6.2 - Binary Logistic Regression with a Single Categorical Predictor

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In these results, the dosage is statistically significant at the significance level of 0. You can conclude that changes in the dosage are associated with changes in the probability that the event binary logistic regression model. Assess the coefficient to determine whether a change in a predictor variable makes the event more likely or less likely.

The relationship between the coefficient and the probability depends on several aspects of the analysis, including the link function. Generally, positive coefficients indicate that the event becomes binary logistic regression model likely as the predictor increases. Negative coefficients indicate that the event becomes less likely as the predictor increases.

For more information, go to Coefficients and Regression equation. The coefficient binary logistic regression model Dose is 3. In these results, the model uses the dosage level of a medicine to predict the presence or absence of bacteria in adults. The odds ratio indicates that for every 1 mg increase in the dosage level, the likelihood that no bacteria is present increases by approximately 38 times. In these results, the response indicates whether a consumer bought a cereal and the categorical predictor indicates whether the consumer saw an advertisement about that cereal.

The odds ratio is 3. To determine how well the model fits your data, examine the statistics in the Model Summary table. For binary binary logistic regression model regression, the data format affects the deviance R 2 statistics but not the AIC. For more information, go to For binary logistic regression model information, go to How data formats affect goodness-of-fit in binary logistic regression.

The higher the deviance R 2the better the model fits your data. Deviance R 2 always increases when binary logistic regression model add additional predictors to a model. For example, the best 5-predictor model will always have an R 2 that is at least as high as the best 4-predictor model.

Therefore, deviance R 2 is most useful when you compare models of the same size. For binary logistic regression, the format of the data affects the deviance R 2 value.

Deviance R 2 values are comparable only between models that use the same data format. Deviance R 2 is just one measure of how well the model fits binary logistic regression model data.

Even when a model has a high R 2you should check the residual plots to assess how well the model fits the data. Use adjusted deviance R 2 to compare models that have different numbers of predictors.

Deviance R 2 always binary logistic regression model when binary logistic regression model add a predictor to the model.

The adjusted deviance R 2 value incorporates the number of predictors in the model to binary logistic regression model you choose the correct model. In these results, the model explains For these data, the Deviance R 2 value indicates the model provides a good fit to the data. If additional models are fit with different predictors, use the adjusted Deviance R 2 value and the AIC value to compare how well the models fit the data. If the deviation is statistically significant, you can try a different link function or change the terms in the model.

In these results, the goodness-of-fit tests are all greater than the significance level of 0. Complete the following steps to interpret a regression analysis.

Key output includes the p-value, the odds ratio, R 2and the goodness-of-fit tests. In This Topic Step 1: Determine whether the association between the response and the term is statistically significant Step 2: Understand the effects of the predictors Step 3: Determine how well the model fits your data Step 4: Determine whether the model does not fit the data.

Determine whether the association between the response and the term is statistically significant To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis.

The null hypothesis is that the term's coefficient is equal to zero, which indicates that there is no association between the term and the response. A significance level of 0. The association is statistically significant If the binary logistic regression model is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.

The association is not statistically significant If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. Binary logistic regression model may want to refit the model without the term.

If there are multiple predictors without a statistically significant association with the response, you must reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction. If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows: If a continuous predictor is significant, you can conclude that the coefficient for the predictor does not equal zero.

If a binary logistic regression model predictor is significant, you can conclude that not all the level means are equal. Understand the effects of the predictors Use the odds ratio to understand the effect of a predictor. Odds Ratios for Continuous Predictors Odds ratios that are greater than 1 indicate that the even is more likely to occur as the predictor increases.

Odds ratios that are less than 1 indicate that the event is less likely to occur as the predictor increases. Odds Ratios for Continuous Predictor. Determine how well the model fits your data To determine how well the model fits your data, examine the statistics binary logistic regression model the Model Summary table.

Deviance R-sq The higher the deviance R 2the better the model fits your data. Deviance R-sq adj Use adjusted deviance R 2 to compare models that have different numbers of predictors. The smaller the AIC, the better the model fits the data. However, the model with the smallest AIC does not necessarily fit the data well.

Also use the residual plots to assess how well the model fits the data. Model Summary Deviance R-sq. Determine whether the model does not fit the data Use the goodness-of-fit tests to determine whether the predicted probabilities deviate from the observed probabilities in a way that the binomial distribution does not predict. If the p-value for the goodness-of-fit test is lower than your chosen significance level, the predicted probabilities deviate from the observed probabilities in a way that the binomial distribution does not predict.

This list provides common reasons for the deviation: Incorrect link function Omitted higher-order term for variables in the model Omitted predictor that is not in the model Overdispersion. For binary logistic regression, the format of the data affects the p-value because it changes the number of trials per row.

