âčïž Skipped - page is already crawled
| Filter | Status | Condition | Details |
|---|---|---|---|
| HTTP status | PASS | download_http_code = 200 | HTTP 200 |
| Age cutoff | PASS | download_stamp > now() - 6 MONTH | 7.1 months ago (distributed domain, exempt) |
| History drop | PASS | isNull(history_drop_reason) | No drop reason |
| Spam/ban | PASS | fh_dont_index != 1 AND ml_spam_score = 0 | ml_spam_score=0 |
| Canonical | PASS | meta_canonical IS NULL OR = '' OR = src_unparsed | Not set |
| Property | Value |
|---|---|
| URL | https://en.wikipedia.org/wiki/Zero-inflated_model |
| Last Crawled | 2025-09-05 04:03:04 (7 months ago) |
| First Indexed | not set |
| HTTP Status Code | 200 |
| Meta Title | Zero-inflated model - Wikipedia |
| Meta Description | null |
| Meta Canonical | null |
| Boilerpipe Text | Statistical model allowing for frequent zero values In statistics , a zero-inflated model is a statistical model based on a zero-inflated probability distribution , i.e. a distribution that allows for frequent zero-valued observations.
Introduction to zero-inflated models [ edit ] Zero-inflated models are commonly used in the analysis of count data, such as the number of visits a patient makes to the emergency room in one year, or the number of fish caught in one day in one lake. [ 1 ] Count data can take values of 0, 1, 2, ⊠(non-negative integer values). [ 2 ] Other examples of count data are the number of hits recorded by a Geiger counter in one minute, patient days in the hospital, goals scored in a soccer game, [ 3 ] and the number of episodes of hypoglycemia per year for a patient with diabetes. [ 4 ] For statistical analysis, the distribution of the counts is often represented using a Poisson distribution or a negative binomial distribution . Hilbe [ 3 ] notes that "Poisson regression is traditionally conceived of as the basic count model upon which a variety of other count models are based." In a Poisson model, "⊠the random variable y {\displaystyle y} is the count response and parameter λ {\displaystyle \lambda } (lambda) is the mean. Often, λ {\displaystyle \lambda } is also called the rate or intensity parameter⊠In statistical literature, λ {\displaystyle \lambda } is also expressed as Ό {\displaystyle \mu } (mu) when referring to Poisson and traditional negative binomial models."
In some data, the number of zeros is greater than would be expected using a Poisson distribution or a negative binomial distribution . Data with such an excess of zero counts are described as Zero-inflated. [ 4 ] Example histograms of zero-inflated Poisson distributions with mean ÎŒ {\displaystyle \mu } of 5 or 10 and proportion of zero inflation Ï {\displaystyle \pi } of 0.2 or 0.5 are shown below, based on the R program ZeroInflPoiDistPlots.R from Bilder and Laughlin. [ 1 ] Examples of zero-inflated count data [ edit ] Fish counts [ 1 ] "⊠suppose we recorded the number of fish caught on various lakes in 4-hour fishing trips to Minnesota. Some lakes in Minnesota are too shallow for fish to survive the winter, so fishing in those lakes will yield no catch. On the other hand, even on a lake where fish are plentiful, we may or may not catch any fish due to conditions or our own competence. Thus, the number of fish caught will be zero if the lake does not support fish, and will be zero, one or more if it does." Number of wisdom teeth extracted. [ 5 ] The number of wisdom teeth that a person has had extracted can range from 0 to 4. Some individuals, about one-third of the population, do not have any wisdom teeth. For these individuals, the number of wisdom teeth extracted will always be zero. For other individuals, the number extracted will be between 0 and 4, where a 0 indicates that the subject has not yet, and may never, have any of their 4 wisdom teeth extracted. Publications by PhD candidates. [ 6 ] Long examined the number of publications by 915 doctoral candidates in biochemistry in the last three years of their PhD studies. The proportion of candidates with zero publications exceeded the number predicted by a Poisson model. "Long [ 6 ] argued that the PhD candidates might fall into two distinct groups: "publishers" (perhaps striving for an academic career) and "non-publishers" (seeking other career paths). One reasonable form of explanation is that the observed zero counts reflect a mixture of the two latent classes â those who simply have not yet published and those who will likely never publish." [ 7 ] Zero-inflated data as a mixture of two distributions [ edit ] As the examples above show, zero-inflated data can arise as a mixture of two distributions. The first distribution generates zeros. The second distribution, which may be a Poisson distribution , a negative binomial distribution or other count distribution, generates counts, some of which may be zeros. [ 7 ] In the statistical literature, different authors may use different names to distinguish zeros from the two distributions. Some authors describe zeros generated by the first (binary) distribution as "structural" and zeros generated by the second (count) distribution as "random". [ 7 ] Other authors use the terminology "immune" and "susceptible" for the binary and count zeros, respectively. [ 1 ] Zero-inflated Poisson [ edit ] Histogram of a zero-inflated Poisson distribution One well-known zero-inflated model is Diane Lambert 's zero-inflated Poisson model, which concerns a random event containing excess zero-count data in unit time. [ 8 ] For example, the number of insurance claims within a population for a certain type of risk would be zero-inflated by those people who have not taken out insurance against the risk and thus are unable to claim. The zero-inflated Poisson (ZIP) model mixes two zero generating processes. The first process generates zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The mixture distribution is described as follows:
Pr ( Y = 0 ) = Ï + ( 1 â Ï ) e â λ {\displaystyle \Pr(Y=0)=\pi +(1-\pi )e^{-\lambda }} Pr ( Y = y i ) = ( 1 â Ï ) λ y i e â λ y i ! , y i = 1 , 2 , 3 , . . . {\displaystyle \Pr(Y=y_{i})=(1-\pi ){\frac {\lambda ^{y_{i}}e^{-\lambda }}{y_{i}!}},\qquad y_{i}=1,2,3,...} where the outcome variable y i {\displaystyle y_{i}} has any non-negative integer value, λ {\displaystyle \lambda } is the expected Poisson count for the i {\displaystyle i} th individual; Ï {\displaystyle \pi } is the probability of extra zeros.
The mean is ( 1 â Ï ) λ {\displaystyle (1-\pi )\lambda } and the variance is λ ( 1 â Ï ) ( 1 + Ï Î» ) {\displaystyle \lambda (1-\pi )(1+\pi \lambda )} .
Estimators of ZIP parameters [ edit ] The method of moments estimators are given by [ 9 ] λ ^ m o = s 2 + m 2 m â 1 , {\displaystyle {\hat {\lambda }}_{mo}={\frac {s^{2}+m^{2}}{m}}-1,} Ï ^ m o = s 2 â m s 2 + m 2 â m , {\displaystyle {\hat {\pi }}_{mo}={\frac {s^{2}-m}{s^{2}+m^{2}-m}},} where m {\displaystyle m} is the sample mean and s 2 {\displaystyle s^{2}} is the sample variance.
The maximum likelihood estimator [ 10 ] can be found by solving the following equation
m ( 1 â e â λ ^ m l ) = λ ^ m l ( 1 â n 0 n ) . {\displaystyle m(1-e^{-{\hat {\lambda }}_{ml}})={\hat {\lambda }}_{ml}\left(1-{\frac {n_{0}}{n}}\right).} where n 0 n {\displaystyle {\frac {n_{0}}{n}}} is the observed proportion of zeros.
A closed form solution of this equation is given by [ 11 ] λ ^ m l = W 0 ( â s e â s ) + s {\displaystyle {\hat {\lambda }}_{ml}=W_{0}(-se^{-s})+s} with W 0 {\displaystyle W_{0}} being the main branch of Lambert's W-function [ 12 ] and
s = m 1 â n 0 n {\displaystyle s={\frac {m}{1-{\frac {n_{0}}{n}}}}} . Alternatively, the equation can be solved by iteration. [ 13 ] The maximum likelihood estimator for Ï {\displaystyle \pi } is given by
Ï ^ m l = 1 â m λ ^ m l . {\displaystyle {\hat {\pi }}_{ml}=1-{\frac {m}{{\hat {\lambda }}_{ml}}}.} In 1994, Greene considered the zero-inflated negative binomial (ZINB) model. [ 14 ] Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model. [ 15 ] Discrete pseudo compound Poisson model [ edit ] If the count data Y {\displaystyle Y} is such that the probability of zero is larger than the probability of nonzero, namely
Pr ( Y = 0 ) > 0.5 {\displaystyle \Pr(Y=0)>0.5} then the discrete data Y {\displaystyle Y} obey discrete pseudo compound Poisson distribution . [ 16 ] In fact, let G ( z ) = â n = 0 â P ( Y = n ) z n {\displaystyle G(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}} be the probability generating function of y i {\displaystyle y_{i}} . If p 0 = Pr ( Y = 0 ) > 0.5 {\displaystyle p_{0}=\Pr(Y=0)>0.5} , then | G ( z ) | â©Ÿ p 0 â â i = 1 â p i = 2 p 0 â 1 > 0 {\displaystyle |G(z)|\geqslant p_{0}-\sum \limits _{i=1}^{\infty }p_{i}=2p_{0}-1>0} . Then from the WienerâLĂ©vy theorem , [ 17 ] G ( z ) {\displaystyle G(z)} has the probability generating function of the discrete pseudo compound Poisson distribution .
