Regression - Poisson Regression

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The Poisson Regression is used to model count data with the assumption that the dependent variable has a Poisson distribution. It is also known as the log-linear model.

The model assumes that the variance of the dependent variable is equal to the mean. If this is not a fair assumption, the NBD regression, which assumes that the variance is proportional to the mean squared, may be a more suitable option.

Data format

The Poisson regression requires a count variable as the dependent variable. In Displayr, the best data format for this type is Numeric. A count variable must only include positive integers.

Count variable.png

The independent variables can be continuous, categorical, or binary — just as with any regression model.

Interpretation

Variable statistics measure the impact and significance of individual variables within a model, while overall statistics apply to the model as a whole. Both are shown in the output.

Variable statistics

Estimate the magnitude of the coefficient indicates the size of the change in the independent variable as the value of the dependent variable changes. A positive number indicates a direct relationship (y increases as x increases), and a negative number indicates an inverse relationship (y decreases as x increases).

The coefficient is colored and bolded if the variable is statistically significant at the 5% level.

Standard Error measures the accuracy of an estimate. The smaller the standard error, the more accurate the predictions.

z-score the estimate divided by the standard error. The magnitude (either positive or negative) indicates the significance of the variable. The values are highlighted based on their magnitude.

p-value expresses the z-score as a probability. A p-value under 0.05 means that the variable is statistically significant at the 5% level; a p-value under 0.01 means that the variable is statistically significant at the 1% level. P-values under 0.05 are shown in bold.

Overall statistics

n the sample size of the model

R-squared & McFadden’s rho-squared assess the goodness of fit of the model. A larger number indicates that the model captures more of the variation in the dependent variable.

AIC Akaike information criterion is a measure of the quality of the model. When comparing similar models, the AIC can be used to identify the superior model.

Example

The example below is a Poisson regression that models a survey respondent’s fast-food consumption based on characteristics like age, gender, and work status.

Create a Poisson Regression Model in Displayr

1. Go to Insert > Regression > Poisson Regression
2. Under Inputs > Outcome, select your dependent variable
3. Under Inputs > Predictor(s), select your independent variables

Object Inspector Options

Outcome The variable to be predicted by the predictor variables.

Predictors The variable(s) to predict the outcome.

Algorithm The fitting algorithm. Defaults to Regression but may be changed to other machine learning methods.

Type: You can use this option to toggle between different types of regression models, but note that certain types are not appropriate for certain types of outcome variable. For a count outcome variable, the other types to consider are Quasi-Poisson and NBD.

Linear See Regression - Linear Regression.
Binary Logit See Regression - Binary Logit.
Ordered Logit See Regression - Ordered Logit.
Multinomial Logit See Regression - Multinomial Logit.
Poisson.
Quasi-Poisson See Regression - Quasi-Poisson Regression.
NBD See Regression - NBD Regression.

Robust standard errors Computes standard errors that are robust to violations of the assumption of constant variance (i.e., heteroscedasticity). See Robust Standard Errors. This is only available when Type is Linear.

Missing data See Missing Data Options.

Output

Summary The default; as shown in the example above.
Detail Typical R output, some additional information compared to Summary, but without the pretty formatting.
ANOVA Analysis of variance table containing the results of Chi-squared likelihood ratio tests for each predictor.
Relative Importance Analysis The results of a relative importance analysis. See here and the references for more information. This option is not available for Multinomial Logit. Note that categorical predictors are not converted to be numeric, unlike in Driver (Importance) Analysis - Relative Importance Analysis.
Effects Plot Plots the relationship between each of the Predictors and the Outcome. Not available for Multinomial Logit.

Correction The multiple comparisons correction applied when computing the p-values of the post-hoc comparisons.

Variable names Displays Variable Names in the output.

Absolute importance scores Whether the absolute value of Relative Importance Analysis scores should be displayed.

Auxiliary variables Variables to be used when imputing missing values (in addition to all the other variables in the model).