The approximation to the chi-square distribution that the Pearson test uses is inaccurate when the expected number of events per row in the data is small.

The Hosmer-Lemeshow test does not depend on the number of trials per row in the data as the other goodness-of-fit tests do. When the data have few trials per row, the Hosmer-Lemeshow test is a more trustworthy indicator of how well the model fits the data. By using this site you agree to the use of cookies for analytics and personalized content.

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Kinds of biological variables. Exact test of goodness-of-fit. Chi-square test of goodness-of-fit. G —test of goodness-of-fit. Chi-square test of independence. G —test of independence. Small numbers in chi-square and G —tests. Repeated G —tests of goodness-of-fit. Linear regression and correlation. Using spreadsheets for statistics. Displaying results in graphs.

Displaying results in tables. Choosing the right test. Use multiple logistic regression when you have one nominal variable and two or more measurement variables, and you want to know how the measurement variables affect the nominal variable.

You can use it to predict probabilities of the dependent nominal variable, or if you're careful, you can use it for suggestions about which independent variables have a major effect on the dependent variable. Use multiple logistic regression when you have one nominal and two or more measurement variables. The nominal variable is the dependent Y variable; you are studying the effect that the independent X variables have on the probability of obtaining a particular value of the dependent variable.

For example, you might want to know the effect that blood pressure, age, and weight have on the probability that a person will have a heart attack in the next year. You can perform multinomial multiple logistic regression, where the nominal variable has more than two values, but I'm going to limit myself to binary multiple logistic regression, which is far more common.

The measurement variables are the independent X variables; you think they may have an effect on the dependent variable. While the examples I'll use here only have measurement variables as the independent variables, it is possible to use nominal variables as independent variables in a multiple logistic regression; see the explanation on the multiple linear regression page.

Epidemiologists use multiple logistic regression a lot, because they are concerned with dependent variables such as alive vs. If you are an epidemiologist, you're going to have to learn a lot more about multiple logistic regression than I can teach you here. If you're not an epidemiologist, you might occasionally need to understand the results of someone else's multiple logistic regression, and hopefully this handbook can help you with that. If you need to do multiple logistic regression for your own research, you should learn more than is on this page.

The goal of a multiple logistic regression is to find an equation that best predicts the probability of a value of the Y variable as a function of the X variables. You can then measure the independent variables on a new individual and estimate the probability of it having a particular value of the dependent variable.

You can also use multiple logistic regression to understand the functional relationship between the independent variables and the dependent variable, to try to understand what might cause the probability of the dependent variable to change. However, you need to be very careful. Please read the multiple regression page for an introduction to the issues involved and the potential problems with trying to infer causes; almost all of the caveats there apply to multiple logistic regression, as well.

As an example of multiple logistic regression, in the s, many people tried to bring their favorite bird species to New Zealand, release them, and hope that they become established in nature. We now realize that this is very bad for the native species, so if you were thinking about trying this, please don't.

They determined the presence or absence of 79 species of birds in New Zealand that had been artificially introduced the dependent variable and 14 independent variables, including number of releases, number of individuals released, migration scored as 1 for sedentary, 2 for mixed, 3 for migratory , body length, etc.

Multiple logistic regression suggested that number of releases, number of individuals released, and migration had the biggest influence on the probability of a species being successfully introduced to New Zealand, and the logistic regression equation could be used to predict the probability of success of a new introduction. While hopefully no one will deliberately introduce more exotic bird species to new territories, this logistic regression could help understand what will determine the success of accidental introductions or the introduction of endangered species to areas of their native range where they had been eliminated.

The main null hypothesis of a multiple logistic regression is that there is no relationship between the X variables and the Y variable; in other words, the Y values you predict from your multiple logistic regression equation are no closer to the actual Y values than you would expect by chance. As you are doing a multiple logistic regression, you'll also test a null hypothesis for each X variable, that adding that X variable to the multiple logistic regression does not improve the fit of the equation any more than expected by chance.

While you will get P values for these null hypotheses, you should use them as a guide to building a multiple logistic regression equation; you should not use the P values as a test of biological null hypotheses about whether a particular X variable causes variation in Y.

Multiple logistic regression finds the equation that best predicts the value of the Y variable for the values of the X variables. The Y variable is the probability of obtaining a particular value of the nominal variable. For the bird example, the values of the nominal variable are "species present" and "species absent. This probability could take values from 0 to 1. If the probability of a successful introduction is 0. In gambling terms, this would be expressed as "3 to 1 odds against having that species in New Zealand.

You find the slopes b 1 , b 2 , etc. Maximum likelihood is a computer-intensive technique; the basic idea is that it finds the values of the parameters under which you would be most likely to get the observed results. You might want to have a measure of how well the equation fits the data, similar to the R 2 of multiple linear regression.

