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