Larger differences in the "-2 Log L" values lead to smaller p-values more evidence against the reduced model in favor of the full model. For our example, \( G^2 = 5176.510 − 5147.390 = 29.1207\) with \(2 − 1 = 1\) degree of freedom. Notice that this matches the deviance we got in the earlier text above.
Also, notice that the \(G^2\) we calculated for this example is equal to 29.1207 with 1df and p-value <.0001 from "Testing Global Hypothesis: BETA=0" section (the next part of the output, see below).
Testing the null hypothesis that the set of coefficients is simultaneously zero. For example, consider the full model
\(\log\left(\dfrac{\pi}{1-\pi}\right)=\beta_0+\beta_1 x_1+\cdots+\beta_k x_k\)
and the null hypothesis \(H_0\colon \beta_1=\beta_2=\cdots=\beta_k=0\) versus the alternative that at least one of the coefficients is not zero. This is like the overall F−test in linear regression. In other words, this is testing the null hypothesis of the intercept-only model:
\(\log\left(\dfrac{\pi}{1-\pi}\right)=\beta_0\)
versus the alternative that the current (full) model is correct. This corresponds to the test in our example because we have only a single predictor term, and the reduced model that removes the coefficient for that predictor is the intercept-only model.
In the SAS output, three different chi-square statistics for this test are displayed in the section "Testing Global Null Hypothesis: Beta=0," corresponding to the likelihood ratio, score, and Wald tests. Recall our brief encounter with them in our discussion of binomial inference in Lesson 2.
Testing Global Null Hypothesis: BETA=0 | |||
---|---|---|---|
Test | Chi-Square | DF | Pr > ChiSq |
Likelihood Ratio | 29.1207 | 1 | <.0001 |
Score | 27.6766 | 1 | <.0001 |
Wald | 27.3361 | 1 | <.0001 |
Large chi-square statistics lead to small p-values and provide evidence against the intercept-only model in favor of the current model. The Wald test is based on asymptotic normality of ML estimates of \(\beta\)s. Rather than using the Wald, most statisticians would prefer the LR test. If these three tests agree, that is evidence that the large-sample approximations are working well and the results are trustworthy. If the results from the three tests disagree, most statisticians would tend to trust the likelihood-ratio test more than the other two.
In our example, the "intercept only" model or the null model says that student's smoking is unrelated to parents' smoking habits. Thus the test of the global null hypothesis \(\beta_1=0\) is equivalent to the usual test for independence in the \(2\times2\) table. We will see that the estimated coefficients and standard errors are as we predicted before, as well as the estimated odds and odds ratios.
Residual deviance is the difference between −2 logL for the saturated model and −2 logL for the currently fit model. The high residual deviance shows that the model cannot be accepted. The null deviance is the difference between −2 logL for the saturated model and −2 logL for the intercept-only model. The high residual deviance shows that the intercept-only model does not fit.
In our \(2\times2\) table smoking example, the residual deviance is almost 0 because the model we built is the saturated model. And notice that the degree of freedom is 0 too. Regarding the null deviance, we could see it equivalent to the section "Testing Global Null Hypothesis: Beta=0," by likelihood ratio in SAS output.
For our example, Null deviance = 29.1207 with df = 1. Notice that this matches the deviance we got in the earlier text above.
An alternative statistic for measuring overall goodness-of-fit is the Hosmer-Lemeshow statistic .
This is a Pearson-like chi-square statistic that is computed after the data are grouped by having similar predicted probabilities. It is more useful when there is more than one predictor and/or continuous predictors in the model too. We will see more on this later.
\(H_0\): the current model fits well \(H_A\): the current model does not fit well
To calculate this statistic:
Warning about the Hosmer-Lemeshow goodness-of-fit test:
In the model statement, the option lackfit tells SAS to compute the HL statistic and print the partitioning. For our example, because we have a small number of groups (i.e., 2), this statistic gives a perfect fit (HL = 0, p-value = 1). Instead of deriving the diagnostics, we will look at them from a purely applied viewpoint. Recall the definitions and introductions to the regression residuals and Pearson and Deviance residuals.
The Pearson residuals are defined as
\(r_i=\dfrac{y_i-\hat{\mu}_i}{\sqrt{\hat{V}(\hat{\mu}_i)}}=\dfrac{y_i-n_i\hat{\pi}_i}{\sqrt{n_i\hat{\pi}_i(1-\hat{\pi}_i)}}\)
The contribution of the \(i\)th row to the Pearson statistic is
\(\dfrac{(y_i-\hat{\mu}_i)^2}{\hat{\mu}_i}+\dfrac{((n_i-y_i)-(n_i-\hat{\mu}_i))^2}{n_i-\hat{\mu}_i}=r^2_i\)
and the Pearson goodness-of fit statistic is
\(X^2=\sum\limits_{i=1}^N r^2_i\)
which we would compare to a \(\chi^2_{N-p}\) distribution. The deviance test statistic is
\(G^2=2\sum\limits_{i=1}^N \left\{ y_i\text{log}\left(\dfrac{y_i}{\hat{\mu}_i}\right)+(n_i-y_i)\text{log}\left(\dfrac{n_i-y_i}{n_i-\hat{\mu}_i}\right)\right\}\)
which we would again compare to \(\chi^2_{N-p}\), and the contribution of the \(i\)th row to the deviance is
\(2\left\{ y_i\log\left(\dfrac{y_i}{\hat{\mu}_i}\right)+(n_i-y_i)\log\left(\dfrac{n_i-y_i}{n_i-\hat{\mu}_i}\right)\right\}\)
We will note how these quantities are derived through appropriate software and how they provide useful information to understand and interpret the models.
