Participants and Data Collection

The data used in this study included LPGA players, falling within the 60^th rank (money leaders), from over a period of 25 years from 1993 to 2017; i.e., the annual average value of 1,500 players (60 players multiplied by 25 years). The data were collected from the LPGA homepage (http://www.lpga.com/) Because the data on the LPGA homepage did not collect private identifier information such as telephone numbers, home addresses, social security numbers, etc., ethical approval was not required for this experimental study. The performance variables chosen were those that were being measured and used in the current LPGA analyses. The variables were reconstituted in this study as independent variables (predicting variables), which were continuous variables, and dependent variables (response variables), which were categorical variables (Table 1).

Experimental Approach to the Problem

The data analysis aimed to determine key performance variables that affected the possibility of winning, the variable that was the most significant, and whether the player would win a game or be in the lead in wins. Four prediction models, i.e., classification tree analysis, logistic regression analysis, discriminant function analysis, and artificial neural network analysis, were employed. The most accurate model was selected, according to the purpose of the study.

Procedures

The player’s accumulated raw data released by the LPGA were arranged using Microsoft Office Excel 2010 (Microsoft Corporation, Redmond, WA, USA), and the result was deduced using the IBM SPSS 22.0 (IBM Corp., Armonk, NY, USA) statistical program. In the first round of analysis, we used classification accuracy as a basis to find the possibility that a certain player could win the game in the LPGA, using the four prediction models (discriminant function analysis, classification tree analysis, logistic regression analysis, and artificial neural network analysis (multilayer perceptron, MLP)). One-way analysis of variance (ANOVA; post-hoc: least significant difference [LSD] test) was used if there was a mean difference in the classification accuracy of the four prediction models.

The input predicting variables of the four prediction models were divided into skill variables (driving accuracy [DA], driving distance [DD], sand saves [SS], GIR, and PA), skill result variables (birdies, eagles, par3 scoring average [P3A], par4 scoring average [P4A], and par5 scoring average [P5A]), and season outcome variables (official money [OM], scoring average [SA], top 10 finish% [T10], 60-strokes average [60SA], and rounds under par [RUP]). When inputting these predicting variables as dependent variables, they were divided into both two groups (victory/no victory) and three groups (no victory/one victory/multiple victories). From the results of the four prediction models, the standardized discriminant function coefficient, normalization importance or Wald value, which are importance indexes linking the independent variable to the dependent variable, could be obtained. Finally, one-way ANOVA and the post-hoc (LSD) test were conducted to examine the mean difference in the classification accuracy of the four models. Statistical significance was set at 0.05.

Statistical Analyses

In the discriminant function analysis, the function to maximize the group difference of an object based on continuous and discrete variables was deduced, and each participant (player) was classified using Fisher’s linear discriminant functions (Mieke et al., 2014; Shehri and Soliman, 2015). It should be known which group, from among the many groups, included each object to be used in this model. When each group was already known, the category to which each object belonged was classified and predicted by calculating the discriminant score of the individuals in each group by finding the discriminant function:

which could classify each group from the measured variables (Kuligowski et al., 2016; Novak, 2016; Schumm, 2006).

Classification tree analysis was used for classification and prediction by tree-structurally schematizing the decision-making rule. A decision-making tree consists of a node, body, and stems that connect different nodes. The decision-making pattern is found at the top of the node if it repetitively classifies the node according to the tree structure forming process. Before the analysis using this decision-making tree, decision trees have an assumption that prior to analysis, the type of variable is precisely specified according to the measurement level. That is, it should be analyzed whether the variables have been accurately designated for the measuring levels (Surucu et al., 2016).

The methods of growing the tree are classified according to the characteristics of the data and purpose of decision making into chi-squared automatic interaction detection (CHAID), exhaustive CHAID, classification and regression tree (CRT), and quick, unbiased, and efficient statistical tree (QUEST). The classification accuracy was found to be high for the CRT basic data (Hayes et al., 2015). The tree structure was formed by designating the standard and pattern (decision trees are classified according to the purpose of the analysis and the structure of the data) as well as classifying for the purpose of analysis and data structuring. The decision tree is to select the predicting variable and to set the standard of the category when forming a low node from a single upper node. A pure low node was formed by most efficiently classifying the distribution of the target (dependent) variables. In this case, purity was defined as the degree of including individuals in a certain category of the target (dependent) variable. It set the predicting model according to the analysis result and interpreted by grasping the meaning of certain parts, as the decision-making tree described the relationships between variables as tree structures (Linda et al., 2008; Neeley et al., 2007).