Weight. Where a weight has been set for the R Output, it will automatically applied when the model is estimated. By default, the weight is assumed to be a sampling weight, and the standard errors are estimated using Taylor series linearization (by contrast, in the Legacy Regression, weight calibration is used). See Weights, Effective Sample Size and Design Effects.

Filter The data is automatically filtered using any filters prior to estimating the model.

Crosstab Interaction Optional variable to test for interaction with other variables in the model. See Linear Regression for more details.

Random seed Seed used to initialize the (pseudo)random number generator for the model fitting algorithm. Different seeds may lead to slightly different answers, but should normally not make a large difference.

Additional options are available by editing the code.

Diagnostics

See Regression Diagnostics.

Additional Properties

When using this feature you can obtain additional information that is stored by the R code which produces the output.

  1. To do so, select Create > R Output.
  2. In the R CODE, paste: item = YourReferenceName
  3. Replace YourReferenceName with the reference name of your item. Find this in the Report tree or by selecting the item and then going to Properties > General > Name from the object inspector on the right.
  4. Below the first line of code, you can paste in snippets from below or type in str(item) to see a list of available information.

For a more in depth discussion on extracting information from objects in R, checkout our blog post here.

Properties which may be of interest are:

  • Summary outputs from the regression model:
item$summary$coefficients # summary regression outputs

Acknowledgements

Uses the glm from the stats R package. If weights are supplied, the svyglm function from the survey R package is used. See also Regression - Generalized Linear Model.

References

Lumley, T., & Scott, A. (2017). Fitting regression models to survey data. Statistical Science, 32(2), 265-278.

Code

form.dropBox({label: "Outcome", 
            types:["Variable: Numeric, Date, Money, Categorical, OrderedCategorical"], 
            name: "formOutcomeVariable",
            prompt: "Independent target variable to be predicted"});
form.dropBox({label: "Predictor(s)",
            types:["Variable: Numeric, Date, Money, Categorical, OrderedCategorical"], 
            name: "formPredictorVariables", multi:true,
            prompt: "Dependent input variables"});

// ALGORITHM
var algorithm = form.comboBox({label: "Algorithm",
               alternatives: ["CART", "Deep Learning", "Gradient Boosting", "Linear Discriminant Analysis",
                              "Random Forest", "Regression", "Support Vector Machine"],
               name: "formAlgorithm", default_value: "Regression",
               prompt: "Machine learning or regression algorithm for fitting the model"}).getValue();
var regressionType = "";
if (algorithm == "Regression")
    regressionType = form.comboBox({label: "Regression type", 
                                        alternatives: ["Linear", "Binary Logit", "Ordered Logit", "Multinomial Logit", "Poisson",
                                                                                                          "Quasi-Poisson", "NBD"], 
                                        name: "formRegressionType", default_value: "Poisson",
                                        prompt: "Select type according to outcome variable type"}).getValue();
form.setHeading((regressionType == "" ? "" : (regressionType + " ")) + algorithm);

// DEFAULT CONTROLS
missing_data_options = ["Error if missing data", "Exclude cases with missing data", "Imputation (replace missing values with estimates)"];

// AMEND DEFAULT CONTROLS PER ALGORITHM
if (algorithm == "Support Vector Machine")
    output_options = ["Accuracy", "Prediction-Accuracy Table", "Detail"];
if (algorithm == "Gradient Boosting") 
    output_options = ["Accuracy", "Importance", "Prediction-Accuracy Table", "Detail"];
if (algorithm == "Random Forest")
    output_options = ["Importance", "Prediction-Accuracy Table", "Detail"];
if (algorithm == "Deep Learning")
    output_options = ["Accuracy", "Prediction-Accuracy Table", "Cross Validation", "Network Layers"];
if (algorithm == "Linear Discriminant Analysis")
    output_options = ["Means", "Detail", "Prediction-Accuracy Table", "Scatterplot", "Moonplot"];

