# S The Difference Between Predictive Modeling and Regression Patricia B. Cerrito, University of Louisville, Louisville, KY

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1 S The Difference Between Predictive Modeling and Regression Patricia B. Cerrito, University of Louisville, Louisville, KY ABSTRACT Predictive modeling includes regression, both logistic and linear, depending upon the type of outcome variable. However, as the datasets are generally too large for a p-value to have meaning, predictive modeling uses other measures of model fit. Generally, too, there are enough observations so that the data can be partitioned into two or more datasets. The first subset is used to define (or train) the model. The second subset can be used in an iterative process to improve the model. The third subset is used to test the model for accuracy. The definition of best model needs to be considered as well. In a regression model, the best model is one that satisfies the criteria of uniform minimum variance unbiased estimator. In other words, it is only best in the class of unbiased estimators. As soon as the class of estimators is expanded, best no longer exists, and we must define the criteria that we will use to determine a best fit. There are several criteria to consider. For a binary outcome variable, we can use the misclassification rate. However, especially in medicine, misclassification can have different costs. A false positive error is not as costly as a false negative error if the outcome involves the diagnosis of a terminal disease. We will discuss the similarities and differences between the types of modeling. INTRODUCTION Regression has been the standard approach to modeling the relationship between one outcome variable and several input variables. Generally, the p-value is used as a measure of the adequacy of the model. There are other statistics, such as the r 2 and the c-statistic (for logistic regression) that are presented, but are not usually considered as important. However, regression has limitations with large samples; all p-values are statistically significant with an effect size of virtually zero. For this reason, we need to be careful when interpreting the model. Instead, we can take a different approach. Because there are so many data values available, we can divide them and create holdout samples. Then, when using predictive modeling, we can use many different models simultaneously, and compare them to find the one that is the best. We can use the traditional regression, but also decision trees and neural network analysis. We can also combine different models. We can focus on accuracy of prediction rather than just identifying risk factors. There is still limited use of predictive modeling in medical research, with the exception of regression models. Most of the use of predictive modeling is fairly recent.(sylvia et al., 2006) While most predictive models are used for examining costs (Powers, Meyer, Roebuck, & Vaziri, 2005), they can be invaluable in improving the quality of care.(hodgman, 2008; Tewari et al., 2001; Weber & Neeser, 2006; Whitlock & Johnston, 2006) In this way, predictive modeling can be used to target the patients at highest risk for more intensive case management.(weber & Neeser, 2006) It has also been used to examine workflow in the healthcare environment.(tropsha & Golbraikh, 2007) Some studies focus on particular types of models such as neural networks.(gamito & Crawford, 2004) In many cases, administrative (billing) data are used to identify patients who can benefit from interventions, and to identify patients who can benefit the most. Most of the use of predictive modeling is fairly recent. In particular, we will discuss some of the issues that are involved when using both linear and logistic regression. Regression requires an assumption of normality. The definition of confidence intervals, too, requires normality. However, most healthcare data are exponential or gamma. According to the Central Limit Theorem, the sample mean can be assumed normal if the sample is sufficiently large. However, if the distribution is exponential, just how large is large enough? If we use nonparametric models, we do not have to be as concerned with the actual population distribution. Also, we want to examine patient-level data rather than group-level data. That will mean that we will want to include information about patient condition in any regression model. Additional assumptions for regression are that the mean of the error term is equal to zero, and that the error term has equal variance for different levels of the input or independent variables. While the assumption of zero mean is almost always satisfied, the assumption of equal variance is not. Often, as the independent variables increase in value, the variance often increases as well. Therefore, modifications are needed to the variables, usually in the form of transformations, substituting the log of an independent variable for the variable itself. Transformations require considerable experience to use properly. In addition, the independent 1