We say that the discrete random variable Y {\displaystyle Y} satisfying probability generating function characterization
G Y ( z ) = â n = 0 â P ( Y = n ) z n = exp ⥠( â k = 1 â α k λ ( z k â 1 ) ) , ( | z | †1 ) {\displaystyle G_{Y}(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}=\exp \left(\sum _{k=1}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)} has a discrete pseudo compound Poisson distribution with parameters
( λ 1 , λ 2 , ⊠) = ( α 1 λ , α 2 λ , ⊠) â R â ( â k = 1 â α k = 1 , â k = 1 â | α k | < â , α k â R , λ > 0 ) . {\displaystyle (\lambda _{1},\lambda _{2},\ldots )=(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left(\sum _{k=1}^{\infty }\alpha _{k}=1,\sum \limits _{k=1}^{\infty }|\alpha _{k}|<\infty ,\alpha _{k}\in \mathbb {R} ,\lambda >0\right).} When all the α k {\displaystyle \alpha _{k}} are non-negative, it is the discrete compound Poisson distribution (non-Poisson case) with overdispersion property.
|
| Markdown | [Jump to content](https://en.wikipedia.org/wiki/Zero-inflated_model#bodyContent)
Main menu
Main menu
move to sidebar
hide
Navigation
- [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]")
- [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia")
- [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events")
- [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]")
- [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works")
- [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia")
Contribute
- [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia")
- [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia")
- [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors")
- [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]")
- [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia")
- [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages)
[  ](https://en.wikipedia.org/wiki/Main_Page)
[Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]")
Appearance
- [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)
- [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Zero-inflated+model "You are encouraged to create an account and log in; however, it is not mandatory")
- [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Zero-inflated+model "You're encouraged to log in; however, it's not mandatory. [o]")
Personal tools
- [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)
- [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Zero-inflated+model "You are encouraged to create an account and log in; however, it is not mandatory")
- [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Zero-inflated+model "You're encouraged to log in; however, it's not mandatory. [o]")
Pages for logged out editors [learn more](https://en.wikipedia.org/wiki/Help:Introduction)
- [Contributions](https://en.wikipedia.org/wiki/Special:MyContributions "A list of edits made from this IP address [y]")
- [Talk](https://en.wikipedia.org/wiki/Special:MyTalk "Discussion about edits from this IP address [n]")
## Contents
move to sidebar
hide
- [(Top)](https://en.wikipedia.org/wiki/Zero-inflated_model)
- [1 Introduction to zero-inflated models](https://en.wikipedia.org/wiki/Zero-inflated_model#Introduction_to_zero-inflated_models)
Toggle Introduction to zero-inflated models subsection
- [1\.1 Examples of zero-inflated count data](https://en.wikipedia.org/wiki/Zero-inflated_model#Examples_of_zero-inflated_count_data)
- [1\.2 Zero-inflated data as a mixture of two distributions](https://en.wikipedia.org/wiki/Zero-inflated_model#Zero-inflated_data_as_a_mixture_of_two_distributions)
- [2 Zero-inflated Poisson](https://en.wikipedia.org/wiki/Zero-inflated_model#Zero-inflated_Poisson)
- [3 Estimators of ZIP parameters](https://en.wikipedia.org/wiki/Zero-inflated_model#Estimators_of_ZIP_parameters)
- [4 Related models](https://en.wikipedia.org/wiki/Zero-inflated_model#Related_models)
- [5 Discrete pseudo compound Poisson model](https://en.wikipedia.org/wiki/Zero-inflated_model#Discrete_pseudo_compound_Poisson_model)
- [6 See also](https://en.wikipedia.org/wiki/Zero-inflated_model#See_also)
- [7 Software](https://en.wikipedia.org/wiki/Zero-inflated_model#Software)
- [8 References](https://en.wikipedia.org/wiki/Zero-inflated_model#References)
Toggle the table of contents
# Zero-inflated model
3 languages
- [CatalĂ ](https://ca.wikipedia.org/wiki/Model_amb_exc%C3%A9s_de_zeros "Model amb excĂ©s de zeros â Catalan")
- [Français](https://fr.wikipedia.org/wiki/Mod%C3%A8le_avec_exc%C3%A8s_de_z%C3%A9ros "ModĂšle avec excĂšs de zĂ©ros â French")
- [äžæ](https://zh.wikipedia.org/wiki/%E9%9B%B6%E8%86%A8%E8%83%80 "é¶èšè â Chinese")
[Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q966010#sitelinks-wikipedia "Edit interlanguage links")
- [Article](https://en.wikipedia.org/wiki/Zero-inflated_model "View the content page [c]")
- [Talk](https://en.wikipedia.org/wiki/Talk:Zero-inflated_model "Discuss improvements to the content page [t]")
English
- [Read](https://en.wikipedia.org/wiki/Zero-inflated_model)
- [Edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit "Edit this page [e]")
- [View history](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=history "Past revisions of this page [h]")
Tools
Tools
move to sidebar
hide
Actions
- [Read](https://en.wikipedia.org/wiki/Zero-inflated_model)
- [Edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit "Edit this page [e]")
- [View history](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=history)
General
- [What links here](https://en.wikipedia.org/wiki/Special:WhatLinksHere/Zero-inflated_model "List of all English Wikipedia pages containing links to this page [j]")
- [Related changes](https://en.wikipedia.org/wiki/Special:RecentChangesLinked/Zero-inflated_model "Recent changes in pages linked from this page [k]")
- [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard "Upload files [u]")
- [Permanent link](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&oldid=1287526710 "Permanent link to this revision of this page")
- [Page information](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=info "More information about this page")
- [Cite this page](https://en.wikipedia.org/w/index.php?title=Special:CiteThisPage&page=Zero-inflated_model&id=1287526710&wpFormIdentifier=titleform "Information on how to cite this page")
- [Get shortened URL](https://en.wikipedia.org/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FZero-inflated_model)
- [Download QR code](https://en.wikipedia.org/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FZero-inflated_model)
Print/export
- [Download as PDF](https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=Zero-inflated_model&action=show-download-screen "Download this page as a PDF file")
- [Printable version](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&printable=yes "Printable version of this page [p]")
In other projects
- [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q966010 "Structured data on this page hosted by Wikidata [g]")
Appearance
move to sidebar
hide
From Wikipedia, the free encyclopedia
Statistical model allowing for frequent zero values
In [statistics](https://en.wikipedia.org/wiki/Statistics "Statistics"), a **zero-inflated model** is a [statistical model](https://en.wikipedia.org/wiki/Statistical_model "Statistical model") based on a zero-inflated [probability distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution"), i.e. a distribution that allows for frequent zero-valued observations.
## Introduction to zero-inflated models
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=1 "Edit section: Introduction to zero-inflated models")\]
Zero-inflated models are commonly used in the analysis of count data, such as the number of visits a patient makes to the emergency room in one year, or the number of fish caught in one day in one lake.[\[1\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-BilderLoughin2015-1) Count data can take values of 0, 1, 2, ⊠(non-negative integer values).[\[2\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-HilbeNBR2014-2) Other examples of count data are the number of hits recorded by a Geiger counter in one minute, patient days in the hospital, goals scored in a soccer game,[\[3\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-HilbeNBR2007-3) and the number of episodes of hypoglycemia per year for a patient with diabetes.[\[4\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-Lachin2011-4)
For statistical analysis, the distribution of the counts is often represented using a [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution "Poisson distribution") or a [negative binomial distribution](https://en.wikipedia.org/wiki/Negative_binomial_distribution "Negative binomial distribution"). Hilbe [\[3\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-HilbeNBR2007-3) notes that "Poisson regression is traditionally conceived of as the basic count model upon which a variety of other count models are based." In a Poisson model, "⊠the random variable y {\\displaystyle y}  is the count response and parameter λ {\\displaystyle \\lambda }  (lambda) is the mean. Often, λ {\\displaystyle \\lambda }  is also called the rate or intensity parameter⊠In statistical literature, λ {\\displaystyle \\lambda }  is also expressed as Ό {\\displaystyle \\mu }  (mu) when referring to Poisson and traditional negative binomial models."
In some data, the number of zeros is greater than would be expected using a [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution "Poisson distribution") or a [negative binomial distribution](https://en.wikipedia.org/wiki/Negative_binomial_distribution "Negative binomial distribution"). Data with such an excess of zero counts are described as Zero-inflated.[\[4\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-Lachin2011-4)
Example histograms of zero-inflated Poisson distributions with mean ÎŒ {\\displaystyle \\mu }  of 5 or 10 and proportion of zero inflation Ï {\\displaystyle \\pi }  of 0.2 or 0.5 are shown below, based on the R program ZeroInflPoiDistPlots.R from Bilder and Laughlin.[\[1\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-BilderLoughin2015-1)
[](https://en.wikipedia.org/wiki/File:Histograms_of_ZIP_distributions.jpg "Histograms of ZIP distributions")
### Examples of zero-inflated count data
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=2 "Edit section: Examples of zero-inflated count data")\]
- Fish counts [\[1\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-BilderLoughin2015-1) "⊠suppose we recorded the number of fish caught on various lakes in 4-hour fishing trips to Minnesota. Some lakes in Minnesota are too shallow for fish to survive the winter, so fishing in those lakes will yield no catch. On the other hand, even on a lake where fish are plentiful, we may or may not catch any fish due to conditions or our own competence. Thus, the number of fish caught will be zero if the lake does not support fish, and will be zero, one or more if it does."
- Number of wisdom teeth extracted.[\[5\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-ChernyavskiyMcmurry-5) The number of wisdom teeth that a person has had extracted can range from 0 to 4. Some individuals, about one-third of the population, do not have any wisdom teeth. For these individuals, the number of wisdom teeth extracted will always be zero. For other individuals, the number extracted will be between 0 and 4, where a 0 indicates that the subject has not yet, and may never, have any of their 4 wisdom teeth extracted.