However, statisticians do not agree on the best measure of fit for multiple logistic regression. Some use deviance, D , for which smaller numbers represent better fit, and some use one of several pseudo- R 2 values, for which larger numbers represent better fit.

You can use nominal variables as independent variables in multiple logistic regression; for example, Veltman et al. See the discussion on the multiple linear regression page about how to do this. Whether the purpose of a multiple logistic regression is prediction or understanding functional relationships, you'll usually want to decide which variables are important and which are unimportant. In the bird example, if your purpose was prediction it would be useful to know that your prediction would be almost as good if you measured only three variables and didn't have to measure more difficult variables such as range and weight.

If your purpose was understanding possible causes, knowing that certain variables did not explain much of the variation in introduction success could suggest that they are probably not important causes of the variation in success.

The procedures for choosing variables are basically the same as for multiple linear regression: The main difference is that instead of using the change of R 2 to measure the difference in fit between an equation with or without a particular variable, you use the change in likelihood.

Otherwise, everything about choosing variables for multiple linear regression applies to multiple logistic regression as well, including the warnings about how easy it is to get misleading results. Multiple logistic regression assumes that the observations are independent. For example, if you were studying the presence or absence of an infectious disease and had subjects who were in close contact, the observations might not be independent; if one person had the disease, people near them who might be similar in occupation, socioeconomic status, age, etc.

Careful sampling design can take care of this. Multiple logistic regression also assumes that the natural log of the odds ratio and the measurement variables have a linear relationship. It can be hard to see whether this assumption is violated, but if you have biological or statistical reasons to expect a non-linear relationship between one of the measurement variables and the log of the odds ratio, you may want to try data transformations.

Multiple logistic regression does not assume that the measurement variables are normally distributed. Some obese people get gastric bypass surgery to lose weight, and some of them die as a result of the surgery. They obtained records on 81, patients who had had Roux-en-Y surgery, of which died within 30 days. They did multiple logistic regression, with alive vs. Manually choosing the variables to add to their logistic model, they identified six that contribute to risk of dying from Roux-en-Y surgery: Instead, they developed a simplified version one point for every decade over 40, 1 point for every 10 BMI units over 40, 1 point for male, 1 point for congestive heart failure, 1 point for liver disease, and 2 points for pulmonary hypertension.

Graphs aren't very useful for showing the results of multiple logistic regression; instead, people usually just show a table of the independent variables, with their P values and perhaps the regression coefficients.

If the dependent variable is a measurement variable, you should do multiple linear regression. There are numerous other techniques you can use when you have one nominal and three or more measurement variables, but I don't know enough about them to list them, much less explain them. There's a very nice web page for multiple logistic regression. It will not do automatic selection of variables; if you want to construct a logistic model with fewer independent variables, you'll have to pick the variables yourself.

Salvatore Mangiafico's R Companion has a sample R program for multiple logistic regression. Here is an example using the data on bird introductions to New Zealand. In the MODEL statement, the dependent variable is to the left of the equals sign, and all the independent variables are to the right.

The summary shows that "release" was added to the model first, yielding a P value less than 0. Next, "upland" was added, with a P value of 0. Next, "migr" was added, with a P value of 0. However, none of the other variables have a P value less than 0. You need to have several times as many observations as you have independent variables, otherwise you can get "overfitting"—it could look like every independent variable is important, even if they're not. A frequently seen rule of thumb is that you should have at least 10 to 20 times as many observations as you have independent variables.

I don't know how to do a more detailed power analysis for multiple logistic regression. Risk factors associated with mortality after Roux-en-Y gastric bypass surgery. Annals of Surgery Correlates of introduction success in exotic New Zealand birds. This page was last revised July 20, Its address is http: It may be cited as: Handbook of Biological Statistics 3rd ed. Sparky House Publishing, Baltimore, Maryland. This web page contains the content of pages in the printed version. You can probably do what you want with this content; see the permissions page for details.

Handbook of Biological Statistics John H. When to use it Use multiple logistic regression when you have one nominal and two or more measurement variables.

Null hypothesis The main null hypothesis of a multiple logistic regression is that there is no relationship between the X variables and the Y variable; in other words, the Y values you predict from your multiple logistic regression equation are no closer to the actual Y values than you would expect by chance.

How it works Multiple logistic regression finds the equation that best predicts the value of the Y variable for the values of the X variables. Using nominal variables in a multiple logistic regression You can use nominal variables as independent variables in multiple logistic regression; for example, Veltman et al. Selecting variables in multiple logistic regression Whether the purpose of a multiple logistic regression is prediction or understanding functional relationships, you'll usually want to decide which variables are important and which are unimportant.