View source for Statistical data type

In [[Statistics|statistics]], groups of individual data points may be classified as belonging to any of various '''statistical data types''', e.g. [[Categorical variable|categorical]] ("red", "blue", "green"), [[Real number|real number]] (1.68, -5, 1.7e+6), odd number (1,3,5) etc.  The data type is a fundamental component of the semantic content of the variable, and controls which sorts of [[Probability distribution|probability distribution]]s can logically be used to describe the variable, the permissible operations on the variable, the type of [[Regression analysis|regression analysis]] used to predict the variable, etc.  The concept of data type is similar to the concept of [[Level of measurement|level of measurement]], but more specific: For example, [[Count data|count data]] require a different distribution (e.g. a [[Poisson distribution]] or [[Binomial distribution|binomial distribution]]) than non-negative real-valued data require, but both fall under the same level of measurement (a ratio scale).

Various attempts have been made to produce a taxonomy of [[Level of measurement|levels of measurement]]. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with [[Earth:Longitude|longitude]] and [[Physics:Temperature|temperature]] measurements in degree [[Celsius]] or degree [[Fahrenheit]]), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation.

Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as [[Categorical variable|categorical variable]]s, whereas ratio and interval measurements are grouped together as [[Variable (mathematics)#Applied statistics|quantitative variables]], which can be either [[Probability distribution#Discrete probability distribution|discrete]] or [[Probability distribution#Continuous probability distribution|continuous]], due to their numerical nature. Such distinctions can often be loosely correlated with [[Data type|data type]] in computer science, in that dichotomous categorical variables may be represented with the [[Boolean data type]], polytomous categorical variables with arbitrarily assigned [[Integer|integer]]s in the [[Integer (computer science)|integral data type]], and continuous variables with the [[Real data type|real data type]] involving floating point computation. But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented. 

Other categorizations have been proposed. For example, [[Biography:Frederick Mosteller|Mosteller]] and [[Biography:John Tukey|Tukey]] (1977)<ref>{{cite book |last1=Mosteller |first1=F. |last2=Tukey |first2=J.W. |date=1977 |title=Data analysis and regression |publisher=Addison-Wesley |isbn=978-0-201-04854-4}}</ref> distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)<ref>{{cite book |last1=Nelder |first1=J.A. |date=1990 |chapter=The knowledge needed to computerise the analysis and interpretation of statistical information |title=Expert systems and artificial intelligence: the need for information about data |publisher=Library Association |location=London |oclc=27042489}}</ref> described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998),<ref>{{cite journal |last=Chrisman |first=Nicholas R. |date=1998 |title=Rethinking Levels of Measurement for Cartography |journal=Cartography and Geographic Information Science |volume=25 |issue=4 |pages=231–242 |doi=10.1559/152304098782383043}}</ref> van den Berg (1991).<ref>{{cite book |last1=van den Berg |first1=G. |date=1991 |title=Choosing an analysis method |location=Leiden |publisher=DSWO Press |isbn=978-90-6695-062-7}}</ref>

The issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values which are not invariant under some transformations. Whether or not a transformation is sensible to contemplate depends on the question one is trying to answer" (Hand, 2004, p.&nbsp;82).<ref>{{cite book |last1=Hand |first1=D.J. |date=2004 |title=Measurement theory and practice: The world through quantification |publisher=Wiley |isbn=978-0-470-68567-9 |page=82}}</ref>

==Simple data types==
The following table classifies the various simple data types, associated distributions, permissible operations, etc.  Regardless of the logical possible values, all of these data types are generally coded using [[Real number|real number]]s, because the theory of [[Random variable|random variable]]s often explicitly assumes that they hold real numbers.

