Modified multiblock partial least squares path modeling algorithm with backpropagation neural networks approach

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1 Modified multiblock partial least squares path modeling algorithm with backpropagation neural networks approach Budi Yuniarto, and Robert Kurniawan Citation: AIP Conference Proceedings 1827, (2017); View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in Modeling relationship between mean years of schooling and household expenditure at Central Sulawesi using constrained B-splines (COBS) in quantile regression AIP Conference Proceedings 1827, (2017); / Mean-Variance portfolio optimization by using non constant mean and volatility based on the negative exponential utility function AIP Conference Proceedings 1827, (2017); / Multilevel poisson regression modelling for determining factors of dengue fever cases in bandung AIP Conference Proceedings 1827, (2017); / Spectral analysis and markov switching model of Indonesia business cycle AIP Conference Proceedings 1827, (2017); / Sequential pattern mining of rawi hadis (Case study: Shahih hadis of Imam Bukhari from software Ensiklopedi Hadis Kitab 9 Imam) AIP Conference Proceedings 1827, (2017); / R programming for parameters estimation of geographically weighted ordinal logistic regression (GWOLR) model based on Newton Raphson AIP Conference Proceedings 1827, (2017); /

2 Modified Multiblock Partial Least Squares Path Modeling Algorithm with Backpropagation Neural Networks Approach Budi Yuniarto a) and Robert Kurniawan b) Department of Computational Statistics, Institute of Statistics (STIS), Jakarta Corresponding author: a) b) Abstract. PLS Path Modeling (PLS-PM) is different from covariance based SEM, where PLS-PM use an approach based on variance or component, therefore, PLS-PM is also known as a component based SEM. Multiblock Partial Least Squares (MBPLS) is a method in PLS regression which can be used in PLS Path Modeling which known as Multiblock PLS Path Modeling (MBPLS-PM). This method uses an iterative procedure in its algorithm. This research aims to modify MBPLS- PM with Back Propagation Neural Network approach. The result is MBPLS-PM algorithm can be modified using the Back Propagation Neural Network approach to replace the iterative process in backward and forward step to get the matrix t and the matrix u in the algorithm. By modifying the MBPLS-PM algorithm using Back Propagation Neural Network approach, the model parameters obtained are relatively not significantly different compared to model parameters obtained by original MBPLS-PM algorithm. Keywords: SEM, Partial Least Squares Path Modeling, Neural Network, PLS, PLS-PM, Multiblock PLS-PM. INTRODUCTION Structural Equation Models (SEM), is one of the multivariate regression models (Fox, 2002). While a classical multivariate linear model only has one predictee variable in the equation, in SEM, the predictee variable of one equation also could had a role as a predictor variable for other predictee variables. Structural equation modeling (SEM) is often used to analyze the causal relationship between latent variables. There are two approaches of SEM methods: covariance based SEM and variance or component-based SEM. Covariancebased SEM (CB-SEM) was first developed by Joreskog (1982), meanwhile variance or component-based SEM with partial least squares (PLS) approach was developed by Wold (1979), and then this approach was known as PLS Path Modeling or PLS-PM (Martens, 1989). Both methods are complementary rather than competitive (Hair, et.al, 2014). PLS method was introduced as a linear regression technique, and the non-linear algorithm Iterative Partial Least Squares or NIPALS (Wold, in Baffi, 1999) is an important key in the PLS method. PLS approach began to be used in the modeling path in 1980 (Wold, 1980). Wangen and Kowalski (1988) introduced a multiblock PLS algorithm for Statistics and its Applications AIP Conf. Proc. 1827, ; doi: / Published by AIP Publishing /$

