Count based data contains events that occur at a certain rate. The rate of occurrence may change over time or from one observation to next. Here are some examples of count based data:. A data set of counts has the following characteristics:. The following table contains counts of bicyclists traveling over various NYC bridges. The counts were measured daily from 01 April to 31 October Here is a time sequenced plot of the bicyclist counts on the Brooklyn bridge:.
The Poisson regression model and the Negative Binomial regression model are two popular techniques for developing regression models for counts. The Poisson distribution has the following P robability M ass F unction. The orange dots predictions are all set to the same value 5. Now we get to the fun part. The job of the regression model is to fit the observed counts y to the matrix of regression values X.
We can also introduce additional regressors such as Month and Day of Month that are derived from Dateand we have the liberty to drop existing regressors such as Date. The following figure illustrates the structure of the Poisson regression model. What might be a good link function f. It turns out the following exponential link-function works great:.
This is a requirement for count based data. In general, we have:. The complete specification of the Poisson regression model for count based data is given as follows:. We reproduce it here:.An Introduction to the Poisson Regression Model
Take a look at the first few rows of this data set:. Our assumption is that the bicyclist counts shown in the red box arise from a Poisson process. Hence we can say that their probabilities of occurrence is given by the Poisson PMF. Here are the probabilities for the first 4 occurrences:.
We can similarly calculate the probabilities for all n counts observed in the training set.
Here is how the joint probability looks like for the entire training set:. Put another way, it is the solution of the equation obtained from differentiating the joint probability equation w. It is easier to differentiate the logarithm of the joint probability equation than the original equation. This logarithmic equation is called the log-likelihood function. For the Poisson regression, the log-likelihood function is given by the following equation:.
As mentioned earlier, we differentiate this log-likelihood equation w. This operation gives us the following equation:.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 or potential follow-up analyses. Example 1. The number of persons killed by mule or horse kicks in the Prussian army per year.
Ladislaus Bortkiewicz collected data from 20 volumes of Preussischen Statistik. These data were collected on 10 corps of the Prussian army in the late s over the course of 20 years. Example 2. The number of people in line in front of you at the grocery store.
Predictors may include the number of items currently offered at a special discounted price and whether a special event e. Example 3. The number of awards earned by students at one high school. Predictors of the number of awards earned include the type of program in which the student was enrolled e.
For the purpose of illustration, we have simulated a data set for Example 3 above. Each variable has valid observations and their distributions seem quite reasonable. In this particular the unconditional mean and variance of our outcome variable are not extremely different. The table below shows the average numbers of awards by program type and seems to suggest that program type is a good candidate for predicting the number of awards, our outcome variable, because the mean value of the outcome appears to vary by prog.
Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable, while others have either fallen out of favor or have limitations. Below we use the poisson command to estimate a Poisson regression model. The i. We use the vce robust option to obtain robust standard errors for the parameter estimates as recommended by Cameron and Trivedi to control for mild violation of underlying assumptions.
This is not a test of the model coefficients which we saw in the header informationbut a test of the model form: Does the poisson model form fit our data? We conclude that the model fits reasonably well because the goodness-of-fit chi-squared test is not statistically significant. If the test had been statistically significant, it would indicate that the data do not fit the model well.
Sometimes, we might want to present the regression results as incident rate ratios, we can use the irr option. These IRR values are equal to our coefficients from the output above exponentiated.The ridge regression model has been consistently demonstrated to be an attractive shrinkage method to reduce the effects of multicollinearity.
The Poisson regression model is a well-known model in application when the response variable is count data. However, it is known that multicollinearity negatively affects the variance of maximum likelihood estimator of the Poisson regression coefficients. To address this problem, a Poisson ridge regression model has been proposed by numerous researchers.
The idea behind the NPRRM is to get diagonal matrix with small values of diagonal elements that leading to decrease the shrinkage parameter, and therefore, the resultant estimator can be better with small amount of bias.
Our Monte Carlo simulation results suggest that the NPRRM estimator can bring significant improvement relative to other existing estimators. In addition, the real application results demonstrate that the NPRRM estimator outperforms both Poisson ridge regression and maximum likelihood estimators in terms of predictive performance.
