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The criteria for inclusion of a variable in the model vary between problems and disciplines. The common approach to statistical model building is minimization of variables until the most parsimonious model that describes the data is found which also results in numerical stability and generalizability of the results. Some methodologists suggest inclusion of all clinical and other relevant variables in the model regardless of their significance in order to control for confounding. This approach, however, can lead to numerically unstable estimates and large standard errors. This paper is based on the purposeful selection of variables in regression methods (with specific focus on logistic regression in this paper) as proposed by Hosmer and Lemeshow [1, 2].

The purposeful selection process begins by a univariate analysis of each variable. Any variable having a significant univariate test at some arbitrary level is selected as a candidate for the multivariate analysis. We base this on the Wald test from logistic regression and p-value cut-off point of 0.25. More traditional levels such as 0.05 can fail in identifying variables known to be important [9, 10]. In the iterative process of variable selection, covariates are removed from the model if they are non-significant and not a confounder. Significance is evaluated at the 0.1 alpha level and confounding as a change in any remaining parameter estimate greater than, say, 15% or 20% as compared to the full model. A change in a parameter estimate above the specified level indicates that the excluded variable was important in the sense of providing a needed adjustment for one or more of the variables remaining in the model. At the end of this iterative process of deleting, refitting, and verifying, the model contains significant covariates and confounders. At this point any variable not selected for the original multivariate model is added back one at a time, with significant covariates and confounders retained earlier. This step can be helpful in identifying variables that, by themselves, are not significantly related to the outcome but make an important contribution in the presence of other variables. Any that are significant at the 0.1 or 0.15 level are put in the model, and the model is iteratively reduced as before but only for the variables that were additionally added. At the end of this final step, the analyst is left with the preliminary main effects model. For more details on the purposeful selection process, refer to Hosmer and Lemeshow [1, 2].

This review introduces logistic regression, which is a method for modelling the dependence of a binary response variable on one or more explanatory variables. Continuous and categorical explanatory variables are considered.

where x is the metabolic marker level for an individual patient. This gives 182 predicted probabilities from which the arithmetic mean was calculated, giving a value of 0.04. This was repeated for all metabolic marker level categories. Table 4 shows the predicted probabilities of death in each category and also the expected number of deaths calculated as the predicted probability multiplied by the number of patients in the category. The observed and the expected numbers of deaths can be compared using a χ2 goodness of fit test, providing the expected number in any category is not less than 5. The null hypothesis for the test is that the numbers of deaths follow the logistic regression model. The χ2 test statistic is given by

The test statistic is compared with a χ2 distribution where the degrees of freedom are equal to the number of categories minus the number of parameters in the logistic regression model. For the example data the χ2 statistic is 2.68 with 9 - 2 = 7 degrees of freedom, giving P = 0.91, suggesting that the numbers of deaths are not significantly different from those predicted by the model.

Like ordinary regression, logistic regression can be extended to incorporate more than one explanatory variable, which may be either quantitative or qualitative. The logistic regression model can then be written as follows:

The method of including variables in the model can be carried out in a stepwise manner going forward or backward, testing for the significance of inclusion or elimination of the variable at each stage. The tests are based on the change in likelihood resulting from including or excluding the variable . Backward stepwise elimination was used in the logistic regression of death/survival on lactate, urea and age group. The first model fitted included all three variables and the tests for the removal of the variables were all significant as shown in Table 6.

In logistic regression no assumptions are made about the distributions of the explanatory variables. However, the explanatory variables should not be highly correlated with one another because this could cause problems with estimation.

This thoroughly expanded Third Edition provides an easily accessible introduction to the logistic regression (LR) model and highlights the power of this model by examining the relationship between a dichotomous outcome and a set of covariables.

The goal of logistic regression is to find the best fitting (yet biologically reasonable) model to describe the relationship between the dichotomous characteristic of interest (dependent variable = response or outcome variable) and a set of independent (predictor or explanatory) variables. Logistic regression generates the coefficients (and its standard errors and significance levels) of a formula to predict a logit transformation of the probability of presence of the characteristic of interest:

Rather than choosing parameters that minimize the sum of squared errors (like in ordinary regression), estimation in logistic regression chooses parameters that maximize the likelihood of observing the sample values.

