File Name: applied logistic regression wiley series in probability and statistics .zip
- Applied logistic regression
- Applied logistic regression.
- Logistic Regression | Stata Data Analysis Examples
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Applied logistic regression
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. Logistic regression provides a method for modelling a binary response variable, which takes values 1 and 0.
For example, we may wish to investigate how death 1 or survival 0 of patients can be predicted by the level of one or more metabolic markers. As an illustrative example, consider a sample of patients whose levels of a metabolic marker have been measured. The proportions of deaths are estimates of the probabilities of death in each category. It suggests that the probability of death increases with the metabolic marker level. However, it can be seen that the relationship is nonlinear and that the probability of death changes very little at the high or low extremes of marker level.
This pattern is typical because proportions cannot lie outside the range from 0 to 1. The relationship can be described as following an 'S'-shaped curve. Proportion of deaths plotted against the metabolic marker group midpoints for the data presented in Table 1. The logit function is defined as the natural logarithm ln of the odds [ 1 ] of death. That is,. The points now follow an approximately straight line.
The relationship between probability of death and marker level x could therefore be modelled as follows:.
Logit p plotted against the metabolic marker group mid-points for the data presented in Table 1. Although this model looks similar to a simple linear regression model, the underlying distribution is binomial and the parameters a and b cannot be estimated in exactly the same way as for simple linear regression. Instead, the parameters are usually estimated using the method of maximum likelihood, which is discussed below.
When the response variable is binary e. The probability of survival is 1 - p. If the probability of death is assumed to be 0. Maximum likelihood estimation involves finding the value s of the parameter s that give rise to the maximum likelihood. For example, again we shall take the seven deaths occurring out of patients and use maximum likelihood estimation to estimate the probability of death, p. From the graph it can be seen that the value of p giving the maximum likelihood is close to 0.
This value is the maximum likelihood estimate MLE of p. In more complicated situations, iterative techniques are required to find the maximum likelihood and the associated parameter values, and a computer package is required. The odds ratio e b has a simpler interpretation in the case of a categorical explanatory variable with two categories; in this case it is just the odds ratio for one category compared with the other.
This indicates that, for example, the odds of death for a patient with a marker level of 3. The model can be used to calculate the predicted probability of death p for a given value of the metabolic marker. For example, patients with metabolic marker level 2.
The corresponding odds of death for these patients are 0. The metabolic marker level at which the predicted probability equals 0. Solving the equation. After estimating the coefficients, there are several steps involved in assessing the appropriateness, adequacy and usefulness of the model.
First, the importance of each of the explanatory variables is assessed by carrying out statistical tests of the significance of the coefficients. The overall goodness of fit of the model is then tested. Additionally, the ability of the model to discriminate between the two groups defined by the response variable is evaluated.
Finally, if possible, the model is validated by checking the goodness of fit and discrimination on a different set of data from that which was used to develop the model. Wald statistics are easy to calculate but their reliability is questionable, particularly for small samples.
For data that produce large estimates of the coefficient, the standard error is often inflated, resulting in a lower Wald statistic, and therefore the explanatory variable may be incorrectly assumed to be unimportant in the model. Likelihood ratio tests see below are generally considered to be superior. The test for the coefficient of the metabolic marker indicates that the metabolic marker contributes significantly in predicting death. The constant has no simple practical interpretation but is generally retained in the model irrespective of its significance.
The likelihood ratio test for a particular parameter compares the likelihood of obtaining the data when the parameter is zero L 0 with the likelihood L 1 of obtaining the data evaluated at the MLE of the parameter. The test statistic is calculated as follows:.
The goodness of fit or calibration of a model measures how well the model describes the response variable. Assessing goodness of fit involves investigating how close values predicted by the model are to the observed values. When there is only one explanatory variable, as for the example data, it is possible to examine the goodness of fit of the model by grouping the explanatory variable into categories and comparing the observed and expected counts in the categories.