- Publications by PhD candidates.[\[6\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-Long1997-6) Long examined the number of publications by 915 doctoral candidates in biochemistry in the last three years of their PhD studies. The proportion of candidates with zero publications exceeded the number predicted by a Poisson model. "Long [\[6\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-Long1997-6) argued that the PhD candidates might fall into two distinct groups: "publishers" (perhaps striving for an academic career) and "non-publishers" (seeking other career paths). One reasonable form of explanation is that the observed zero counts reflect a mixture of the two latent classes â those who simply have not yet published and those who will likely never publish."[\[7\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-FriendlyMeyer2016-7)
### Zero-inflated data as a mixture of two distributions
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=3 "Edit section: Zero-inflated data as a mixture of two distributions")\]
As the examples above show, zero-inflated data can arise as a [mixture](https://en.wikipedia.org/wiki/Mixture_distribution "Mixture distribution") of two distributions. The first distribution generates zeros. The second distribution, which may be a [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution "Poisson distribution"), a [negative binomial distribution](https://en.wikipedia.org/wiki/Negative_binomial_distribution "Negative binomial distribution") or other count distribution, generates counts, some of which may be zeros.[\[7\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-FriendlyMeyer2016-7)
In the statistical literature, different authors may use different names to distinguish zeros from the two distributions. Some authors describe zeros generated by the first (binary) distribution as "structural" and zeros generated by the second (count) distribution as "random".[\[7\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-FriendlyMeyer2016-7) Other authors use the terminology "immune" and "susceptible" for the binary and count zeros, respectively.[\[1\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-BilderLoughin2015-1)
## Zero-inflated Poisson
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=4 "Edit section: Zero-inflated Poisson")\]
[](https://en.wikipedia.org/wiki/File:Zero-inflated-poisson-distribution.png)
Histogram of a zero-inflated Poisson distribution
One well-known zero-inflated model is [Diane Lambert](https://en.wikipedia.org/wiki/Diane_Lambert "Diane Lambert")'s zero-inflated Poisson model, which concerns a random event containing excess zero-count data in unit time.[\[8\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-8) For example, the number of [insurance claims](https://en.wikipedia.org/wiki/Insurance_claim "Insurance claim") within a population for a certain type of risk would be zero-inflated by those people who have not taken out insurance against the risk and thus are unable to claim. The zero-inflated Poisson (ZIP) model [mixes](https://en.wikipedia.org/wiki/Mixture_distribution "Mixture distribution") two zero generating processes. The first process generates zeros. The second process is governed by a [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution "Poisson distribution") that generates counts, some of which may be zero. The [mixture distribution](https://en.wikipedia.org/wiki/Mixture_distribution "Mixture distribution") is described as follows:
Pr
(
Y
\=
0
)
\=
Ï
\+
(
1
â
Ï
)
e
â
λ
{\\displaystyle \\Pr(Y=0)=\\pi +(1-\\pi )e^{-\\lambda }}
Pr
(
Y
\=
y
i
)
\=
(
1
â
Ï
)
λ
y
i
e
â
λ
y
i
\!
,
y
i
\=
1
,
2
,
3
,
.
.
.
{\\displaystyle \\Pr(Y=y\_{i})=(1-\\pi ){\\frac {\\lambda ^{y\_{i}}e^{-\\lambda }}{y\_{i}!}},\\qquad y\_{i}=1,2,3,...}
where the outcome variable y i {\\displaystyle y\_{i}}  has any non-negative integer value, λ {\\displaystyle \\lambda }  is the expected Poisson count for the i {\\displaystyle i} th individual; Ï {\\displaystyle \\pi }  is the probability of extra zeros.
The mean is ( 1 â Ï ) λ {\\displaystyle (1-\\pi )\\lambda }  and the variance is λ ( 1 â Ï ) ( 1 \+ Ï Î» ) {\\displaystyle \\lambda (1-\\pi )(1+\\pi \\lambda )} .
## Estimators of ZIP parameters
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=5 "Edit section: Estimators of ZIP parameters")\]
The method of moments estimators are given by[\[9\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-9)
λ
^
m
o
\=
s
2
\+
m
2
m
â
1
,
{\\displaystyle {\\hat {\\lambda }}\_{mo}={\\frac {s^{2}+m^{2}}{m}}-1,}
Ï
^
m
o
\=
s
2
â
m
s
2
\+
m
2
â
m
,
{\\displaystyle {\\hat {\\pi }}\_{mo}={\\frac {s^{2}-m}{s^{2}+m^{2}-m}},}
where m {\\displaystyle m}  is the sample mean and s 2 {\\displaystyle s^{2}}  is the sample variance.
The maximum likelihood estimator[\[10\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-10) can be found by solving the following equation
m
(
1
â
e
â
λ
^
m
l
)
\=
λ
^
m
l
(
1
â
n
0
n
)
.
{\\displaystyle m(1-e^{-{\\hat {\\lambda }}\_{ml}})={\\hat {\\lambda }}\_{ml}\\left(1-{\\frac {n\_{0}}{n}}\\right).}
where n 0 n {\\displaystyle {\\frac {n\_{0}}{n}}}  is the observed proportion of zeros.
A closed form solution of this equation is given by[\[11\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-11)
λ
^
m
l
\=
W
0
(
â
s
e
â
s
)
\+
s
{\\displaystyle {\\hat {\\lambda }}\_{ml}=W\_{0}(-se^{-s})+s}
with W 0 {\\displaystyle W\_{0}}  being the main branch of Lambert's W-function[\[12\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-12) and
s
\=
m
1
â
n
0
n
{\\displaystyle s={\\frac {m}{1-{\\frac {n\_{0}}{n}}}}}
.
Alternatively, the equation can be solved by iteration.[\[13\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-13)
The maximum likelihood estimator for Ï {\\displaystyle \\pi }  is given by
Ï
^
m
l
\=
1
â
m
λ
^
m
l
.
{\\displaystyle {\\hat {\\pi }}\_{ml}=1-{\\frac {m}{{\\hat {\\lambda }}\_{ml}}}.}
## Related models
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=6 "Edit section: Related models")\]
In 1994, Greene considered the zero-inflated [negative binomial](https://en.wikipedia.org/wiki/Negative_binomial_distribution "Negative binomial distribution") (ZINB) model.[\[14\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-14) Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model.[\[15\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-15)
## Discrete pseudo compound Poisson model
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=7 "Edit section: Discrete pseudo compound Poisson model")\]
If the count data Y {\\displaystyle Y}  is such that the probability of zero is larger than the probability of nonzero, namely
Pr
(
Y
\=
0
)
\>
0\.5
{\\displaystyle \\Pr(Y=0)\>0.5}
then the discrete data Y {\\displaystyle Y}  obey discrete pseudo [compound Poisson distribution](https://en.wikipedia.org/wiki/Compound_Poisson_distribution "Compound Poisson distribution").[\[16\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-16)
In fact, let G ( z ) \= â n \= 0 â P ( Y \= n ) z n {\\displaystyle G(z)=\\sum \\limits \_{n=0}^{\\infty }P(Y=n)z^{n}}  be the [probability generating function](https://en.wikipedia.org/wiki/Probability_generating_function "Probability generating function") of y i {\\displaystyle y\_{i}} . If p 0 \= Pr ( Y \= 0 ) \> 0\.5 {\\displaystyle p\_{0}=\\Pr(Y=0)\>0.5} , then \| G ( z ) \| â©Ÿ p 0 â â i \= 1 â p i \= 2 p 0 â 1 \> 0 {\\displaystyle \|G(z)\|\\geqslant p\_{0}-\\sum \\limits \_{i=1}^{\\infty }p\_{i}=2p\_{0}-1\>0} . Then from the [WienerâLĂ©vy theorem](https://en.wikipedia.org/wiki/Wiener%E2%80%93L%C3%A9vy_theorem "WienerâLĂ©vy theorem"),[\[17\]](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_note-17) G ( z ) {\\displaystyle G(z)}  has the [probability generating function](https://en.wikipedia.org/wiki/Probability_generating_function "Probability generating function") of the discrete pseudo [compound Poisson distribution](https://en.wikipedia.org/wiki/Compound_Poisson_distribution "Compound Poisson distribution").
We say that the discrete random variable Y {\\displaystyle Y}  satisfying [probability generating function](https://en.wikipedia.org/wiki/Probability_generating_function "Probability generating function") characterization
G
Y
(
z
)
\=
â
n
\=
0
â
P
(
Y
\=
n
)
z
n
\=
exp
âĄ
(
â
k
\=
1
â
α
k
λ
(
z
k
â
1
)
)
,
(
\|
z
\|
â€
1
)
{\\displaystyle G\_{Y}(z)=\\sum \\limits \_{n=0}^{\\infty }P(Y=n)z^{n}=\\exp \\left(\\sum \_{k=1}^{\\infty }\\alpha \_{k}\\lambda (z^{k}-1)\\right),\\quad (\|z\|\\leq 1)}
has a discrete pseudo [compound Poisson distribution](https://en.wikipedia.org/wiki/Compound_Poisson_distribution "Compound Poisson distribution") with parameters
(
λ
1
,
λ
2
,
âŠ
)
\=
(
α
1
λ
,
α
2
λ
,
âŠ
)
â
R
â
(
â
k
\=
1
â
α
k
\=
1
,
â
k
\=
1
â
\|
α
k
\|
\<
â
,
α
k
â
R
,
λ
\>
0
)
.