2 variables are assumed to be independent of each other. While the model can tolerate some correlation between these variables, too much correlation will result in a poor model that cannot be used effectively on fresh data. A similar problem occurs if the independent variables have different range scales. If most of the variables are 0-1 indicator functions with patient s age as a scale of 0-100, the value of age will completely dominate the regression equation. The variable scales should be standardized before the model is developed. Probably the most worrisome is the assumption that the error terms are identically distributed. In order for this assumption to be valid, we must assume the uniformity of data entry. That means that all providers must enter poorly defined values in exactly the same way. Unfortunately, such an assumption cannot possibly be valid. Consider, for example, the condition of uncontrolled diabetes, which is one coded patient condition. The term, uncontrolled is not defined. Therefore, the designation remains at the discretion of the provider to define the term. For this reason, different providers will define it differently. LOGISTIC REGRESSION We want to see if we can predict mortality in patients using a logistic regression model. There is considerable incentive to increase the number of positive indicators, called upcoding. The value, α + X + α X +... α X increases as the number of nonzero X s increases. The greater this value, the greater the likelihood that it will cross the threshold value that predicts mortality. However, consider for a moment that just about every patient condition has a small risk of mortality. Once the threshold value is crossed, every patient with similar conditions are predicted to die. Therefore, the more patients who can be defined over the threshold value, the higher the predicted mortality rate, decreasing the difference between predicted and actual mortality. There is considerable incentive to upcode patient diagnoses to increase the likelihood of crossing this threshold vlaue. To simplify, we start with just one input variable to the logistic regression; the occurrence of pneumonia. Table 1 gives the chi-square table for the two variables. Table 1. Chi-square Table for Mortality by Pneumonia Table of pneumonia by DIED pneumonia Frequency Row Pct Col Pct DIED Total Total Frequency Missing = 3041 Approximately 7% of the patients with pneumonia died compared to just under 2% generally. However, if we consider the classification table (Table 2) for a logistic regression with pneumonia as the input and mortality as the outcome variable, the accuracy rate is above 90% for any choice of threshold value of less than 1.0, where 100% of the values are to predict non-mortality. Therefore, even though patients with pneumonia are almost 4 times as likely to die compared to patients without pneumonia, pneumonia by itself is a poor predictor of mortality because of the rare occurrence. 2

3 Table 2. Classification Table for Logistic Regression Classification Table Prob Level Correct Incorrect Percentages Correct Specificity POS NEG E E E E3 394E E E3 394E E E3 394E E E We now add a second patient diagnosis to the regression. Table 3 gives the chi-square table for pneumonia and septicemia. Table 3. Chi-square Table for Pneumonia and Septicemia Controlling for septicemia=0 Controlling for septicemia=1 pneumonia Died Total DIED Total Frequency Row Pct Col Pct Total Of the patients with septicemia only (pneumonia=0), 20% died, increasing to 28% with both septicemia and pneumonia. For patients without septicemia but with pneumonia, 5% died. The classification table for the logistic regression is given in Table 4. Table 4. Classification Table for Logistic Regression With Pneumonia and Septicemia Classification Table Prob Level Correct Incorrect Percentages Sensitivity Correct Sensitivity Specificity POS NEG E E E E E E E E E E E E E E E E

4 Prob Level Classification Table Correct Incorrect Percentages Correct Specificity POS NEG E E E E E E E E E E E E3 149E E E3 149E E E3 149E E E3 149E E E3 149E E E3 149E E E3 517E E E3 517E E E Again, for any threshold value below 98%, the logistic regression model will be over 90% accurate by identifying most of the observations as non-occurrences so that the false negative rate is over 70%. In other words, adding a second input variable did not change the problems with the regression, which are caused by attempting to predict a rare occurrence. We add Immune Disorder to the model (Table 5). Table 5. Classification Table Adding Immune Disorder Classification Table Prob Level Correct Incorrect Percentages Sensitivity Correct Sensitivity Specificity POS NEG E E E E E E E E E E E E E E E E E E E E E E E E