The merit of this study is that the process is simpler than the other methods (artificial neural network analysis, discriminant function analysis, regression analysis, and so on), as prediction or classification is described based on the induction rule of the tree structure. In this study, CRTs of four tree-growing methods were used. Homogeneity within nodes was maximized by dividing the parent node for maximum homogeneity of the dependent variable within the child node (Hayes et al., 2015). In the splitting criterion of the classification tree, the status to merge the input variable selection and category when each parent branch formed a child branch was a criterion, and it was processed from the input variable, grasping distribution of the target variable, and child branch forming in sequence (i.e., first from the input variable, then from the grasping distribution, etc.). The degree of classifying the distribution of target variables was measured in terms of the purity or impurity. The purity of the child branch was very high, compared to that of the parent branch. Pruning removed the branch that had high risk of misclassification or inappropriate induction rules.

There is cross-validation and split-sample validation for the validity evaluation. Namely, cross-validation and split-sample validation existed in the assessment of validity. The analytical sample was divided into m (= 2, 3, 4 ...) parts, and the remaining part of the sample was excluded. Thus, each part of the data was used to generate m-1 trees, and 1 was used to evaluate trees. That is, this study used cross-validation that divided analysis samples into parts of m values, made the tree with the rest of certain parts of m values, and conducted model assessment with the remaining one part. Split-sample validation divided the observation samples into training samples (training: 70%) and test samples (test: 30%) and conducted an assessment of the tree with the test samples after making the tree with the training samples. This means that the produced tree, without just being a sample, can perform expended application to a population, which is the origin of the analysis sample. Model assessment could be described with profit charts or risk charts. Namely, the decision tree found the hidden pattern and useful correlations using data and could be used as a reference for decision making in the future, as well as for finding associations between data that were difficult to quantify accurately (Duan et al., 2015).

In logistic regression analysis, variables measured by nominal, ordinal, interval, and ratio scales could be used as independent variables; however, the dependent variables had to be categorical variables that were measured in a nominal scale to analyze and predict whether an individual observation belonged to a certain group. The functional formula of the logistic technique was

which was expressed as P(X)when the Yvalue predicted using X was E(YIX) and E(YIX) had a probability concept when Y was a discrete variable (Pang et al., 2017; Sperandei, 2014). This model was not a linear function, but an S-curve logistic function with an upper limit of 1 and a lower limit of 0, with a problem in analysis as it could not be described as a linear function (Agga and Scott, 2015). The upper and lower limits could be avoided if this probability was converted to logit. The logit relationship with the independent variable can be described by a linear function (Cenker et al., 2009; Zhao et al., 2015)

resulting in ability possibility for linear regression analysis. Thus, the natural log value in brackets, which is on the left-hand side of this logit linear function is an odds-ratio; p, which is the numerator, is the probability that an individual belongs to a certain group; and 1 − p, which is the denominator, is the probability that an individual does not belong to a certain group. Thus, as a result of calculation using n predicting variables (X) in the right-hand side, the bigger the logit value, the higher is the possibility it belongs to the group (Curtis, 2019).

Artificial neural network analysis, by using learning materials in computers, aims to learn the optimum result, apply that result of learning to new data or conditions, and deduce an expected result such as how a human behaves, through learning (Chen and Liu, 2014). The neural network used in this study was composed of three layers (input layer, hidden layer, and output layer), and each layer included several neurons (Chen and Liu, 2014; Nair et al., 2016). The neurons in the hidden layer received the stimulation (every type of information) from the neurons in the input layer and the linear combination

was connected as a weighted value. The bigger this linear combination, the higher the activation the neuron received; it was deactivated in the opposite case (Almassri et al., 2018; Nair et al., 2016).

If the degree of this activation value was S, the activation {logistic functions: S=eL1+eL,(0≤S ≤ 1)} and hyperbolic tangent functions: {S = eL+eLeL+e−L,(−1≤S≤1)were intervened to S=f(L),a conversion from L to S to enable S to take a limited range (0≤S≤1,−1≤S≤1).The output node produced the final response by combining signals from the hidden neuron as weighted values. It applied the weighted value combination of the signal when the target variable was continuous, but it was calculated after converting to probability value for softmax conversion  softmax: Ok=expLk∑j=1kexpLjto enable all categoricaloutput values to show the probability value when S was categorical value (Nair et al., 2016), where k was the output range index and k was the output range (Li et al., 2017). In this study, the output group was two (victory/no victory) or three (no victory, one victory, multiple victories); thus, k was 2 or 3.