{\\displaystyle (\\lambda \_{1},\\lambda \_{2},\\ldots )=(\\alpha \_{1}\\lambda ,\\alpha \_{2}\\lambda ,\\ldots )\\in \\mathbb {R} ^{\\infty }\\left(\\sum \_{k=1}^{\\infty }\\alpha \_{k}=1,\\sum \\limits \_{k=1}^{\\infty }\|\\alpha \_{k}\|\<\\infty ,\\alpha \_{k}\\in \\mathbb {R} ,\\lambda \>0\\right).}
When all the α k {\\displaystyle \\alpha \_{k}}  are non-negative, it is the discrete [compound Poisson distribution](https://en.wikipedia.org/wiki/Compound_Poisson_distribution "Compound Poisson distribution") (non-Poisson case) with [overdispersion](https://en.wikipedia.org/wiki/Overdispersion "Overdispersion") property.
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=8 "Edit section: See also")\]
- [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution "Poisson distribution")
- [Zero-truncated Poisson distribution](https://en.wikipedia.org/wiki/Zero-truncated_Poisson_distribution "Zero-truncated Poisson distribution")
- [Compound Poisson distribution](https://en.wikipedia.org/wiki/Compound_Poisson_distribution "Compound Poisson distribution")
- [Sparse approximation](https://en.wikipedia.org/wiki/Sparse_approximation "Sparse approximation")
- [Hurdle model](https://en.wikipedia.org/wiki/Hurdle_model "Hurdle model")
## Software
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=9 "Edit section: Software")\]
- [pscl](https://cran.r-project.org/web/packages/pscl/index.html), [glmmTMB](https://cran.r-project.org/web/packages/glmmTMB/index.html) and [brms](https://paul-buerkner.github.io/brms/) R packages
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&action=edit§ion=10 "Edit section: References")\]
1. ^ [***a***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-BilderLoughin2015_1-0) [***b***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-BilderLoughin2015_1-1) [***c***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-BilderLoughin2015_1-2) [***d***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-BilderLoughin2015_1-3)
Bilder, Christopher; Loughin, Thomas (2015), *Analysis of Categorical Data with R* (First ed.), CRC Press / Chapman & Hall, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1439855676](https://en.wikipedia.org/wiki/Special:BookSources/978-1439855676 "Special:BookSources/978-1439855676")
2. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-HilbeNBR2014_2-0)**
Hilbe, Joseph M. (2014), *Modeling Count Data* (First ed.), Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1107611252](https://en.wikipedia.org/wiki/Special:BookSources/978-1107611252 "Special:BookSources/978-1107611252")
3. ^ [***a***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-HilbeNBR2007_3-0) [***b***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-HilbeNBR2007_3-1)
Hilbe, Joseph M. (2007), *Negative Binomial Regression* (Second ed.), Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0521198158](https://en.wikipedia.org/wiki/Special:BookSources/978-0521198158 "Special:BookSources/978-0521198158")
4. ^ [***a***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-Lachin2011_4-0) [***b***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-Lachin2011_4-1)
Lachin, John M. (2011), *Biostatistical Methods: The Assessment of Relative Risks* (Second ed.), Wiley, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0470508220](https://en.wikipedia.org/wiki/Special:BookSources/978-0470508220 "Special:BookSources/978-0470508220")
5. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-ChernyavskiyMcmurry_5-0)**
["Biostatistics II. 1.3 - Zero-inflated Models"](https://www.youtube.com/watch?v=14B5QUUmqts). *[YouTube](https://en.wikipedia.org/wiki/YouTube "YouTube")*. Retrieved July 1, 2022.
6. ^ [***a***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-Long1997_6-0) [***b***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-Long1997_6-1)
Long, J. Scott (1997), *Regression Models for Categorical and Limited Dependent Variables* (First ed.), Sage Publications, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0803973749](https://en.wikipedia.org/wiki/Special:BookSources/978-0803973749 "Special:BookSources/978-0803973749")
7. ^ [***a***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-FriendlyMeyer2016_7-0) [***b***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-FriendlyMeyer2016_7-1) [***c***](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-FriendlyMeyer2016_7-2)
Friendly, Michael; David, Thomas (2016), *Discrete Data Analysis with R* (First ed.), CRC Press / Chapman & Hall, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1498725835](https://en.wikipedia.org/wiki/Special:BookSources/978-1498725835 "Special:BookSources/978-1498725835")
8. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-8)**
[Lambert, Diane](https://en.wikipedia.org/wiki/Diane_Lambert "Diane Lambert") (1992). "Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing". *Technometrics*. **34** (1): 1â14\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/1269547](https://doi.org/10.2307%2F1269547). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [1269547](https://www.jstor.org/stable/1269547).
9. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-9)**
Beckett, Sadie; Jee, Joshua; Ncube, Thalepo; Washington, Quintel; Singh, Anshuman; Pal, Nabendu (2014). ["Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to model data from natural calamities"](https://doi.org/10.2140%2Finvolve.2014.7.751). *Involve*. **7** (6): 751â767\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2140/involve.2014.7.751](https://doi.org/10.2140%2Finvolve.2014.7.751).
10. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-10)**
Johnson, Norman L.; Kotz, Samuel; [Kemp, Adrienne W.](https://en.wikipedia.org/wiki/Adrienne_W._Kemp "Adrienne W. Kemp") (1992). *Univariate Discrete Distributions* (2nd ed.). Wiley. pp. 312â314\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-471-54897-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-54897-3 "Special:BookSources/978-0-471-54897-3")
.
11. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-11)**
Dencks, Stefanie; Piepenbrock, Marion; Schmitz, Georg (2020). ["Assessing Vessel Reconstruction in Ultrasound Localization Microscopy by Maximum-Likelihood Estimation of a Zero-Inflated Poisson Model"](https://doi.org/10.1109%2FTUFFC.2020.2980063). *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/TUFFC.2020.2980063](https://doi.org/10.1109%2FTUFFC.2020.2980063).
12. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-12)**
Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W Function". *Advances in Computational Mathematics*. **5** (1): 329â359\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1809\.07369](https://arxiv.org/abs/1809.07369). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF02124750](https://doi.org/10.1007%2FBF02124750).
13. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-13)**
Böhning, Dankmar; Dietz, Ekkehart; Schlattmann, Peter; Mendonca, Lisette; Kirchner, Ursula (1999). "The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology". *Journal of the Royal Statistical Society, Series A*. **162** (2): 195â209\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1111/1467-985x.00130](https://doi.org/10.1111%2F1467-985x.00130).
14. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-14)**
Greene, William H. (1994). "Some Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models". *Working Paper EC-94-10: Department of Economics, New York University*. [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [1293115](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1293115).
15. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-15)**
Hall, Daniel B. (2000). "Zero-Inflated Poisson and Binomial Regression with Random Effects: A Case Study". *Biometrics*. **56** (4): 1030â1039\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1111/j.0006-341X.2000.01030.x](https://doi.org/10.1111%2Fj.0006-341X.2000.01030.x).
16. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-16)**
Huiming, Zhang; Yunxiao Liu; Bo Li (2014). "Notes on discrete compound Poisson model with applications to risk theory". *Insurance: Mathematics and Economics*. **59**: 325â336\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.insmatheco.2014.09.012](https://doi.org/10.1016%2Fj.insmatheco.2014.09.012).
17. **[^](https://en.wikipedia.org/wiki/Zero-inflated_model#cite_ref-17)**
Zygmund, A. (2002). [*Trigonometric Series*](https://en.wikipedia.org/wiki/Trigonometric_Series "Trigonometric Series"). Cambridge: Cambridge University Press. p. 245.
