Climate variations and salmonellosis transmission in Adelaide, South Australia: a comparison between regression models

Abstract

This is the first study to identify appropriate regression models for the association between climate variation and salmonellosis transmission. A comparison between different regression models was conducted using surveillance data in Adelaide, South Australia. By using notified salmonellosis cases and climatic variables from the Adelaide metropolitan area over the period 1990–2003, four regression methods were examined: standard Poisson regression, autoregressive adjusted Poisson regression, multiple linear regression, and a seasonal autoregressive integrated moving average (SARIMA) model. Notified salmonellosis cases in 2004 were used to test the forecasting ability of the four models. Parameter estimation, goodness-of-fit and forecasting ability of the four regression models were compared. Temperatures occurring 2 weeks prior to cases were positively associated with cases of salmonellosis. Rainfall was also inversely related to the number of cases. The comparison of the goodness-of-fit and forecasting ability suggest that the SARIMA model is better than the other three regression models. Temperature and rainfall may be used as climatic predictors of salmonellosis cases in regions with climatic characteristics similar to those of Adelaide. The SARIMA model could, thus, be adopted to quantify the relationship between climate variations and salmonellosis transmission.

Appendix

• 1. Standard poisson regression.

  In the standard Poisson regression analysis, the dependent variable is weekly salmonellosis cases. The Poisson distribution has only one parameter, ν, which equals the mean (or variance). The distribution is given by the formula (Weisstein 2004):

  $$ {\text{Pr}}{\left( {\text{Y}} \right)} = \frac{{\nu ^{n} e^{{ - \nu }} }} {{n!}} $$

  The estimation of the parameters is obtained by a maximum likelihood function. The model can be summarized as: ln (ν _t) = α + β ₁ temperature_t + β ₂ rainfall_t + sin(2πt/52) + cos(2πt/52)

• 2. Autoregressive adjusted poisson regression.

  According to our previous analysis, there is autocorrelation between both dependent and independent variables. Therefore, the Poisson regression model adjusted for autocorrelation can be given as:

  $$ \begin{aligned} & {\text{ln }}{\left( {\nu _{{\text{t}}} } \right)} = \alpha + \beta _{1} {\text{ Y}}_{{{\text{t}} - {\text{1}}}} + \beta _{2} \,{\text{Y}}_{{{\text{t}} - {\text{2}}}} + \,\,\beta _{3} {\text{ Y}}_{{{\text{t}} - {\text{3}}}} + {\text{ }}\beta _{4} {\text{ Y}}_{{{\text{t}} - {\text{4}}}} \\ & + \beta _{5} {\text{temperature}}_{{\text{t}}} + {\text{ }}\beta _{6} {\text{temperaruep}}_{{{\text{t}} - {\text{1}}}} \\ & + {\text{ }}\beta _{7} {\text{temperaturep}}_{{{\text{t}} - {\text{2}}}} {\text{ }} + {\text{ }}\beta _{8} {\text{rainfall}}_{{\text{t}}} + \beta _{9} {\text{rainfall}}_{{{\text{t}} - {\text{1}}}} \\ & + \beta _{{10}} {\text{rainfall}}_{{{\text{t}} - {\text{2}}}} {\text{ }} + {\text{sin}}{\left( {{{\text{2}}\pi {\text{t}}} \mathord{\left/ {\vphantom {{{\text{2}}\pi {\text{t}}} {52}}} \right. \kern-\nulldelimiterspace} {52}} \right)}\, + {\text{cos}}{\left( {{{\text{2}}\pi {\text{t}}} \mathord{\left/ {\vphantom {{{\text{2}}\pi {\text{t}}} {52}}} \right. \kern-\nulldelimiterspace} {52}} \right)} \\ \end{aligned} $$

• 3. Multiple linear regression.

  The regression model is:

  $$ \begin{aligned} & {\sqrt {Yt} } = {\text{ }}\alpha + \beta _{1} {\sqrt {Y_{{t - 1}} } } + \beta _{2} {\sqrt {Y_{{t - 2}} } } + \beta _{3} {\sqrt {Y_{{t - 3}} } } + \beta _{4} {\sqrt {Y_{{t - 4}} } } \\ & + \beta _{5} {\text{temperature}}_{{\text{t}}} + \beta _{6} {\text{temperaute}}_{{{\text{t}} - {\text{1}}}} \\ & + \beta _{7} {\text{temperature}}_{{{\text{t}} - {\text{2}}}} + \beta _{8} {\text{rainfall}}_{{\text{t}}} + \beta _{9} {\text{rainfall}}_{{{\text{t}} - {\text{1}}}} \\ & + \beta _{{10}} {\text{rainfall}}_{{{\text{t}} - {\text{2}}}} {\text{ }} + {\text{sin}}{\left( {{{\text{2}}\pi {\text{t}}} \mathord{\left/ {\vphantom {{{\text{2}}\pi {\text{t}}} {52}}} \right. \kern-\nulldelimiterspace} {52}} \right)}\, + {\text{cos}}{\left( {{{\text{2}}\pi {\text{t}}} \mathord{\left/ {\vphantom {{{\text{2}}\pi {\text{t}}} {52}}} \right. \kern-\nulldelimiterspace} {52}} \right)} + Et \\ \end{aligned} $$

  Where Et is independent random error with mean 0.

• 4. Seasonal autoregressive moving average (SARIMA) models.

  An autoregression moving average (ARMA) model predicts the outcome variable from the values of the outcome at previous time points. It is an approach to handle time-series modelling and forecasting based on the landmark work of Box and Jenkins (1976). The theory and application of the models have been described in many papers and books (Shumway and Stoffer 2000; Brockwell and Davis 1991; Tobias and Saez 2004). The ARMA model is suitable only for stationary processes. The SARIMA model allows for a trend and seasonal effects by differencing. The model described here SARIMA(4,0,0)*(1,0,0)₅₂, which stands for 4-order autoregression and 1-order seasonal autoregression with a period of 52 weeks, has the following form. Let X_t be the square root of Y_t. Define Wt by W_t = X_t−X_t−52. Then

  $$ \begin{aligned} & Wt = \alpha _{0} + \alpha _{1} {\text{ }}W_{{{\text{t}} - {\text{1}}}} + \alpha _{2} \,W_{{{\text{t}} - {\text{2}}}} + \alpha _{3} \,W_{{{\text{t}} - {\text{3}}}} \, + \alpha _{4} \,W_{{{\text{t}} - {\text{4}}}} \\ & + \beta {\rm E}_{{{\text{t}} - 52}} + \gamma _{1} {\text{Temperature}}_{{\text{t}}} + {\text{ }}\gamma _{2} {\text{Temperaute}}_{{{\text{t}} - {\text{1}}}} \\ & + {\text{ }}\gamma _{3} {\text{Temperature}}_{{{\text{t}} - {\text{2}}}} {\text{ }} + {\text{ }}\gamma _{4} {\text{rainfall}}_{{\text{t}}} + \gamma _{5} {\text{rainfall}}_{{{\text{t}} - {\text{1}}}} \\ & + \gamma _{6} {\text{rainfall}}_{{{\text{t}} - {\text{2}}}} {\text{ }}E_{{\text{t}}} \\ \end{aligned} $$

  Where E_t is independent random error with mean 0.

  The Aikake information criterion (AIC) is used to compare models fit to one same series. The model with the smaller AIC fits the data better. AIC = −2 ln(L) + 2 k , where L is the likelihood function and k is the number of free parameters.