The goodness-of-fit of the neural network was obtained by maximizing the corresponding likelihood function using the back-propagation algorithm. Conceptually, this algorithm attempts efficient calculation by combining the learning rate (the intensity in the direction where the slope is the highest) and moment (the intensity in the direction until now) (Jida and Jie, 2015; Smaoui et al., 2018). Namely, the neural-network fitting algorithm was started from a random location, and it actively explored the highest point using a high learning rate at the beginning. It gradually lowered the learning rate to reach the highest point (Sun and Lo, 2018; Xi et al., 2013). This process was repeated at the other locations. The point finally reached by repeating this process dozens of times was not the local highest point, but the global highest point (Nair et al., 2016). It found a weight parameter for which the probability became the maximum. The predicting variable was set to skill (DA, DD, SS, GIR, PA), skill result (birdies, eagles, P3A, P4A, P5A), and season outcome variables (OM, SA, T10, 60SA, RUP), and the dependent variable was categorized to no victory and victory or no victory, one victory, and multiple victories.

Influence of Skill Variable on Achieving Victory

The type of an athlete that belongs to a certain group can be predicted using different models. Namely, it is possible to predict which athlete will belong to which group using a prediction model. Table 2 solves this problem when it comes to the probability of victory between an LPGA rookie and a veteran. Table 2 categorizes the dependent variables according to victory (Yes/No) from the results of four prediction model tables, when the independent prediction variable was set to a skill variable such as DA, DD, SS, GIR or PA.

This discriminant function was significant as the Wilks'λ test statistic was 0.883 (p < 0.001). The classification accuracy of this discriminant function was 74.1% and the importance of the prediction variables was in the order of SS < DA < DD < PA < GIR. The validity evaluation of classification tree analysis, the second model, was described by risk estimates. The misclassification rate was 26.4% and 27.2%, when the classification tree model included training data of the sample and cross-validation, respectively. Namely, this misclassification is a value divided by the misclassified values ((59+335) / 1500), and the total classification accuracy of this model was 73.7%. The importance of the prediction variables was in the order of DA < SS < DD < PA < GIR.

In the goodness-of-fit test of the third model, the binomial logistic regression model, the model was found to be better than the base model, as chi-square ( x² ) was 186.83, which was significant (p < 0.001). The classification accuracy of this model was 74.2%, and the importance of the predicting variables was in the order of SS < DA < DD < PA < GIR.

The goodness-of-fit of the fourth model, the artificial neural network analysis model, was determined by the area under the curve (AUC), and the model improved as the AUC became closer to 1. The AUC of this study model under the receiver operating characteristic (ROC) curve could fall in two categories: 0.736, a group with winning experience, and 0.736, a group without it. With higher accuracy of prediction, the shape of the ROC curve moved further up from the 45° line. The AUC was the area under the ROC curve, the 45° line was a curve corresponding to the random classification ratio, and the AUC was 0.5.

Thus, the AUC was in the range of 0.5 to 1.0, if it was superior to the random classification, and it became close to 1 for a more accurate model. The probability value was calculated by applying the importance index of each predicting variable to the hyperbolic tangent function between the input and hidden layers. If the hidden layer was formed and the weight coefficient value of the variable that belonged to the hidden layer was applied to the softmax function that was applied between the hidden and output layers, the probability value that corresponded to each category (Yes/No) of the finally calculated dependent variable changed from 0 to 1, and group classification criteria could be applied to the classification standard of the group by estimating the sum of probability to 1.0.

The classification accuracy rate from these repeated processes was 75.3%. The importance of predicting variables in this model was in the order of SS <DD < DA < PA < GIR. To sum up, artificial neural network analysis showed a higher prediction accuracy rate than the other three models; i.e., prediction accuracy rates were as follows: classification tree model (73.7%) < discriminant model (74.1%) < binominal logistic regression model (74.2%) < artificial neural network model (75.3%). Moreover, predicting variables that were most significant for determining victory included GIR and PA in all four prediction models (Table 2).

Influence of Skills on Victory

If an LPGA player needs to determine the possibility of victory in a tour, the results in Table 3 will help solve this problem (or will help provide this information). Table 3 is a result table for the four prediction models, based on the category of victory (Yes/No) and the predicting variable, which is an independent variable composed of the skill variables: birdies, eagles, P3A, P4A, and P5A. The discriminant model discriminated between the groups to which each participant belonged, using the coefficient value of the discriminant function. This discriminant function was significant as the test statistic Wilks'λ was 0.879 (p < 0.001).

The classification accuracy of this discriminant function was 74.1% and the importance of the predicting variables was in the order of eagles < P4A < P3A < P5A < birdies. The feasibility study of the second model, the classification tree model, is described by the risk estimate. In the training data of the samples, the misclassification rate of this model was 25.6% and cross-validation showed 26.1% misclassification. Namely, this misclassification was a value divided by misclassified (112+272)/1500, and the total classification accuracy of this model was 74.4%. The importance of predicting variables was in the order of eagles < P5A < P4A < P3A < birdies. The goodness-of-fit of the binominal logistic regression model was better than that of the base model, as the chi-square value (x²) was 188.04, which was significant (p < 0.001).