| [v](https://en.wikipedia.org/wiki/Template:Statistics "Template:Statistics") [t](https://en.wikipedia.org/wiki/Template_talk:Statistics "Template talk:Statistics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Statistics "Special:EditPage/Template:Statistics")[Statistics](https://en.wikipedia.org/wiki/Statistics "Statistics") | |
|---|---|
| [Outline](https://en.wikipedia.org/wiki/Outline_of_statistics "Outline of statistics") [Index](https://en.wikipedia.org/wiki/List_of_statistics_articles "List of statistics articles") | |
| [Descriptive statistics](https://en.wikipedia.org/wiki/Descriptive_statistics "Descriptive statistics") | |
| | |
| [Continuous data](https://en.wikipedia.org/wiki/Continuous_probability_distribution "Continuous probability distribution") | |
| | |
| [Center](https://en.wikipedia.org/wiki/Central_tendency "Central tendency") | [Mean](https://en.wikipedia.org/wiki/Mean "Mean") [Arithmetic](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean") [Arithmetic-Geometric](https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean "Arithmeticâgeometric mean") [Contraharmonic](https://en.wikipedia.org/wiki/Contraharmonic_mean "Contraharmonic mean") [Cubic](https://en.wikipedia.org/wiki/Cubic_mean "Cubic mean") [Generalized/power](https://en.wikipedia.org/wiki/Generalized_mean "Generalized mean") [Geometric](https://en.wikipedia.org/wiki/Geometric_mean "Geometric mean") [Harmonic](https://en.wikipedia.org/wiki/Harmonic_mean "Harmonic mean") [Heronian](https://en.wikipedia.org/wiki/Heronian_mean "Heronian mean") [Heinz](https://en.wikipedia.org/wiki/Heinz_mean "Heinz mean") [Lehmer](https://en.wikipedia.org/wiki/Lehmer_mean "Lehmer mean") [Median](https://en.wikipedia.org/wiki/Median "Median") [Mode](https://en.wikipedia.org/wiki/Mode_\(statistics\) "Mode (statistics)") |
| [Dispersion](https://en.wikipedia.org/wiki/Statistical_dispersion "Statistical dispersion") | [Average absolute deviation](https://en.wikipedia.org/wiki/Average_absolute_deviation "Average absolute deviation") [Coefficient of variation](https://en.wikipedia.org/wiki/Coefficient_of_variation "Coefficient of variation") [Interquartile range](https://en.wikipedia.org/wiki/Interquartile_range "Interquartile range") [Percentile](https://en.wikipedia.org/wiki/Percentile "Percentile") [Range](https://en.wikipedia.org/wiki/Range_\(statistics\) "Range (statistics)") [Standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") [Variance](https://en.wikipedia.org/wiki/Variance#Sample_variance "Variance") |
| [Shape](https://en.wikipedia.org/wiki/Shape_of_the_distribution "Shape of the distribution") | [Central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem") [Moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") [Kurtosis](https://en.wikipedia.org/wiki/Kurtosis "Kurtosis") [L-moments](https://en.wikipedia.org/wiki/L-moment "L-moment") [Skewness](https://en.wikipedia.org/wiki/Skewness "Skewness") |
| [Count data](https://en.wikipedia.org/wiki/Count_data "Count data") | [Index of dispersion](https://en.wikipedia.org/wiki/Index_of_dispersion "Index of dispersion") |
| Summary tables | [Contingency table](https://en.wikipedia.org/wiki/Contingency_table "Contingency table") [Frequency distribution](https://en.wikipedia.org/wiki/Frequency_distribution "Frequency distribution") [Grouped data](https://en.wikipedia.org/wiki/Grouped_data "Grouped data") |
| [Dependence](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Pearson product-moment correlation](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient "Pearson correlation coefficient") [Rank correlation](https://en.wikipedia.org/wiki/Rank_correlation "Rank correlation") [Kendall's Ï](https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient "Kendall rank correlation coefficient") [Spearman's Ï](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient "Spearman's rank correlation coefficient") [Scatter plot](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") |
| [Graphics](https://en.wikipedia.org/wiki/Statistical_graphics "Statistical graphics") | [Bar chart](https://en.wikipedia.org/wiki/Bar_chart "Bar chart") [Biplot](https://en.wikipedia.org/wiki/Biplot "Biplot") [Box plot](https://en.wikipedia.org/wiki/Box_plot "Box plot") [Control chart](https://en.wikipedia.org/wiki/Control_chart "Control chart") [Correlogram](https://en.wikipedia.org/wiki/Correlogram "Correlogram") [Fan chart](https://en.wikipedia.org/wiki/Fan_chart_\(statistics\) "Fan chart (statistics)") [Forest plot](https://en.wikipedia.org/wiki/Forest_plot "Forest plot") [Histogram](https://en.wikipedia.org/wiki/Histogram "Histogram") [Pie chart](https://en.wikipedia.org/wiki/Pie_chart "Pie chart") [QâQ plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot "QâQ plot") [Radar chart](https://en.wikipedia.org/wiki/Radar_chart "Radar chart") [Run chart](https://en.wikipedia.org/wiki/Run_chart "Run chart") [Scatter plot](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") [Stem-and-leaf display](https://en.wikipedia.org/wiki/Stem-and-leaf_display "Stem-and-leaf display") [Violin plot](https://en.wikipedia.org/wiki/Violin_plot "Violin plot") |
| [Data collection](https://en.wikipedia.org/wiki/Data_collection "Data collection") | |
| | |
| [Study design](https://en.wikipedia.org/wiki/Design_of_experiments "Design of experiments") | [Effect size](https://en.wikipedia.org/wiki/Effect_size "Effect size") [Missing data](https://en.wikipedia.org/wiki/Missing_data "Missing data") [Optimal design](https://en.wikipedia.org/wiki/Optimal_design "Optimal design") [Population](https://en.wikipedia.org/wiki/Statistical_population "Statistical population") [Replication](https://en.wikipedia.org/wiki/Replication_\(statistics\) "Replication (statistics)") [Sample size determination](https://en.wikipedia.org/wiki/Sample_size_determination "Sample size determination") [Statistic](https://en.wikipedia.org/wiki/Statistic "Statistic") [Statistical power](https://en.wikipedia.org/wiki/Statistical_power "Statistical power") |
| [Survey methodology](https://en.wikipedia.org/wiki/Survey_methodology "Survey methodology") | [Sampling](https://en.wikipedia.org/wiki/Sampling_\(statistics\) "Sampling (statistics)") [Cluster](https://en.wikipedia.org/wiki/Cluster_sampling "Cluster sampling") [Stratified](https://en.wikipedia.org/wiki/Stratified_sampling "Stratified sampling") [Opinion poll](https://en.wikipedia.org/wiki/Opinion_poll "Opinion poll") [Questionnaire](https://en.wikipedia.org/wiki/Questionnaire "Questionnaire") [Standard error](https://en.wikipedia.org/wiki/Standard_error "Standard error") |
| [Controlled experiments](https://en.wikipedia.org/wiki/Experiment "Experiment") | [Blocking](https://en.wikipedia.org/wiki/Blocking_\(statistics\) "Blocking (statistics)") [Factorial experiment](https://en.wikipedia.org/wiki/Factorial_experiment "Factorial experiment") [Interaction](https://en.wikipedia.org/wiki/Interaction_\(statistics\) "Interaction (statistics)") [Random assignment](https://en.wikipedia.org/wiki/Random_assignment "Random assignment") [Randomized controlled trial](https://en.wikipedia.org/wiki/Randomized_controlled_trial "Randomized controlled trial") [Randomized experiment](https://en.wikipedia.org/wiki/Randomized_experiment "Randomized experiment") [Scientific control](https://en.wikipedia.org/wiki/Scientific_control "Scientific control") |
| Adaptive designs | [Adaptive clinical trial](https://en.wikipedia.org/wiki/Adaptive_clinical_trial "Adaptive clinical trial") [Stochastic approximation](https://en.wikipedia.org/wiki/Stochastic_approximation "Stochastic approximation") [Up-and-down designs](https://en.wikipedia.org/wiki/Up-and-Down_Designs "Up-and-Down Designs") |
| [Observational studies](https://en.wikipedia.org/wiki/Observational_study "Observational study") | [Cohort study](https://en.wikipedia.org/wiki/Cohort_study "Cohort study") [Cross-sectional study](https://en.wikipedia.org/wiki/Cross-sectional_study "Cross-sectional study") [Natural experiment](https://en.wikipedia.org/wiki/Natural_experiment "Natural experiment") [Quasi-experiment](https://en.wikipedia.org/wiki/Quasi-experiment "Quasi-experiment") |
| [Statistical inference](https://en.wikipedia.org/wiki/Statistical_inference "Statistical inference") | |
| | |
| [Statistical theory](https://en.wikipedia.org/wiki/Statistical_theory "Statistical theory") | [Population](https://en.wikipedia.org/wiki/Population_\(statistics\) "Population (statistics)") [Statistic](https://en.wikipedia.org/wiki/Statistic "Statistic") [Probability distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") [Sampling distribution](https://en.wikipedia.org/wiki/Sampling_distribution "Sampling distribution") [Order statistic](https://en.wikipedia.org/wiki/Order_statistic "Order statistic") [Empirical distribution](https://en.wikipedia.org/wiki/Empirical_distribution_function "Empirical distribution function") [Density estimation](https://en.wikipedia.org/wiki/Density_estimation "Density estimation") [Statistical model](https://en.wikipedia.org/wiki/Statistical_model "Statistical model") [Model specification](https://en.wikipedia.org/wiki/Model_specification "Model specification") [L*p* space](https://en.wikipedia.org/wiki/Lp_space "Lp space") [Parameter](https://en.wikipedia.org/wiki/Statistical_parameter "Statistical parameter") [location](https://en.wikipedia.org/wiki/Location_parameter "Location parameter") [scale](https://en.wikipedia.org/wiki/Scale_parameter "Scale parameter") [shape](https://en.wikipedia.org/wiki/Shape_parameter "Shape parameter") [Parametric family](https://en.wikipedia.org/wiki/Parametric_statistics "Parametric statistics") [Likelihood](https://en.wikipedia.org/wiki/Likelihood_function "Likelihood function") [(monotone)](https://en.wikipedia.org/wiki/Monotone_likelihood_ratio "Monotone likelihood ratio") [Locationâscale family](https://en.wikipedia.org/wiki/Location%E2%80%93scale_family "Locationâscale family") [Exponential family](https://en.wikipedia.org/wiki/Exponential_family "Exponential family") [Completeness](https://en.wikipedia.org/wiki/Completeness_\(statistics\) "Completeness (statistics)") [Sufficiency](https://en.wikipedia.org/wiki/Sufficient_statistic "Sufficient statistic") [Statistical functional](https://en.wikipedia.org/wiki/Plug-in_principle "Plug-in principle") [Bootstrap](https://en.wikipedia.org/wiki/Bootstrapping_\(statistics\) "Bootstrapping (statistics)") [U](https://en.wikipedia.org/wiki/U-statistic "U-statistic") [V](https://en.wikipedia.org/wiki/V-statistic "V-statistic") [Optimal decision](https://en.wikipedia.org/wiki/Optimal_decision "Optimal decision") [loss function](https://en.wikipedia.org/wiki/Loss_function "Loss function") [Efficiency](https://en.wikipedia.org/wiki/Efficiency_\(statistics\) "Efficiency (statistics)") [Statistical distance](https://en.wikipedia.org/wiki/Statistical_distance "Statistical distance") [divergence](https://en.wikipedia.org/wiki/Divergence_\(statistics\) "Divergence (statistics)") [Asymptotics](https://en.wikipedia.org/wiki/Asymptotic_theory_\(statistics\) "Asymptotic theory (statistics)") [Robustness](https://en.wikipedia.org/wiki/Robust_statistics "Robust statistics") |