3 PLS regression. Arteaga et al (2010) developed an algorithm of multiblock Partial Least Squares Path Modeling (MBPLS-PM), which is adapting the multiblock PLS regression method. In the real world, data often exhibit nonlinear properties (Wold et al, 2001 and Li et al, 2007 in Abdel-Rahman and Lim, 2009), so the technique of non-linear PLS was developed. According Vinzi et al (2010b), there are several options for using non-linear regression PLS in PLS path modeling, among other quadratic forms, smoothing procedure, spline functions, and neural networks. Therefore, in this paper, we modified the algorithm of multiblock Partial Least Squares Path Modeling (MBPLS-PM) with back-propagation feed forward neural networks. OVERVIEW PLS Path Modeling Wold (in Ghozali, 2008) developed Partial Least Square as a general method for estimating path models using the multiple indicator latent constructs. In contrast to covariance based SEM, PLS-PM does not aim to reproduce the sample covariance matrix. PLS-PM is a soft-modeling approach that does not require stringent assumptions in relation to the distribution, the sample size, and the scale of measurement (Vinzi et al, 2010). PLS-PM is the estimation method based on the component (Tenenhaus, 2008). PLS-PM is an iterative algorithm which independently makes the solution on measurement models and then estimates the structural path coefficients in the model. With PLS approach, it is assumed that all the size of the variance is a useful variance to be explained. The approach to estimate the latent variables is considered as a linear combination of indicators (measurement variables) so that it can avoid the trouble factor indeterminacy and provide an exact definition of the component scores (Wold, 1982). There are three categories of estimated parameters obtained with the PLS-PM. The first is a weight estimate that is used to create the latent variable score. The second estimate is the path estimate that links among latent variables and between latent variables and the indicators. And the third is the measurement coefficient between indicators and latent variables. PLS-PM has three stages iteration process in obtaining the three estimated parameters (weight estimates, path estimates, and measurement coefficient), where each stage estimates each parameter. The first stage is an important part of the PLS-PM algorithm which provides iterative procedure that will generates a stable estimated weight. As in SEM, we must specify a path model in PLS-PM, which is consisted of structural models and measurement models. Thus, in PLS-PM we have three kinds of models: an inner model, an outer model, and weight relation. The inner model or the structural model involve only the latent variables. Relationships among these latent variables basically can be represented by multi-linear equation. The general equation for inner models is (1) with (2) where is path coefficient parameter of latent variable i to latent variable j, is the inner residual, β 0j is a constant, and a is number of latent variable. From specification, we have (3)

4 which has mean that the expected value of inner residual equal to zero and does not correlate with latent variable. The outer model or the measurement model formed the relation between the blocks of indicators (measurement variables) and the latent variables. There are three ways to represent this relationship in the outer models, i.e reflective, formative, and MIMIC (multiple effect indicators for multiple causes). In the reflective outer model, the block indicator is a manifestation of the latent variable and assumed as a linear function of the latent variable ξj, the equation is given below: (4) where x jk is the k-th indicator variable of the j-th block (latent variable), λ jk is the loading coefficient of j-th block towards k-th indicator variable, is the outer residual of k-th indicator variable of j-th block. Thus, we have Then (5) (6) which means the expected value of residual is equal to zero and uncorrelated with the latent variable. with In the formative outer model, latent variable is assumed as a linear function of the indicator. (7) (8) (9) where is regression coefficient of k-th indicator in j-th block, δ j is residual, and p is number of latent variable in j-th block. Meanwhile, the MIMIC outer model is a mixture of reflective outer models and a formative outer model. Although the outer model describes the relationship between the latent variables and the block of indicators, the actual value of the latent variable cannot be known. Therefore, the weight relations should be defined to complete it. The estimation of the latent variables is defined as follow: (10) PLS Path Modeling Algorithm PLS path modeling algorithm (Wold, 1980) to estimate the parameters of the model can be explained as follow: Stage 1: Weight estimation 1) Define the initial outer weights 2) Estimate the latent variables scores using the outer weights

