både tränings- och testdata. Totalt fyra olika modeller med varierande komplexitet undersöktes. Det konstaterades att prestandan på träningsdata var likvärdig mellan modellerna, men när det gällde testdata presterade modeller med tidsvarierande parametrar bättre än sina statiska motsvarigheter, med avseende på osäkerhet. Dessutom påverkade den funktionella formen av interceptet osäkerheten hos prognoserna över längre tidshorisonter.

Longitudinal assessments are crucial in evaluating the disease state and trajectory in patients of neurodegenerative diseases. Neuropsychological outcomes measured over time often have a non-linear trajectory with autocorrelated residuals and skewed distributions. Due to these issues, statistical analysis and interpretation involving longitudinal cognitive outcomes can be a difficult and controversial task, thus hindering most convenient transformations (e.g. logarithmic) to avoid the assumption violations of common statistical modelling techniques. We propose the Adjusted Local Linear Trend (ALLT) model, an extended state space model in lieu of the commonly-used linear mixed-effects model (LMEM) in modeling longitudinal neuropsychological outcomes. Our contributed model has the capability to utilize information from the stochasticity of the data while accounting for subject-specific trajectories with the inclusion of covariates and unequally-spaced time intervals. The first step of model fitting involves a likelihood maximization step to estimate the unknown variances in the model before parsing these values into the Kalman Filter and Kalman Smoother recursive algorithms. Results from simulation studies showed that the ALLT model is able to attain lower bias, lower standard errors and high power, particularly in short longitudinal studies with equally-spaced time intervals, as compared to the LMEM. The ALLT model also outperforms the LMEM when data is missing completely at random (MCAR), missing at random (MAR) and, in certain cases, even in data with missing not at random (MNAR). In terms of model selection, likelihood-based inference is applicable for the ALLT model. Although a Chi-Square distribution with k degrees of freedom, where k is the number of parameter lost during estimation, was not the asymptotic distribution in the case of ALLT, we were able to derive an asymptotic distribution approximation of the likelihood ratio test statistics using the power transformation method for the utility of a Gaussian distribution to facilitate model selections for ALLT. In light of these findings, we believe that our proposed model will shed light into longitudinal data analysis not only in the neuropsychological data realm but also on a broader scale for statistical analysis of longitudinal data. / 2023-10-05T00:00:00Z