{| class="wikitable"
! Data Type !! Possible values !! Example usage !! Level of measurement !! Distribution !! Scale of relative differences !! Permissible statistics !! Regression analysis
|-
! [[Binary variable|binary]]
| 0, 1 (arbitrary labels)
| binary outcome ("yes/no", "true/false", "success/failure", etc.)
| rowspan=2|[[Nominal scale|nominal scale]]
| [[Bernoulli distribution|Bernoulli]]
| rowspan=2|[[Comparability|incomparable]]
| rowspan=2|[[Mode (statistics)|mode]], [[Chi-squared test|Chi-squared]]
| [[Logistic regression|logistic]], probit
|-
! [[Categorical variable|categorical]]
| "name1", "name2", "name3", ... "nameK" (arbitrary labels)
| categorical outcome with names or places like "Rome", "Amsterdam", "Madrid", "London", "Washington" (specific [[Biology:Blood type|blood type]], [[Social:Political party|political party]], word, etc.)
| [[Categorical distribution|categorical]]
| multinomial logit, [[Multinomial probit|multinomial probit]]
|-
! ordinal
| ordering categories or [[Integer|integer]] or [[Real number|real number]] (arbitrary scale)
| Ordering adverbs like "Small", "Medium", "Large", relative score, significant only for creating a ranking
| ordinal scale
| [[Categorical distribution|categorical]]
| relative comparison
| 
| [[Ordinal regression|ordinal regression]] ([[Ordered logit|ordered logit]], [[Ordered probit|ordered probit]])
|-
! binomial
| 0, 1, ..., N
| number of successes (e.g. yes votes) out of ''N'' possible
| interval scale
| [[Binomial distribution|binomial]], [[Beta-binomial distribution|beta-binomial]], etc.
| additive
| [[Mean|mean]], [[Median|median]], [[Mode (statistics)|mode]], [[Standard deviation|standard deviation]], [[Correlation|correlation]]
| [[Binomial regression|binomial regression]] ([[Logistic regression|logistic]], probit)
|-
! count
| nonnegative [[Integer|integer]]s (0, 1, ...)
| number of items (telephone calls, people, molecules, births, deaths, etc.) in given interval/area/volume
| [[Ratio scale|ratio scale]]
| [[Poisson distribution|Poisson]], [[Negative binomial distribution|negative binomial]], etc.
| multiplicative
| All statistics permitted for interval scales plus the following: [[Geometric mean|geometric mean]], [[Harmonic mean|harmonic mean]], [[Coefficient of variation|coefficient of variation]]
| [[Poisson regression|Poisson]], negative binomial regression
|-
! real-valued additive
| [[Real number|real number]]
| temperature in degree Celsius or degree Fahrenheit, relative distance, [[Location parameter|location parameter]], etc. (or approximately, anything not varying over a large scale)
| interval scale
| [[Normal distribution|normal]], etc. (usually symmetric about the [[Mean|mean]])
| additive
| [[Mean|mean]], [[Median|median]], [[Mode (statistics)|mode]], [[Standard deviation|standard deviation]], [[Correlation|correlation]]
| standard [[Linear regression|linear regression]]
|-
! real-valued multiplicative
| positive [[Real number|real number]]
| temperature in [[Kelvin|kelvin]], price, income, size, [[Scale parameter|scale parameter]], etc. (especially when varying over a large scale)
| [[Ratio scale|ratio scale]]
| [[Log-normal distribution|log-normal]], [[Gamma distribution|gamma]], [[Exponential distribution|exponential]], etc. (usually a skewed distribution)
| multiplicative
| All statistics permitted for interval scales plus the following: [[Geometric mean|geometric mean]], [[Harmonic mean|harmonic mean]], [[Coefficient of variation|coefficient of variation]]
| [[Generalized linear model|generalized linear model]] with [[Logarithm|logarithm]]ic link
|}

==Multivariate data types==
Data that cannot be described using a single number are often shoehorned into random vectors of real-valued [[Random variable|random variable]]s, although there is an increasing tendency to treat them on their own.  Some examples:

*Random vectors.  The individual elements may or may not be correlated.  Examples of distributions used to describe correlated random vectors are the [[Multivariate normal distribution|multivariate normal distribution]] and [[Multivariate t-distribution|multivariate t-distribution]].  In general, there may be arbitrary correlations between any elements and any others; however, this often becomes unmanageable above a certain size, requiring further restrictions on the correlated elements.
*[[Random matrix|Random matrices]].  Random matrices can be laid out linearly and treated as random vectors; however, this may not be an efficient way of representing the correlations between different elements.  Some probability distributions are specifically designed for random matrices, e.g. the [[Matrix normal distribution|matrix normal distribution]] and [[Wishart distribution]].
*[[Random sequence]]s.  These are sometimes considered to be the same as random vectors, but in other cases the term is applied specifically to cases where each random variable is only correlated with nearby variables (as in a [[Markov model]]).  This is a particular case of a Bayes network and often used for very long sequences, e.g. gene sequences or lengthy text documents.  A number of models are specifically designed for such sequences, e.g. [[Hidden Markov model|hidden Markov model]]s.
*Random processes.  These are similar to random sequences, but where the length of the sequence is indefinite or infinite and the elements in the sequence are processed one-by-one.  This is often used for data that can be described as a [[Time series|time series]], e.g. the price of a stock on successive days.  Random processes are also used to model values that vary continuously (e.g. the temperature at successive moments in time), rather than at discrete intervals.
*Bayes networks.  These correspond to aggregates of random variables described using [[Graphical model|graphical model]]s, where individual random variables are linked in a [[Graph (discrete mathematics)|graph]] structure with conditional distributions relating variables to nearby variables.
**[[Multilevel model]]s are subclasses of Bayes networks that can be thought of as having multiple levels of [[Linear regression|linear regression]].
**Random trees.  These are a subclass of Bayes network, where the variables are linked in a [[Tree structure|tree structure]].  An example is the problem of [[Parsing|parsing]] a sentence, when statistical parsing techniques are used, such as [[Probabilistic context-free grammar|probabilistic context-free grammar]]s (PCFG's).
*[[Random field]]s.  These represent the extension of random processes to multiple dimensions, and are common in [[Physics:Physics|physics]], where they are used in [[Physics:Statistical mechanics|statistical mechanics]] to describe properties such as [[Physics:Force|force]] or [[Physics:Electric field|electric field]] that can vary continuously over three dimensions (or four dimensions, when time is included).

These concepts originate in various scientific fields and frequently overlap in usage.  As a result, it is very often the case that multiple concepts could potentially be applied to the same problem.

==References==
{{Reflist}}

[[Category:Statistical data types| ]]

{{Sourceattribution|Statistical data type}}