5 3) Re-estimate each latent variable using others latent variables which is connected to it. where e ji is an inner weights, which obtained by three scheme, i.e: a. Scheme of centroid: eji = sign[cor(yi,yj)] b. Scheme of factor: eji = cor(yi,yj) c. Scheme of path: eji = cor(yi,yj) if Yi predicts Yj eji = regression coefficient, if Yi predicted by Yj 4) Update outer weights w jk Mode A: = (Z j Zj) -1 Z j xjk, for reflective outer model Mode B: = (X j Xj) -1 X j Zj, for formative outer model 5) Check, if is convergence then proceed to next stage, else, return to step 2) and use new outer weight Stage 2: Estimate path coefficient 6) Do an OLS regression on structural model to obtain path coefficients. Stage 3: Estimate measurement coefficient 7) Do an OLS regression on measurement model to obtain measurement coefficient. Multiblock PLS Path Modeling In the two blocks or multiblocks PLS, Xj denotes predictor variables, while the block of respond variables are denoted by Y, the latent variables of block Xj are denoted by tj and the latent variables of block Y are denoted by u. The multiblock PLS Path Modeling algorithm (Arteaga, 2010) with j blocks variables can be explained as follow: Step 0. Initialization. Set tj and uj = first column of Xj for j increase from 1 to J Step 1. Backward stage. For j decrease from J to 1: - If Xj predicts no blocks, then set tj = uj. - If Xj predicts only one block Xj then wj = Xj T uj tj = (Xjwj) - If Xj predicts more than one blocks (say h blocks), then Uj = [uj1, uj2,, ujh] cuj = Uj T tj uuj = Ujcuj wj = Xj T uuj tj = (Xjwj). Step 2. Forward stage. For j increase from 1 to J: - If Xj not predicted by any blocks: set uj = tj - If Xj predicted by only one block Xj : cj = Xj T tj uj = (Xjcj) - If Xj predicted by more than one block (h blocks): Tj = [tj1, tj2,, tjh] wtj = Tj T uj ttj = Tjwtj wj = Xj T ttj tj = (Xjcj). Then check the convergence of uj, if, within desired precision, these uj are convergence, go to step 3, otherwise return to step

6 Step 3. Compute path coefficient using OLS regression. Feed forward back-propagation neural networks Basic model of neural networks (NN) composed of neurons, and arranged in some layers. In biology, neuron is a cell that has the ability to receive and forward the neural signals. While in NN, neuron is defined as an algorithm that implements a mathematical model inspired by the properties of neurons in the biological sciences (Fahey, 2003). Feed forward back-propagation neural networks (BPNN) needs a lot of pairs of input and target for data training, the internal procedures mapping is hard to understand and there are no indications that the entire system can generate acceptable solution (Anderson and McNeill, in Jeatrakul and Wong, 2009). However, BPNN is a robust NN and easily applied to various problems (Chen et al, in Jeatrakul and Wong, 2009). Figure 1 shows the general architecture of BPNN. Performance of BPNN depends on the number of neurons, networks design and methods used in the learning process. FIGURE 1. The backpropagation network structure In the BPNN algorithm, the backpropagation is used to update the weights and biases of the neural networks. Weights and biases are updated using a variety of gradient descent algorithms. The gradient is determined by propagating the backwards computation from output layer to first hidden layer. MULTIBLOCK PLS-PM WITH BACKPROPAGATION NEURAL NETWORKS APPROACH The basic algorithm of multiblock methods PLS Path Modeling involves an iterative procedure both in forward or backward stages. The basic idea of the multiblock PLS path modeling with BPNN in this study is to map the outer relations or measurement model in PLS-PM using back-propagation neural networks instead of using iterative procedure in the original multiblock PLS-PM algorithm

7 Modified multiblock PLS-PM with back-propagation neural networks algorithm that we proposed is given below: a. Initialization Set tj and uj = first column of Xj for j increase from 1 to J. b. Backward stage. For j decrease from J to 1: - If Xj predicts no blocks, then set tj = uj. - If Xj predicts only one block Xj then 1) train the data on uj to Xj with BPNN, 2) Compute tj = (Xjwj) where wj is the result of data training. - If Xj predicts more than one blocks (say h blocks), then 1) Uj = [uj1, uj2,, ujh] 2) do a data training on Uj toward Xj using BPNN 3) Compute tj = (Xjwj) where wj is the result of data training. c. Forward stage. For j increase from 1 to J: - If Xj not predicted by any blocks: set uj = tj - If Xj predicted by only one block Xj : 1) train the data on tj to Xj using BPNN, 2) Compute uj = (Xjcj) where cj is the result of data training. - If Xj predicted by more than one block (h blocks): 1) Tj = [tj1, tj2,, tjh] 2) do a data training on Tj toward Xj using BPNN 3) Compute uj = (Xjcj) where cj is the result of data training. d. Compute path coefficient and factor loading using OLS regression. MODEL EVALUATION Evaluation of PLS path modeling consists of a two-stage process, i.e the assessment on the measurement model and the assessment on the structural model. To assess the measurement models, it is important to check whether the indicators are reflective or formative. For reflective indicators, we can evaluate three aspects: unidimensionality of indicators, the indicators can be explained by the latent variables, and the difference of latent variables. One of the tools for evaluating the unidimensionality of a latent variable is Cronbach's alpha coefficient. Cronbach's alpha coefficient evaluates how well a block indicators measure the related latent variables. As an alternative, we can use Dillon Goldstein (DG) rho to evaluate unidimensionality of the latent variable. Evaluation of unidimensionality of the latent variables was performed to assess the composite reliability. Seidel & Back (2009) suggests a limit value of DG rho is at least 0.7 as moderate composite reliability. To assess the indicators can be explained by the latent variables or not, we can use communality and average variance extracted (AVE). The communality measures how the latent variable explain the variance from the existing manifest variables. Meanwhile, AVE measures the variance that can be captured by the latent variables from their indicators. If the value of the square root of AVE from each construct is greater than the value of the correlation between the construct and others construct in the model, then we can say that the model has a good discriminant validity (Fornell and Larcker, 1981 in Ghozali, 2008)