| [Frequentist inference](https://en.wikipedia.org/wiki/Frequentist_inference "Frequentist inference") | |
| | |
| [Point estimation](https://en.wikipedia.org/wiki/Point_estimation "Point estimation") | [Estimating equations](https://en.wikipedia.org/wiki/Estimating_equations "Estimating equations") [Maximum likelihood](https://en.wikipedia.org/wiki/Maximum_likelihood "Maximum likelihood") [Method of moments](https://en.wikipedia.org/wiki/Method_of_moments_\(statistics\) "Method of moments (statistics)") [M-estimator](https://en.wikipedia.org/wiki/M-estimator "M-estimator") [Minimum distance](https://en.wikipedia.org/wiki/Minimum_distance_estimation "Minimum distance estimation") [Unbiased estimators](https://en.wikipedia.org/wiki/Bias_of_an_estimator "Bias of an estimator") [Mean-unbiased minimum-variance](https://en.wikipedia.org/wiki/Minimum-variance_unbiased_estimator "Minimum-variance unbiased estimator") [RaoâBlackwellization](https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem "RaoâBlackwell theorem") [LehmannâScheffĂ© theorem](https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem "LehmannâScheffĂ© theorem") [Median unbiased](https://en.wikipedia.org/wiki/Median-unbiased_estimator "Median-unbiased estimator") [Plug-in](https://en.wikipedia.org/wiki/Plug-in_principle "Plug-in principle") |
| [Interval estimation](https://en.wikipedia.org/wiki/Interval_estimation "Interval estimation") | [Confidence interval](https://en.wikipedia.org/wiki/Confidence_interval "Confidence interval") [Pivot](https://en.wikipedia.org/wiki/Pivotal_quantity "Pivotal quantity") [Likelihood interval](https://en.wikipedia.org/wiki/Likelihood_interval "Likelihood interval") [Prediction interval](https://en.wikipedia.org/wiki/Prediction_interval "Prediction interval") [Tolerance interval](https://en.wikipedia.org/wiki/Tolerance_interval "Tolerance interval") [Resampling](https://en.wikipedia.org/wiki/Resampling_\(statistics\) "Resampling (statistics)") [Bootstrap](https://en.wikipedia.org/wiki/Bootstrapping_\(statistics\) "Bootstrapping (statistics)") [Jackknife](https://en.wikipedia.org/wiki/Jackknife_resampling "Jackknife resampling") |
| [Testing hypotheses](https://en.wikipedia.org/wiki/Statistical_hypothesis_testing "Statistical hypothesis testing") | [1- & 2-tails](https://en.wikipedia.org/wiki/One-_and_two-tailed_tests "One- and two-tailed tests") [Power](https://en.wikipedia.org/wiki/Power_\(statistics\) "Power (statistics)") [Uniformly most powerful test](https://en.wikipedia.org/wiki/Uniformly_most_powerful_test "Uniformly most powerful test") [Permutation test](https://en.wikipedia.org/wiki/Permutation_test "Permutation test") [Randomization test](https://en.wikipedia.org/wiki/Randomization_test "Randomization test") [Multiple comparisons](https://en.wikipedia.org/wiki/Multiple_comparisons "Multiple comparisons") |
| [Parametric tests](https://en.wikipedia.org/wiki/Parametric_statistics "Parametric statistics") | [Likelihood-ratio](https://en.wikipedia.org/wiki/Likelihood-ratio_test "Likelihood-ratio test") [Score/Lagrange multiplier](https://en.wikipedia.org/wiki/Score_test "Score test") [Wald](https://en.wikipedia.org/wiki/Wald_test "Wald test") |
| [Specific tests](https://en.wikipedia.org/wiki/List_of_statistical_tests "List of statistical tests") | |
| | |
| [*Z*\-test (normal)](https://en.wikipedia.org/wiki/Z-test "Z-test") [Student's *t*\-test](https://en.wikipedia.org/wiki/Student%27s_t-test "Student's t-test") [*F*\-test](https://en.wikipedia.org/wiki/F-test "F-test") | |
| [Goodness of fit](https://en.wikipedia.org/wiki/Goodness_of_fit "Goodness of fit") | [Chi-squared](https://en.wikipedia.org/wiki/Chi-squared_test "Chi-squared test") [*G*\-test](https://en.wikipedia.org/wiki/G-test "G-test") [KolmogorovâSmirnov](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test "KolmogorovâSmirnov test") [AndersonâDarling](https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test "AndersonâDarling test") [Lilliefors](https://en.wikipedia.org/wiki/Lilliefors_test "Lilliefors test") [JarqueâBera](https://en.wikipedia.org/wiki/Jarque%E2%80%93Bera_test "JarqueâBera test") [Normality (ShapiroâWilk)](https://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test "ShapiroâWilk test") [Likelihood-ratio test](https://en.wikipedia.org/wiki/Likelihood-ratio_test "Likelihood-ratio test") [Model selection](https://en.wikipedia.org/wiki/Model_selection "Model selection") [Cross validation](https://en.wikipedia.org/wiki/Cross-validation_\(statistics\) "Cross-validation (statistics)") [AIC](https://en.wikipedia.org/wiki/Akaike_information_criterion "Akaike information criterion") [BIC](https://en.wikipedia.org/wiki/Bayesian_information_criterion "Bayesian information criterion") |
| [Rank statistics](https://en.wikipedia.org/wiki/Rank_statistics "Rank statistics") | [Sign](https://en.wikipedia.org/wiki/Sign_test "Sign test") [Sample median](https://en.wikipedia.org/wiki/Sample_median "Sample median") [Signed rank (Wilcoxon)](https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test "Wilcoxon signed-rank test") [HodgesâLehmann estimator](https://en.wikipedia.org/wiki/Hodges%E2%80%93Lehmann_estimator "HodgesâLehmann estimator") [Rank sum (MannâWhitney)](https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test "MannâWhitney U test") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_statistics "Nonparametric statistics") [anova](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [1-way (KruskalâWallis)](https://en.wikipedia.org/wiki/Kruskal%E2%80%93Wallis_test "KruskalâWallis test") [2-way (Friedman)](https://en.wikipedia.org/wiki/Friedman_test "Friedman test") [Ordered alternative (JonckheereâTerpstra)](https://en.wikipedia.org/wiki/Jonckheere%27s_trend_test "Jonckheere's trend test") [Van der Waerden test](https://en.wikipedia.org/wiki/Van_der_Waerden_test "Van der Waerden test") |
| [Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference "Bayesian inference") | [Bayesian probability](https://en.wikipedia.org/wiki/Bayesian_probability "Bayesian probability") [prior](https://en.wikipedia.org/wiki/Prior_probability "Prior probability") [posterior](https://en.wikipedia.org/wiki/Posterior_probability "Posterior probability") [Credible interval](https://en.wikipedia.org/wiki/Credible_interval "Credible interval") [Bayes factor](https://en.wikipedia.org/wiki/Bayes_factor "Bayes factor") [Bayesian estimator](https://en.wikipedia.org/wiki/Bayes_estimator "Bayes estimator") [Maximum posterior estimator](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation "Maximum a posteriori estimation") |
| [Correlation](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | |
| | |
| [Correlation](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Pearson product-moment](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient "Pearson product-moment correlation coefficient") [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Confounding variable](https://en.wikipedia.org/wiki/Confounding "Confounding") [Coefficient of determination](https://en.wikipedia.org/wiki/Coefficient_of_determination "Coefficient of determination") |
| [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | [Errors and residuals](https://en.wikipedia.org/wiki/Errors_and_residuals "Errors and residuals") [Regression validation](https://en.wikipedia.org/wiki/Regression_validation "Regression validation") [Mixed effects models](https://en.wikipedia.org/wiki/Mixed_model "Mixed model") [Simultaneous equations models](https://en.wikipedia.org/wiki/Simultaneous_equations_model "Simultaneous equations model") [Multivariate adaptive regression splines (MARS)](https://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines "Multivariate adaptive regression splines") |
| [Linear regression](https://en.wikipedia.org/wiki/Linear_regression "Linear regression") | [Simple linear regression](https://en.wikipedia.org/wiki/Simple_linear_regression "Simple linear regression") [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares") [General linear model](https://en.wikipedia.org/wiki/General_linear_model "General linear model") [Bayesian regression](https://en.wikipedia.org/wiki/Bayesian_linear_regression "Bayesian linear regression") |
| Non-standard predictors | [Nonlinear regression](https://en.wikipedia.org/wiki/Nonlinear_regression "Nonlinear regression") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_regression "Nonparametric regression") [Semiparametric](https://en.wikipedia.org/wiki/Semiparametric_regression "Semiparametric regression") [Isotonic](https://en.wikipedia.org/wiki/Isotonic_regression "Isotonic regression") [Robust](https://en.wikipedia.org/wiki/Robust_regression "Robust regression") [Homoscedasticity and Heteroscedasticity](https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity "Homoscedasticity and heteroscedasticity") |
| [Generalized linear model](https://en.wikipedia.org/wiki/Generalized_linear_model "Generalized linear model") | [Exponential families](https://en.wikipedia.org/wiki/Exponential_family "Exponential family") [Logistic (Bernoulli)](https://en.wikipedia.org/wiki/Logistic_regression "Logistic regression") / [Binomial](https://en.wikipedia.org/wiki/Binomial_regression "Binomial regression") / [Poisson regressions](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") |