COMMENTS
The alternative hypothesis states that not every coefficient is simultaneously equal to zero. The following examples show how to decide to reject or fail to reject the null hypothesis in both simple logistic regression and multiple logistic regression models.
Testing a single logistic regression coefficient using LRT logit(πi) = β0 + β1x1i + β2x2i We want to test H0 : β2 = 0 vs. HA : β2 6= 0 Our model under the null hypothesis is
12.1 - Logistic Regression. Logistic regression models a relationship between predictor variables and a categorical response variable. For example, we could use logistic regression to model the relationship between various measurements of a manufactured specimen (such as dimensions and chemical composition) to predict if a crack greater than 10 ...
Logistic regression is a statistical method for modeling the dependence of a categorical (binomial) outcome variable on one or more categorical and continuous predictor variables (Bewick et al 2005). The logistic function may be used to transform a sigmoidal curve to a more or less straight line while also changing the range of the data from ...
10.5 Hypothesis Test In logistic regression, hypotheses are of interest: the null hypothesis, which is when all the coefficients in the regression equation take the value zero, and the alternate hypothesis that the model currently under consideration is accurate and differs significantly from the null of zero, i.e. gives significantly better than the chance or random prediction level of the ...
In this handout, we'll examine hypothesis testing in logistic regression and make comparisons between logistic regression and OLS. A separate handout provides more detail about using Stata.
We reject the null hypothesis and conclude at least one of the variables is useful. STATA also gives an equivalent of R2-adjusted called the "Pseudo R-squared".
The Null hypothesis \(\left(H_{O}\right)\) is a statement about the comparisons, e.g., between a sample statistic and the population, or between two treatment groups. The former is referred to as a one-tailed test whereas the latter is called a two-tailed test. The null hypothesis is typically "no statistical difference" between the ...
13.2 - Logistic Regression. Logistic regression models a relationship between predictor variables and a categorical response variable. For example, we could use logistic regression to model the relationship between various measurements of a manufactured specimen (such as dimensions and chemical composition) to predict if a crack greater than 10 ...
Extension of logistic regression to more than 2 categories Suppose Y takes values in {1, 2, …, K}, then we can use a linear model for the log odds against a baseline category (e.g. 1): for j ≠ 1
Notes on Chapter 5: Logistic Regression, a course by MITx that covers the basics of logistic regression.
When testing the null hypothesis that there is no association between vomiting and age we reject the null hypothesis at the 0.05 alpha level ( z = -3.89, p-value = 9.89e-05). On average, the odds of vomiting is 0.98 times that of identical subjects in an age group one unit smaller.
The answer is the P P -value for the test of the null hypothesis that the logistic regression slope is zero. If this P P -value is not small, our graph may be misleading.
The null hypothesis, which is when all the coefficients in the regression equation take the value zero, and The alternate hypothesis that the model currently under consideration is accurate and differs significantly from the null of zero, i.e. gives significantly better than the chance or random prediction level of the null hypothesis.
When we run a logistic regression on Serena's polling data the output indicates a log odds of 1.21. We look at the "Z-Value" and see a large value (15.47) which leads us to reject the null hypothesis that household incomes does not tell us anything about the log odds of voting for Serena.
Like other regression techniques, logistic regression involves the use of two hypotheses: 1.A Null hypothesis: null hypothesis beta coefficient is equal to zero, and, 2. Alternative hypothesis: Alternative hypothesis assumes that beta coefficient is not equal to zero.
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 ...
18.1 What is logistic regression used for? Logistic regression is useful when we have a response variable which is categorical with only two categories. This might seem like it wouldn't be especially useful, however with a little thought we can see that this is actually a very useful thing to know how to do. Here are some examples where we might use logistic regression.
We introduce a projection-based test for assessing logistic regression models using the empirical residual marked empirical process and suggest a model-based bootstrap procedure to calculate critical values. We comprehensively compare this test and Stute and Zhu's test with several commonly used goodness-of-fit (GoF) tests: the Hosmer-Lemeshow test, modified Hosmer-Lemeshow test, Osius ...
This page offers all the basic information you need about logistic regression analysis. It is part of Statkat's wiki module, containing similarly structured info pages for many different statistical methods. The info pages give information about null and alternative hypotheses, assumptions, test statistics and confidence intervals, how to find p values, SPSS how-to's and more. To compare ...
This tutorial explains how to interpret logistic regression coefficients, including an example.
This involves two aspects, as we are dealing with the two sides of our logistic regression equation. First, consider the link function of the outcome variable on the left hand side of the equation. We assume that the logit function (in logistic regression) is the correct function to use. Secondly, on the right hand side of the equation, we ...
Logistic Regression | SPSS Annotated Output This page shows an example of logistic regression with footnotes explaining the output. These data were collected on 200 high schools students and are scores on various tests, including science, math, reading and social studies (socst). The variable female is a dichotomous variable coded 1 if the student was female and 0 if male.
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.
71 As far as I understand the Wald test in the context of logistic regression is used to determine whether a certain predictor variable X is significant or not. It rejects the null hypothesis of the corresponding coefficient being zero.
The slope line represents a simple linear regression analysis for the indicated cancer type. ... p-value is for the test with the null hypothesis that the effect of the doubling time is zero from ...
6.2.3 - More on Model-fitting. Suppose two models are under consideration, where one model is a special case or "reduced" form of the other obtained by setting k of the regression coefficients (parameters) equal to zero. The larger model is considered the "full" model, and the hypotheses would be. H 0: reduced model versus H A: full model.