For example, for each of the patients with metabolic marker level less than one the predicted probability of death was calculated using the formula. This gives predicted probabilities from which the arithmetic mean was calculated, giving a value of 0. This was repeated for all metabolic marker level categories. The null hypothesis for the test is that the numbers of deaths follow the logistic regression model.
The Hosmer—Lemeshow test is a commonly used test for assessing the goodness of fit of a model and allows for any number of explanatory variables, which may be continuous or categorical. The observations are grouped into deciles based on the predicted probabilities. Further checks can be carried out on the fit for individual observations by inspection of various types of residuals differences between observed and fitted values.
These can identify whether any observations are outliers or have a strong influence on the fitted model. For further details see, for example, Hosmer and Lemeshow [ 2 ]. Most statistical packages provide further statistics that may be used to measure the usefulness of the model and that are similar to the coefficient of determination R 2 in linear regression [ 3 ].
The values for the example data are 0. The R 2 statistics do not measure the goodness of fit of the model but indicate how useful the explanatory variables are in predicting the response variable and can be referred to as measures of effect size. The value of 0. The discrimination of a model — that is, how well the model distinguishes patients who survive from those who die — can be assessed using the area under the receiver operating characteristic curve AUROC [ 4 ]. The value of the AUROC is the probability that a patient who died had a higher predicted probability than did a patient who survived.
When the goodness of fit and discrimination of a model are tested using the data on which the model was developed, they are likely to be over-estimated.
If possible, the validity of model should be assessed by carrying out tests of goodness of fit and discrimination on a different data set from the original one.
We may wish to investigate how death or survival of patients can be predicted by more than one explanatory variable. As an example, we shall use data obtained from patients attending an accident and emergency unit. Serum metabolite levels were investigated as potentially useful markers in the early identification of those patients at risk for death. Two of the metabolic markers recorded were lactate and urea.
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 [ 2 ]. Tests for the removal of the variables for the logistic regression on the accident and emergency data. Therefore all the variables were retained. For these data, forward stepwise inclusion of the variables resulted in the same model, though this may not always be the case because of correlations between the explanatory variables. Several models may produce equally good statistical fits for a set of data and it is therefore important when choosing a model to take account of biological or clinical considerations and not depend solely on statistical results.
The Wald tests also show that all three explanatory variables contribute significantly to the model. This is also seen in the confidence intervals for the odds ratios, none of which include 1 [ 5 ]. Coefficients and Wald tests for logistic regression on the accident and emergency data. Because there is more than one explanatory variable in the model, the interpretation of the odds ratio for one variable depends on the values of other variables being fixed.
The interpretation of the odds ratio for age group is relatively simple because there are only two age groups; the odds ratio of 4. The odds ratio for the quantitative variable lactate is 1. However, the Nagelkerke R 2 value was 0. Although the contribution of the three explanatory variables in the prediction of death is statistically significant, the effect size is small.
The logistic transformation of the binomial probabilities is not the only transformation available, but it is the easiest to interpret, and other transformations generally give similar results. 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.
Large sample sizes are required for logistic regression to provide sufficient numbers in both categories of the response variable.
The more explanatory variables, the larger the sample size required. With small sample sizes, the Hosmer—Lemeshow test has low power and is unlikely to detect subtle deviations from the logistic model. Hosmer and Lemeshow recommend sample sizes greater than The choice of model should always depend on biological or clinical considerations in addition to statistical results.
Logistic regression provides a useful means for modelling the dependence of a binary response variable on one or more explanatory variables, where the latter can be either categorical or continuous. The fit of the resulting model can be assessed using a number of methods. National Center for Biotechnology Information , U. Journal List Crit Care v. Crit Care. Published online Jan
Applied logistic regression.
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Logistic Regression | Stata Data Analysis Examples
Logistic regression, also called a logit model, is used to model dichotomous outcome variables. In the logit model the log odds of the outcome is modeled as a linear combination of the predictor variables. Please note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and checking, verification of assumptions, model diagnostics and potential follow-up analyses.