| [Partition of variance](https://en.wikipedia.org/wiki/Partition_of_sums_of_squares "Partition of sums of squares") | [Analysis of variance (ANOVA, anova)](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [Analysis of covariance](https://en.wikipedia.org/wiki/Analysis_of_covariance "Analysis of covariance") [Multivariate ANOVA](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") [Degrees of freedom](https://en.wikipedia.org/wiki/Degrees_of_freedom_\(statistics\) "Degrees of freedom (statistics)") |
| [Categorical](https://en.wikipedia.org/wiki/Categorical_variable "Categorical variable") / [multivariate](https://en.wikipedia.org/wiki/Multivariate_statistics "Multivariate statistics") / [time-series](https://en.wikipedia.org/wiki/Time_series "Time series") / [survival analysis](https://en.wikipedia.org/wiki/Survival_analysis "Survival analysis") | |
| | |
| [Categorical](https://en.wikipedia.org/wiki/Categorical_variable "Categorical variable") | [Cohen's kappa](https://en.wikipedia.org/wiki/Cohen%27s_kappa "Cohen's kappa") [Contingency table](https://en.wikipedia.org/wiki/Contingency_table "Contingency table") [Graphical model](https://en.wikipedia.org/wiki/Graphical_model "Graphical model") [Log-linear model](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") [McNemar's test](https://en.wikipedia.org/wiki/McNemar%27s_test "McNemar's test") [CochranâMantelâHaenszel statistics](https://en.wikipedia.org/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics "CochranâMantelâHaenszel statistics") |
| [Multivariate](https://en.wikipedia.org/wiki/Multivariate_statistics "Multivariate statistics") | [Regression](https://en.wikipedia.org/wiki/General_linear_model "General linear model") [Manova](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") [Principal components](https://en.wikipedia.org/wiki/Principal_component_analysis "Principal component analysis") [Canonical correlation](https://en.wikipedia.org/wiki/Canonical_correlation "Canonical correlation") [Discriminant analysis](https://en.wikipedia.org/wiki/Linear_discriminant_analysis "Linear discriminant analysis") [Cluster analysis](https://en.wikipedia.org/wiki/Cluster_analysis "Cluster analysis") [Classification](https://en.wikipedia.org/wiki/Statistical_classification "Statistical classification") [Structural equation model](https://en.wikipedia.org/wiki/Structural_equation_modeling "Structural equation modeling") [Factor analysis](https://en.wikipedia.org/wiki/Factor_analysis "Factor analysis") [Multivariate distributions](https://en.wikipedia.org/wiki/Multivariate_distribution "Multivariate distribution") [Elliptical distributions](https://en.wikipedia.org/wiki/Elliptical_distribution "Elliptical distribution") [Normal](https://en.wikipedia.org/wiki/Multivariate_normal_distribution "Multivariate normal distribution") |
| [Time-series](https://en.wikipedia.org/wiki/Time_series "Time series") | |
| | |
| General | [Decomposition](https://en.wikipedia.org/wiki/Decomposition_of_time_series "Decomposition of time series") [Trend](https://en.wikipedia.org/wiki/Trend_estimation "Trend estimation") [Stationarity](https://en.wikipedia.org/wiki/Stationary_process "Stationary process") [Seasonal adjustment](https://en.wikipedia.org/wiki/Seasonal_adjustment "Seasonal adjustment") [Exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing "Exponential smoothing") [Cointegration](https://en.wikipedia.org/wiki/Cointegration "Cointegration") [Structural break](https://en.wikipedia.org/wiki/Structural_break "Structural break") [Granger causality](https://en.wikipedia.org/wiki/Granger_causality "Granger causality") |
| Specific tests | [DickeyâFuller](https://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test "DickeyâFuller test") [Johansen](https://en.wikipedia.org/wiki/Johansen_test "Johansen test") [Q-statistic (LjungâBox)](https://en.wikipedia.org/wiki/Ljung%E2%80%93Box_test "LjungâBox test") [DurbinâWatson](https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic "DurbinâWatson statistic") [BreuschâGodfrey](https://en.wikipedia.org/wiki/Breusch%E2%80%93Godfrey_test "BreuschâGodfrey test") |
| [Time domain](https://en.wikipedia.org/wiki/Time_domain "Time domain") | [Autocorrelation (ACF)](https://en.wikipedia.org/wiki/Autocorrelation "Autocorrelation") [partial (PACF)](https://en.wikipedia.org/wiki/Partial_autocorrelation_function "Partial autocorrelation function") [Cross-correlation (XCF)](https://en.wikipedia.org/wiki/Cross-correlation "Cross-correlation") [ARMA model](https://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model "Autoregressiveâmoving-average model") [ARIMA model (BoxâJenkins)](https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method "BoxâJenkins method") [Autoregressive conditional heteroskedasticity (ARCH)](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity") [Vector autoregression (VAR)](https://en.wikipedia.org/wiki/Vector_autoregression "Vector autoregression") |
| [Frequency domain](https://en.wikipedia.org/wiki/Frequency_domain "Frequency domain") | [Spectral density estimation](https://en.wikipedia.org/wiki/Spectral_density_estimation "Spectral density estimation") [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis") [Least-squares spectral analysis](https://en.wikipedia.org/wiki/Least-squares_spectral_analysis "Least-squares spectral analysis") [Wavelet](https://en.wikipedia.org/wiki/Wavelet "Wavelet") [Whittle likelihood](https://en.wikipedia.org/wiki/Whittle_likelihood "Whittle likelihood") |
| [Survival](https://en.wikipedia.org/wiki/Survival_analysis "Survival analysis") | |
| | |
| [Survival function](https://en.wikipedia.org/wiki/Survival_function "Survival function") | [KaplanâMeier estimator (product limit)](https://en.wikipedia.org/wiki/Kaplan%E2%80%93Meier_estimator "KaplanâMeier estimator") [Proportional hazards models](https://en.wikipedia.org/wiki/Proportional_hazards_model "Proportional hazards model") [Accelerated failure time (AFT) model](https://en.wikipedia.org/wiki/Accelerated_failure_time_model "Accelerated failure time model") [First hitting time](https://en.wikipedia.org/wiki/First-hitting-time_model "First-hitting-time model") |
| [Hazard function](https://en.wikipedia.org/wiki/Failure_rate "Failure rate") | [NelsonâAalen estimator](https://en.wikipedia.org/wiki/Nelson%E2%80%93Aalen_estimator "NelsonâAalen estimator") |
| Test | [Log-rank test](https://en.wikipedia.org/wiki/Log-rank_test "Log-rank test") |
| [Applications](https://en.wikipedia.org/wiki/List_of_fields_of_application_of_statistics "List of fields of application of statistics") | |
| | |
| [Biostatistics](https://en.wikipedia.org/wiki/Biostatistics "Biostatistics") | [Bioinformatics](https://en.wikipedia.org/wiki/Bioinformatics "Bioinformatics") [Clinical trials](https://en.wikipedia.org/wiki/Clinical_trial "Clinical trial") / [studies](https://en.wikipedia.org/wiki/Clinical_study_design "Clinical study design") [Epidemiology](https://en.wikipedia.org/wiki/Epidemiology "Epidemiology") [Medical statistics](https://en.wikipedia.org/wiki/Medical_statistics "Medical statistics") |
| [Engineering statistics](https://en.wikipedia.org/wiki/Engineering_statistics "Engineering statistics") | [Chemometrics](https://en.wikipedia.org/wiki/Chemometrics "Chemometrics") [Methods engineering](https://en.wikipedia.org/wiki/Methods_engineering "Methods engineering") [Probabilistic design](https://en.wikipedia.org/wiki/Probabilistic_design "Probabilistic design") [Process](https://en.wikipedia.org/wiki/Statistical_process_control "Statistical process control") / [quality control](https://en.wikipedia.org/wiki/Quality_control "Quality control") [Reliability](https://en.wikipedia.org/wiki/Reliability_engineering "Reliability engineering") [System identification](https://en.wikipedia.org/wiki/System_identification "System identification") |
| [Social statistics](https://en.wikipedia.org/wiki/Social_statistics "Social statistics") | [Actuarial science](https://en.wikipedia.org/wiki/Actuarial_science "Actuarial science") [Census](https://en.wikipedia.org/wiki/Census "Census") [Crime statistics](https://en.wikipedia.org/wiki/Crime_statistics "Crime statistics") [Demography](https://en.wikipedia.org/wiki/Demographic_statistics "Demographic statistics") [Econometrics](https://en.wikipedia.org/wiki/Econometrics "Econometrics") [Jurimetrics](https://en.wikipedia.org/wiki/Jurimetrics "Jurimetrics") [National accounts](https://en.wikipedia.org/wiki/National_accounts "National accounts") [Official statistics](https://en.wikipedia.org/wiki/Official_statistics "Official statistics") [Population statistics](https://en.wikipedia.org/wiki/Population_statistics "Population statistics") [Psychometrics](https://en.wikipedia.org/wiki/Psychometrics "Psychometrics") |
| [Spatial statistics](https://en.wikipedia.org/wiki/Spatial_analysis "Spatial analysis") | [Cartography](https://en.wikipedia.org/wiki/Cartography "Cartography") [Environmental statistics](https://en.wikipedia.org/wiki/Environmental_statistics "Environmental statistics") [Geographic information system](https://en.wikipedia.org/wiki/Geographic_information_system "Geographic information system") [Geostatistics](https://en.wikipedia.org/wiki/Geostatistics "Geostatistics") [Kriging](https://en.wikipedia.org/wiki/Kriging "Kriging") |
| **[Category](https://en.wikipedia.org/wiki/Category:Statistics "Category:Statistics")** **[](https://en.wikipedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg) [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics "Portal:Mathematics")** **[Commons](https://commons.wikimedia.org/wiki/Category:Statistics "commons:Category:Statistics")**  **[WikiProject](https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Statistics "Wikipedia:WikiProject Statistics")** | |
| [v](https://en.wikipedia.org/wiki/Template:Least_squares_and_regression_analysis "Template:Least squares and regression analysis") [t](https://en.wikipedia.org/wiki/Template_talk:Least_squares_and_regression_analysis "Template talk:Least squares and regression analysis") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Least_squares_and_regression_analysis "Special:EditPage/Template:Least squares and regression analysis")[Least squares](https://en.wikipedia.org/wiki/Least_squares "Least squares") and [regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | |
|---|---|
