Link Function for Binomial Model in Estimating Knockdown Time (KT95 and KT50) of Mosquito Repellents

Estimation of knockdown time (KT) is useful in determining bio-effectiveness of mosquito repellents. Knockdown or not knockdown is a binary variable thus, analysis is done by fitting generalized linear models, based on binomial distribution. Use of appropriate link function in fitting a generalized linear model is crucial especially when estimating quantities such as KT50 and KT95. This study was done to determine the most appropriate link function in fitting generalized linear models to estimate KT50 and KT95. Knockdown activity of metofluthrin 0.005% (w/w) and d-trans-allethrin 0.12% (w/w) was tested under two different physiological conditions (blood fed and sucrose fed) using wild-caught female Culex tritaeniorhynchus mosquitoes from an agro-farming area of the north-western province of Sri Lanka. Coefficient of variation of the observed KT50 and KT95 was less than 5.5%. Both KT50 and KT95 values were estimated by fitting altogether 120 binomial distribution-based generalized liner models with three different link functions namely, logit, probit, and complementary log–log. The G2 statistics was used to test the goodness of fit of the models. However, in order to evaluate the accuracy of all estimated KT50 and KT95 values obtained using the above three link functions, they were compared against corresponding observed values using ANOVA followed by Dunnett mean separation procedure. The probit and logit link functions were found to be appropriate in the estimation of KT50. As the logit link function is commonly used in modeling binary responses, out of the two, logit link function is recommended. Complementary log–log link function was found to be the most appropriate in estimation of KT95. Thus, one link function cannot be recommended in estimating both KT parameters.


INTRODUCTION
The efficacy of an insecticide against particular insect is determined under laboratory and field conditions using various parameters. Out of them estimation of 50% cumulative knockdown (KT 50 ) and 50% lethal dose (LD 50 ) are widely applied parameters. Knockdown is the rapid paralysis of insects causing them to fall down and remain in a state as to be incapable of co-ordinate movements and apparently dead (SLS, 2001). Although KT 50 is the popular concept in the comparison of knockdown patterns among different mosquito species, KT 95 indicates the accepted maximum tolerance limit of the target insect species against particular concentration of an active ingredient. The generalized liner model based on binomial model with probit procedure is recommended for the calculation of KT values for the testing of the efficacy of mosquito coils (WHO, 2013). Sri Lanka Standards (SLS) 453:2001 section E.5.1 instructs to analyze the obtained knock-down data using PROBIT procedure, either implemented in a computer programme or PROBIT graph paper by plotting proportion of knockdown verses knockdown time in minutes. Bliss (1934) reported using probit link function but logit and complementary log-log link functions have also been used in fitting of binomial models. Although it is well established that quantities such as LD 50 is estimated by fitting binomial models based on probit link function, no adequate literature is available on appropriate link function for binomial models when establishing KT in general. Estimating quantities KT 50 and KT 95 is crucial because bio-effectiveness of mosquito repellent product are adjusted specifically based on KT 95 . The objective of this study was to recommend the best link function for binomial models when estimating KT 50 and KT 95 using three types of link functions namely probit, logit and complementary log-log.

Data Collection
Two types of commercially available (bio-efficacy approved) mosquito coils containing metofluthrin 0.005%w/w and d-trans-allethrin 0.12%w/w as active ingredient were used for the study. A rural area with large paddy fields in Kuliyapitiya of Kurunegala district was selected for the collection of mosquitoes. Cattle baited net trap was used as the sole method of sampling the test mosquitoes. Mosquitoes belonged to Culex tritaeniorhynchus (a known vector of Japanese encephalitis in Sri Lanka) found within the cattle traps was used for the study. From these mosquitoes samples of 20 blood fed and 20 sucrose fed mosquitoes were exposed to a coil (without active ingredient) to ensure the suitability of them for the efficacy testing. KT 50 and KT 95 was estimated following standard procedure (SLS, 2001) against two active ingredients metofluthrin 0.005%w/w and d-trans-allethrin 0.12%w/w. Thus KT was measured under four conditions viz: (i) metofluthrin-blood fed (ii) metofluthrin-sucrose fed (iii) d-trans-allethrin-blood fed (iv) d-trans-allethrin-sucrose fed under each condition 10 packs were tested. Accordingly three were 40 KT sets.

Model fitting
The general form of the models fitted was where denotes the link function. Link functions considered in the study were logit, probit, and complementary loglog and they are respectively of the form , where p is proportion knockdown and Φ -1 indicates inverse cumulative standard normal distribution. The variances of the cumulative distribution functions are not same. In fact the means and the variance of the three distributions, probit, logit and complementary log-log are respectively (0, 1), (0, π 2 /3) and ( , π 2 /6) where is the Euler constant (Bilder, 2010;Gourdon & Sebah, 2004).
Models were fitted for the data using above three link functions and thus altogether 120 models were fitted. The goodness of fit of the fitted models was evaluated using G 2 statistics (McCullagh & Nelder, 1989). Mean KT 50 and KT 95 values estimated from the fitted models. These estimates were compared with the observed mean KT 50 and KT 95 values using one way ANOVA followed by Dunnett mean separation technique using observed mean as the control.

RESULTS AND DISCUSSION
The summary of the goodness of fit (G 2 ) with 19 th degree of freedom (df) for 120 fitted models are represented in Table 1. According to the Table 1, all fitted models were adequate (P>0.05) and thus models with any of the three link functions is able to capturing the variability of the response variable. Thus a model with any of those link functions can be considered in estimating important quantities.

Fig. 1. Coefficient of variability of observed KT 50 and KT 95
The coefficient of variability (CV) of the observed KT 50 and KT 95 are shown in Fig 1. All CVs are below 5.5%, indicating that data have been generated under well controlled conditions and thus even a minor effect can be detected. The CV of the blood fed mosquitoes was relatively lower than that of the sucrose fed mosquitoes. The CV of the KT 95 is higher than that of KT 50 , for a given feed type and for a given active ingredient.  With estimated KT 95 under same four conditions using three different link functions also shows that there is a significant mean difference (P <0.0001) except for condition MT/SF (P=0.1903). According to Dunnett mean separation results the means of estimated KT 95 using probit and logit link functions were significantly different from the observed mean KT 95.

CONCLUSION
The estimate of KT values is comparatively an easy practice but not routinely applied for monitoring of susceptibility except in the specialized entomological laboratory testing (in the field and under insectary conditions). To achieve the recommended accuracy and precision of KT of specific vector mosquitoes, it is necessary to have both specialized entomological skills and the appropriate statistical procedures. The KT 95 indicates the accepted maximum tolerance limit of the target insect species against to a particular concentration of an active ingredient and it is more sensitive to the development of insecticide resistance. Therefore the accurate estimation of KT 95 is important in early detection of insecticide resistance. In general, KT values are estimated by fitting binary regression models with probit link function. However, from this study it can be concluded that complementary log-log link function is more appropriate to estimate KT 95 for C. tritaeniorhynchus. Both probit and logit link functions are appropriate in the estimation of KT 50 for the same mosquito population. However, out of the two, logit link function is recommended due to the reasons mentioned earlier. Therefore a single link function is not recommended for calculation of KT values under different conditions.