Material and Methods
Yearly notifications of brucellosis were extracted from the Annual Health Reports on the Health of the Maltese islands. A comprehensive description of the reporting of brucellosis can be found in Tripp and Sawchuk (2015). Age and sex notification was only available for the sister smaller island of Gozo from the Health Office in Gozo, the records for Malta were unfortunately destroyed.
Monthly numbers of births and stillbirths by sex from April 1919 to June 1954 were drawn from the Maltese Gazette that was published under the auspices of the Medical Officer of Health. These records are housed at the National Archives of Malta (NAM), and The National Archives in Kew, England. Data on brucellosis cases was published in the monthly Gazette reports under the heading of ‘Return of cases of infectious diseases reported by month and location.’ The few missing values were estimated by linear interpolation (Moritz, 2016).
Beyond reporting the basic undulant fever rates, the primary goal was to investigate the relationship between the number of brucellosis cases and the proportion of stillbirths. Multiple regression was used to model a relationship between a dependent variable and one or more independent variables. These models assume the error terms to be independent. This assumption is frequently violated when applied to time series data. A model that allows correlated errors, is the regression model with integrated autoregressive moving average (ARIMA) errors. Consequently, in order to study the relationship between the number of brucellosis cases and the proportion of stillbirths, we used a regression model that allows auto correlated errors and has an ARIMA process. Ethical approval was not required for this study human subjects were not included in the study and the data was aggregated information that did not reveal any personnel identifiers. The graphs were created in Statistica (Statsoft, 2011) and the regression model was completed in R statistical software.
For the regression model, we used a logit transformation on the proportion of stillbirths, in particular the logarithms of the odds of stillbirths (i.e. \(\log\left(\frac{p}{1-p}\right))\), wherep denotes the proportion of stillbirths as the dependent variable, and time and brucellosis are the explanatory variables.
The data showed significant autocorrelations and cross correlations at many lags indicating serial correlations in the time series (see Figure 1). or each sex, we first fitted a logistic regression model with the logit of the stillbirth rate as the dependent variable and the time, month, and the number of brucellosis cases as the independent variables. The models were tested for multicollinearity, based on variance inflation factors (VIF). They are all well below 10, the usual critical value, indicating no serious multicollinearity exist between independent variables. We then fitted the best suited ARIMA models for the resulting residuals (Hyndman & Khandakar, 2008), which were used as the process generating the errors of the regression models (Shumway, Stoffer, & Stoffer, 2000; Stoffer, 2016). The residual plots for the models fitted indicate no serious violations of the assumptions. The residuals of the white ARIMA processes for errors appear to be approximately white noise having an approximate normal distribution. The residuals of the white ARIMA processes for errors appear to be approximately white noise having an approximate normal distribution. In other words, the number of stillbirths and undulant fever taken at different months were random in nature taking on an approximate normal distribution.
Figure 1. Time series plots of logit transformed stillbirth prevalence for males and females, and for the independent variable of undulant fever cases