5 Classification Table Prob Level Correct Incorrect Percentages Correct Sensitivity Specificity POS NEG E E E E E E E E E E E E E E E E E E E E3 149E E E3 258E E E3 258E E E3 258E E4 103E E E E The problem still persists, and will continue to persist regardless of the number of input variables. We need to change the sample size so that the group sizes are close to equal. PREDICTIVE MODELING IN SAS ENTERPRISE MINER Figure 1 gives a diagram of a predictive model in SAS Enterprise Miner. Enterprise Miner includes the standard types of regression, artificial neural networks, and decision trees. The regression model will choose linear or logistic automatically, depending upon the type of outcome variable. Figure 1 shows that many different models can be used. Once defined, the models are compared and the optimal model chosen based upon pre-selected criteria. Then, additional data can be scored so that patients, in this example, at high risk for adverse events can be identified for more aggressive treatment. The purpose of the partition node in Figure 1 is to divide the data into training, validation, and testing subsets, by default, a 40/30/30 split in the data. Usually, the datasets are large enough that such a partitioning is possible. The training set is used to define the model; the testing set is a holdout sample used as fresh data to test the accuracy of the model. The validation set is not needed for regression; it is needed for neural networks and any model that is defined iteratively. The model is examined on the validation set, and adjustments are made to the model if necessary. This process is repeated until no more changes are necessary. 5

6 Figure 1. Predictive Modeling of Patient Outcomes For predicting a rare occurrence, one more node is added to the model in Figure 1, the sampling node (Figure 2). This node uses all of the observations with the rare occurrence, and then takes a random sample of the remaining data. While the sampling node can use any proportional split, we recommend a 50:50 split. Figure 3 shows how the defaults are modified in the sampling node of SAS Enterprise Miner to make predictions. 6

7 Figure 2. Addition of Sampling Node 7

8 Figure 3. Change to Defaults in Sampling Node The first arrow indicates that the sampling is stratified, and the criterion is level based. The rarest level (in this case, mortality) is sampled so that it will consist of half (50% sample proportion) of the sample. Consider the problem of predicting mortality that was discussed in the previous section on logistic regression. We use just the same three patient diagnoses of pneumonia, septicemia, and immune disorder that we used previously. However, in this case, we use the sampling node to get a 50/50 split in the data. We use all of the models depicted in Figure 1. According to the model comparison, the rule induction provides the best fit, using the misclassification rate as the measure of best. We first look at the regression model, comparing the results to those in the previous chapter when a 50/50 split was not performed. The overall misclassification rate is 28%, with the divisions as shown in Table 6. Table 6. Misclassification in Regression Model Target Outcome Target Percentage Outcome Percentage Count Total Percentage Training Data Validation Data

9 Note that the misclassification becomes more balanced between false positives and false negatives with a 50/50 split in the data. The model gives heavier weight to false positives than it does to false negatives. We also want to examine the decision tree model. While it is not the most accurate model, it is one that clearly describes the rationale behind the predictions. This tree is given in Figure 4. The tree shows that the first split occurs on the variable, Septicemia. Patients with Septicemia are more likely to suffer mortality compared to patients without Septicemia. As shown in the previous chapter, the Immune Disorder has the next highest level of mortality followed by Pneumonia. Figure 4. Decision Tree Results Since rule induction is identified as the best model, we examine that one next. The misclassification rate is only slightly smaller compared to the regression model. Table 7 gives the classification table. Table 7. Misclassification in Rule Induction Model Target Outcome Target Percentage Outcome Percentage Count Total Percentage Training Data Validation Data 9

10 Target Outcome Target Percentage Outcome Percentage Count Total Percentage Training Data The results look virtually identical to those in Table 6. For this reason, the regression model, although not defined as the best, can be used to predict outcomes when only these three variables are used. The similarities in the models can also be visualized in the ROC (received-operating curve) that graphs the sensitivity versus one minus the specificity (Figure 5). The curves for rule induction and regression are virtually the same. Figure 5. Comparison of ROC Curves MANY VARIABLES IN LARGE SAMPLES There can be hundreds if not thousands of variables collected for each patient. These are far too many to include in any predictive model. The use of too many variables can cause the model to over-fit the results, inflating the outcomes. Therefore, there needs to be some type of variable reduction method. In the past, factor analysis has been used to reduce the set of variables prior to modeling the data. However, there is now a more novel method available (Figure 6). In our example, there are many additional variables that can be considered in this analysis. Therefore, we use the variable selection technique to choose the most relevant. We first use the decision tree followed by regression, and then regression followed by the decision tree. 10