This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Algamal ZY Diagnostic in poisson regression models. Electron J Appl Stat Anal — Algamal ZY a A new method for choosing the biasing parameter in ridge estimator for generalized linear model. Chemom Intell Lab Syst — Algamal ZY b Shrinkage parameter selection via modified cross-validation approach for ridge regression model.
Commun Stat Simul Comput. Mod Appl Sci — Algamal ZY, Lee MH b Applying penalized binary logistic regression with correlation based elastic net for variables selection. J Mod Appl Stat Method Pak J Stat Oper Res — J Chemom e Commun Stat Simul Comput — Surv Math Appl — Cambridge University Press, Cambridge.
Google Scholar. Technometrics — Commun Stat Theory Method — Kibria BMG Performance of some new ridge regression estimators. J Mod Appl Stat Method — Econ Model — Wiley, New York.One key criterion is the relative value of the variance to the mean after accounting for the effect of the predictors. A previous article discussed the concept of a variance that is larger than the model assumes: overdispersion.
The Pearson Chi 2 dispersion statistic for the model run in that article was 2. If the variance is equal to the mean, the dispersion statistic would equal one. When the dispersion statistic is close to one, a Poisson model fits.
If it is larger than one, a negative binomial model fits better. Plotting the standardized deviance residuals to the predicted counts is another method of determining which model, Poisson or negative binomial, is a better fit for the data. Here is the plot using a Poisson model when regressing the number of visits to the doctor in a two week period on gender, income and health status. The series of waves in the graph is not an unusual structure when graphing count model residuals and predicted outcomes.
Our primary focus is on the scale of the y axis.
A good fitting model will have the majority of the points between negative 2 and positive 2. There should be few points below negative 3 and above positive 3. Adding more predictors to the model can have an impact on improving the plot but the Poisson model is clearly a very poor fitting model for these data.
If we use the same predictors but use a negative binomial model, the graph improves significantly. Notice now the maximum value for the standardized deviance residual is now 4 as compared to 8 for the Poisson model.
The model still has room for improvement.
That would require, if they are available, selecting better predictors of the outcome. We will now regress the count of rabbits per square yard plots on shrub coverage, density of shrubbery and variety of shrubbery.
The Pearson Chi 2 dispersion for this model is 1. As you have seen, graphing the standardized deviance residuals by the predicted outcomes can help us verify which type of model is a better fit for your data. Jeff Meyer is a statistical consultant with The Analysis Factor, a stats mentor for Statistically Speaking membership, and a workshop instructor.
Read more about Jeff here. Tagged as: Count modelsdispersion statisticModel Fitnegative binomialoverdispersionpoissonpredicted countresidual plot. Hi, thanks a lot for this interesting article? What are the commands for the residual plots? Thanks for your kindest support. I created the graphs using Stata.I start with the packages we will need. Then I move into data cleaning and assumptions. The model itself is possibly the easiest thing to run.
Logistic regression, also known as logit regression, is what you use when your outcome variable dependent variable is dichotomous. Do you have a disease, yes or no? Did you use [insert your favorite illicit substance here] in the past week, yes or no?
Rather than estimate beta sizes, the logistic regression estimates the probability of getting one of your two outcomes i. Poisson regression, also known as a log-linear model, is what you use when your outcome variable is a count i. The Poisson distribution is unique in that its mean and its variance are equal. This is often due to zero inflation. Sometimes two processes may be at work: one that determines whether or not an event happens at all and another that determines how many times the event happens when it does.
Using our count variables from above, this could be a sample that contains individuals with and without heart disease: those without heart disease cause a disproportionate amount of zeros in the data and those with heart disease trail off in a tail to the right with increasing amounts of heart attacks. This is why logistic and Poisson regressions go together in research: there is a dichotomous outcome inherent in a Poisson distribution.
Usually, once you install a package, library should be sufficient to load it in the future. I basically string together things available in several places online so that we have everything we need for logistic regression analysis here in one chapter. See endnotes for links and references. We start with the logistic ones. Because we will be using multiple datasets and switching between them, I will use attach and detach to tell R which dataset each block of code refers to.
Attach is a function that brings the dataset into action and detach phases it out again. Location is listed above.Analyzed the data: GM. The package also includes all datasets analyzed in this article. This work is about assessing model adequacy for negative binomial NB regression, particularly 1 assessing the adequacy of the NB assumption, and 2 assessing the appropriateness of models for NB dispersion parameters.