| [Computational statistics](https://en.wikipedia.org/wiki/Computational_statistics "Computational statistics") | [Least squares](https://en.wikipedia.org/wiki/Least_squares "Least squares") [Linear least squares](https://en.wikipedia.org/wiki/Linear_least_squares_\(mathematics\) "Linear least squares (mathematics)") [Non-linear least squares](https://en.wikipedia.org/wiki/Non-linear_least_squares "Non-linear least squares") [Iteratively reweighted least squares](https://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares "Iteratively reweighted least squares") |
| [Correlation and dependence](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Pearson product-moment correlation](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient "Pearson product-moment correlation coefficient") [Rank correlation](https://en.wikipedia.org/wiki/Rank_correlation "Rank correlation") ([Spearman's rho](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient "Spearman's rank correlation coefficient") [Kendall's tau](https://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient "Kendall tau rank correlation coefficient")) [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Confounding variable](https://en.wikipedia.org/wiki/Confounding "Confounding") |
| [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares") [Partial least squares](https://en.wikipedia.org/wiki/Partial_least_squares_regression "Partial least squares regression") [Total least squares](https://en.wikipedia.org/wiki/Total_least_squares "Total least squares") [Ridge regression](https://en.wikipedia.org/wiki/Tikhonov_regularization "Tikhonov regularization") |
| Regression as a [statistical model](https://en.wikipedia.org/wiki/Statistical_model "Statistical model") | |
| | |
| [Linear regression](https://en.wikipedia.org/wiki/Linear_regression "Linear regression") | [Simple linear regression](https://en.wikipedia.org/wiki/Simple_linear_regression "Simple linear regression") [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares") [Generalized least squares](https://en.wikipedia.org/wiki/Generalized_least_squares "Generalized least squares") [Weighted least squares](https://en.wikipedia.org/wiki/Weighted_least_squares "Weighted least squares") [General linear model](https://en.wikipedia.org/wiki/General_linear_model "General linear model") |
| Predictor structure | [Polynomial regression](https://en.wikipedia.org/wiki/Polynomial_regression "Polynomial regression") [Growth curve (statistics)](https://en.wikipedia.org/wiki/Growth_curve_\(statistics\) "Growth curve (statistics)") [Segmented regression](https://en.wikipedia.org/wiki/Segmented_regression "Segmented regression") [Local regression](https://en.wikipedia.org/wiki/Local_regression "Local regression") |
| Non-standard | [Nonlinear regression](https://en.wikipedia.org/wiki/Nonlinear_regression "Nonlinear regression") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_regression "Nonparametric regression") [Semiparametric](https://en.wikipedia.org/wiki/Semiparametric_regression "Semiparametric regression") [Robust](https://en.wikipedia.org/wiki/Robust_regression "Robust regression") [Quantile](https://en.wikipedia.org/wiki/Quantile_regression "Quantile regression") [Isotonic](https://en.wikipedia.org/wiki/Isotonic_regression "Isotonic regression") |
| Non-normal errors | [Generalized linear model](https://en.wikipedia.org/wiki/Generalized_linear_model "Generalized linear model") [Binomial](https://en.wikipedia.org/wiki/Binomial_regression "Binomial regression") [Poisson](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") [Logistic](https://en.wikipedia.org/wiki/Logistic_regression "Logistic regression") |
| [Decomposition of variance](https://en.wikipedia.org/wiki/Partition_of_sums_of_squares "Partition of sums of squares") | [Analysis of variance](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [Analysis of covariance](https://en.wikipedia.org/wiki/Analysis_of_covariance "Analysis of covariance") [Multivariate AOV](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") |
| Model exploration | [Stepwise regression](https://en.wikipedia.org/wiki/Stepwise_regression "Stepwise regression") [Model selection](https://en.wikipedia.org/wiki/Model_selection "Model selection") [Mallows's *Cp*](https://en.wikipedia.org/wiki/Mallows%27s_Cp "Mallows's Cp") [AIC](https://en.wikipedia.org/wiki/Akaike_information_criterion "Akaike information criterion") [BIC](https://en.wikipedia.org/wiki/Bayesian_information_criterion "Bayesian information criterion") [Model specification](https://en.wikipedia.org/wiki/Model_specification "Model specification") [Regression validation](https://en.wikipedia.org/wiki/Regression_validation "Regression validation") |
| Background | [Mean and predicted response](https://en.wikipedia.org/wiki/Mean_and_predicted_response "Mean and predicted response") [GaussâMarkov theorem](https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem "GaussâMarkov theorem") [Errors and residuals](https://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics "Errors and residuals in statistics") [Goodness of fit](https://en.wikipedia.org/wiki/Goodness_of_fit "Goodness of fit") [Studentized residual](https://en.wikipedia.org/wiki/Studentized_residual "Studentized residual") [Minimum mean-square error](https://en.wikipedia.org/wiki/Minimum_mean-square_error "Minimum mean-square error") [FrischâWaughâLovell theorem](https://en.wikipedia.org/wiki/Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem "FrischâWaughâLovell theorem") |
| [Design of experiments](https://en.wikipedia.org/wiki/Design_of_experiments "Design of experiments") | [Response surface methodology](https://en.wikipedia.org/wiki/Response_surface_methodology "Response surface methodology") [Optimal design](https://en.wikipedia.org/wiki/Optimal_design "Optimal design") [Bayesian design](https://en.wikipedia.org/wiki/Bayesian_experimental_design "Bayesian experimental design") |
| [Numerical](https://en.wikipedia.org/wiki/Numerical_analysis "Numerical analysis") [approximation](https://en.wikipedia.org/wiki/Approximation_theory "Approximation theory") | [Numerical analysis](https://en.wikipedia.org/wiki/Numerical_analysis "Numerical analysis") [Approximation theory](https://en.wikipedia.org/wiki/Approximation_theory "Approximation theory") [Numerical integration](https://en.wikipedia.org/wiki/Numerical_integration "Numerical integration") [Gaussian quadrature](https://en.wikipedia.org/wiki/Gaussian_quadrature "Gaussian quadrature") [Orthogonal polynomials](https://en.wikipedia.org/wiki/Orthogonal_polynomials "Orthogonal polynomials") [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials "Chebyshev polynomials") [Chebyshev nodes](https://en.wikipedia.org/wiki/Chebyshev_nodes "Chebyshev nodes") |
| Applications | [Curve fitting](https://en.wikipedia.org/wiki/Curve_fitting "Curve fitting") [Calibration curve](https://en.wikipedia.org/wiki/Calibration_curve "Calibration curve") [Numerical smoothing and differentiation](https://en.wikipedia.org/wiki/Numerical_smoothing_and_differentiation "Numerical smoothing and differentiation") [System identification](https://en.wikipedia.org/wiki/System_identification "System identification") [Moving least squares](https://en.wikipedia.org/wiki/Moving_least_squares "Moving least squares") |
| [Regression analysis category](https://en.wikipedia.org/wiki/Category:Regression_analysis "Category:Regression analysis") [Statistics category](https://en.wikipedia.org/wiki/Category:Statistics "Category:Statistics") [](https://en.wikipedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg) [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics "Portal:Mathematics") [Statistics outline](https://en.wikipedia.org/wiki/Outline_of_statistics "Outline of statistics") [Statistics topics](https://en.wikipedia.org/wiki/List_of_statistics_articles "List of statistics articles") | |
Retrieved from "<https://en.wikipedia.org/w/index.php?title=Zero-inflated_model&oldid=1287526710>"
[Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"):
- [Generalized linear models](https://en.wikipedia.org/wiki/Category:Generalized_linear_models "Category:Generalized linear models")
- [Categorical data](https://en.wikipedia.org/wiki/Category:Categorical_data "Category:Categorical data")
- [Poisson point processes](https://en.wikipedia.org/wiki/Category:Poisson_point_processes "Category:Poisson point processes")
Hidden categories:
- [Articles with short description](https://en.wikipedia.org/wiki/Category:Articles_with_short_description "Category:Articles with short description")
- [Short description matches Wikidata](https://en.wikipedia.org/wiki/Category:Short_description_matches_Wikidata "Category:Short description matches Wikidata")
- This page was last edited on 26 April 2025, at 20:39 (UTC).
- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License "Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License"); additional terms may apply. By using this site, you agree to the [Terms of Use](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use "foundation:Special:MyLanguage/Policy:Terms of Use") and [Privacy Policy](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy "foundation:Special:MyLanguage/Policy:Privacy policy"). WikipediaÂź is a registered trademark of the [Wikimedia Foundation, Inc.](https://wikimediafoundation.org/), a non-profit organization.
- [Privacy policy](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy)
- [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About)
- [Disclaimers](https://en.wikipedia.org/wiki/Wikipedia:General_disclaimer)
- [Contact Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:Contact_us)
- [Code of Conduct](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct)
- [Developers](https://developer.wikimedia.org/)
- [Statistics](https://stats.wikimedia.org/#/en.wikipedia.org)
- [Cookie statement](https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement)
- [Mobile view](https://en.m.wikipedia.org/w/index.php?title=Zero-inflated_model&mobileaction=toggle_view_mobile)
- [](https://www.wikimedia.org/)
- [](https://www.mediawiki.org/)
Search
Toggle the table of contents
Zero-inflated model
3 languages
[Add topic](https://en.wikipedia.org/wiki/Zero-inflated_model) |
| Readable Markdown | null |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Zero-inflated_model s443 |