SAS SPSS STATA Test Answers Bibliography Index Contents xi Recognize that to obtain an odds ratio from a logistic. This new introduction to statistics integrated with STATA and SPSS offers an accessible overview for students in sociology, political science.
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Probabilities range between 0 and 1. Odds are determined from probabilities and range between 0 and infinity. Odds are defined as the ratio of the probability of success and the probability of failure. The odds of success are. This looks a little strange but it is really saying that the odds of failure are 1 to 4. The odds of success and the odds of failure are just reciprocals of one another, i. Next, we will add another variable to the equation so that we can compute an odds ratio.
This example is adapted from Pedhazur Suppose that seven out of 10 males are admitted to an engineering school while three of 10 females are admitted. The probabilities for admitting a male are,. If you are male, the probability of being admitted is 0. If you are female it is just the opposite, the probability of being admitted is 0.
Thus, for a male, the odds of being admitted are 5. Here are the Stata logistic regression commands and output for the example above. In this example admit is coded 1 for yes and 0 for no and gender is coded 1 for male and 0 for female. In Stata, the logistic command produces results in terms of odds ratios while logit produces results in terms of coefficients scales in log odds.
There is a direct relationship between the coefficients produced by logit and the odds ratios produced by logistic. However, most people find risk ratios easier to interpret than odds ratios. In randomized studies it is of course easy to estimate the risk ratio comparing the two treatment intervention groups.
With observational data, where the exposure or treatment is not randomly allocated, estimating the risk ratio for the effect of the treatment is somewhat trickier. The ideal situation — randomized treatment assignment Ideally the assignment to treatment groups would be randomized, as in a randomized controlled trial.
To illustrate the methods to come, we first simulate in Stata a large dataset which could arise in a randomized trial:. This code generates a dataset for 10, individuals. Each has a value of a baseline variable x, which is simulated from a standard N 0,1 distribution. Next, as per a randomized study, we simulated a binary variable z with probability 0. The risk ratio is estimated as 1. Estimating risk ratios from observational data Let us now consider the case of observational data.
To do so we simulate a new dataset, where now the treatment assignment depends on x:. Using a log-link generalized linear model The most obvious approach is to add x to our GLM command:. This however fails to converge, with Stata giving us repeated not concave warnings. This problem, of log link GLMs failing to converge, is well known, and is an apparent road block to estimating a valid risk ratio for the effect of treatment, adjusted for the confounder x.
Estimating the risk ratio via a logistic working model A relatively easy alternative is to use a logistic working model to estimating a risk ratio for treatment which adjusts for x. To do this we first fit an appropriate logistic regression model for y, with x and z as predictors:. This of course gives us an odds ratio for the treatment effect, not a risk ratio. This code first generates a new variable, zcopy, which keeps a copy of the original treatment assignment variable.
By using a logistic regression working model to come up with the predictions, we overcome the numerical difficulties which are often encountered when one instead attempts to directly fit a GLM adjusting for the confounders with a log link and binomial response. The approach we have described here is not new - see this paper by Sander Greenland. However, the approach is still perhaps not widely used. Confidence intervals We have found a point estimate for the risk ratio, but we would of course also like a confidence interval, to indicate the precision of the estimate.
One could bootstrap the whole procedure. An alternative is based on the theory of estimating equations, and is implemented in Stata's teffects command. Thanks to David Drukker, of Stata Corp. The first part, y x, logit , tells Stata that the outcome model for y is a logistic regression with x as a predictor. To calculate the risk ratio and a confidence interval, we first use teffects ra , coeflegend to find the names that Stata has saved the estimates in:.
We can now calculate the risk ratio and its confidence interval using the nlcom. However, since this will give us a symmetric Wald based confidence interval, it is preferable to find this interval for the log risk ratio, and then to back transform the resulting interval to the risk ratio scale:.
Assumptions and alternatives The preceding procedure relies on an assumption that the logistic regression working model is correctly specified. That is, if we use teffects ra, we assume that in each treatment group, y follows a logistic regression model given x.
With additional confounders these can be added to the outcome model, with suitable interactions and non-linear terms if deemed necessary. The teffects command offers a number of alternative approaches to the regression adjustment approach we have taken here. The first is inverse probability weighting IPW by the propensity score, using teffects ipw. Here, rather than modelling the distribution of the outcome conditional on the confounders, we specify a model for the treatment assignment mechanism.
The validity of estimates then relies on the model for treatment assignment being correctly specified. For our simple setup above, this is performed by typing:. See here for a nice paper on the propensity score approach, and some discussion on its merits relative to the regression adjustment approach.
A further approach combines the regression adjustment and IPW approaches teffects ipwra. Here we specify both a model for the outcome and a model for the treatment assignment mechanism.
In this example admit is coded 1 for yes and 0 for no and gender is coded 1 for male and 0 for female. In Stata, the logistic command produces results in terms of odds ratios while logit produces results in terms of coefficients scales in log odds. There is a direct relationship between the coefficients produced by logit and the odds ratios produced by logistic. The range is negative infinity to positive infinity. In regression it is easiest to model unbounded outcomes.
Logistic regression is in reality an ordinary regression using the logit as the response variable. The logit transformation allows for a linear relationship between the response variable and the coefficients:. This means that the coefficients in a simple logistic regression are in terms of the log odds, that is, the coefficient 1. Equation [3] can be expressed in odds by getting rid of the log. This is done by taking e to the power for both sides of the equation.
The odds ratio for gender is defined as the odds of being admitted for males over the odds of being admitted for females:. For this particular example which can be generalized for all simple logistic regression models , the coefficient b for a two category predictor can be defined as. Using the inverse property of the log function, you can exponentiate both sides of the equality [7a] to result in [6]:. In our particular example, e 1. Click here to report an error on this page or leave a comment.
Your Name required. Your Email must be a valid email for us to receive the report! Economists especially refer to what others call the odds as the odds ratio. Below, we will be careful to define our terms. The language here is sometimes confusing because some authors call this the odds ratio. Englishwise, they are correct: it is the odds and the odds are based on a ratio calculation. It is not , however, the odds ratio that is talked about when results are reported.
The odds ratio when results are reported refers to the ratio of two odds or, if you prefer, the ratio of two odds ratios. In particular, we want to consider the ratio of the odds for a one-unit change in one of the components of X. Let us now write. This is the standard result.
The ratio of the odds for a one-unit increase in X i is exp b i. This ratio is constant: it does not change according to the value of the other X s because they cancel out in the calculation. It is the language, and not the math, that leads to the confusion. When we say that in a logistic model, the odds ratio is constant, we mean. Here is an example of computing the odds ratio and the odds with a logistic regression.
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