Minitab provides three link functions, which lets you to fit a broad class of Poisson response models. You want to choose a link function that results in a good fit to your data. Examine the goodness-of-fit statistics in the output to compare how well the model fits your data using different link functions. You can also choose link functions for historical reasons or because they have a special meaning in your discipline. For more information, go to What is a link function?.

In general, link functions are useful for a response that describes the number of times an event occurs for 3 reasons:

- The relationship between the predictors and the response is a nonlinear function.
- The variability of the response increases as the mean of the response increases.
- The response must be greater than 0.

- Link function
- Select one of the following link functions:
- Log: Use the log link function in most cases because it has better mathematical properties than the square root link function. The log-link function and the square root link function both ensure that all predicted values are nonnegative.
- Square root: Use the square root link function when the log link function does not fit but the nature of the response variable requires a link function for an adequate fit. The square root link function ensures that all predicted values are nonnegative and has coefficients with asymptotically equal standard errors.
- Identity: Use the identity link function for a model that does not need a link function for an adequate fit. This link function is useful when the range of response values is restricted enough that the 3 general advantages of using a link function do not apply. When the identity link function fits, the interpretation of the model is simpler.
###### Note

The results of Fit Poisson Model with the identity link function will not match the results of Fit Regression Model. Fit Poisson Model uses the maximum likelihood estimation method. Fit Regression Model uses the least squares estimation method.

In Weights, enter a numeric column of weights to perform weighted regression. The weights must be greater than or equal to zero. The weights column must have the same number of rows as the response column. For more information about determining the appropriate weight, go to Weighted regression.

Enter the level of confidence for the confidence intervals for the coefficients and the fitted values.

Usually, a confidence level of 95% works well. A 95% confidence level indicates that, if you took 100 random samples from the population, the confidence intervals for approximately 95 of the samples would contain the mean response. For a given set of data, a lower confidence level produces a narrower interval, and a higher confidence level produces a wider interval.

To display the confidence intervals, you must go to the Results sub-dialog box, and from Display of results, select Expanded tables.

You can select a two-sided interval or a one-sided bound. For the same confidence level, a bound is closer to the point estimate than the interval. The upper bound does not provide a likely lower value. The lower bound does not provide a likely upper value.

For example, the mean number of patients who come to a clinic in a given hour is 4.58. The 95% confidence interval for the mean number of events for multiple future observations is 2.7 to 6.5. The 95% upper bound for the mean is 6.2, which is more precise because the bound is closer to the predicted mean.

- Two-sided: Use a two-sided confidence interval to estimate both likely lower and upper values for the mean number of events.
- Lower bound: Use a lower confidence bound to estimate a likely lower value for the mean number of events.
- Upper bound: Use an upper confidence bound to estimate a likely higher value for the mean number of events.

The deviance and Pearson residuals help identify patterns in the residual plots and outliers. Observations that are poorly fit by the model have high deviance and Pearson residuals. Minitab calculates the residual values for each distinct factor/covariate pattern.

- Deviance: Deviance residuals are a measure of how well the model predicts the observation. Deviance residuals are often preferred for a logistic regression that uses the logit link function because the distribution of the residuals is more like the distribution of residuals from least squares models. The logit link function is the most common link function.
- Pearson: Pearson residuals are also a measure of how well the model predicts the observation. A common approach for identifying outliers is to plot the Pearson residuals by the order of the observations in the worksheet.

Select a deviance for calculating the chi-square values and the p-values. It is most common to use the adjusted deviance. Use the sequential deviance to determine the significance of terms by the order that they enter the model.

- Adjusted (Type III): Measures the reduction in the deviance for each term relative to a model that contains all the remaining terms.
- Sequential (Type I): Measures the reduction in the deviance when a term is added to a model that contains only the terms before it.