11 Figure 6. Variable Selection Using the decision tree to define the variables, Figure 7 shows the ones that remain for the modeling. Figure 7. Decision Tree Variables This tree shows that age, length of stay, having septicemia, immune disorder, and total charges are related to mortality. The remaining variables have been rejected from the model. The rule induction is the best model, and the misclassification rate decreases to 22% with the added variables. The ROC curve looks considerably improved (Figure 8). 11

12 Figure 8. ROC Curves for Models Following Decision Tree The ROC curve is much higher compared to that in Figure 5. If we use regression to perform the variable selection, the results remain the same. In addition, a decision tree is virtually the same when it follows the regression compared to when it precedes (Figure 9). Figure 9. Decision Tree Following Regression The above example only used three possible diagnosis codes. We want to expand upon the number of diagnosis codes, and also to use a number of procedure codes. In this example, we restrict our attention to 12

13 patients with a primary diagnosis of COPD (chronic obstructive pulmonary disease resulting primarily from smoking). There are approximately 245,000 patients in the NIS dataset. Table 8 gives the list of diagnosis codes used; Table 9 gives a list of procedure codes used as well. Table 8. Diagnosis Codes Used to Predict Mortality Condition ICD9 Codes Acute myocardial 410, 412 infarction Congestive heart 428 failure Peripheral 441,4439,7854,V434 vascular disease Cerebral vascular accident Dementia 290 Pulmonary 490,491,492,493,494,495,496,500,501,502,503,504,505 disease Connective 7100,7101,7104,7140,7141,7142,7148,5171,725 tissue disorder Peptic ulcer 531,532,533,534 Liver disease 5712,5714,5715,5716 Diabetes 2500,2501,2502,2503,2507 Diabetes 2504,2505,2506 complications Paraplegia 342,3441 Renal disease 582,5830,5831,5832,5833,5835,5836,5837,5834,585,586,588 Cancer 14,15,16,17,18,170,171,172,174,175,176,179,190,191,193, 194,1950,1951,1952, 1953,1954,1955,1958,200,201,202,203, 204,205,206,207,208 Metastatic 196,197,198,1990,1991 cancer Severe liver 5722,5723,5724,5728 disease HIV 042,043,044 Table 9. Procedure Codes Used to Predict Mortality 13

14 pr Procedure Translation Frequency Percent 9904 Transfusion of packed cells Venous catheterization, not elsewhere classified Continuous mechanical ventilation for less than 96 consecutive hours Closed [endoscopic] biopsy of bronchus Continuous mechanical ventilation for 96 consecutive hours or more Thoracentesis Hemodialysis Insertion of endotracheal tube Injection of antibiotic Respiratory medication administered by nebulizer Diagnostic ultrasound of heart Esophagogastroduodenoscopy [EGD] with closed biopsy Continuous positive airway pressure Closed endoscopic biopsy of lung Computerized axial tomography of thorax Other endoscopy of small intestine If we perform standard logistic regression without stratified sampling, the false positive rate remains small (approximately 3-4%), but with a high false negative rate (minimized at 38%). Given the large dataset, almost all of the input variables are statistically significant. The percent agreement is 84% and the ROC curve looks fairly good (Figure 10). Figure 10. ROC Curve for Traditional Logistic Regression 14

15 If we perform predictive modeling, the accuracy rate drops to 75%, but the false negative rate is considerably improved. Figure 11 gives the ROC curve from predictive modeling. Figure 11. ROC From Predictive Modeling CHANGE IN SPLIT IN THE DATA All of the analyses in the previous section assumed a 50/50 split between mortality and non-mortality. We want to look at the results if mortality composes only 25% of the data, and 10% of the data. Table 10 gives the regression classification breakdown for a 25% sample; Table 11 gives the breakdown for a 10% sample. Table 10. Misclassification Rate for a 25% Sample Target Outcome Target Percentage Outcome Percentage Count Total Percentage Training Data Validation Data Note that the ability to classify mortality accurately is decreasing with the decrease of the split; almost all of the observations are classified as non-mortality. The decision tree (Figure 12) is considerably different from that in Figure 9 with a 50/50 split. Now, the procedure of Esophagogastroduodenoscopy gives the first leaf of the tree. Figure 12. Decision Tree for 25/75 Split in the Data 15

