Chapter 7 Modeling of the joint determinants of malaria Rapid Diagnosis
7.4 Evaluation of malaria RDT result using joint models approach
173 ρ(β) = v£v£τ
[v£τ+v£[v£τ+v£. (7.20)
Equation (7.20) can be performed with general random effects design matrices and for more than two components.
Full joint distribution is not necessary for the general model formulation. A full joint model specification needs full bivariate model specification, conditional upon the random effects. Furthermore, the generalized linear mixed model formulation can be extended to the hierarchical cases. The hierarchical cases include repeated measures, meta-analysis, cluster data, correlated data, etc.
Model i = m+ ∈ is sufficient to generate marginal and random effects models. For shared parameters between models of different types, it is important to ensure the models to be meaningful. For correlations in the model
with random effects, the correlation structure can be derived from
= $(i) ≃ ∆ÜV∆V+ Ʃ. In general, the parameters from joint models can be estimated using numerical approximation method. These methods include Gaussian quadrature and Laplace approximation. Estimation based on data using pseudo-likelihood where pseudo data created based on a linearization of the mean. Furthermore, the pseudo-likelihood approach can be used to estimate parameters in marginal models and random effects with or without correlations. But, quadrature or Laplace approximations can only estimate parameters in the conditional independence random effects models.
174 individual. The effect of predictor variables on malaria RDT result was explored on the previous chapters (Ayele et al., 2012, Ayele et al., 2013a, Ayele et al., 2013b). In this Chapter, the aim is to further investigate the joint effect of these predictor variables (socio-economic, demographic and geographic factors) on malaria RDT result, use of mosquito nets and use of anti- mosquito spray in the last twelve months. More specifically, it is important to assess whether the explanatory variables that were found to be significantly related with RDT result in the previous studies would still have a significant effect on malaria RDT result even when use of mosquito nets and use of anti- mosquito spray is accounted for. Also assessing the association between the two outcomes (malaria RDT result and use of mosquito nets) and (malaria RDT result and use of anti- mosquito spray) is of interest. The advantages of fitting a joint model over a separate model that would contain use of mosquito nets and use of ant- malaria spray in a linear predictor include possible gains in efficiency of the parameter estimates (Gueorguieva, 2001). The respondent’s malaria RDT result status (positive/negative) has been modelled as a binary variable that follows Bernoulli distribution.
To evaluate the association between malaria RDT result, use of mosquito nets and use indoor residual spray in the last twelve months, the generalized multivariate mixed effects model was fitted. The three response variables could be taken to be completely independent at any point. In this model, the correlation between the three outcomes as well as the correlation coming from the structure of the data is specified through the random effects structure. This is done by assuming separate random intercepts for each outcome variable and then combining them by imposing a joint multivariate distribution on the random intercepts. The SAS procedure GLIMMIX (SAS 9.3) was used to fit the marginal model. This procedure allows us to jointly model outcomes with different distributions and/or different link functions. The estimates from GLIMMIX were used as initial estimates for NLIMIXED procedure.
175 Table 7. 1: Parameter estimates for a joint marginal model for malaria
RDT result, use of mosquito nets and use of indoor residual spray for main effects
Effects
Malaria RDT result Use of mosquito nets Use of indoor residual spray
Est SE P-
value Est SE P-value Est SE P-value Intercept 0.68 0.67 0.94 -9.21 2.70 0.00 -9.21 2.70 0.0001 Age -1.01 0.00 <.0001 -1.04 0.01 <.0001 -1.04 0.01 <.0001
Gender (ref. Male)
Female 2.99 0.61 0.53 4.10 0.26 <.0001 -4.78 0.73 0.99 Family size 0.07 0.01 0.02 0.01 0.01 0.14 0.04 0.01 0.01
Region (ref. SNNP)
Amhara 0.09 0.05 0.10 -0.01 0.07 0.91 0.07 0.12 0.58 Oromiya 0.09 0.06 0.99 0.09 0.08 0.30 0.16 0.14 0.27 Altitude 0.00 0.00 0.00 -0.01 0.00 0.01 0.00 0.00 0.31 Main source of drinking water (Ref. protected water) Tap water -0.36 0.23 <.0001 -2.52 0.30 <.0001 -0.56 0.16 0.0001 Unprotected
water 2.46 0.28 <.0001 2.43 0.38 <.0001 0.55 0.13 <.0001
Time to collect water (ref. > 90 minutes)
< 30 minutes -1.21 0.01 0.0001 -1.23 0.08 0.00 -1.64 0.49 0.0001 between 30 to
40 minutes 0.45 0.28 0.11 -0.12 0.05 0.03 -2.43 0.50 <.0001 between 40 -
90 minutes -0.12 0.09 <.0001 0.47 0.36 0.19 -0.85 0.58 0.15
Toilet facility (ref. No facility)
Pit Latrine -0.74 0.23 <.0001 -0.01 0.29 0.97 -0.11 0.18 0.03 Toilet with
flush -0.92 0.23 <.0001 -0.54 0.29 0.06 -1.90 0.77 0.02
Availability of electricity (ref. no)
Yes 2..072 0.08 <.0001 2.30 0.30 <.0001 2.04 0.12 <.0001
Availability of television (ref. no)
Yes -0.43 0.16 <.0001 0.25 0.16 <.0001 0.03 0.06 0.64
Availability of radio (ref. no)
Yes -0.63 0.03 0.72 -0.03 0.05 0.54 -0.60 0.16 0.0001 Total number of
rooms -0.23 0.04 0.0001 -0.49 0.14 0.00 -0.18 0.05 0.0001 Main material of room's wall (ref. Cement Blocks) Corrugated
