Can be used to compare different models, even models that are non-nested. Sample size: The small sample behavior of ML estimators for count models is largely unknown. It is risky to use ML with samples smaller than Samples over seem adequate. Count models need some sort of mechanism to deal with the fact that counts can be made over different observation periods. For example, the number of accidents are recorded for 50 different intersections.
However, the number of vehicles that pass through the intersections can vary greatly. Fifteen accidents for 30, vehicles is very different from 15 accidents for 1, vehicles. Count models account for these differences by including the log of the exposure variable in model with coefficient constrained to be one. The use of exposure is superior in many instances to analyzing rates as response variables because it makes use of the correct probability distributions.
It should be noted that exposure is used to adjust counts on the response variable and that it is possible to various kinds of rates, indexes or per capita measures as predictors. The response variable is the number of deaths recorded at each of five different age-group and two smoker categories.
The difference in the number of patient years will be accounted for with an exposure variable pyears. Below, note that rows 1 and 10 have almost identical numbers of deaths but have very different values for patient years. The predictor variables are four age-group dummy variables and a dummy variable to indicate smokers. These data can be analyzed with either a Poisson regression model or a negative binomial regression model.
There is not much difference between the two models based on the log-likelihood and the BIC but the Poisson model has a slightly better BIC. Basically, I want to compare counts taken in spring to counts taken in the fall, to see if the number of deer observed differs between the seasons. The count surveys were all done at the same location and multiple surveys were done in each season of each year.
The program I'm using is R. This sounds like a relatively simple stats question to me, but I'm new to R and am not sure what kind of a statistical analysis I should run.
Any advice would be greatly appreciated! Edit for clarity: There is only one location. All surveys were conducted on the same ranch along the same stretch of road. I'd recommend starting off using a Poisson regression model, which is well suited for count models. Since you seem to have multiple counts at different locations, you will need to use a method that takes into account the correlation of these observations within their clusters?
If you aren't interested in examining differences between measurement sites, then I'd recommend going with GEE since it offers population average estimates.
One simplified approach would entail pooling the counts for spring over the three-year interval, and the counts for fall over the same three years, separately.
You can approach this as a goodness-of-fit GoF chi-squared test. The idea is that the number of counts would under the null hypothesis of no difference between seasons follow a uniform distribution across seasons. A more complete approach would entail setting up a Poisson regression model in which the explanatory variables are season and year :.
A basic statistical test you can do is the unpaired t-test. R code below:. If your t-value falls outside the confidence interval, you can safely reject the null hypothesis and accept the alternative hypothesis. The range of the data is Included next to its graph is the graph of the Poisson variable with a mean and variance of 2. There is a large difference in the number of unique observations 4, for the continuous set and 9 for the discrete Poisson set. Examine your outcome variable to determine whether it is discrete or continuous.
If it is discrete, find the probability distribution function that best matches its make-up. 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. Your email address will not be published. Skip to primary navigation Skip to main content Skip to primary sidebar by Jeff Meyer, MBA, MPA One of the most important concepts in data analysis is that the analysis needs to be appropriate for the scale of measurement of the variable.
The Poisson Distribution The Poisson distribution often fits count data. But many count variables fail these tests. The Normal Distribution If the mean of a Poisson or negative binomial variable is high enough, it will be symmetric and bell-shaped. Note the following: The ranges differ a lot values are less than zero for the continuous variable.
The take-away here? Learn when you need to use Poisson or Negative Binomial Regression in your analysis, how to interpret the results, and how they differ from similar models. Take Me to The Video! Leave a Reply Cancel reply Your email address will not be published. The Analysis Factor uses cookies to ensure that we give you the best experience of our website. Line 1 is the stochastic part of this specification. Line 3 is the systematic part. The specification of a generalized linear model has both stochastic and systematic parts but adds a third part, which is a link function connecting the stochastic and systematic parts.
A GLM models the response with a distribution specified in the stochastic part. The probability distributions introduced in this chapter are the Poisson and Negative Binomial. When modeling counts using the Poisson or negative binomial distributions with a log link, the link scale is linear, and so the effects are additive on the link scale, while the response scale is nonlinear it is the exponent of the link scale , and so the effects are multiplicative on the response scale. The inverse of the link function backtransforms the parameters from the link scale back to the response scale.
The example is an experiment measuring the effect of the parasitic tapeworm Schistocephalus solidus infection on the susceptibility of infection from a second parasite, the trematode Diplostomum pseudospathaceum , in the threespine stickleback fish Gasterosteus aculeatus 8.
The response is the number of trematode larvae counted in the eyes right and left combined of the fish. A histogram of the counts is shown in Figure Instead of testing assumptions of a model using formal hypothesis tests before fitting the model, a better strategy is to 1 fit a model, and then do 2 model checking using diagnostic plots , diagnostic statistics, and simulation.
With these data, a researcher would typically fit a GLM with a Poisson or negative binomial distribution and log link. Here, I start with a linear model to illustrate the interpretation of diagnostic plots with non-normal data.
The plot shows that the residuals are clumped at the negative end of the range, which suggests that a model with a normally distributed conditional outcome or normal error is not well approximated.
A Distribution of the residuals of the fit linear model. B Normal Q-Q plot of the residuals of the fit linear model. A better way to investigate this is with the Normal Q-Q plot in Figure If the conditional outcome approximates a normal distribution, the points should roughly follow the line. Instead, for the worm data, the points are above the line at both ends. Remembering that this plot is of residuals, if we think about this as counts, this means that our smallest counts are not as small as we would expect given the mean and a normal distribution.
At the positive end, the sample values are again more positive than the theoretical values. Thinking about this as counts, this means that are largest counts are larger than expected given the mean and a normal distribution. A quantile or percentile of a vector of numbers is the value of the point at a specified percentage rank. In a Normal Q-Q plot, we want to plot the quantiles of the residuals against a set of theoretical quantiles.
Now simply plot the observed against theoretical quantiles. Often, the standardized quantiles are plotted. A standardized variable has a mean of zero and a standard deviation of one and is computed by 1 centering the vector at zero by subtracting the mean from every value, and 2 dividing each value by the standard deviation of the vector.
This plot also nicely shows how the residuals of the worm data deviate from that expected if these had a normal distribution. The plot nicely shows that the most negative observed quintiles are not as negative as expected given a normal distribution, which again makes sense because this would imply negative counts since the mean is close to zero.
And it nicely shows that the most positive observed quantiles are more positive than expected given a normal distribution, again this makes sense in right skewed count data.
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