Regression - NBD Regression

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The Negative Binomial Distribution (NBD) Regression is a generalization of the Poisson regression, in which the Negative Binomial distribution replaces the Poisson distribution. The NBD regression is used when modeling an overdispersed count variable.

The Poisson model assumes that the variance is equal to the mean, which is not always a fair assumption. When the variance is greater than the mean, a Negative Binomial model, which assumes that the variance is proportional to the mean squared, is more appropriate.

Data format

The NBD model 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.


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.

See also Regression Diagnostics.


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

Create a NBD Regression Model in Displayr

1. Go to Insert > Regression > NBD 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.


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


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.

Automated outlier removal percentage Optional control to remove possible outliers in the data. See Linear Regression for more details on the general methodology. The specific residual used in the case of NBD is a type of deviance residual in an unweighted regression and the Pearson residual in a weighted regression (see Davison and Snell (1991), the rstudent function in the stats R package and Linear Regression for more related information).

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.


Cook's distance plot Creates a line/rug plot showing Cook's Distance for each observation.

Cook's distance vs leverage plot Creates a scatterplot showing Cook's distance vs leverage for each observation.

Influence index plot Creates index plots of studentized residuals, hat values, and Cook's distance.

Multicollinearity (VIF) table Creates a table containing variance inflation factors (VIF) to diagnose multicollinearity.

Normal Q-Q plot Creates a normal Quantile-Quantile (QQ) plot to reveal departures of the residuals from normality.

Prediction-accuracy table Creates a table showing the observed and predicted values, as a heatmap.

Residual normality (Shapiro-Wilk) test Conducts a Shapiro-Wilk test of normality on the (deviance) residuals.

Residuals vs fitted plot Creates a scatterplot of residuals versus fitted values.

Residuals vs leverage plot Creates a plot of residuals versus leverage values.

Scale-location plot Creates a plot of the square root of the absolute standardized residuals by fitted values.

Serial correlation (Durbin-Watson) test Conducts a Durbin-Watson test of serial correlation (auto-correlation) on the residuals.


Save fitted values Creates a new variable containing fitted values for each case in the data.

Save predicted values Creates a new variable containing predicted values for each case in the data.

Save residuals Creates a new variable containing residual values for each case in the data.

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

More Information

Getting Started with NBD Regression


Uses the function glm.nb from the MASS package. See also Regression - Generalized Linear Model.


Davison, A. C. and Snell, E. J. (1991) Residuals and diagnostics. In: Statistical Theory and Modelling. In Honour of Sir David Cox, FRS, eds. Hinkley, D. V., Reid, N. and Snell, E. J., Chapman & Hall.


var controls = [];

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"});

algorithm = algorithm.getValue();

var regressionType = "";
if (algorithm == "Regression")
    regressionTypeControl = form.comboBox({label: "Regression type", 
                                           alternatives: ["Linear", "Binary Logit", "Ordered Logit", "Multinomial Logit", "Poisson",
                                                          "Quasi-Poisson", "NBD"], 
                                           name: "formRegressionType", default_value: "NBD",
                                           prompt: "Select type according to outcome variable type"});
    regressionType = regressionTypeControl.getValue();

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

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", "Jaccard Coefficient", "Correlation", "Effects Plot"];
        output_options = ["Summary", "Detail", "ANOVA", "Relative Importance Analysis", "Effects Plot"];

var outputControl = form.comboBox({label: "Output", prompt: "The type of output used to show the results",
                                   alternatives: output_options, name: "formOutput",
                                   default_value: output_options[0]});
var output = outputControl.getValue();

if (algorithm == "Regression") {
    if (regressionType == "Linear") {
        if (output == "Jaccard Coefficient" || output == "Correlation")
            missing_data_options = ["Error if missing data", "Exclude cases with missing data", "Use partial data (pairwise correlations)"];
            missing_data_options = ["Error if missing data", "Exclude cases with missing data", "Dummy variable adjustment", "Use partial data (pairwise correlations)", "Multiple imputation"];
        missing_data_options = ["Error if missing data", "Exclude cases with missing data", "Dummy variable adjustment", "Multiple imputation"];

var missingControl = 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"});
var missing = missingControl.getValue();
controls.push(form.checkBox({label: "Variable names", name: "formNames", default_value: false, prompt: "Display names instead of labels"}));


