ftp:// Wiley Series in Probability and Statistics. Applied Logistic. Regression. Second Edition. David W. Hosmer. Stanley Lemeshow com o -. Applied Logistic Regression, Third Edition. Author(s). David W. Hosmer, Jr. Stanley Lemeshow · Rodney X. Sturdivant. First published View Table of Contents for Applied Logistic Regression Hosmer and Lemeshow have used very little mathematics, have presented difficult.
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Applied logistic regression. Home · Applied logistic regression Author: David W . Hosmer | Stanley Lemeshow. downloads Views 3MB Size Report. python_data_analysis/Statistics eBook - Hosmer, Lemeshow - Applied Logistic smigabovgrisus.tk Find file Copy path. Fetching contributors Cannot retrieve. by Rodney X. Sturdivant, Stanley Lemeshow, David W. Hosmer, Jr. Applied Logistic Regression, Third Edition emphasizes applications in the health sciences.
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Applied logistic regression
Tools Get online access For authors. Email or Customer ID. Forgot password? In recent years there has been a shift away from deterministic methods for model building to purposeful selection of variables.
However, in the analysis the socially important variables associated with the response variable are selected under purposeful selection. The most common method used to estimate the unknown parameters in linear regression is the Ordinary Least Squares OLS. Under usual assumptions, least square estimations have some desirable properties. But when the OLS method is applied to estimate a model with dichotomous outcome the estimators no longer have these properties.
In such a situation, the most commonly used method of estimating the parameters of a logistic regression model is the method of Maximum Likelihood ML. In logistic regression, the likelihood equations are non-linear explicit function of unknown parameters. Therefore, we use a very effective and well known Newton-Raphson iterative method to solve the equations which is known as iteratively reweighted least square algorithm.
In general, the sample likelihood function is defined as the joint probability function of the random variables. Specifically, suppose y1, y2 yn be the n independent random observations corresponding to the random variables Y1, Y2 n. The solution of the likelihood equations requires special software that is available in most, but not all, statistical packages.
In this study, SPSS In order to determine the worth of the individual regressor in logistic regression, the Wald statistic denoted as Hosmer and Lemeshow, ; Hauck and Donner, : However, several researchers have designed: which was different but equivalent form Rao , Wald, and Jennings The Wald chi square statistics Table 1 agree reasonably well with the assumption that all the individual predictors have significant contribution to predict the response variable.
Table 1: Analysis of maximum likelihood estimates The binary logistic regression model has been fitted under the assumption that we are at least preliminarily satisfied with our efforts at the model building stage.
By this we mean that, to the best of our knowledge, the model contains those variables that should be in the model and the variables have been entered in the correct functional form. Hence, we may like to know how effectively the model has described the outcome variable.
Once, the particular multiple logistic regression model has been fitted, we begin the process of model assessment. The likelihood ratio test is performed to test the overall significance of all coefficients in the model on the basis of test statistic: 3 where, L0 is the likelihood of the null model and L1is the likelihood of the saturated model. The statistic G plays the same role in logistic regression as the numerator of the partial F-test does in linear regression. In order to find the overall goodness-of-fit, Hosmer and Lemeshow and Lemeshow and Hosmer proposed grouping based on the values of the estimated probabilities.
This test is more reliable and robust than the traditional chi-square test Agresti, The large p-value signifies that there is no significant difference between the observed and the predicted values of the outcome.
This indicates that the model seems to fit quite reasonable.
Model building strategy for logistic regression: purposeful selection
A comparison of the observed and expected frequencies in each of the 20 cells Table 3 also shows close agreement within each decile. Hosmer et al. They recommended that overall assessment of fit be examined using a combination of Hosmer-Lemeshow goodness-of-fit test, Osius and Rojek normal approximation test and Stukel test as adjunct of their tests.
The large sample normal approximation to the distribution of the Pearson chi-square statistic derived by Osius and Rojek may be easily computed in any package that has the option to save the fitted values from the logistic regression model and do a weighted linear regression.
Importance of Assessing the Model Adequacy of Binary Logistic Regression
The essential steps in the procedure when we have J covariate patterns are as follows: Step 1: Save the fitted values from the model, denoted as: Step 2:Create the variable Step 3:Create the variable: Step 4:Compute the Pearson chi-square statistic defined as: Step 5:Perform a weighted linear regression of c, defined in step 3, on the model covariates X1, X2, X3, X4 and X5, using weights v, defined in step 2.
It is important to note that the sample size for this linear regression is J, the number of covariate patterns.