The option to plot a graph that shows the logistic regression curve is only available when there is just one single independent variable. Results After you click OK the following results are displayed:

Cox & Snell R2 and Nagelkerke R2 are other goodness of fit measures known as pseudo R-squareds. Note that Cox & Snell's pseudo R-squared has a maximum value that is not 1. Nagelkerke R2 adjusts Cox & Snell's so that the range of possible values extends to 1. Regression coefficients The logistic regression coefficients are the coefficients b0, b1, b2, ... bk of the regression equation:

An independent variable with a regression coefficient not significantly different from 0 (P>0.05) can be removed from the regression model (press function key F7 to repeat the logistic regression procedure). If P

The Hosmer-Lemeshow test is a statistical test for goodness of fit for the logistic regression model. The data are divided into approximately ten groups defined by increasing order of estimated risk. The observed and expected number of cases in each group is calculated and a Chi-squared statistic is calculated as follows:

The classification table is another method to evaluate the predictive accuracy of the logistic regression model. In this table the observed values for the dependent outcome and the predicted values (at a user defined cut-off value, for example p=0.50) are cross-classified. In our example, the model correctly predicts 74% of the cases.

Another method to evaluate the logistic regression model makes use of ROC curve analysis. In this analysis, the power of the model's predicted values to discriminate between positive and negative cases is quantified by the Area under the ROC curve (AUC). The AUC, sometimes referred to as the C-statistic (or concordance index), is a value that varies from 0.5 (discriminating power not better than chance) to 1.0 (perfect discriminating power).

Sample size calculation for logistic regression is a complex problem, but based on the work of Peduzzi et al. (1996) the following guideline for a minimum number of cases to include in your study can be suggested.

This thoroughly expanded Third Edition provides an easily accessible introduction to the logistic regression (LR) model and highlights the power of this model by examining the relationship between a dichotomous outcome and a set of covariables.

In recent years, outcome prediction models using artificial neural network and multivariable logistic regression analysis have been developed in many areas of health care research. Both these methods have advantages and disadvantages. In this study we have compared the performance of artificial neural network and multivariable logistic regression models, in prediction of outcomes in head trauma and studied the reproducibility of the findings.

1000 Logistic regression and ANN models based on initial clinical data related to the GCS, tracheal intubation status, age, systolic blood pressure, respiratory rate, pulse rate, injury severity score and the outcome of 1271 mainly head injured patients were compared in this study. For each of one thousand pairs of ANN and logistic models, the area under the receiver operating characteristic (ROC) curves, Hosmer-Lemeshow (HL) statistics and accuracy rate were calculated and compared using paired T-tests.

Almost all of the published articles indicate that the performance of ANN models and logistic regression models have been compared only once in a dataset and the essential issue of internal validity (reproducibility) of the models has not been addressed.

The objective of this study was to compare the performance of ANN and multivariate logistic regression models for prediction of mortality in head trauma based on initial clinical data and whether these models are reproducible. We used different variables even if they were interdependent.

Since we were trying to build and compare models for prediction of outcome mainly based on the initial clinical data, only data related to the GCS, tracheal intubation status, age, systolic blood pressure (SBP), respiratory rate(RR), pulse rate(PR), injury severity score (ISS)(upon admission) and outcome were used in our study. In order to prepare the data for the Neural Network software and to enhance the reliability of the data, three variables of systolic blood pressure, respiratory rate and pulse rate were transformed to dichotomous(1,0) variables. Low systolic blood pressure was defined according to the following cutoff points: up to 5 years of age, less than 80 mmHg; and 5 years of age or older, less than 90 mmHg. Respiratory rate of 35 per min and pulse rate of 90 per min were selected as limits for definition of tachypnea and tachycardia. Other variables including GCS, age, systolic blood pressure (SBP) and injury severity score (ISS) variables were also converted from decimal (Base 10) to binary (Base 2). This conversion was carried out in order to render the input data suitable for processing by our ANN software with its default settings. The data and the data format were similar for both ANN and logistic regression models.

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