if (algorithm == "CART") {
    output_options = ["Sankey", "Tree", "Text", "Prediction-Accuracy Table", "Cross Validation"];
    missing_data_options = ["Error if missing data", "Exclude cases with missing data",
                             "Use partial data", "Imputation (replace missing values with estimates)"]
}
if (algorithm == "Regression") {
    if (regressionType == "Multinomial Logit")
        output_options = ["Summary", "Detail", "ANOVA"];
    else if (regressionType == "Linear")
        output_options = ["Summary", "Detail", "ANOVA", "Relative Importance Analysis", "Shapley regression", "Effects Plot"];
    else
        output_options = ["Summary", "Detail", "ANOVA", "Relative Importance Analysis", "Effects Plot"]
    if (regressionType == "Linear")
        missing_data_options = ["Error if missing data", "Exclude cases with missing data", "Use partial data (pairwise correlations)", "Multiple imputation"];
    else
        missing_data_options = ["Error if missing data", "Exclude cases with missing data", "Multiple imputation"];
}

// COMMON CONTROLS FOR ALL ALGORITHMS
var output = form.comboBox({label: "Output", 
              alternatives: output_options, name: "formOutput", default_value: output_options[0]}).getValue();
var missing = form.comboBox({label: "Missing data", 
              alternatives: missing_data_options, name: "formMissing", default_value: "Exclude cases with missing data",
              prompt: "Options for handling cases with missing data"}).getValue();
form.checkBox({label: "Variable names", name: "formNames", default_value: false, prompt: "Display names instead of labels"});

// CONTROLS FOR SPECIFIC ALGORITHMS

if (algorithm == "Support Vector Machine")
    form.textBox({label: "Cost", name: "formCost", default_value: 1, type: "number",
                  prompt: "High cost produces a complex model with risk of overfitting, low cost produces a simpler mode with risk of underfitting"});

if (algorithm == "Gradient Boosting") {
    form.comboBox({label: "Booster", 
                  alternatives: ["gbtree", "gblinear"], name: "formBooster", default_value: "gbtree",
                  prompt: "Boost tree or linear underlying models"})
    form.checkBox({label: "Grid search", name: "formSearch", default_value: false,
                   prompt: "Search for optimal hyperparameters"});
}

if (algorithm == "Random Forest")
    if (output == "Importance")
        form.checkBox({label: "Sort by importance", name: "formImportance", default_value: true});

if (algorithm == "Deep Learning") {
    form.numericUpDown({name:"formEpochs", label:"Maximum epochs", default_value: 10, minimum: 1, maximum: 1000000,
                        prompt: "Number of rounds of training"});
    form.textBox({name: "formHiddenLayers", label: "Hidden layers", prompt: "Comma delimited list of the number of nodes in each hidden layer", required: true});
    form.checkBox({label: "Normalize predictors", name: "formNormalize", default_value: true,
                   prompt: "Normalize to zero mean and unit variance"});
}

if (algorithm == "Linear Discriminant Analysis") {
    if (output == "Scatterplot")
    {
        form.colorPicker({label: "Outcome color", name: "formOutColor", default_value:"#5B9BD5"});
        form.colorPicker({label: "Predictors color", name: "formPredColor", default_value:"#ED7D31"});
    }
    form.comboBox({label: "Prior", alternatives: ["Equal", "Observed",], name: "formPrior", default_value: "Observed",
                   prompt: "Probabilities of group membership"})
}

