Count data

countcount type response variable datacounting statisticscounts
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.wikipedia
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Statistical data type

type
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.
The concept of data type is similar to the concept of level of measurement, but more specific: For example, count data require a different distribution (e.g. a Poisson distribution or binomial distribution) than non-negative real-valued data require, but both fall under the same level of measurement (a ratio scale).

Binary data

binarybinary variablebinary random variable
The statistical treatment of count data is distinct from that of binary data, in which the observations can take only two values, usually represented by 0 and 1, and from ordinal data, which may also consist of integers but where the individual values fall on an arbitrary scale and only the relative ranking is important.
However, binary data is frequently converted to count data by considering one of the two values as "success" and representing the outcomes as 1 or 0, which corresponds to counting the number of successes in a single trial: 1 (success) or 0 (failure); see.

Poisson regression

PoissonDiscrete RegressionNegative binomial regression
The Poisson distribution can form the basis for some analyses of count data and in this case Poisson regression may be used.
In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.

Overdispersion

underdispersionover-dispersionoverdispersed
This is a special case of the class of generalized linear models which also contains specific forms of model capable of using the binomial distribution (binomial regression, logistic regression) or the negative binomial distribution where the assumptions of the Poisson model are violated, in particular when the range of count values is limited or when overdispersion is present.
For example, Poisson regression analysis is commonly used to model count data.

Index of dispersion

Variance-to-mean ratioCoefficient of dispersionRelative variance
It is only defined when the mean \mu is non-zero, and is generally only used for positive statistics, such as count data or time between events, or where the underlying distribution is assumed to be the exponential distribution or Poisson distribution.

Generalized linear model

generalized linear modelslink functiongeneralised linear model
This is a special case of the class of generalized linear models which also contains specific forms of model capable of using the binomial distribution (binomial regression, logistic regression) or the negative binomial distribution where the assumptions of the Poisson model are violated, in particular when the range of count values is limited or when overdispersion is present.
Another example of generalized linear models includes Poisson regression which models count data using the Poisson distribution.

Statistics

statisticalstatistical analysisstatistician
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.

Data

statistical datascientific datadatum
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.

Integer

integersintegralZ
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.

Counting

inclusive countinginclusivecount
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.

Ranking

rankrankedrankings
In statistics, count data is a statistical data type, a type of data in which the observations can take only the non-negative integer values {0, 1, 2, 3, ...}, and where these integers arise from counting rather than ranking.

Ordinal data

ordinalordinal variableordered categorical data
The statistical treatment of count data is distinct from that of binary data, in which the observations can take only two values, usually represented by 0 and 1, and from ordinal data, which may also consist of integers but where the individual values fall on an arbitrary scale and only the relative ranking is important.

Categorical variable

categoricalcategorical datadichotomous
Statistical analyses involving count data includes simple counts, such as the number of occurrences of thunderstorms in a calendar year, and categorical data in which the counts represent the numbers of items falling into each of several categories.

Random variable

random variablesrandom variationrandom
When such a variable is treated as a random variable, the Poisson, binomial and negative binomial distributions are commonly used to represent its distribution.

Poisson distribution

PoissonPoisson-distributedPoissonian
The Poisson distribution can form the basis for some analyses of count data and in this case Poisson regression may be used. When such a variable is treated as a random variable, the Poisson, binomial and negative binomial distributions are commonly used to represent its distribution. In particular, the square root transformation might be used when data can be approximated by a Poisson distribution (although other transformation have modestly improved properties), while an inverse sine transformation is available when a binomial distribution is preferred.

Binomial distribution

binomialbinomial modelBinomial probability
This is a special case of the class of generalized linear models which also contains specific forms of model capable of using the binomial distribution (binomial regression, logistic regression) or the negative binomial distribution where the assumptions of the Poisson model are violated, in particular when the range of count values is limited or when overdispersion is present. When such a variable is treated as a random variable, the Poisson, binomial and negative binomial distributions are commonly used to represent its distribution. In particular, the square root transformation might be used when data can be approximated by a Poisson distribution (although other transformation have modestly improved properties), while an inverse sine transformation is available when a binomial distribution is preferred.

Negative binomial distribution

negative binomialGamma-Poisson distributioninverse binomial distribution
This is a special case of the class of generalized linear models which also contains specific forms of model capable of using the binomial distribution (binomial regression, logistic regression) or the negative binomial distribution where the assumptions of the Poisson model are violated, in particular when the range of count values is limited or when overdispersion is present. When such a variable is treated as a random variable, the Poisson, binomial and negative binomial distributions are commonly used to represent its distribution.

Data transformation (statistics)

data transformationtransformationData Transformations
Graphical examination of count data may be aided by the use of data transformations chosen to have the property of stabilising the sample variance. These can be adapted to deal with count data by using data transformations such as the square root transformation, but such methods have several drawbacks; they are approximate at best and estimate parameters that are often hard to interpret.

Square root

square rootssquareradical
In particular, the square root transformation might be used when data can be approximated by a Poisson distribution (although other transformation have modestly improved properties), while an inverse sine transformation is available when a binomial distribution is preferred. These can be adapted to deal with count data by using data transformations such as the square root transformation, but such methods have several drawbacks; they are approximate at best and estimate parameters that are often hard to interpret.

Dependent and independent variables

dependent variableindependent variableexplanatory variable
Here the count variable would be treated as a dependent variable.

Least squares

least-squaresmethod of least squaresleast squares method
Statistical methods such as least squares and analysis of variance are designed to deal with continuous dependent variables.

Analysis of variance

ANOVAanalysis of variance (ANOVA)corrected the means
Statistical methods such as least squares and analysis of variance are designed to deal with continuous dependent variables.

Parameter

parametersparametricargument
These can be adapted to deal with count data by using data transformations such as the square root transformation, but such methods have several drawbacks; they are approximate at best and estimate parameters that are often hard to interpret.