16 Table 11. Misclassification Rate for a 10% Sample Target Outcome Target Percentage Outcome Percentage Count Total Percentage Training Data Validation Data Note that the trend shown in the 25% is even more exaggerated in the 10% sample. Figure 13 shows that the decision tree has changed yet again. It now includes the procedure of continuous positive airway pressure and the diagnosis of congestive heart failure. AT a 1% sample, the misclassification becomes even more disparate. Figure 13. Decision Tree for 10% Sample 16

17 ADDITION OF WEIGHTS FOR DECISION MAKING In most medical studies, a false negative is more costly to the patient compared to a false positive. This occurs because a false positive generally leads to more invasive tests; however, a false negative means that a potentially life-threatening illness will go undiagnosed, and hence, untreated. Therefore, we can weight a false negative at higher cost, and then change the definition of a best model to one that minimizes costs. The problem is to determine which costs to use. The best thing to do is to experiment with magnitudes of difference in cost between the false positive and false negative to see what happens. At a 1:1 ratio, the best model is still based upon the misclassification rate. A change to a 5:1 ratio indicates that a false negative is five times as costly compared to a false positive. A 10:1 ratio makes it ten times as costly. We need to determine if changes to this ratio result in changes to the optimal model. INTRODUCTION TO LIFT Lift allows us to find the patients at highest risk for occurrence, and with the greatest probability of accurate prediction. This is especially important since these are the patients we would want to take the greatest care for. Using lift, true positive patients with highest confidence come first, followed by positive patients with lower confidence. True negative cases with lowest confidence come next, followed by negative cases with highest confidence. Based on that ordering, the observations are partitioned into deciles, and the following statistics are calculated: 17

18 The Target density of a decile is the number of actually positive instances in that decile divided by the total number of instances in the decile. The Cumulative target density is the target density computed over the first n deciles. The lift for a given decile is the ratio of the target density for the decile to the target density over all the test data. The Cumulative lift for a given decile is the ratio of the cumulative target density to the target density over all the test data. Given a lift function, we can decide on a decile cutpoint so that we can predict the high risk patients above the cutpoint, and predict the low risk patients below a second cutpoint, while failing to make a definite prediction for those in the center. In that way, we can dismiss those who have no risk, and aggressively treat those at highest risk. Lift allows us to distinguish between patients without assuming a uniformity of risk. Figure 14 shows the lift for the testing set when we use just the three input variables of pneumonia, septicemia, and immune disorder. Figure 14. Lift Function for Three-Variable Input Random chance is indicates by the lift value of 1.0; values that are higher than 1.0 indicate that the observations are more predictable compared to random chance. In this example, 40% of the patient records have a higher level of prediction than just chance. Therefore, we can concentrate on these 4 deciles of patients. If we use the expanded model that includes patient demographic information plus additional diagnosis and procedure codes for COPD, we get the lift shown in Figure 15. The model can now predict the first 5 deciles of patient outcomes. Figure 15. Lift Function for Complete Model Therefore, we can predict accurately those patients most at risk for death; we can determine which patients can benefit from more aggressive treatment to reduce the likelihood that this outcome will occur. DISCUSSION Given large datasets and the presence of outliers, the traditional statistical methods are not always applicable or meaningful. Assumptions can be crucial to the applicability of the model, and assumptions are 18

19 not always carefully considered. The assumptions of a normal distribution and uniformity of data entry are crucial and need to be considered carefully. The data may not give high levels of correlation, and regression may not always be the best way to measure associations. It must also be remembered that associations in the data as identified in regression models do not demonstrate cause and effect. CONTACT INFORMATION Patricia Cerrito Department of Mathematics University of Louisville Louisville, KY (fax) SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are trademarks of their respective companies. 19

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