if (algorithm == "CART") {
    form.comboBox({label: "Pruning", alternatives: ["Minimum error", "Smallest tree", "None"], 
                   name: "formPruning", default_value: "Minimum error",
                   prompt: "Remove nodes after tree has been built"})
    form.checkBox({label: "Early stopping", name: "formStopping", default_value: false,
                   prompt: "Stop building tree when fit does not improve"});
    form.comboBox({label: "Predictor category labels", alternatives: ["Full labels", "Abbreviated labels", "Letters"],
                   name: "formPredictorCategoryLabels", default_value: "Abbreviated labels",
                   prompt: "Labelling of predictor categories in the tree"})
    form.comboBox({label: "Outcome category labels", alternatives: ["Full labels", "Abbreviated labels", "Letters"],
                   name: "formOutcomeCategoryLabels", default_value: "Full labels",
                   prompt: "Labelling of outcome categories in the tree"})
    form.checkBox({label: "Allow long-running calculations", name: "formLongRunningCalculations", default_value: false,
                   prompt: "Allow predictors with more than 30 categories"});
}

if (algorithm == "Regression") {
    if (missing == "Multiple imputation")
        form.dropBox({label: "Auxiliary variables",
            types:["Variable: Numeric, Date, Money, Categorical, OrderedCategorical"], 
            name: "formAuxiliaryVariables", required: false, multi:true,
            prompt: "Additional variables to use when imputing missing values"});
    form.comboBox({label: "Correction", alternatives: ["None", "False Discovery Rate", "Bonferroni"], name: "formCorrection",
                   default_value: "None", prompt: "Multiple comparisons correction applied when computing p-values of post-hoc comparisons"});
    var is_RIA_or_shapley = output == "Relative Importance Analysis" || output == "Shapley regression";
    if (regressionType == "Linear" && missing != "Use partial data (pairwise correlations)" && missing != "Multiple imputation")
        form.checkBox({label: "Robust standard errors", name: "formRobustSE", default_value: false,
                       prompt: "Standard errors are robust to violations of assumption of constant variance"});
    if (is_RIA_or_shapley)
        form.checkBox({label: "Absolute importance scores", name: "formAbsoluteImportance", default_value: false,
                       prompt: "Show absolute instead of signed importances"});
    if (regressionType != "Multinomial Logit" && (is_RIA_or_shapley || output == "Summary"))
        form.dropBox({label: "Crosstab interaction", name: "formInteraction", types:["Variable: Numeric, Date, Money, Categorical, OrderedCategorical"],
                      required: false, prompt: "Categorical variable to test for interaction with other variables"});
}

form.numericUpDown({name:"formSeed", label:"Random seed", default_value: 12321, minimum: 1, maximum: 1000000,
                    prompt: "Initializes randomization for imputation and certain algorithms"});
library(flipMultivariates)

model <- MachineLearning(formula = QFormula(formOutcomeVariable ~ formPredictorVariables),
                                    algorithm = formAlgorithm,
                                    weights = QPopulationWeight, subset = QFilter,
                                    missing = formMissing, output = formOutput, show.labels = !formNames,
                                    seed = get0("formSeed"),
                                    cost = get0("formCost"),
                                    booster = get0("formBooster"),
                                    grid.search = get0("formSearch"),
                                    sort.by.importance = get0("formImportance"),
                                    hidden.nodes = get0("formHiddenLayers"),
                                    max.epochs = get0("formEpochs"),
                                    normalize = get0("formNormalize"),
                                    outcome.color = get0("formOutColor"),
                                    predictors.color = get0("formPredColor"),
                                    prior = get0("formPrior"),
                                    prune = get0("formPruning"),
                                    early.stopping = get0("formStopping"),
                                    predictor.level.treatment = get0("formPredictorCategoryLabels"),
                                    outcome.level.treatment = get0("formOutcomeCategoryLabels"),
                                    long.running.calculations = get0("formLongRunningCalculations"),
                                    type = get0("formRegressionType"),
                                    auxiliary.data = get0("formAuxiliaryVariables"),
                                    correction = get0("formCorrection"),
                                    robust.se = get0("formRobustSE", ifnotfound = FALSE),
                                    importance.absolute = get0("formAbsoluteImportance"),
                                    interaction = get0("formInteraction"))