Again on the basis of large p-value, we cannot reject the null hypothesis that the model fits quite well. Stukel proposed a two degree-of-freedom test to ascertain whether a generalized logistic model is better than a standard model fit to the data.
Her test determines whether two parameters in a generalized logistic model are equal to zero. Briefly, the two additional parameters allow the tails of the logistic regression model that is the small and large probabilities to be either heavier or lighter than the standard logistic regression model.
This test is not actually a goodness-of-fit test since, it does not compare observed and fitted values. However, it does provide a test of the basic logistic regression model assumption and in that sense it may be considered as a useful adjunct to the Hosmer-Lemeshow and Osius-Rojek goodness of fit tests. Application of the four step procedure given by Stukel to the fitted model in Table 1 yields a value for the partial likelihood ratio test of 0.
Again large p-value signifies we cannot reject the null hypothesis that the logistic regression model is the correct model. An intuitively appealing way to summarize the results of a fitted logistic regression model is via a classification table. This table is the result of cross-classifying the outcome variable with a dichotomous variable whose values are derived from the estimated logistic probabilities.
If the estimated probability exceeds c then we let the derived variable be equal to 1 otherwise it is equal to 0. The most commonly used value for c is 0. The terms sensitivity and specificity come from the classification table.
Sensitivity is the ability of the model to predict an event correctly and specificity is the ability of the model to predict a nonevent correctly. Sensitivity and specificity rely on a single cut point to classify a test result as positive. A more complete description of classification accuracy is given by the area under the ROC Receiver Operating Characteristic curve. This curve, originating from signal detection theory, shows how the receiver operates the existence of signal in the presence of noise.
But ROC curve analysis has more recently been used as a test of model adequacy in Medicine, Psychology, Demography and other areas like data mining. So, it has been considered as a statistical tool to evaluate the performance of a model adequacy. In the social sciences, ROC curve analysis is often called ROC accuracy ratio, a common technique for judging the accuracy of fitted binary logistic regression model.
It plots the probability of detecting true signal sensitivity and false signal 1-specificity for an entire range of possible cut points. The area under the ROC curve gives a quantitative indication of how good the test is. The ideal curve has an area of 1, the worst case scenario is 0. This area provides a measure of the models ability to discriminate between those subjects who experience the outcome of interest versus those who do not.
In this study, for the given model Table 1 , a plot of sensitivity versus 1-specificity over all possible cut points is shown Fig. The curve generated by these points is called the ROC curve and the area under this curve is determined by Mann-Whitney U statistic and is 0.
The accuracy of the test depends on how well the test separates the group being tested into those with or without the criteria in the model. Accuracy is measured by the area under the ROC curve.
If the area lies 0. So, on the basis of the area under the curve indicates our model performance is excellent. Because much of the lack of efforts in assessing the fit of any statistical model and logistic regression in particular, may be traced to a general misunderstanding and confusion by many users of regression methods as to what goodness-of-fit is and what its role is in the modeling process.
Thus, inferences from analysis where there has been no assessment of goodness-of-fit should be viewed with some skepticism.
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Despite the frequent use of logistic regression in the social sciences, considerable confusion exists about its use and interpretation in the modeling process. This conclusion is attributed to a lack of fit, inadequate teaching materials and to unfamiliarity with logistic regression by many instructors Lottes et al. Some of the earlier study illustrated the basic concept of binary logistic regression, analogies between ordinary least square regression and logistic regression but the research did not quantifying the predictive ability of the fitted model.
But the present study illustrated clearly the fitting process of binary logistic regression model under standard assumptions indicating evaluation process of optimum summary measures of goodness-of-fit to determine its predictive ability. Tsiatis had been used maximum partial likelihood test or score test to assess the goodness-of-fit but the disadvantage of this test is that actual values of the observed and estimated expected frequencies need not be obtained.
Thus, the lack of fit was remained hidden there. On the contrary, the advantages of different summary measures in the present study provide observed and expected frequencies that can be easily interpretable and adequate to assess the fit of the model.Overall assessment of fit be examined using a combination of Likelihood Ratio Test, Hosmer-Lemeshow goodness of fit test, Osius and Rojek normal approximation to the distribution of the Pearson chi-square statistic, Stukels test and ROC curve analysis for adequacy of the fitted model.
However, it does provide a test of the basic logistic regression model assumption and in that sense it may be considered as a useful adjunct to the Hosmer-Lemeshow and Osius-Rojek goodness of fit tests.
For more information on interpreting odds ratios see our FAQ page How do I interpret odds ratios in logistic regression? New and updated features include: Applied Logistic Regression, Third Edition emphasizes applications in the health sciences and handpicks topics that best suit the use of modern statistical software.
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