The typically small number of biological samples and large number of genes in RNA-Seq analysis motivate us to address the trade-offs between robustness and statistical power using NB regression models.
One widely-used power-saving strategy, for example, is to assume some commonalities of NB dispersion parameters across genes via simple models relating them to mean expression rates, and many such models have been proposed. As RNA-Seq analysis is becoming ever more popular, it is appropriate to make more thorough investigations into power and robustness of the resulting methods, and into practical tools for model assessment.
In this article, we propose simulation-based statistical tests and diagnostic graphics to address model adequacy. We provide simulated and real data examples to illustrate that our proposed methods are effective for detecting the misspecification of the NB mean-variance relationship as well as judging the adequacy of fit of several NB dispersion models. The negative binomial NB model has been widely adopted for regression of count responses because of its convenient implementation and flexible accommodation of extra-Poisson variability.
Let Y represent a univariate count response variable and X a p -dimensional vector of known explanatory variables.
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The NB distribution can be derived as a Poisson-gamma mixture model. The NB2 probability mass function p. Other NB parameterizations follow from different parameterizations for the gamma mixing distribution.
RNA-Seq analysis [ 4 ] may be performed on biological units from any of the traditional forms of life science study, such as randomized experiments with multiple treatments and covariates, or observational studies with multiple observed explanatory variables.
Although much of the statistical attention to RNA-Seq analysis has so far been focused on the two-group problem—and, therefore, on identification of differentially expressed genes—there is a clear need for regression analysis for identifying differential expression after accounting for other variables, and for identifying patterns of expression and differential expression as a function of explanatory variables.
Future statistical techniques might be derived for the multivariate regression on all genes simultaneously, but the problem is currently tackled by the simpler univariate regression on each gene individually, with appropriate attention to false discovery rate.
The response for a single gene is the number of RNA-Seq reads corresponding to that gene Y out of a total number of reads for the particular biological unit s. Practically, the NB model is both flexible and convenient.
The primary statistical challenge involves simultaneous regression fitting for tens of thousands of genes from fairly small numbers of biological samples e. More flexible models are much more likely to fit the data, of course, but at the expense of tens of thousands of nuisance parameters.
Because of the very large number of hypothesis tests performed in a single RNA-Seq study and the very large number of RNA-Seq studies being performed world-wide, even a small improvement in power can have an important impact on the overall rate of scientific learning from the RNA-Seq technology.
Traditional tools for model diagnostics in generalized linear models GLMsuch as deviance and Pearson residuals and goodness-of-fit GOF tests, are suitable for binomial and Poisson regression if the means are large, i. Such GOF tests are not appropriate for small means which are typical for the majority of genes in RNA-Seq analysisand the theory for the null sampling distribution of the residuals and GOF test statistics does not extend to NB regression.
In this article, we propose a goodness-of-fit test statistic for NB regression based on Pearson residuals, and the calculation of a p -value using Monte Carlo-estimated null sampling distributions. The same simulations are used to estimate expected ordered residuals for an empirical probability plot. Suppose also that.
Deviance goodness of fit test for Poisson regression
We also introduce here a new approach, in which the dispersion parameter trend is quadratic on the log scale:.Variable selection in count data using penalized Poisson regression is one of the challenges in applying Poisson regression model when the explanatory variables are correlated.
To tackle both estimate the coefficients and perform variable selection simultaneously, elastic net penalty was successfully applied in Poisson regression. However, elastic net has two major limitations. First it does not encouraging grouping effects when there is no high correlation. Second, it is not consistent in variable selection. To address these issues, a modification of the elastic net AEN and its adaptive modified elastic net AAEMare proposed to take into account the small and medium correlation between explanatory variables and to provide the consistency of the variable selection simultaneously.
Our simulation and real data results show that AEN and AAEN have advantage with small, medium, and extremely correlated variables in terms of both prediction and variable selection consistency comparing with other existing penalized methods. Algamal, Z. Diagnostic in poisson regression models.
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Goodness-of-Fit Tests and Model Diagnostics for Negative Binomial Regression of RNA Sequencing Data
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