8 Evaluation of inner structural model is performed by check the R-squared value of the dependent latent variables and also by check the path coefficient of structural model. R-squared is the coefficient of determination that has the interpretation exactly same as in regression analysis, which is the percentage of variance explained by the model. To assess the overall model, we may use the global criterion of goodness of fit (GoF), where GoF is the geometric average of the average communality and the average R-squared (Amato et al, 2004). BOOTSTRAPPING Because of PLS path modeling does not depend on any assumption of statistical distribution, the classical theory based of significance test of model parameter estimators could not be applied. Therefore, resampling procedure such as blindfolding, jackknifing and bootstrapping, had been developed to obtain information about the estimation of the variability of the parameters. The bootstrapping technique is more superior than the other two resampling methods (Temme et al, 2006). In the bootstrap method, we generate some random samples with replacement from the original sample. Then we can compute the bootstrap standard error of the estimator of the parameter θ, i.e the standard deviation of the estimated parameter resulted by each random sample generated. TESTING THE ALGORITHM ON POVERTY MODEL OF EAST JAVA To test the modified multiblock PLS-PM algorithm with back-propagation neural networks approach, and to compare it to the original multiblock PLS-PM algorithm, we use Anuraga (2013) conceptual model on his research about poverty modeling in East Java using SEM-PLS. The conceptual model is shown in Figure 2. This research used the national socio-economic survey (SUSENAS) 2013 data from Baand Pusat Statistik (Indonesia Statistics). This data will be evaluated using two algorithms, then compare the results from both algorithms. FIGURE 2. Conceptual model of poverty

9 In the conceptual model, there are three models of the relationship among blocks of latent variables, i.e Absolute Poverty (LV1) modeled by three other blocks as a predictor, Economic (LV2) modeled by blocks Health (LV4) and Human Resources (LV3) as a predictor, and Human Resources modeled by Health as a predictor. The indicator variables for each latent variables are given below (Table 1). TABLE 1. Latent variables and its indicators No Latent Variables Indicators 1 Absolute poverty Percentage of the poor (Y1) Poverty gap index (Y2) Poverty severity index (Y3) 2 Economy Percentage of the poor age 15 and over who are not working (X1). Percentage of the poor age 15 and over who work in the agricultural sector (X2). Percentage of households that never bought subsidiary rice (X3). Percentage of non-food expenditure per capita (X4). 3. Human resources Percentage of the poor age 15 and over who not finish primary school (X5). Literacy Rate of poor people aged years (X6). School participation rate of the poor aged years (X7). Mean year school (X8). 4. Health Percentage of poor women who use contraception (X9). Percentage of toddler in poor households that the delivery process assisted by health worker (X10). Percentage of toddler in poor households who have had immunization (X11). Percentage of poor housholds with floor area per capita 8 msq (X12). Percentage of poor households with access to clean water (X13) Percentage of poor households with lattrine (their s own or joint) (X14). Percentage of poor households that receive public health insurance services (X15). Life expectancy (X16) Model Evaluation Table 2 shows the assesment of the unidimensionality of the latent variables for both algorithms. Cronbach's Alpha shows the Economy and Human Resources has a value less than 0.70, that means these blocks is less qualified. If we use the DG rho value, the Economy is the only block that less qualified. TABLE 2. Assesment of the unidimensionality of the latent variables Block Cronbach's DG rho 1st Eigen value 1 Health Human res Economy Abs. poverty nd