The classification accuracy of this model was 74.3%, and the importance of predicting variables was in the order of P4A < eagles < P3A < P5A < birdies. In the artificial neural network analysis goodness-of-fit test, the AUC, which was the area under the ROC curve, could take values in two groups: a group with winning experience (0.733) and a group without any experience of victory (0.733). If it was superior to random classification, the AUC was between 0.5 and 1.0, and the model improved as the AUC increased and reached closer to 1; the AUC was 0.5 for this model. If the hidden layer was formed and the weight coefficient value of the variable that belonged to the hidden layer was applied to softmax function that was applied between the hidden and output layers, the probability value that corresponded to each category (Yes/No) of the finally calculated dependent variable changed from 0 to 1, and could be applied to the classification standard of the group by estimating the sum of probability to 1.0.

The classification accuracy rate from these repeated processes was 75.7%. The importance of predicting variables in this model was in the order of eagles < P3A < P5A < P4A < birdies. To sum up, artificial neural network analysis showed higher prediction accuracy rates than the other three models, as in the discriminant model (74.1%) < binominal logistic regression model (74.3%) < classification tree model (74.4%) < artificial neural network model (75.7%). Moreover, the predicting variable that was most important in determining the victory was found to be birdies in all four predicting models (Table 3).

Influence of the Season Outcome on Victory

The data in Table 4 help a player determine the possibility of victory during the LPGA tour. Table 4 is a result table of the four prediction models and the predicting variable is a season outcome such as OM, SA, T10, 60SA, and RUP. The dependent variable is victory (Yes/No).

The discriminant model discriminated between the groups to which each participant belonged, based on the coefficient value of the discriminant function. This discriminant function was significant as the Wilks' λ test statistic was 0.717 (p < 0.001). The classification accuracy of this discriminant function was 78.5% and the importance of the predicting variables was in the order of SA < RUP < 60SA < OM < T10. The evaluation of the validity of the second model, the classification tree model, was described by risk estimates. The misclassification ratio of the model when the sample was training data was 20.3% and cross-validation showed 21.3% misclassification. Namely, this misclassification was a value divided by the wrongly classified (137+167) / 1500, and the total classification accuracy of this model was 79.7%. The importance of predicting variables was in the order of 60SA < RUP < SA < OM < T10. In the goodness-of-fit test of the binominal logistic regression model, the model fit improved compared to the base model as the chi-square (x²) of the analysis model was 477.262, which was significant (p < 0.001).

The classification accuracy of this model was 78.7%, and the importance of the predicting variables was in the order of 60SA < SA < RUP < T10 < OM. In the artificial neural network analysis goodness-of-fit test, the AUC could be in two different groups: a group with winning experience (0.844) and a group without any winning experience (0.844). If it were superior to random classification, the AUC would be between 0.5 and 1.0, and the model improved as the AUC increased and reached closer to 1; the AUC was 0.5 for this model. If the hidden layer was formed and the weight coefficient value of a variable that belonged to the hidden layer was applied to the softmax function between the hidden and output layers, the probability value that corresponded to each category (Yes/No) of the finally calculated dependent variable changed from 0 to 1. Furthermore, this value could be applied to the classification standard of the group by estimating the sum of the probability to 1.0.

The classification accuracy rate from these repeated processes was 80.2%. The importance of predicting variables in this model was in the order of 60SA < RUP < T10 < SA < OM. To sum up, the artificial neural network analysis showed a higher prediction accuracy rate than the other three models, as in the discriminant model (78.5%) < binominal logistic regression model (78.7%) < classification tree model (79.7%) < artificial neural network model (80.2%). Moreover, predicting variables that were most significant in determining victory were T10 and OM in the discriminant model and classification tree, and OM, T10, and SA in the binominal logistic regression model and artificial neural network model (Table 4).

Test of Mean Difference of Classification Accuracy of Prediction Models

Table 5 shows the best model in terms of the classification accuracy from the four prediction models, showing the mean difference in the classification accuracy of the statistic models, arising from the change in the number of independent variables according to the change in the dependent variable level (2 or 3). The test of mean difference of the classification accuracy ratio was conducted by one-way ANOVA and it was significant (p < 0.05). The post-hoc test was necessary to determine the exact difference between the prediction models. The LSD post-hoc test showed that the artificial neural network model had higher classification accuracy than the other three models.