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

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

if (algorithm == "Deep Learning") {
    controls.push(form.numericUpDown({name:"formEpochs", label:"Maximum epochs", default_value: 10, minimum: 1, maximum: 1000000,
                                      prompt: "Number of rounds of training"}));
    controls.push(form.textBox({name: "formHiddenLayers", label: "Hidden layers", prompt: "Comma delimited list of the number of nodes in each hidden layer", required: true}));
    controls.push(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")
        controls.push(form.colorPicker({label: "Outcome color", name: "formOutColor", default_value:"#5B9BD5"}));
        controls.push(form.colorPicker({label: "Predictors color", name: "formPredColor", default_value:"#ED7D31"}));
    controls.push(form.comboBox({label: "Prior", alternatives: ["Equal", "Observed",], name: "formPrior", default_value: "Observed",
                                 prompt: "Probabilities of group membership"}));

if (algorithm == "CART") {
    controls.push(form.comboBox({label: "Pruning", alternatives: ["Minimum error", "Smallest tree", "None"], 
                                 name: "formPruning", default_value: "Minimum error",
                                 prompt: "Remove nodes after tree has been built"}));
    controls.push(form.checkBox({label: "Early stopping", name: "formStopping", default_value: false,
                                 prompt: "Stop building tree when fit does not improve"}));
    controls.push(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"}));
    controls.push(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"}));
    controls.push(form.checkBox({label: "Allow long-running calculations", name: "formLongRunningCalculations", default_value: false,
                                 prompt: "Allow predictors with more than 30 categories"}));

var stacked_check = false;
if (algorithm == "Regression") {
    if (missing == "Multiple imputation")
        controls.push(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"}));
    controls.push(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";
    var is_Jaccard_or_Correlation = output == "Jaccard Coefficient" || output == "Correlation";
    if (regressionType == "Linear" && missing != "Use partial data (pairwise correlations)" && missing != "Multiple imputation")
        controls.push(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)
        controls.push(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 || is_Jaccard_or_Correlation || output == "Summary"))
        controls.push(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"}));
    if (regressionType !== "Multinomial Logit")
        controls.push(form.numericUpDown({name : "formOutlierProportion", label:"Automated outlier removal percentage", default_value: 0, 
                                          minimum:0, maximum:49.9, increment:0.1,
                                          prompt: "Data points removed and model refitted based on the residual values in the model using the full dataset"}));
    stacked_check_box = form.checkBox({label: "Stack data", name: "formStackedData", default_value: false,
                                       prompt: "Allow input into the Outcome control to be a single multi variable and Predictors to be a single grid variable"})
    stacked_check = stacked_check_box.getValue();

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

var outcome = form.dropBox({label: "Outcome", 
                            types: [ stacked_check ? "VariableSet: BinaryMulti, NominalMulti, OrdinalMulti, NumericMulti" : "Variable: Numeric, Date, Money, Categorical, OrderedCategorical"], 
                            multi: false,
                            name: "formOutcomeVariable",
                            prompt: "Independent target variable to be predicted"});
var predictors = form.dropBox({label: "Predictor(s)",
                               types:[ stacked_check ? "VariableSet: BinaryGrid, NumericGrid" : "Variable: Numeric, Date, Money, Categorical, OrderedCategorical"], 
                               name: "formPredictorVariables", multi: stacked_check ? false : true,
                               prompt: "Dependent input variables"});


form.setHeading((regressionType == "" ? "" : (regressionType + " ")) + algorithm);

model <- MachineLearning(formula = if (isTRUE(get0("formStackedData"))) as.formula(NULL) else QFormula(formOutcomeVariable ~ formPredictorVariables),
                         algorithm = formAlgorithm,
                         weights = QPopulationWeight, subset = QFilter,
                         missing = formMissing,
                         output = if (formOutput == "Shapley Regression") "Shapley regression" else formOutput,
                         show.labels = !formNames,
                         seed = get0("formSeed"),
                         cost = get0("formCost"),
                         booster = get0("formBooster"),
                = get0("formSearch"),
                = 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"),
                = get0("formAuxiliaryVariables"),
                         correction = get0("formCorrection"),
                = get0("formRobustSE", ifnotfound = FALSE),
                         importance.absolute = get0("formAbsoluteImportance"),
                         interaction = get0("formInteraction"),
                = if (get0("formRegressionType", ifnotfound = "") != "Multinomial Logit") get0("formOutlierProportion")/100 else NULL,
                = get0("formStackedData"),
                = if (isTRUE(get0("formStackedData"))) list(Y = get0("formOutcomeVariable"), X = get0("formPredictorVariables")) else NULL)