Metal -0.53 0.11 <.0001 -3.30 0.34 <.0001 -0.76 0.03 <.0001 Mud Blocks 0.27 0.26 <.0001 3.05 2.65 0.25 -11.5 0.05 <.0001
Main material of room's roof (ref. Corrugate)
Thatch 0.43 0.052 <.0001 0.51 0.07 <.0001 0.16 0.05 <.0001 Sticks and
mud 1.21 0.12 <.0001 0.61 0.18 0.0001 0.24 0.18 <.0001 Main material of room's roof (ref. earth/Local dung plaster) Cement -0.26 1.29 0.25 -3.83 2.87 <.0001 -6.13 0.41 0.25 Wood -0.45 1.02 <.0001 -3.67 2.67 <.0001 -5.92 0.33 <.0001
176 The conditional independence random effects model was fitted with SAS 9.3 PROC NLMIXED using the general log-likelihood option. The NLMIXED procedure using the general log-likelihood function allows one to impose a joint multivariate distribution on the random effects from separate models. All statistical tests were conducted at a 5% level of significance.
The linear predictors which were used for fitted models consists the same variables which were used in the previous studies (Ayele et al., 2012, Ayele et al., 2013a, Ayele et al., 2013b). The following socio-economic, demographic and geographic variables were considered as explanatory variables. The socio- economic variables are main source of drinking water, time to collect water, toilet facilities, availability of electricity, radio and television, total number of rooms, main material of the room's wall, main material of the room's roof and main material of the room's floor. Geographic variables are region and altitude, and demographic variables are gender, age and family size. In addition to the main effects, some two-way and three-way interaction effects which were significant in the previous studies were included in the model. These two-way and three-way interaction effects are drinking water and roof material, time to collect water and floor material, time to collect water and main material of room's roof, age and gender, gender and main source of drinking water, gender and availability of electricity, gender and floor material, age, gender and main source of drinking water, age, gender and electricity, and age, gender and floor material.
For this study, malaria RDT result, use of mosquito nets and use of indoor residual spray the last twelve months are binary outcome variables. Therefore, malaria RDT result, use of mosquito nets and use of indoor residual spray in the last twelve months will jointly be modelled using generalized linear mixed models. For this model, it is assumed uncorrelated random intercepts with correlated residual errors. The results from the generalized linear mixed model
177 analysis are given in Tables 7.1 and 7.2. The result from joint models for malaria RDT result, use of mosquito nets and use of indoor residual spray in the last twelve months confirm the results obtained from other models in the previous Chapters.
Table 7. 2: Parameter estimates and their corresponding standard errors of a joint marginal model for malaria RDT result, use of mosquito nets and use of indoor residual spray for interaction effects
Effects Malaria RDT result Use of mosquito nets
Est SE P-value Est SE P-value
Age and gender (ref. male)
Female 1.426 0.215 <.0001 0.988 0.006 <.0001 Gender and main source of drinking water (ref. Male & protected water)
Female and Tap water -2.107 0.114 <.0001 -2.390 0.447 <.0001 Female and Unprotected
water 0.534 0.162 <.0001 -1.592 0.483 0.001
Gender and availability of electricity (ref. Male & yes)
Female and No -2.152 0.291 <.0001 -3.256 0.593 <.0001 Age, gender and main source of drinking water (ref. Male & protected water)
Female and Tap water -0.335 0.159 0.017 -0.024 0.008 0.004 Female and Unprotected
water 2.480 0.263 0.014 -0.008 0.008 0.286
Age and gender and material of room's floor (ref. Male and earth/Local dung plaster)
Female and Cement -0.468 1.026 <.0001 1.076 0.023 <.0001 Female and Wood 0.353 0.039 <.0001 1.064 0.000 <.0001
Effects use of indoor residual spray
Est SE P-value
Age and gender (ref. male)
Female 0.988 0.006 <.0001
Gender and main source of drinking water (ref. Male & protected water)
Female and Tap water -2.39 0.447 <.0001
Female and Unprotected
water -1.592 0.483 0.001
Gender and availability of electricity (ref. Male & yes)
Female and No -3.256 0.593 <.0001
Age, gender and main source of drinking water (ref. Male & protected water)
Female and Tap water -0.024 0.008 0.004
Female and Unprotected
water -0.008 0.008 0.286
Age and gender and material of room's floor (ref. Male and earth/Local dung plaster)
Female and Cement 1.076 0.023 <.0001
Female and Wood 1.064 0 <.0001
178 The main significant socio-economic, demographic and geographic factors which were found from the joint model of malaria RDT result, use of mosquito nets and use of indoor residual spray in the last twelve months are age, family size, altitude, main source of drinking water, time to collect water, toilet facility, availability of radio, television and radio, total number of rooms, main material of room's wall, main material of room's roof and main material of room's floor. The two-way significant effects were drinking water and roof material, age and gender, gender and main source of drinking water; and gender and availability of electricity. Age, gender and main source of drinking water; and age, gender and floor material were found to be significant three- way interaction effects (Tables 7.1 and 7.2).