10 Fornnel and Larcker (1981) states that the average variance extracted (AVE) can be used to measure the reliability of component score of the latent variables. Recommended value for AVE is must be greater than AVE of both algorithms can be seen in Table 3. TABLE 3. Average Variance Extracted (AVE) Block AVE Predictor Predictee MBPLS-PM 1 Health Human res Economy Abs. poverty Modified MBPLS- 1 Health PM 2 Human res Economy Abs. poverty The evaluation of structural models uses R-squared, which R-squared indicates how is the model is able to explain the variance of the data. Table 4 presents the R squared value of a predictive model for both algorithms. TABLE 4. R-squared value of predictive model R-squared Predicted Latent MBPLS- Modified MBPLS- Variable PM PM 1 Health Human res Economy Abs. poverty Referring to Chin (1998), the predictor blocks of Health, Human Resources, and Economic in both algorithm is greater then 0.50, that means the blocks can explain the predicted of block Absolute Poverty moderately, then the predictor blocks of Health and Human Resources is greater then 0.70, that means the blocks are can explain the predicted block Economic substantially, and the predictor block Health able to explain predicted block Human Resource substantially. In Table 5, we can see the comparison of path coefficient parameters, where path coefficient is the regression coefficient of predicted blocks and their predictor blocks. Predicted Blocks TABLE 5. Path Coefficient of sample Path Coefficient Predictor Modified MBPLS- Block MBPLSPM PM Abs. poverty Health Human res Economy Economy Health Human res Human res. Health

11 Meanwhile, to evaluate the overall model, we use the global criterion of goodness of fit (GoF). The value of global criterion of Goodness of Fit from the model resulted by the MBPLS-PM algorithm is , then from the model resulted by modified MBPLS-PM algorithms, the value of global criterion of Goodness of Fit is Bootstrap The significance test for the path coefficient and loading factor performed by t-test using the bootstrap method. Bootstraping is done by taking 100 samples bootstrap. The results are shown in Table 6, and the results indicate that all path coefficient are significant for both algorithm and Table 6 also shows us that bootstrap path coefficients resulted by both algorithm is not different. TABLE 6. Mean of bootstrap path coefficient, t-statistic, and confidence interval Predicted Blocks Predictor Block Mean of Bootstrap t-statistic Conf. Interval lower bound upper bound MBPLS-PM Health Abs. Poverty Human Res Economy Economy Health Human Res Human Res. Health Modified MBPLS-PM Health Abs. Poverty Human Res Economy Economy Health Human Res Human Res. Health

12 Meanwhile, Table 7 shows the results of significance tests of factor loading for predictor latent variable, and Table 8 shows the results of significance tests of factor loading for predicted latent variable. Table 7 and table 8 show that all parameters of factor loading for each variable in all latent variables are significant, both for MBPLSPM algorithm or modified MBPLS-PM. The negative signs in front of the factor loading means the direction of the correlation between latent variable and its indicator is inverse. TABLE 7. Mean of bootstrap of factor loading, t-statistic and confidence interval (predictor latent variables) Predictor Latent Variables Indicators Mean of bootstrap t-statistic lower bound Conf. Interval upper bound MBPLS-PM X Health Human Res. X X X X X X X X X X X X Economy X X X Modified MBPLS-PM X Health Human Res. Economy X X X X X X X X X X X X X X X

13 TABLE 8. Mean of bootstrap of factor loading, t-statistic and confidence interval (predicted latent variables) Predicted Latent Variables MBPLS-PM Human Res. Economy Miskin Indicators Mean of bootstrap t-statistic Conf. Interval lower bound upper bound X X X X X X X X Y Y NA NA Y Modified MBPLS-PM X Human Res. Economy Poverty X X X X X X X Y Y NA NA Y CONCLUSION MBPLS-PM algorithm can be modified using Backpropagation Neural Network approach to replace the iterative process in backward step and forward to get the matrix t and the matrix u. The model parameters obtained by modified MBPLS-PM relatively similar to the parameter obtained by original MBPLS-PM algorithm. The GoF for the overall model for both algorithms provide moderate results and is not much different, that means the modified MBPLS-PM algorithm is not better than the original. However, the modified MBPLSPM algorithm still require a lot of testing with different condition of data and perhaps MBPLSPM algorithm can be developed using different computational method of optimization

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