Furthermore, among the main effects age, gender, main source of drinking water, main material of room's roof and availability of electricity were involved in the interaction effects (Table 7.2). The estimates of the significant effects are given in Tables 7.1 and 7.2. Based on the results for a unit increase in family size, the odds of positive rapid diagnosis test increases by 7.6% (OR = 1.076, P- value = 0.02). With reference to individuals with no toilet facilities, the odds of a positive malaria rapid diagnosis test is lower for those individuals using a flushing toilet to those who have septic tanks (OR = 0.397, P-value <0.0001) or pit latrine slabs (OR = 0.477, P - value <0.0001). Moreover, for a unit increase in the number of total rooms, the odds of malaria diagnosis test for an individual decreased by 20.1% (OR = 0.799, P-value = 0.0001). With reference to individuals with no access to radio, the odds of a positive malaria rapid diagnosis test is lower for those individuals who have access to radio (OR = 0.535, P - value <0.0001). Similarly, for those households who have electricity, the odd of malaria RDT result to be positive is increased (OR=7.937, P – value <
0.0001) compared to households who have no electricity. Moreover, for households who have access to television, the odds of positive rapid diagnosis test increases (OR = 0.651, P - value <0.0001).
179 Interaction effects
From Table 7.2, it can be seen that there are significant two-way and three-way interaction effects. The estimates of these significant effects are given in Table 7.2. As the result indicates one of the three-way interaction effects which was found to be significant is age, gender and main source of drinking water. The result is presented in Figure 7.1 and 7.2.
Figure 7. 1: Log odds associated with rapid diagnosis test and age for male respondents with source of drinking water
Figure 7. 2: Log odds associated with rapid diagnosis test and age for female respondents with source of drinking water
-14 -12 -10 -8 -6 -4 -2 0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Logit(Positive_RDT)
Source of drinking water
Unprotected water Protected water Tap water
-10 -5 0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Logit(Positive_RDT)
Source of drinking water
Unprotected water Protected water Tap water
180 From the figures it can be seen that as age increased, positive malaria diagnosis was less likely for males than for females who were using protected, unprotected and tap water for drinking. Furthermore, as age of respondents increased, malaria RDT was less likely to be positive for individuals who used tap water for drinking for males and for females. More specifically, positive malaria diagnosis rates increased with age for females whereas it decreased for males as age increased (Figures 7.1 and 7.2). The Figures further show that the gap in the malaria RDT Test between respondents using unprotected, protected and tap water for drinking widens with increasing age for females.
Figure 7. 3: Log odds associated with rapid diagnosis test and age for male respondents with material for room’s floor
Figure 7. 4: Log odds associated with rapid diagnosis test and age for female respondents with material for room’s floor
0 10 20 30 40 50 60 70 80 90
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Cement Wood earth
0 10 20 30 40 50 60 70 80 90
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Cement Wood earth
181 The other three-way significant interaction effect is between age, gender and material of room's floor (Table 7.2). The results are presented in Figures 7.3 and 7.4 show the interaction between age, gender and material of room's floor for male and female respectively. From the figures it can be seen that as age increased, positive malaria diagnosis was also increased for males for all kinds of material used for roof construction. As can be seen from the figures, individuals who has cement floor has less risk to be positive for malaria RDT result followed by wood and earth. Furthermore, as age of respondents increased, malaria RDT test was also increasing for females. Unlike males, for females the risk of malaria is the same for all type of house construction.
Figure 7. 5: Log odds associated with rapid diagnosis test and availability of electricity with gender
Figure 7.5 presents the interaction effect between availability of electricity and gender for individuals. Prevalence of malaria was significantly higher for female than for male respondents who were living in a house with electricity. Similarly, a female living in a house, which has no electricity, the positive malaria result was significantly higher than it was for males.
The random effects for malaria RDT result and use of mosquito nets are significantly negatively associated i.e., -0.468 (p-value <.0001) (Table 7.3). This indicates a negative correlation between malaria RDT result and use of mosquito nets. This means that increasing the use of mosquito nets tends to
0.4 0.5 0.6 0.7 0.8 0.9
Male Female
Yes No